Has anyone here actually read the paper? Heavily clickbait headline, completely exaggerating the study results, which themselves are not very rigorous (to be generous)
Tiny sample size - 36 people completed
Numeracy has R^2 = .27
Language has R^2 = .31
They then run stepwise regression to determine variance contributions, seemingly ignoring their earlier results, and this leads to almost no contribution from numeracy. Why? Because they have ~10% shared variance and stepwise regression is greedy - will just take whatever you give it first.
I can't mention this part enough. If you got a second, very similar language test, and added it to the model, you would _also_ find it has almost no unique variance added.
Every thing they measure is incredibly noisy and they do not once attempt to deal with this. Human based reviewers, time-to-completion, etc.
p-value for "language learning is more significant than numeracy" on the values they give (Steiger test) gives 0.772. Utterly insignificant.
Also, to beat the point home, just think about the argument here:
* Numeracy contributes 27% of variance
* Language skills contribute 31% of variance
* After regression step, numeracy contributes only 2% of unique variance. Because you added a correlated variable!
To be somewhat fair, their numeracy scatter plot visually looks more like a dart board than any real process. The fact that you got a positive-sloping line out at all from the regression has to do more with the positions of the outliers than anything else. You're at least able to visually examine the language aptitude plot and see an up-and-to-the right connection between the two variables.
I agree, but also via some napkin math, the chance of getting those different results is something like 40%.
(Ie, if you sample the same signal twice for the numbers they did in the study, there’s a 40% chance it’ll be >0.04 away from the original sample, as numeracy was from language)
Good code doesn’t just solve a problem, it solves it in a way that’s readable and modular.
I think the problem-solving part of coding requires math skills, while the organization part requires writing skills. The organization part affects the problem-solving part, because if you write messy code (that you can’t reread once you forget or extend without rewriting) you’ll quickly get overwhelmed.
Writing large math proofs also requires organization skills, since you’ll refer to earlier sections of your proof and may have to modify it when you encounter issues. But to me, math seems to have more “big steps”: sudden insights that can’t be derived from writing (“how did you discover this?”), and concepts that are intrinsically complicated so one can’t really explain them no matter how well they can write. Whereas programming has more “small steps”: even someone who’s not smart (but has grit) can write an impressive program, if they write one component at a time and there aren’t too many components that rely on each other.
Most coding doesn't need much of any math past boolean logic and very basic set operations. I'm much more likely to spend my time studying DB and interface schemas to understand how something works than doing a lot of mathy fiddling. Sure, some people write game engines and such, but even much of 3D graphics doesn't need anything more complicated than the first half of a linear algebra course.
> Most coding doesn't need much of any math past boolean logic and very basic set operations
Coding IS math.
Not "coding uses math", I mean it is math.
Mathematicians do not deal in objects, but in relations among objects; they are free to replace some object by others so long as the relations remain unchanged. Content to them is irrelevant; they are interested in form only.
- Poincare[0]
I don't know how you code, but I don't think I'm aware of code that can't be reasonably explained as forming relationships between objects. The face we can trace a program seems to necessitate this.
Programming is an expression of logic, which is absolutely mathematics.
But then we also have to think about naming variables and classes, structuring our code so that it is more readable by other developers, and so on. That's less about formal reasoning and more about communication.
There is an engineering aspect to programming (prototyping, architecture, optimization, etc). It's a combination of mathematics and engineering. Software Engineering.
The murky world of software patents would like a word about that.
For me, coding also feels artistic at times. But we see artistic beauty in mathematics all the time, so no wonder. But often I look at ugly code, and maybe that is subjective, but fixing that code makes it also feel prettier.
It is clear all developers have their owm artistic style, and probably that is why there is so much disagreement in the industry. Maybe we are lacking the pure mathemetical language to describe our intent in a more beutiful and precise way that is more clearly the-right-way. As in how we find beauty in simple physics equations.
The presumably-mathematicians who wrote my Algorithms textbook (CLRS) didn't seem to think giving the variables in their algorithms useful names. They just use i, x, etc. all over the place and don't even consider giving things actual names. This makes the book much less accessible than it would be if they treated it more like most people write code.
I do agree with your point btw, but I did want to note that there are good conventions around symbols. The brevity is heavily influenced by the medium. Variable names sucked when you had punch cards. It's still burdensome to write long names when using paper, chalkboard, whiteboard, or any system that doesn't have autocomplete.
In general, lower case letters are used as constants, excluding x,y,z,t,i,j,k (sometimes u,v,w). It isn't a hard rule, but strong preference to begin at the beginning of the alphabet for these. Capital letters usually are held for things like Variable Sets (like random variables). Greek letters need context for constants or variables. BB and Cal typefaces for sets (e.g. Real Numbers, Integers). And much more.
I think a lot of the difficulty in it is that these "rules" or patterns are generally learned through usage and often not explicitly stated. But learning them can really help read unfamiliar topics and is why "notation abuse" leads to confusion. But after all, math is all about abstraction so technically any symbol will do, but no doubt some are (significantly) better than others for communicating.
There are two hard things in Computer Science:
- Cache Invalidation
- Naming Things
- Off-by-One Errors
That's how it is in applied mathematics. That's a hardcore computer science textbook. If your job is computer science research, inventing new algorithms, that is the kind of book you will get used to. There are better options for practical learning.
Of course they do, but that part is not "mathematics". It is communication. If they use the English language to write the proof that doesn't mean English is also mathematics.
Structuring sentences and naming variables so that it is easier for other people to understand is less about formal mathematical reasoning, and more about communication.
You could name a variable x, y, or Banana, but it doesn't change the logic.
Neither is it "the point" in programming. You should be concerned with communication and have every right to get upset when someone is being needlessly convoluted but that's as much of a point in programming as it is in math, physics, or any domain.
I mean the reason we get mad at this is because it is someone destroying "society" in some sense. Even if that society is your team or just the set of programmers. It would be a pretty dick move were I to just use a word that significantly diverged from conventional meaning and expected you to mull it over. Similarly if I drop a unknown out of context math equation. It would be meaningless.
And I'm on your side, really! I strongly advocate for documenting. And let's be real, the conclusion of your argument more strongly argues for documentation than good variable names. Because variable names are much more constrained and much more easily misinterpreted considering how any word has multiple definitions. Surrounding code is often insufficient to derive necessary contextualization.
I don't think that quote really supports coding and math being equivalent. To me, the quote provides an abstraction of math through a structuralist perspective. Language can also be viewed through this abstraction. I think coding could share the abstraction, but that doesn't make the three of these fields equivalent.
Well a lot of people did wildly misunderstand the Poincare quote. To me is is obviously about abstraction and I think this is true for any mathematician. I thought it would also be natural for programmers considering we use "object" quite similarly, if not identically. So... maybe it is or maybe this is the joke.
"There are those who tell us that any choice from among theoretically-equivalent alternatives is merely a question of taste. These are the people who bring up the Strong Church-Turing Thesis in discussions of programming languages meant for use by humans. They are malicious idiots. The only punishment which could stand a chance at reforming these miscreants into decent people would be a year or two at hard labor. And not just any kind of hard labor: specifically, carrying out long division using Roman numerals." — Stanislav Datskovskiy
That sounds like you're backing off from your original claim, probably because it is impossible to defend.
That you can use mathematics to describe code doesn't seem very different from using math to describe gravity, or the projected winner in an election, or how sound waves propagate.
Isn't the primary purpose of math to describe the world around us?
Then it shouldn't be surprising that it can also be used to describe programming.
In the real world, however, software engineering has nothing to do with mathematical abstractions 99% of the time
A programmer constructs a function from some data type to another while a mathematician constructs a function from witnesses of some proposition to another?
Though interpreting a CRUD app as a theorem (or collection of theorems) doesn’t result in an interesting theorem, and interpreting a typical theorem as a program… well, sometimes the result would be a useful program, but often it wouldn’t be.
Interpreting a CRUD apps (or fragments of them) as theorems is interesting (given a programming language and culture that doesn't suck)! e.g. if you have a function `A => ZIO[Any,Nothing,B]`, then you have reasonable certainty that barring catastrophic events like the machine going OOM or encountering a hardware failure (essentially, things that happen outside of the programming model), that given an A, you can run some IO operation that will produce a B and will not throw an exception or return an error. If you have an `A => B`, then you know that given an A, you can make a B. Sounds simple enough but in practice this is extremely useful!
It's not the type of thing that gets mathematicians excited, but from an engineering perspective, such theorems are great. You can often blindly code your way through things by just following the type signatures and having a vague sense of what you want to accomplish.
It's actually the halting problem that I find is not relevant to practical programming; in practice, CRUD apps are basically a trivial loop around a dispatcher into a bunch of simple functions operating on bounded data. The hard parts have been neatly tidied away into databases and operating systems (which for practical purposes, you can usually import as "axioms").
> It's not the type of thing that gets mathematicians excited
Says who? I've certainly seen mathematicians get excited about these kinds of things. Frequently they study Programming Languages and will talk your ear off about Category Theory.
> You can often blindly code your way through things by just following the type signatures and having a vague sense of what you want to accomplish.
Sounds like math to me. A simple and imprecise math, but still math via Poincare's description.
> in practice, CRUD apps are basically a trivial loop around a dispatcher into a bunch of simple functions operating on bounded data
In common settings. But those settings also change. You may see those uncommon settings as not practical or useful but I'd say that studying those uncommon settings is necessary for them to become practical and useful (presumably with additional benefits that the current paradigm doesn't have).
I think we're in agreement. My comment about the halting problem was meant to refer to Rice's theorem (the name was slipping my mind), which I occasionally see people use to justify the idea that you can't prove interesting facts about real-world programs. In practice, real-world programming involves constantly proving small, useful theorems. Your useful theorem (e.g. `Map[User,Seq[Account]] => Map[User,NetWorth]`) is probably not that interesting to even the category theorists, but that's fine, and there's plenty you can learn from the theorists about how to factor the proof well (e.g. as _.map(_.map(_.balance).sum)).
> Isn't the primary purpose of math to describe the world around us?
No, that's Physics[0]. I joke that "Physics is the subset of mathematics that reflects the observable world." This is also a jab at String Theorists[1].
Physicists use math, but that doesn't mean it is math. It's not the only language at their disposal nor do they use all of math.
> software engineering has nothing to do with mathematical abstractions 99% of the time
I'd argue that 100% of the time it has to do with mathematical abstractions. Please read the Poincare quote again. Take a moment to digest his meaning. Determine what an "object" means. What he means by "[content] is irrelevant" and why only form matters. I'll give you a lead: a class object isn't the only type of object in programming, nor is a type object. :)
[0] Technically a specific (class of) physics, but the physics that any reasonable reader knows I'm referencing. But hey, I'll be a tad pedantic.
[1] String Theory is untestable, therefore doesn't really reflect the observable world. Even if all observable consequences could be explained through this theory it would still be indistinguishable from any other alternative theory which could do so. But we're getting too meta and this joke is rarely enjoyed outside mathematician and physicist communities.
Going on a total tangent, if you'll forgive me, and I ask purely as a curious outsider: do you think math could have ever come into being if it weren't to fill the human need of describing and categorizing the world?
What would have been the very beginning of math, the first human thought, or word or action, that could be called "math"? Are you able to picture this?
> do you think math could have ever come into being if it weren't to fill the human need of describing and categorizing the world?
I'm a bit confused. What exactly is the counterfactual[0] here? If it is hyper-specific to categorizing and describing then I think yes, those creatures could still invent math.
But my confusion is because I'm having a difficult time thinking where such things aren't also necessary consequences of just being a living being in general. I cannot think of a single creature that does not also have some world model, even if that model is very poor. My cat understands physics and math, even though her understandings are quite naive (also Wittgenstein[1] is quite wrong. I can understand my cat, even if not completely and even though she has a much harder time understanding me). More naive than say the Greeks, but they were also significantly more naive than your average math undergrad and I wouldn't say the Greeks "didn't do math".
It necessitates a threshold value and I'm not sure that this is useful framing. At least until we have a mutual understanding of what threshold we're concerned with. Frankly, we often place these contrived thresholds/barriers in continuous processes. They can be helpful but they also lead to a lot of confusion.
> What would have been the very beginning of math
This too is hard to describe. Mull over the Poincare quote a bit. There's many thresholds we could pick from.
I could say when the some of the Greeks got tired of arguing with people who were just pulling shit out of their asses, but that'd ignore many times other civilizations independently did the same.
I could say when the first conscious creature arose (I don't know when this was). It needed to understand itself (an object) and its relationship to others. Other creatures, other things, other... objects.
I could also say the first living creature. As I said above, even a bad world model has some understanding that there are objects and relationships between them.
I could also say it always was. But then we get into a "tree falls in a forest and no one is around to hear it" type of thing (also with the prior one). Acoustic vibrations is a fine definition, but so is "what one hears".
I'd more put the line closer to "Greeks" (and probably conscious). The reason for this is formalization, and I think this is a sufficient point where there's near universal agreement. In quotes because I'll accept any point in time that can qualify with the intended distinction, which is really hard to pin-point. I'm certainly not a historian nor remotely qualified to point to a reasonable time lol. But this also seems to be a point in history often referenced as being near "the birth" and frankly I'm more interested in other questions/topics than really getting to the bottom of this one. It also seems unprovable, and I'm okay with that. I'm not so certain it matters when that happened.
To clarify, I do not think life itself necessitates this type of formalization though. I'm unsure what conditions are necessary for this to happen (as an ML researcher I am concerned with this question though), but it does seem the be a natural consequence of a sufficient level of intelligence.
I'll put it this way, if we meet an alien creature I would be astonished if they did not have math. I have no reason to believe that their math would look remotely similar to ours, and I do think there would be difficulties in communicating, but if we both understand Poincare's meaning then it'll surely make that process easier.
Sorry, I know that was long and probably confusing. I just don't have a great answer. Certainly I don't know the answer either. So all I can give are some of my thoughts.
>Even if all observable consequences could be explained through this theory it would still be indistinguishable from any other alternative theory which could do so.
That's not unique, all quantitative theories allow small modifications. Then you select parsimonious theory.
Coding is math in the sense that coding is a subset of math.
Mathematics is a very extensive field, and covers a vast amount of subjects.
For the same reason it can't be said that mathematics is equivalent to coding, as there are many things in mathematics that are not relevant for coding.
However, by far the most interesting parts of coding are definitely related to mathematics.
Working with different objects doesn't make it any less of math. Just because you can derive calculus from set theory (analogous to assembly or even binary here) doesn't make calculus "not math".
Math is about abstractions and relations. See the Poincare quote again.
Plus, the Programming Languages people would like to have a word with you. Two actually: Category Theory. But really, if you get them started they won't shut up. That's either a great time or a terrible time, but I think for most it is the latter.
Wheeler: It from bit. Otherwise put, every it—every particle, every field of force, even the space-time continuum itself—derives its function, its meaning, its very existence entirely—even if in some contexts indirectly—from the apparatus-elicited answers to yes-or-no questions, binary choices, bits. It from bit symbolizes the idea that every item of the physical world has at bottom—at a very deep bottom, in most instances—an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes–no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and that this is a participatory universe.
That is not in contention with what I said, now is it?
Wheeler is arguing that if a tree falls in the forest, and there is nobody to hear it, that it still makes a sound because there are things that interact with the sound. But if the tree fell in a forest and there was nothing else in the universe then there is no sound because there is no observation.
It helps to read the whole thing[0] and to understand the context of the discussion. This is meta-physics and a deep discussion into what the nature of reality is. Ian Hacker has a good introduction to the subject but I find develop grave misunderstandings when they also do not have the strong math and physics background necessary to parse the words. Even people who understand the silliness of "The Secret" and that an observer need not be human often believe that this necessitates a multi-verse. A wildly convoluted solution to the problem of entropy not being invertible. Or closer to computer terms, a solution that insists that P = NP. There is information lost.
If you wanna argue that there's no difference between the word cup and a cup itself because there is no word without the observer who has the language, then yeah.
But that doesnt necessarily mean successful programmers are good at conventional math. This is why certain people in the department are identified as "math people".
I'm not sure why you'd think I disagree. It seems you understood I argued that it's unhelpful to make the distinction between math and "conventional" math
Arguably writing a novel is math, if you use the right definition of math. But sometimes its more helpful to use more informal definitions that capture what people mean then what is technically accurate.
By that same logic you could also say that language is math. In fact I think your quote kind of disproves your point because the content/state of a program is super important in coding more than the form.
Coding used to be very close to pure math (many early computer science classes were taught in the Math Department in universities) but it has been so far abstracted from that to the point that it is its own thing and is as close to math as any other subject is.
> By that same logic you could also say that language is math
Not quite, but the inverse is true. The language to math direction doesn't work because a lack of formalism. I can state incomprehensible sentences or words. (There's an advantage to that in some cases!) but when you do that with code you get errors and even you do it with math its just that there's no compiler or interpreter that tells at you
I think you misunderstand what "paradox" means. While it can mean "self-contradictory" it can also mean "contrary to one's expectation." Math uses both, but in very different contexts.
The contradiction is used in proof formulation, specifically to invalidate some claim. I don't think this is what you're implying.
The latter is what it contextually sounds like you're stating; things like the Banach-Tarksi Paradox. There's no self-contradiction in that, but it is an unexpected result and points to the need to refine certain things like the ZFC set theory.
I'd also stress that there are true statements which cannot be proven through axiomatic systems. The Halting Problem is an example of what Godel proved. But that's not contradictory, even if unexpected or frustrating.
This is vague and doesn't mean anything. People can't even agree what 'objects' are and people did a lot of programming before the term 'object' was invented.
Programming is about is fundamentally about instructions and data. Yin and yang of two things that are completely different, even if people can occasionally mix them and conflate them.
Category theory is often jokingly called the study of dots and arrows. What's a dot? Anything. What's an arrow? A relationship.
I'm surprised I hit a nerve with so many people. I'm quoting someone who's considered one of the greatest mathematicians. Obviously I don't know the backgrounds of people but it seems like programmers have strong opinions on what math is that disagrees with what mathematicians say they do.
Now it's not math it's 'category theory' and that's dots and arrows and dots are anything.
At what point does the abstract description just not offer any useful insight anymore?
I'm surprised I hit a nerve with so many people.
A lot of people had very good explanations for why you're pretty far off in trying to say two things are the same.
Programming is much more like building a machine than any sort of straight math. There is state, there is interactivity, there is performance, there is IO and there are real world implications to all of it.
Saying they are the same is like saying gardening is plumbing just because you sometimes use a hose.
That's a wild claim considering Category Theory is a branch of mathematics
| Category theory is a general theory of mathematical structures and their relations.
- https://en.wikipedia.org/wiki/Category_theory
It is necessary that you provide an alternative definition as to what "category theory" is, though I suspect it will make many category theorists and mathematicians upset.
That's a wild claim considering Category Theory is a branch of mathematics
It's not a wild claim since you misquoted me.
A lot of non-mathematicians disagreed with mathematicians.
Mathematicians can claim whatever they want, when it comes to programming, programmers understand it better and they're trying to explain to you why this is nonsense. Vanderbilt claims to be "the harvard of the south" but wouldn't you know it, harvard doesn't claim to be "the vanderbilt of the north".
Show me programming languages designed by mathematicians to be 'mathematically pure' and I'll show you a language that hasn't been used to ship software that people want to use.
No, not always. Quite a lot of high-level code doesn't require any math at all. It doesn't take math to perform CRUD operations, which account for a lot of programming work. Sure, the underlying machine code is all based on math, but the higher level programming doesn't need to involve a single math equation for it to be useful. Let's see where the goalposts move now...
Mathematics is not "equations". Most of mathematics is not related with calculus either.
For anything CRUD, you put the data into a database. The underlying database is relational, or a key->store.
If it's relational, that's one branch of mathematics. If it's another kind of database, it's another branch of mathematics. Mathematics is extensive, and covers more things you can imagine at a glance.
The main difference between writing mathematics and programming, and this applies to any form of programming, is that in mathematics writing a formal proof is amazingly close to: you write the program, and you are also the compiler, and you are the CPU, performing all operations yourself. With programming you only have to write the program.
Source: On one hand I have studied pure mathematics (not the simplified applied mathematics that are taught in engineering, which is mostly equations), on the other hand I have been working as a software developer for over 15 years.
Yes it does. Just because the objects you are working with aren't numbers doesn't mean it isn't math. In fact, that's my entire point. It is why I quoted Poincare in the first place. He didn't say "numbers" he said "objects".
Sorry but you are wrong. "CREATE, READ, UPDATE, DELETE" operations are not mathematical in nature. It doesn't matter what the data is or how esoteric you want to get with it - a programmer doesn't need any math at all for simple CRUD. You're trying to move the goalposts all the way back to the 1800s to win a pointless internet argument.
A while back I made this[0] as a quick demo of generic batching for CRUD requests. It's very much practically oriented (it's for performance optimization while making the code nicely reusable/not muddying up the business logic), but also felt quite a bit like the same sort of things I did in my math degree.
Actually I'm starting to wonder whether the thing that made university math easy for me was I quickly developed a good internal "type checker"/"type inference engine" in my head, and that helped make the next steps of proofs seem straightforward.
I understand you have strong opinions, I just don't understand why.
In math are highly concerned with structures and abstraction. We have things called operators. They aren't just addition and multiplication. We also use those words to describe different operations. They have things like groups, rings, fields, and algebras. Yes, plural.
The purpose of these things is to create logical frameworks. It matters not what the operators are. Nor does it matter what objects we operate on. Poincaré is explicitly saying this.
The part you're not understanding is the abstraction. This is what math is about. It is also why the Programming Language people are deeper in the math side and love Category Theory (I have a few friends who's PL dissertations are more math heavy than some math dissertations I've seen). It's no surprise. What's a function? How do you abstract functions? How do you define structure? These are shared critical questions. PL people are more concerned with types but I wouldn't say that makes them any less of a mathematician than a set theorist.
We can perfectly describe the CRUD operations with set theory. Do most programmers need concern themselves with this? Absolutely not. But is it something people designing those operations and systems is thinking about? Yes.
I'd encourage you to learn some set theory, abstract algebra, and maybe a bit of cat theory. It'll make these things pretty clear. But I'd caution about having strong opinions on topics you aren't intimately familiar with. Especially when arguing with those that are. Frankly, CRUD is a terrible example. I'm confident a quick google search (or asking a GPT) would quickly point you to relational algebra. It's discussed in most database classes. It certainly was in the one I was an assistant for.
> I understand you have strong opinions, I just don't understand why.
As someone that is "on the other side of the fence", and getting flamed for it, maybe I can shed some light since I have the other perspective as well. IMO, The reason for not seeing eye to eye is (for example) akin to saying "word problems are math". (Thinking of a grade school word problems for common reference). Yes, they are readily mapped to mathematical models that can solve the word problem & perhaps almost indistinguishably so. Though no - word problems are not math. Word problems are a series of phrases and words. That's where the talking past each other comes in... Different interpretations of "word problems are math", or "code is math". It's seemingly not clear whether we are talking about logical 'implies', 'element of', or 'equals'.
Which goes to "We can perfectly describe the CRUD operations with set theory.", we all agree there. That is not readily conveyed though when writing things like "code is math".
That is a good analogy, except programming languages are formal languages, not natural languages.
> they are readily mapped to mathematical models
With code we are not talking about something that is mapped to mathematical models. Code is not modelled by mathematics, it is defined by mathematics (denotational semantics and operational semantics).
"Code is math" is true in a theoretical sense. Practically, coding doesn't always feel like what is commonly thought of as "doing mathematics", but it is essentially mathematical.
> That's where the talking past each other comes in...
Then sorry, but that's your own damn fault. I was clear about my definition and quoted a famous mathematician to give some authority, to not be "trust me even though I'm some rando". The way to respond to that is not "you're wrong, trust me, I'm some rando".
Yes, I agree we're misunderstanding each other because we're using different definitions but "you've" rejected "mine" without defining "yours" and expecting everyone to understand. Of course that'll lead to confusion. You can reject the mathematicians definition of math, but you sure gotta say more than "trust me" and it's a pretty wild thing to do, especially as non mathematicians.
The problem here is one side who's dabbled in cooking says "a chef makes desserts" and chefs are responding "we do a lot more than that". Maybe there's a few chefs that go "yeah all I do is dessert" and everyone points to that while ignoring the second part of their sentence is "but that's just my speciality." Wouldn't you think that conversation is insane? The only reason it's obviously so is because we all know what a chef is and agree on the definition. But who is better qualified to define the chef's job? The chef or consumer?
The way I'm responding I'd more characterize as: "wait, if what you are saying is true, then this other thing should be true too, but it does not seem to be. That would indicate what you are saying is not true."
In another thread, you characterized my response as stating: " ¬(A ↦ B) ⟹ ¬(B ↦ A)" (and this is a great example of language not being math, but math being language!). That was not at all my claim.
My claim is "I believe you are saying 'A = B'. It appears that 'B != A', therefore 'A != B'." My only claims are
(1) I believe you are writing to convey that you mean Math IS Language in the sense they are equal, identical, interchangeable, and fully equivalent, and bi-directionally so
(2) that: B != A
The only results can either be:
- "yeah, because B != A, the statement A = B is not true"
- Your claim (1) is false, I'm not actually saying "A = B"
- Your claim (2) is false, "B = A" is in fact true. I would find that to be an interesting assertion and would have loved to explore more why you think that.
You could say that math is behind the structure of a leaf, but the farmer doesn't care about that. Keep moving the goalpost all you want, I don't have time to argue these pointless things, and I stopped reading after the first sentence of your comment. I'm done here.
CREATE, READ, UPDATE, DELETE are fundamentally mathematical in nature.
In a typical implementation these are database operations. That involves relational algebra operations, state transitions, boolean logic.
The READ part can be a very complex SQL query (composed of many algebraic operations) but even the simplest query (SELECT * from t where ID = 1) is filtering a set based on a predicate. That is mathematics.
No one is moving goalposts. Set theory and logic are at the foundations of mathematics.
It's almost the argument of programming vs computer science coming out here.
This is math:
{x | x.id = 1}
OTOH, a SQL query is a SQL query.
This thread is hilarious though. It's like
- Cashier: here is your change.
- Customer: you did math!
- Cashier, no, I gave you change.
- Customer: that IS math!
- Cashier: You mean, I used math to give you change?
- Customer: No, giving change doesn't use math, it IS math!!!!" [2]
= D
Moving along.. FWIW, indeed SQL was created to model set theory (relational algebra and tuple calculus), the close relationship is no accident of course [0][1])
> No one is moving goalposts
I feel too they are.
First goal post:
> Coding IS math. >> No, not always. Quite a lot of high-level code doesn't require any math at all. It doesn't take math to perform CRUD operations
Second goal post:
> CREATE, READ, UPDATE, DELETE are fundamentally mathematical in nature
CRUD is closely related to SQL, and SQL is closely related to various mathematics. Are they identical and therefore equivalent? No - because your database is not going to like it when you write "{x | x.id = 1}", and the Oracle DB might not like something that you can write for your Postgres DB.
The problem you're running into is that people who have some "mathematical maturity" don't get bogged down in notation, so it's difficult for them to see the distinction you're trying to draw between e.g. `{ x∈S | x.id = 1}` and `select x from S where x.id = 1`[0]. You say "a SQL query is a SQL query" and they just think "yes, which is also obviously a mathematical expression".
Computer programs are proofs[1]. This is intuitively and also formally true. You would agree writing proofs is doing math, yeah? Then obviously writing a computer program is also doing math.
Like I have a degree in math and have been a software engineer for over a decade. I do not know what distinction people are trying to get at. It's like trying to argue about the distinction between U(1), the complex numbers with magnitude 1, and the unit circle, and getting upset when the mathematicians say "those are 3 names for the same thing". Or saying that writing C is programming but writing in a functional language like Scala or Haskell (or Lean) is not.
[0] Modulo details like NULLs and multi-set semantics, but surely that's not the distinction?
>All of that code is a series of logical statements and expressions. Mathematical logic.
Which code? The machine code that underlies everything? Or the lines of simple high-level CRUD that don't even need a single number or logic statement to function? Not all programming has to be mathematical or even logical at a high enough level, and I say this as someone who's been coding assembly language for 40 years.
Storing a string in some abstraction is not a mathematical operation. I'm done with this thread, it's going way too far down too many rabbit holes. The quarks that make up the matter that the computers are made of are pure "math". There, now I've moved the goalposts.
That is a narrow perspective of mathematics and computer science.
Assigning a variable is a mathematical operation. It is a logical state transition. This is formalized in Programming Language Theory.
We are not talking about quarks. We are talking about lines of code which are objectively mathematical statements, no matter how non-mathematical they seem to you subjectively.
You start with a practical problem to solve, and a computing machine that can perform logical operations on data. Your job is to figure out a correct sequence of logical operations that will solve the problem. The program that you have written is a mathematical structure. It is mathematics.
You confuse map with the territory. By the same logic you can say quarks are mathematics, because they are modeled by a theory that has some mathematics in it.
Our mathematical models of reality (i.e. Physics) is not the same as reality. So yes, by the same logic "quarks" are mathematics, but only in the same way that a "cup" is English. The quark was a great example, considering we can't observe it and purely rely on mathematical models, but the same argument would still hold true for a bowling ball. Physics is our description of reality, not reality itself. Our description of reality highly leverages the language of math, as that language was developed to enforce rules of consistency. Much the same way we design our programming languages, which is why programming language people study so much more math than your typical CS person.
If you're going to accuse someone of confusing the map with the territory, you really should make sure you aren't making the same error.
How math helps with programming languages? What math says about zero based indexes? How do you prevent off by one errors? How do you prevent buffer overflows? It's ergonomics problems.
It is hard to answer because of exactly what ndriscoll said[0]
> It's like trying to argue about the distinction between U(1), the complex numbers with magnitude 1, and the unit circle, and getting upset when the mathematicians say "those are 3 names for the same thing". Or saying that writing C is programming but writing in a functional language like Scala or Haskell (or Lean) is not.
As ndriscoll suggests, it is tautological. I mean look at what I said. I really need you to hear it. I said that coding is math. So what I hear is "How programming languages helps with programming languages?" Why are you expecting me to hear anything different?
> What math says about zero based indexes?
Start at 0? Start at 1? Who cares, it is the same thing. The natural numbers, non-negative integers, integers, even integers, who cares? They're the same thing. And who cares about indexing at 0 or 1 in programming? That's always been a silly argument that's inconsequential.
> How do you prevent off by one errors?
By not being off by one? What's the question? Like being confused about if you start at 0 or start at 1 and how to get the right bound? It is a shift from one to the other, but they are isomorphic. We can perfectly map. But I really don't get the question. You can formalize these relationships with equations you know. I know it isn't "cool" but you can grab a pen and paper (or a whiteboard) and write down your program structure if you are often falling for these mistakes. This seems more about the difficulties of keeping track of a lot of things in your head all at once.
> How do you prevent buffer overflows?
By not going over your bounds? I'm so confused. I mean you are asking something like "if f(x) = inf when x > 10, how does math help you prevent the output of the function from being infinite?"
Maybe what will help is seeing what some of the Programming Languages people do and why they like Haskell[1].
Or maybe check out Bartosz Milewski[2,3]. His blog[2], is titled "Bartosz Milewski's Programming Cafe: Category Theory, Haskell, Concurrency, C++". It may look very mathy, and you'd be right(!), but it is all about programming! Go check out his Category Theory Course[3], it is _for programmers_.
Don't trust me, go look at papers published in programming language conferences [4]. You'll find plenty of papers that are VERY mathy as well as plenty that are not. It really depends on the topic and what is the best language for the problems they're solving. But you'll certainly find some of the answers you're looking for.
Seriously, don't trust me, verify these things yourself. Google them. Ask an LLM. I don't know what to tell you because these are verifiable things (i.e. my claims are falsifiable!). The only thing you need to do is look.
Nonmathematical problems are difficult to answer when you try to find mathematical answers. It should be obvious, why it doesn't work.
Mentioning Scala is ironic, it's very light on math spik and to begin with was created to unify object oriented with functional programming, which is mathematically meaningless, because both are Turing complete and thus equivalent, tautological.
>So what I hear is "How programming languages helps with programming languages?"
Oh, right, in mathematics axiom is argument, but in reality it isn't. In programming you should assert what you assume, otherwise your assumptions can be wrong due to divergence from reality, but there no reality in mathematics, only fantasy, so you can't understand this with mathematics alone.
That may be true to some extent, but I think you are missing the point. Quarks are physical in nature, and code is logical in nature. Programs themselves are formal systems. Code isn’t just modelled by mathematics, it is defined by mathematics.
In the case of code, we can argue that the map is the territory.
Code is logical if you define logic as reasoning in general, broader than mathematics, and since it runs in physical environment, it now interacts with a messy world. Code is defined by business requirements, and there's no mathematics for that.
The discussion started about whether human needs math skills to write code. That's what I mean when I say programming isn't mathematics. Meaning of code is defined by human, how do you intend code to be defined by mathematics? The human first imagines mathematical formulas, then by some unknown process they become code? I don't think anybody does it like that. You start with requirements, intuitively guess architecture, then decompose it, it works more like Cauchy problem (but less numeric, more conceptual, you have an owl, now draw it), but I don't think anybody models it like that.
>Once the code is actually written, that exists as a formal logical system, defined by mathematics
I still think that's not code, but your favorite model of code. For spellchecker language is defined by mathematics too: it splits text into words by whitespace, then for each word not found in dictionary it selects best matches and sorts them by relevance. Oh and characters are stored as numbers.
> The discussion started about whether human needs math skills to write code.
Writing code IS a math skill. When writing code you are writing logic in a formal system. Logic is mathematics.
You may be thinking that mathematics is just like doing arithmetic or solving equations. It is way deeper and broader than that.
> I still think that's not code, but your favourite model of code
Code is not just modelled through mathematics, it is actually defined by mathematics. It is fundamentally a mathematical construct, grounded in formal semantics.
It's definitely a gray area. Is a DAG traversal algo "math", or is it more computer-sciencey? What if you do it in SQL? Certainly there's a mix of more or less concentrated logic/math vs glue code, and most of that is very dependent on the domain you're working in.
I find this distinction useful in the abstract, that one can engage different parts of the brain for different components of development. This probably explains why a well-written DSL can be so powerful in the right context.
Computer science is math in the same way that physics is philosophy, in that computer science certainly started as math, just like the natural sciences used to be subdisciplines of philosophy.
But it's hardly a useful grouping any more. You can study and do well in computer science with minimal knowledge of most of the core mathematical subjects.
While graph theory certainly crosses over into math, you can cover most of the parts of it relevant to most computer science as a discussion of algorithms that would not be the natural way of dealing with them for most mathematicians.
> You can study and do well in computer science with minimal knowledge of most of the core mathematical subjects.
You will fail at Theoretical Computer Science without mathematical proficiency. Go read some textbooks and papers in theoretical CS. It is a subfield of mathematics. Theorems and proofs. Rigorous and difficult mathematics.
I've read plenty of theoretical computer science papers over the last 30+ years, and while some of it requires "rigorous and difficult mathematics" that is by no means universal.
I wrote my MSc thesis on the use of statistical methods for reducing error rates for OCR, and most of the papers in my literature review hardly required more than basic arithmetic and algebra.
So I stand by my statement.
Sure, there are subsets of computer science where you need more maths, just like in any field there are sub fields where you will need to understand other subjects as well, but that does not alter what I claimed.
EDIT:
Some authors are quicker to pull out the maths than others, and frankly in a lot of CS papers maths is used to obscure lack of rigor rather than to provide it. E.g the problem I ran across when writing my thesis was that once you unpacked the limited math into code you'd often reveal unstated assumptions that were less than obvious if you just read their formulas.
Is it possible your mental model of what CS is more aligned with software engineering rather than actual CS? Could you share some examples of what you consider to be CS but lacks any mathematical relation?
I agree is not a useful grouping in practice. I'm just interested in what makes you think like you do.
I did categorically not claim, nor even suggest, that any CS "lacks any mathematical relation".
What I claimed was that in computer science we often discuss things in terms that would not be the natural way of dealing with it in maths. We do that because our focus is different, and our abstractions are different.
It doesn't mean it's not math. It means it's not useful to insist that it isn't a different field, and its obtuse when people insist it's all the same.
Now we're getting into the "define maths" part of the discussion which is always where these discussions die. It can be argued that turning a kettle on and boiling some water is "maths" or it can be as narrow as "everything above basic arithmetic is logic, not maths."
So how much of programming is maths? Before we answer that, let's answer: How much of maths is actually maths? Because first we define maths, and then we define programming based on whatever that is, but until we have that first concrete definition this discussion cannot occur.
I will add that "it is taught by the maths department in college" is a flimsy argument, and frankly one the Physics department in particular would mock.
I think for this discussion, "math is stuff you'd learn in a math department" is a pretty useful definition, even if it's not a very good one. There's a lot of math involved in the design and manufacture of the kettle, electrical grid, and water utilities, but a person's ability to put a kettle on isn't going to be improved by math classes. In that way, programming probably is a bit mathy, but good programming is more like good technical writing than it is like math.
> I think for this discussion, "math is stuff you'd learn in a math department" is a pretty useful definition
That means that definition shifts over time. For example, courses on numerical analysis, graph algorithms, programming, and on compilers used to be part of “what you’d learn in a math department”.
It likely also even today will show geographical variation.
Overlap between programming and math is quite small, e.g. halting problem is solvable in programming (by flow analysis) but isn't solvable in math. Programming deals only with practical problems and can be completely guided by practical considerations, while math requires abstract outlandish skills - exact opposite. Why talk about math at all if it's already well known that programming is engineering?
Probability is math and there you have caching, hashing, etc. Then there are permutations, combinations, etc. with a lot of usecases in software. Distributing graphs across nodes? More math.
Boolean logic and set theory are very important to computer logic, and graph theory is not far behind. But you can also learn Boolean logic in the philosophy dept. Which I unfortunately learned the hard way by taking Philosophy 101 after already having the CS class on logic. Took a semester to go over what we did for the midterm. Got a lot of naps that semester.
It's funny, reading the post you're replying to, I basically read it as
> I don't need math, I need <math, but by another name>
My teenage daughter used to complain about math, and I spent some time trying to explain to her that we use math every day... EVERY day. Now, when I see her do something that was math (even if it's not obvious it was math), I say "Math... every day". I say that a lot.
Also, yes, my daughter finds me annoying. But also funny; but likely not for the math thing.
DBs aren't math. In math it's perfectly ok for computation to take infinite time, you can just assume it already completed, but DBs don't work like that.
> Good code doesn’t just solve a problem, it solves it in a way that’s readable and modular.
Strongly disagree.
There are plenty of cases where non-modular code is more readable and faster than modular code (littered, presumably, with invocations of modular logic).
There are also countless cases, particularly in low-level languages or languages with manual memory management, where the best solution -- or the correct solution -- is far from readable.
Readability is anyways in the eye of the beholder (My code is readable. Isn't yours?) and takes a back seat to formal requirements.
Just as a matter of personal style, I prefer long, well-organized functions that are minimally modular. In my experience, context changes -- file boundaries, new function contexts, etc -- are the single greatest source of complexity and bugs, once one has shipped a piece of code and needs to maintain it or build on it. Modular code, by multiplying those, tends to obscure complexity and augment organizational problems by making them harder to reason about and fix. Longer functions certainty feel less readable initially but I'd wager they produce better, clearer mental models, leading to better solutions.
I think the important concept that "readable and modular" is trying to get at is how easy is it to continue working on the code in future. There's definitely codebases that are easier to work on than others, even when the domain is the same.
I'd say that readability, which often boils down to consistency, and modularity are ways to do this, but they aren't the only ways. And as you say, sometimes there's a need for "unreadable" code, so not everything can be easy.
I'm not a mathematician, but I figure mathematicians aim for clean, composable abstractions the same way programmers do. Something complicated, not just complex in its interactions with other things, seems more useful as a bespoke tool (e.g. in a proof) than as a general purpose object?
> Whereas programming has more “small steps”: even someone who’s not smart (but has grit) can write an impressive program, if they write one component at a time and there aren’t too many components that rely on each other.
This is well put. I often wonder if a merely average working memory might be a benefit to (or at least may place a lower bound on the output quality of) a programmer tasked with writing maintainable code. You cannot possibly deliver working spaghetti if you can't recall what you wrote three minutes ago.
> You cannot possibly write or deliver spaghetti, working or otherwise, if you're not capable of remembering what you wrote three minutes ago.
Forth programmers make a similar point. Forth is stack based; you typically use stack operations rather than local variables. This is ok when your 'words' (analogous to functions/procedures) have short and simple definitions, but code can quickly become unreadable if they don't. In this way, the language strongly nudges the programmer toward developing composable words with simple definitions.
(Of course, Forth sees little use today, and it hasn't won over the masses with its approach, but the broader point stands.)
> Whereas programming has more “small steps”: even someone who’s not smart (but has grit) can write an impressive program, if they write one component at a time and there aren’t too many components that rely on each other.
... In my experience, learning to write one component at a time (and try the code, and make sure it works before proceeding) is itself a skill that many struggle to develop. Similarly for avoiding unnecessary dependencies between components. Oh, and also being able to analyze the problem and identify separable components.
One of the most frustrating things about teaching programming, for me, is the constant insistence from other teachers that you have to maintain an "absolutely everyone can learn to program" attitude at all times. Many people who start to learn programming have misguided or confused reasons for doing so and - to say the least - could make much more effective use of their time developing other skills. (It's not a question of elitism; I'd surely flounder at some tasks that others find natural.)
I dislike "everyone can learn to program" because it conflates many, many levels of skills and expertise.
I very much think many people could learn the more advanced Excel Formulas, Power Automate and even simple Bash/PowerShell scripting to make their work more effective. I've met quite a few folks who had been intimidated out of trying who could do it.
On the other hand, how many people on this site could bootstrap a linux kernel on either very new or very old hardware? I know there are some, but they are certainly not the majority. I certainly won't be the first person to get linux and doom to run on a quantum computer.
But that is similar to other professions. Everyone with a largely functioning body can learn to turn a few planks and some metal parts into a functional shed door with some basic tools or to put up a decent brick wall that won't topple over in a month.
That doesn't mean everyone is qualified to pour concrete for a dam or a bridge foundation, or to re-do some historical work in original style.
> Everyone with a largely functioning body can learn to turn a few planks and some metal parts into a functional shed door
It's shocking how little physical and spatial ability some people have - that is definitely not true. Sometimes it might be a personal discount or lack of confidence, but this remains true regardless of the cause.
> That doesn't mean everyone is qualified to pour concrete for a dam or a bridge foundation, or to re-do some historical work in original style.
Exactly!
I think statements like that are more concerned with philosophy than reality. Any discussion surrounding topics like this typically ends up being a discussion around definitions.
I believe the vast majority of human beings are capable of learning how to program in the most extreme elementary sense of the word. As in, outside of severe disabilities or complete and utter inaccessibility to circumstances in which one could learn program, then I think the remaining population of people could learn to program to some degree. Obviously, not everyone will learn to program due to a near infinite number of reasons.
I would argue it's like music. Anyone can make 'music.' Just make a sound -- any sound. The difference between noise and music is subjective. I would not argue that everyone could be the next Lovelace, Turning, Ritchie, Thompson, Torvalds, etc..
Now, for my jaded opinion, I think a lot of the "everyone can learn to program" talk does not come from a place of desire to share the gift of knowledge and joy of programming. I think it's more of a subtle way to encourage people to go into programming so that they may be hired by mega corps. In order to keep the Capitalist machine running. It's like the National Hockey League's slogan, "Hockey is for everyone." That is just a fancy way of saying, "everyone's money can be spent on the NHL."
I recall reading about a professor at uni which carefully designed his tests throughout the year so he could determine if a student had grasped a certain programming topic. He also tracked effort, through handins and such.
After collecting data for a few semesters he concluded his students could be clearly divided into three categories: those who just "got" programming, those who understood it after working hard, and a small group that just didn't grasp regardless of effort.
I guess it depends on what is meant by "everyone can learn to program". It's a bit like saying everyone can learn to write, or to do math, or play tennis, etc.
I'm sure everyone is capable of learning some basic level of programming, just as they are able to learn a basic (high school) level of any subject. However, not everyone is going to have the aptitude to take that to an advanced professional level, however hard they try. We're not all cut out to be artists, or writers, or doctors, or scientists, or developers, etc.
Personally I've always considered a solid grasp of algebra to be the minimum bar for being able to program, at least for anything that isn't utterly trivial. Being able to take a word problem and turn it into a system of equations and solve it is a pretty close analog to being able to take some sort of problem or business requirement and turn it into code.
And the sad truth is that a huge percentage of the population struggle with just arithmetic, let alone algebra.
I mean - not "absolutely everyone" can learn to do just about anything. There's a wide distribution of intelligence, aptitude/ability and desire amongst folks amongst all of the various things that you might learn (not to mention learning disabilities, impairments, etc).
I might be capable of learning advanced accounting, but that sounds like torture to me and I'll be damned if I'll ever take that on! I'm sure programming feels like that to a wide variety of people, and I don't see any need for us to try to pretend otherwise - outside of a bizarre ideological desire for equivalent outcomes from disparate groups.
What do you mean by math? Abstractions or calculations? The effort it takes me to do long division in my head matches the effort it takes me to follow some obtuse spaghetti code. See, I can get good and fast at long division in my head, but I may never understand the fundamental theorem of calculus. Some people are really good at mucking around garbage code (they have no choice, they get paid to), but what part of programming did they get good at? Obviously, some part of it, but nothing to write home about. Whenever I sense that I'm just getting practice at doing the equivalent of mental long division at work, that's when I always seek a new job. No amount of money is worth falling behind like that.
I'm thinking of "computation", "intuition", and "organization".
Computation is following an algorithm. e.g. long division or computing a derivative.
Intuition, AKA brilliance, is finding a non-obvious solution. Think "solving an NP problem without brute force"*. e.g. solving an integral (in a form that hasn't already been memorized) or discovering an interesting proof.
Organization is recording information in a way that a) is easy for you to recall later on (and get insights from) and b) is digestible by others**. e.g. explaining how to compute a derivative, solve an integral, or anything else.
Math, programming, and writing each require all skills. The kind of math taught in school (e.g. long division) and your boring jobs are primarily computation. I believe advanced math (e.g. calculus) is primarily intuition; it requires some organization because big theories are broken into smaller steps, but seems to mostly involve smart people "banging their head against the wall" to solve problems that are still quite unclear***. Programming is primarily organization. It requires some intuition (I think this is why some people seemingly can't learn to code), but in contrast to math, most programs can be broken into many relatively-simple features. IMO implementing all the features and interactions between them without creating a buggy, verbose, and unmaintainable codebase is programming's real challenge. Writing is also primarily organization, but finding interesting ideas requires intuition, and worldbuilding requires computation (even in fiction, there must be some coherence or people won't like your work).
> Some people are really good at mucking around garbage code (they have no choice, they get paid to), but what part of programming did they get good at? Obviously, some part of it, but nothing to write home about.
I agree that work you find boring should be avoided, and I also try to avoid working with it. But some people really seem to like working on esoteric code, and I think there are some skills (beyond computation) developed from it, that even apply when working with good code. Building a mental model of a spaghetti codebase involves organization, and if the codebase uses "genius hacks", intuition. Moreover, the same techniques to discern that two code segments in completely different locations are tightly coupled, may also discern that two seemingly-separate ideas have some connection, leading to an "intuitive" discovery. There's an MIT lecture somewhere that describes how a smart student found interesting work in a factory, and I think ended up optimizing the factory; the lesson was that you can gain some amount of knowledge and growth from pretty much any experience, and sometimes there's a lot of opportunity where you'd least expect it.
* Or maybe it is just brute force but people with this skill ("geniuses") do it very fast.
** These are kind of two separate skills but they're similar. Moreover, b) is more important because it's necessary for problems too large for one person to solve, and it implies a).
*** And whatever method solves these problems doesn't seem to be simplification, because many theories and proofs were initially written down very obtuse, then simplified later.
A lot of people in this type of threads always makes the same mistake: confusing what math is with branches of math, or rather, ways in which math is used. The way the education system is built certainly contributes to this.
I've always found the car metaphor to work very good to understand this: A car is a machine that can transport itself to point A to B (some other rules apply). There are different types of cars, but you certainly haven't understood the definition of you say that something is not a car because is not a Volvo, or because it doesn't look like a Ford, when it's clearly able to transport itself.
Math is the study of ideal objects and the way they behave or are related to each others. We have many branches of mathematics because people have invented so many objects and rules to play with them. Programming is nothing if not this very definition. The fact that you don't have to "use math" when programming is not really addressing the point, it's like saying a car is not a car because it has no discernible brand.
"Time is not a thing in math" is not understanding what math is. Time is another ideal object following certain rules under a given domain. Programming is coming up with objects of different size, with different characteristics, with interact at different points in time, i.e. following certain rules.
Its very rare imo that computational problems emerge fully formed & ready to be tackled like proofs.
Usually even deciding what the problem is is in part an art, requires an act of narrativization, to shape and form concepts of origin, movement, and destination.
A good problem solver has a very wide range of abstract ideas and concepts and concrete tools they can use to model and explain problem, solution, & destination. Sometimes raw computational intellect can arrive at stunningly good proposals, can see brilliant paths through. But more often, my gut tells me it's about having a breadth of exposure, to different techniques and tools, and being someone who can both see a vast number of ways to tackle a situation, and being able to see tradeoffs in approaches, being able to weight long and short term impacts.
> Its very rare imo that computational problems emerge fully formed & ready to be tackled like proofs.
In my generation, the perfect example is Python's Timsort. It is an modest improvement upon prior sorting algorithms, but it has come to dominate. And, frankly, in terms of computer science history, it was discovered very late. The paper was written in 1993, but the first major, high-impact open source implementation was not written until 2003. Ref: https://en.wikipedia.org/wiki/Timsort
It has been reimplemented in a wide variety of languages today. I look forward to the next iteration: WolfgangSort or FatimaSort or XiaomiSort or whatever.
The a capital example of an exception that prove the rule.
I absolutely value & have huge respect for the deeply computational works that advance us all along!
But this is an exceedingly rare event. Most development is more glue work than advancing computional fundamentals. Very very very little of the industry is paid to work on honing data structures so generally.
+1. Excellent description of how the skills relate. I often read, listen to poetry when stuck in a math or programming problem. ... or just talk to the rubber duck. :-)
The problem with doing things right the first time is some people look at it and just say, “well of course it should work that way.” Yes but did you think of doing it that way?
Although the study by Prat et al. (2020) suggests that linguistic aptitude is a better predictor than numeracy for learning programming in Python, it should be read carefully, as it can easily be oversimplified.
To begin with, the study measures functional numeracy: the ability to solve everyday numerical problems. This is quite different from the kind of advanced mathematics often associated with programming, such as formal logic, symbolic abstraction, or the use of formal languages (as found in denotational semantics or type theory).
These more abstract skills—not basic arithmetic—are essential for understanding recursion, type inference, or algorithm design. That functional numeracy has low predictive power in this study does not imply that deep mathematical reasoning is irrelevant to programming.
Moreover, the language used in the study is Python, which was explicitly designed to be readable and semantically close to natural language. This may give an advantage to individuals with strong verbal skills, but the results don’t necessarily generalize to languages like C, Lisp, or Haskell, where symbolic and logical density is much higher.
Finally, language and mathematics are not opposing domains. They share cognitive underpinnings, such as working memory, executive attention, and hierarchical structure processing. The key is not which one "wins," but how they interact and complement each other in different programming contexts.
> These more abstract skills—not basic arithmetic—are essential for understanding recursion, type inference, or algorithm design.
No they're not. Academia has spent decades trying to formalize many aspects of programming and continues to be confused by the lack of correlation between comp sci grads and innovative programmers. Why is it that the drop-outs are succeeding so wildly?
Recursion, for example, is learned by most of us real world achievers when we hit a brick wall in programming that other methods won't solve, and we have that aha moment of "this is why this exists". Not because we studied advanced math with symbolic abstraction, denotational semantics and type theory.
The uncomfortable truth is that almost all of professional programming and innovative programming (creating useful stuff never before seen) never uses any of the advanced math skills that are prerequisites in every degree program. I think much of the sadism around teaching this is perpetuated by "I did it so you have to" and academic gatekeeping.
When you get really really good at programming and hit the most productive zone in your life, it feels like language. That you have the ability to just say it.
The drop-outs are "succeeding wildly" because people like Turing, Church, and Hilbert invented the sand pit they're playing in.
Knuth created LaTeX. Pandoc is written in Haskell, famous for being a completely useless academic language with no real purpose beyond torturing undergraduates (it says here.) Efficient search and data compression algorithms aren't hacked together in late night hobby coding sessions.
Cryptography, digital signal processing for images, sound, and video, and ML core algorithms are all mathematical inventions. The digital world literally runs on them.
"Real world achievers" might want to try being a little less parochial and a little more educated about the originators of the concepts and environments they take for granted.
Vibe coding "Social AI chatbot network with ads = $$profit$$" or "Cat videos as a service" is only possible because the entire field stands on the shoulders of mathematical giants.
Well said. I guess such ignorance is not that surprising given that most "programming" today is glueing code together with scripting languages. This is really just a form of configuration. It certainly requires domain knowledge and a tolerance for digital bureaucracy, but not always maths. I do wish schools would do a better job of teaching that the entire fabric of western civilization rests upon mathematics.
It sounds like these advanced skills might be needed to build the platforms that the dropouts build on top of. But I agree that, as a professional programmer of 20 years with a 4-year business degree from a state school with a weak math background (and also as someone with a talent for writing and language), I have very little need for advanced math in my day-to-day.
You are making a rather wild assumption that dropouts don't have the abstract math skills that correlate with programming skill.
I'd argue that if they can figure out recursion after hitting a brick wall like you describe, then that's a good indication they did have abstract math aptitude to begin with.
In my personal experience the hardest part of mathematics is it's grammar and language. It's very different from natural language, whereas programming is much closer. You can take nearly any math problem and convert it to pseudo code and it'll be much more understandable for those programmers who never studied (or struggled with) advanced math.
Programming requires a base level of natural language aptitude that nearly all adults have, there's diminishing returns for anything approaching the levels of a poet or novelist for example.
When you get good at math, it also feels like language. Mathematicians see expressions like `|x-y|` and the read it as synonymous with "distance" just as they would "distancia". Physicists pick out fragments of large equations and say "this term acts sort of like an 'effective' mass", etc. There is a sort of vocabulary to mathematical patterns.
Picking up a skill without intentional study is great, but you still learned the skill. Programming languages are formal languages. Most mathematicians don't study foundations either.
Professional programming doesn't often make use of specific advanced mathematical knowledge, but I find it makes everyday use of the skills.
My own definition of mathematics for the sake of this comparison:
- If you can write a formal proof, starting from some assumptions and proving some result, it is mathematics. The assumptions are the axioms. Then you use logic, which is the same programming logic, and then you get to some result.
It can be an invariant in a programming function, it can be a more general result, if you can write a proof, it is mathematics. Most algorithms involve proofs, so they are mathematics.
It has nothing to do with it being "sadism" or academic gatekeeping.
These people are doing mathematics without knowing it is mathematics. That's all.
> Why is it that the drop-outs are succeeding so wildly?
Here is where you can learn about confirmation bias and educate yourself.
This totally makes sense to me. I've always been a very good / fast reader, which has been incredibly useful in my programming career. I had a good SAT math score (710) but got a perfect score (800) on the SAT verbal (in the late 90s).
I remember when I first started working on my Master's project on wireless sensor networks, my advisor sat me down and said "I think I know a good project for you. I want you to print out the source code for TinyOS, study it for a week, and come back to me when you think you know enough to make these changes." This was a sort of formative experience for me, and ever since when joining a new project I made sure to take the time to read through the code to understand how things fit together.
Also checking in as someone with strong language skills. When I was a kid in school a million years ago I was actually rather weak in math and crapped out around trig, but excelled in English, Spanish and C. Now I'm a senior web dev, and I still can't do much math beyond the basics required for everyday living (though 18 years in food service means I'm really quick at estimating percentages to within delta)
When I tell you that I was a server and bartender it should all become very clear. Though your guess isn't far off, it's a skill that's served me well since I've picked up breadmaking as a hobby
> I've always been a very good / fast reader, which has been incredibly useful in my programming career. I had a good SAT math score (710) but got a perfect score (800) on the SAT verbal (in the late 90s).
This is me and also many of the CS students in my cohort, and AFAIK something that universities actively selected for, and students also self-selected around in an era of RTFM/MUDs/IRC before LLMs or youtube. The best programmers I've worked with are still always very linguistically brained.. polyglots even when they didn't have to be, or with a long track record of engaging with difficult literature. If nothing else.. just very witty in that certain way that's meta-cognitive, meta-linguistic.
This is still true I think but it's much harder see out in the wild due to the degree/career trajectory popularity. Plus as long we're optimizing for leet-coding even though built-from-scratch algorithms are a very rare need compared to skills with good design/exposition.. naturally the math-brain is favored.
Remember people whining about weak ecosystems? That's a lot of opportunity to build things from scratch instead of making another - for real this time - IRC clone with Electron.
Early, strong reader here. Off-the-charts spatial reasoning, as measured by tests.
Terrible at math, I hate it and feel dyslexic trying to read most mathematical writing. I excelled at it in elementary school, then quickly came to feel frustratingly stupid at it as it became less about algorithms (more on that in a bit...) and all about equations and abstract stuff with unknown applications.
However, programming was natural and easy to pick up. I've repeatedly had to take more time convincing myself I actually understand some supposedly "hard" thing, like pointers or recursion, than it took to learn them in the first place, because they were in fact very easy to understand so I kept second-guessing myself—"I must not get it, because that was easy". I've been the go-to guy for "hard" and "low-level" problems basically everywhere I've worked.
What I've noticed is that when I must read math, the only way I can get any headway is to turn everything into steps, through which some example values may pass and affect one another. I have to turn it all into algorithms. Algorithms, I can get. Attempts to express meaning through equations and proofs, no, I have to painstakingly turn every single boundary between every symbol into a step and "walk through" it to have any hope of understanding it, and once I do, this new understanding only barely illuminates the original representation.
I think programming clicked for me because, as typically encountered and taught, it's very heavy on algorithms and very light on other varieties of mathematical presentation. Plus, having so very much more context available about what variables represent and what routines do, than a jumble of letters and symbols. FFS, if we say Perl is line noise, what's mathematical writing? Straight gibberish from a brain-wrecked Cthulhu cultist? Perl's the clearest thing in the world by comparison!
... where I do run into trouble is languages with "mathy" syntax, where the idiomatic style favors single-letter variables and accomplishing-things-by-asserting-equality. I can't read Haskell to save my life. Put the same supposedly-tricky concepts (monads, type classes) in any more-ordinary language, and it's easy, but when I tried to learn them using Haskell, I couldn't get anywhere at all. Shit, it takes me forever just to understand fizzbuzz-level programs written in Haskell.
Very interesting, as I am nearly the opposite. Bad spatial reasoning though pretty good at most other things. In school, geometry was fun and interesting - I enjoyed proofs. And I majored in English in college with a CS minor, so I think there must be a variety of in-roads for programming.
I think one "verbal" skill that has served me well is fast reading. When I had to read a 300 page novel once a week you learn how to skim for key elements, which is immensively useful for getting up to speed in a new codebase/language or locating a bug.
Same but I am literally math dyslexic (formal diagnosis of dyscalculia).
It prevented me from having a CS degree, I was unable to complete the math courses, but as far as actual programming and "software engineering" goes (design, etc) it's never hindered me. I can work out the logic and I let the computer do the math.
Edit: I'm downvoted below zero for this comment. I don't know what people are so offended by?
Yeah, to be clear, I'm not diagnosed with anything like that, and am only likening the experience to what I imagine it's like for dyslexics of humans language as a ready metaphor—meanwhile, the concepts aren't hard for me, like, a lot of people with no diagnosis of anything fall off the math-wagon right around the time operations on fractions are introduced, but the concepts have always posed no trouble to me and I breezed through early potential trip-ups like that. The style of presentation, specifically, is what gives me such a hard time and is what makes trying to approach even fairly easy "real" mathematics so hard for me.
> It prevented me from having a CS degree, I was unable to complete the math courses, but as far as actual programming and "software engineering" goes (design, etc) it's never hindered me. I can work out the logic and I let the computer do the math.
This is what's wild to me: I have a long, successful career in a "STEM" field that's allegedly math-heavy, while being practically incapable of working with math. Like, it's never even been slightly a problem. I can't relate at all to characterizations of programming as heavy on math. It's never been my experience of it, and at this rate, probably never will be. If it were, I'd for-sure be in a different job.
They probably suspect you to be a slow coder who writes code with bad abstractions because they percieve the maths skills as how one does things in software engineering.
Oh wow, are you me - I had an almost identical SAT experience. Oddly enough compared to another poster in this thread, I love functional programming and Haskell and studying things like dependent types in a programming context has helped me patch up my crappy understanding of "actual math", proofs, etc.
I came here to comment the same -- also ready to share my SAT scores from the 1980s. I was also much highter on the verbal side and I have had an awesome software engineering career using many, many languages (often together) on large, professional projects at large companies anybody here would recognize.
There is no such thing as a "language brain" or a "math brain" unless you show experimentally that those bunch of neurons can be grouped into two non-overlapping regions dedicated to "language" and "mathematics".
Mathematics itself is a human-made formal language that can be bootstrapped from definitions and axioms of logics and set theory, which have to be given in human language first.
Experienced mathematicians read formal theorems written in Greek letters off their blackboards as if it was normal English, suggesting they think about it like it was just normal English. This is not to say they cannot also view in front of their mental eye visual representations isomorphic with that language if they chose to.
Dyscalculia, dysgraphia and dyslexia can all present independent of one another, heavily implying that different processes govern math, reading and writing. To call them "language brain" and "math brain" is simply a useful fiction, whether they're actually separate piles of meat is irrelevant.
In higher math (typically 2nd year and onwards at US universities) there is fairly little explicit calculation, and math is more about studying the (formal) relationships between mathematical objects. For example, we can abstractly define topological spaces and abstractly define algebraic “groups,” and algebraic topology studies the relationships between topological spaces and algebraic groups. This is somewhat like having a “data type” defined by some library, examining ways it can be transformed into data types defined by some other library, and studying what invariants hold under those transformations.
In some sense, an undergraduate math education is akin to learning the “standard library” (in the software engineering sense) of higher mathematics. Most courses start with basic abstractions of some mathematical object and repeatedly construct more and more abstractions on top of those. The structure of those abstractions is similar to how you might build a library. A professional mathematician is expected to be fluent in the mathematical standard library, just like how you might expect an experienced software engineer to be fluent in Python’s standard library.
If this analogy is true, people who can learn Python relatively quickly might be able to also learn higher mathematics relatively quickly under the right pedagogical environment.
One personal project I'd love to find the time for is to try to write out a sequence of math notes that essentially builds up a Lean repository, intentionally "refactoring" as it goes along. So you do some explicit coordinate vector or system-of-equation calculations with both real and complex numbers, eventually define interfaces for fields and vectors, do your linear algebra proofs, realize that linear endomorphisms are almost-but-not-quite a field, so define rings and go back and refactor your field proofs to use them, realize vector spaces are "K[T] not-quite vector spaces", define modules, refactor your vector space code, etc.
This was kind of how math classes worked, but without that explicit phrasing. It would certainly make the analogy between the two activities more obvious. I also wonder whether people would have less trouble with quantifiers if they were phrased in programming terms: a proof of "forall x, p(x)" is a function x=>p(x), and a proof of "there exists x such that p(x)" is a pair (x, p(x)). e.g.
proof of UC => C = (epsilon, h, x_0) => let delta, h2 = UC(epsilon,h) in (delta, h2(_,x_0,_))
So when you're trying to figure out how to do the proof, it's clear what kind of type you need to return and your IDE could help you with autocomplete based on type inference.
It is hugely judgmental of people making claims and doing studies who don't understand that elementary arithmetic is a tiny part of math. I judge them. It's ok to be ignorant of a topic, but they shouldn't discuss it or make suggestions.
The comment below/above sort of explains it, except I'd argue covering it starts/can start/should start much earlier than at university. Basically, the vast majority of most branches of math, especially pure math, involves very little in the way of calculation. And the way math gets taught (and stupid claims about "math brain") means that many kids who aren't in the top few % of "doing calculation" never get to do the classes where it's less important.
LLMs, trained on words/tokens and symbols and logical combinations of them, are proving to be good at math, and bad at calculation/arithmetic. If an LLM went to school we'd never let it train on real math tokens. It would get shoved in the corner as a "model bad at math" because it was a "model bad at arithmetic".
Most math is more about definition, proof and structuring/relationships than about numbers. It is a bit of a joke among math students that numbers above 5 are effectively infinity, they crop up that rarely.
This. And we do a huge disservice to tons of people by basically instilling the opposite idea. So many of the people I know that either claim they are "not math people" or who claim they "dislike math" say this because they were subject to nothing but tedious calculation in school and were never show the logical, conceptual, and actually interesting parts of mathematics. I despised math until I read Frege's foundations of arithmetic in college since it was the first time I ever had someone actually show the point and make it interesting.
The other thing I don't like about this framing is the implicit false dichotomy. The best programmers I've met often have strong language and math skills.
Poetry is the art of giving different names to the same thing. Mathematics is the art of giving the same name to different things. (Henri Poincaré)
It's basically being good at "naming things", in particular abstractions.
> Poetry is the art of giving different names to the same thing.
Did Robert Frost say something like, "all language is poetry?" A chair is the term I learned to describe the object I am currently sitting on while typing this comment. Germans might call the same object a 'Stuhl', the French might call it a 'chaise', etc.. My point being that the object isn't technically any of those words, but rather those words symbolically map to that object.
Yes, exactly, just watch how little babies recognize and communicate about hundreds of objects, emotions and situations before they're able to utter momma or dadda.
I wanted to say something like this, but in the opposite direction. I think perhaps more so there’s a difference between abstract and concrete thinking? But tests like the SAT in my experience (which is now 20 years out of date) treat the verbal section as more of a set of abstractions (analogies, for instance) and the math the same. I got almost equal marks in each (660 verbal/650 math) on my only try.
Yeah, the brain is a sparse MoE. There is a lot of overlap in the hardware of the "language brain" and the "math brain". That being said, I can discuss software concepts in a foreign language, but struggle with basic arithmetic in anything but English. So while the hardware might be the same, the virtualization layer that sits on top might have some kind of compartmentalization.
Yes, but there is a visual/spatial ability and a verbal/mathematical, and you can do maths with the visual/spatial ability or with the verbal/mathematical ability.
It leads to different styles of thinking and problem solving.
> There is no such thing as a "language brain" or a "math brain"
Tell that to the people who designed the study, the people that approved the study, and the people that funded the study.
At the end of the day, the study set out certain criteria for assessing certain skill types/competencies and divided the people by those well defined criteria. I think it’s pretty hard to argue against the idea that people might have at least some level of aptitude for different types of activity and skill.
I think its worth noting the study did not actually make this claim, the news article did. I suspect the study authors would agree the term "language brain" or "math brain" is misleading.
What about Wernicke's area and Broca's area? If damaged they produce causal changes to our use and processing of language. I agree it's not as simple "language brain" but there are physical structures in the brain that are directly related to language.
It think this is silly on multiple accounts. I'll claim that there's not real thing such as a "language brain" or "math brain." I'll also claim that most people don't know what math is, and that their evidence supports a "math brain".
Math isn't about calculations/computations, it is about patterns. You get to algebra and think "what are these letters doing in my math" but once you get further you think "what are these numbers doing in my math?"
A great tragedy we have in math education is that we focus so much on calculation. There's tons of useful subjects that are only taught once people get to an undergraduate math degree or grad school despite being understandable by children. The basics of things like group theory, combinatorics, graphs, set theory, category theory, etc. All of these also have herculean levels of depth, but there's plenty of things that formalize our way of thinking yet are easily understandable by children. If you want to see an example, I recommend Visual Group Theory[0]. Math is all about abstraction and for some reason we reserve that till "late in the game". But I can certainly say that getting this stuff accelerates learning and has a profound effect on the way I think. Though an important component of that is ensuring that you really take to heart the abstraction, not getting in your own way by thinking these tools only apply in very specific applications. A lot of people struggle with word problems, but even though they might involve silly situations like having a cousin named Throckmorton or him wanting to buy 500 watermelons, they really are part of that connection from math to reality.
This is why "advanced" math accelerating my learning, because that "level" of math is about teaching you abstractions. Ways to think. These are tremendously helpful even if you do not end up writing down equations. Because, math isn't really about writing down equations. But we do it because it sure helps, especially when shit gets complicated.
Or just your regular 8/9th grade Algebra course where you should have learned systems of equations. That's where I first encountered using matrices for systems of equations at least. (maybe it was Algebra 2+ but it was definitely before taking Linear Algebra in college).
I'm not sure how to interpret this comment, it could easily be read in multiple ways. (Depends on priors)
Regardless, it's a shockingly common occurrence. I'd agree it's not the right way, but it is also a common way. That perspective might define those priors
When growing up, I'm sure my fellow students and I were not the only ones get responses to questions like "Why can't one divide by zero?" or "Why is Pi necessary to compute to area of a circle?" answered like "because that's just how things work" or "because it just is."
I'm not going to sit here and act like I was a star student or anything. I was more of a class clown type. I absolutely hate math and all things math. That was, until I went to college. A switch flipped when I was in a Calculus II class.
Our professor asked, "What is a 100 divided by 0?" People in the class responded with, "You can't divide by 0 because it's undefined." To which our professor responded with, "Why?" Then a student took out a calculator and showed the professor that the answer as indeed undefined. To which he responded with, "Ok, how do you that calculator is correct? Do you just believe it because people told you that dividing by zero is undefined? Ok, the answer is undefined... but why?"
Right then and there, a switch flipped in my brain. I realized that I was basically institutionalized to just not question math and to just accept that things work a certain way "just because." It was at that point I actually started to become interested in math, and it completely changed my outlook on math in a positive manner. I love math now (despite being horrible at it).
This reminds me of a lot of what I see in grad school and academia. Something akin to what Tom Wolf talks about here[0]
> I’ve always been a straight-A student.
> if something was not written in a book I could not invent it unless it was a rather useless variation of a known theory. More annoyingly, I found it very hard to challenge the status-quo, to question what I had learned.
I don't think this part is isolated to math, but there's a lot of acceptance for "because" being an answer. Being in an authoritative position and being challenged can be frustrating, but I think a lot of that frustration is self-generated. We stifle creativity when young and it should be no surprise that frequently when people challenge, they don't have the tools to do so (or be receptive) effectively. Truthfully, "I don't know"[1] still shuts down the conversation.
But I think people themselves are uncomfortable with not knowing. I know I am! But that isn't a feeling of shame, it is a feeling that creates drive.
[1] Alternatively, variations like: "That's a good question, I don't know" or "Great question, but we don't have the tools to address that yet". The last one need not undermine your authority either. Truthfully, the common responses resulted in me becoming strongly anti-authoritarian. Which, also has greatly benefited my career as a researcher. So, thanks lol
I remember my high school math teacher responding "that's an interesting question, I don't know" and then coming back with an answer the next lesson after researching the subject.
Yeah, I was worried it might come off as sarcastic or something. Hopefully the original commenter doesn’t take it that way, or reads this follow up. Unfortunately, adding stuff like “honestly” often makes a comment look even more sarcastic. It is a real pain.
But yes, I am expressing honest admiration—it was a bad move on the part of the teacher I think, which the poster seems to have overcome!
Well I appreciate clarifying! I originally took it as sarcastic but second guessed myself. I figured math is pedantic, so maybe being a little pedantic here could help haha.
I went through the same thing, it really isn't easy. But it is also why I don't blame others for not seeing it. It would be hypocritical to do so. I'm just not sure what's the best way to share, I'm open to ideas. Unfortunately we have to contend with priors that we see here, though I'm happy if my efforts even make one more person able to share in this beauty.
Even if we narrow math to numbers, what many people thing of as "math", we quickly move from arithmetic to patterns.
The patterns of repetitions of single things, repetitions over different things, repetitions between different things, repetitions over repetitions, ...
Natural/counting numbers are just the simplest patterns of repetition.
If we taught music the way we teach math. No wonder so many people are "bad at math".
“Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.”
Frankly, with all the mess that music notation is and the way it's often being taught, this may actually be a slight upgrade over status quo. I have never experienced more "that's just how it is" than in music.
> It think this is silly on multiple accounts. I'll claim that there's not real thing such as a "language brain" or "math brain."
It seems plainly obvious that this language just means “areas of brain that activate when dealing with math problems” vs “areas of brain that activate when dealing with language problems” and yes there is hard evidence that there is a difference between them.
Please reread my comment in full. I'm willing to bet we disagree on the definition of math. I'll strongly insist that the one I'm using is common among mathematicians. If you'd like to retort by saying it's about semantics then congrats, we're on the same page (and can be verified by reading you sibling comments and/or my replies to some of them)
It has nothing to do with the definition of math. If you put people in an MRI and give them random problems then you’ll notice a distribution in the patterns of brain activity that are distinct. Some of these groupings we’ll call “math like” because the problems are more like mathematics than linguistics. And in the other distribution we’ll call them “word like” because the problems fall into the category we associate with word problems.
The fact that we see separate patterns of brain activity is the interesting part not any semantic argument about what you or mathematicians feel should be defined as being math or not.
I am willing to bet good money that teaching pure mathematics like group theory, abstract algebra, or stuff like big Oh and combinatronics will do our kids a lot more good and advance them far more than calculus or geometry as taught in schools will.
I have the same experience as you did, in that studying by myself abstract algebra accelerated my learning and reasoning skills.
I think differential equations was where the switch from "I'm good at math" to "I think I want to actually know more of this" happened. I changed my major from engineering to physics. But agreed, abstract algebra was another big change and after that I took every class my university would let me and the chair of the department even made some new ones for me and some others.
I don't think this experience is uncommon (among people who get to these levels). Which is why it is really sad. Especially given how linear algebra and abstract algebra are in a lot of ways easier than calculus. I also think they should be taught earlier purely due to the fact that they teach abstraction and reasoning.
Just anecdotally based on my 2000something kid high school, the copious amount of tiering of math classes seemed to indicate to me that either we're really bad at teaching math to 80% of people, or only 20% of people will be able to handle precalculus.
We had basically 4 tracks: one ended with you doing algebra 1 in senior year, another ended with you doing trig in sr year, yet another that ended with trig (no precalculus), and then one that ended with you doing trig and precalc. That final class then had further subdivisions that were too small to have their own full classes: some kids just did precalc, some did calc 1 and took the AP Calculus AB exam and/or IB Math SL, while even even smaller group took AP Calculus BC and/or IB Math HL. The total number of kids who took the AP Calc AB exam in my year was 20ish, out of a graduating class of 500-600.
> the copious amount of tiering of math classes seemed to indicate to me that either we're really bad at teaching math to 80% of people, or only 20% of people will be able to handle precalculus
Or maybe it indicates that the people designing this system should be fired? Job security through complexity?
(Or maybe I’m just biased by the system I know… I’m just asking questions)
I'm not disagreeing, but just wanted to point out that this phrasing is commonly used by bad faith actors. I'm not saying you're using it this way and I legitimately do not think you are. But I wanted to point it out because I think your comment could be interpreted another way and this phrase might be evidence for someone to make that conclusion.
The classic example is conspiracy theorists. But lots of bad faith actors also use it to create leading questions that generally ignore or lead away from important context. Again, I do not think you're doing this. The rest of your comment makes me think you're acting in good faith.
I went to a small high school; my graduating class was around 30 people.
Math classes weren't separated by grade. So, I took Algebra 2 in 8th grade alongside a cross section of the school; there was one other 8th grader, two seniors, and a selection of people in between.
There was no path to take the AP Calculus AB. Trig / Calc A was offered as two semesters, and then Calc B / Calc C was offered as two more semesters, after which you'd take the BC test. There was also no such thing as "precalculus". Trig followed Algebra II.
In my Calc C class, there were probably 8ish people, of which one or two (besides me) would have been in my grade.
To be fair, the original study uses "numeracy", and the correlation numbers are almost exactly like the ones for "language aptitude".
At the same time, the study excluded "five participants ... due to attrition (not completing the training sessions), and one participant ... because he was an extreme outlier in learning rate (>3 sd away from the mean)." I mean, if you are to exclude 15% of your subjects without looking at their aptitude (maybe they didn't do it because it was too hard to pass the training tests to move to the next lesson, yet their language aptitude is high?), with only 36 subjects of which 21 are female (it's obvious programming is male dominated, so they only had 15 males: maybe it doesn't matter, but maybe it does), how can you claim any statistical significance with such small numbers?
That's fair, although I don't think it changes my response. And the article still really leads to the wrong conclusions. You want to teach children abstraction and reasoning? You teach them math. Not numeracy, math.
100%. I have 2 kids and what they learn at school is absurd... I have to put them in an extra class outside school where they learn a lot of the more abstract stuff, albeit as you say at a level which makes sense (not too deep).
Yeah, certainly we'd need to adapt classes to match their level, but we've tried before[0]. While it "failed" it also seemed pretty successful, especially in France and the USSR.
I mean is it any surprise kids get bored in math? They spend years learning the same thing. You spend years learning addition and multiplication (subtraction and division are the same operators). I sure as hell hated math as a kid. Especially doesn't help that it is always taught by someone who's also dispassionate about the subject. You really can get a process going where most middle schoolers are doing calculus and linear algebra (and advance ones are doing this in elementary). It isn't as far fetched as many would believe.
> I mean is it any surprise kids get bored in math?
Indeed. Kids did not evolve to learn things because they would be useful years down the road. But they did evolve to inhale knowledge at very high rates wherever it extends their capabilities in the moment.
My take on this is that math should be tied to kids crafting, experimenting, and learned directly in the context of its fun and creative uses.
Geometry screams out to be developed within a context of design, crafting, art and physical puzzles. Algebra, trigonometry, calculus ... they all have direct (and fun) uses, especially at the introduction stage.
The availability for graphics software, sim worlds, 3D printing, etc. should make math, from the simplest to most advanced, more fun and immediately applicable than ever.
Boredom with math is a failure to design education for human beings.
(Then there is the worst crime of all - moving kids through math in cohorts, pushing individuals both faster and slower than they are able to absorb it. Profound failure by design.)
> Indeed. Kids did not evolve to learn things because they would be useful years down the road. But they did evolve to inhale knowledge at very high rates wherever it extends their capabilities in the moment.
I strongly disagree with this. Children play. Most animals play. It is not hard to see how the skills learned through play lead to future utility despite potentially none at the time. We invent new games, new rules, and constantly imagine new worlds. The skills gained from these often have no utility beyond that that is self-constructed. Though many do have future rewards. I think we often miss those though because that path is through generalization. Learning how to throw a ball can help with learning physics, writing, driving, and much more. They don't perfectly transfer but it'd be naive to conclude that there aren't overlaps.
I think the truth is that as long as we are unable to predict the future, we really can't predict what skills will be useful and what specific knowledge should be pursued. We can do this in a more abstract sense as we can better predict the near future than far but that also means we should learn creativity and abstraction, as these allow us to adapt to changes. And that is why I believe we evolved these methods, and why you see things like play in most creatures.
> I strongly disagree with this. Children play. Most animals play.
I am not arguing against this. On the contrary, I agree with this completely.
Pairing the opportunity to learn and use highly general long term knowledge, toward progression in their own intrinsic interests (crafting, whatever) is the point I am making.
The alternative, a long line of math for math's sake, which neglects to harness and support each child's need and motivation to continually advance their own natural interests in all kinds of idiosyncratic directions.
What is the question asking? Why do you have Aphasia? I'm not sure what the relationship here is. Or are you suggesting you're good at math but not good at language?
I feel weird answering even if I infer the right question because it feels tautological. If you have Wernicke's Aphasia does it not create the possibility that I already have but your condition resulted in misunderstanding. Given the condition does it not create a high probability that a response will similarly be misunderstood? Is not Anosognosia quite common?
Maybe I'm really misunderstanding but honestly I'm not sure what you're asking
By the very expansive definition that you advocate for, what exactly do we do that isn't math?
One could argue that all we do is turn thoughts, senses, and memories into further thoughts and appropriate actions, which is applying a pattern, which is math. But at that point the definition is too broad to be helpful for anything but playing word games.
There are no rules. On the other hand, code is very formal. It is a set of rules. It has structures. There is wrongness and you can do verification that is not the result of opinion nor is it subjective.
But yes, math is very broad. But it doesn't cover everything. At least not at this time.
Math major here. 'is' is a reflexive relationship. If math is language, therefore language is math. I believe clearly language is not math, therefore math is not language. Math is described with language, Math itself is not language. It is a 'has-a' relationship vs a 'is-a' relationship.
Forgive me if I doubt, but your comment would strongly suggest otherwise, along with this one[0].
The reason for doubt is a failure in fairly basic logic. Your claim is:
¬(A ↦ B) ⟹ ¬(B ↦ A)
I'd expect anyone willing to claim the title of "math major" is aware of structures other than bijections and isomorphisms. The mapping operator doesn't form an abelian group.
The word "is", typically refers to the logical equals operator (written as "="). Let A = Math, and let B = Language, the phrase "Math is Language" is therefore "A = B". My claim is that "B != A", which implies "A != B" because "!=" is reflexive. "A != B" then contradicts the claim "A = B". In words: because "language is not math", therefore it cannot be true that "math is language".
This is not number theory, not set theory, just logic. I don't see how the properties of Abelian groups applies. I do suspect that non-abelian groups is a case where "¬(A ↦ B) ⟹ ¬(B ↦ A)" is false, and would be a contadiction to my counter claim if "¬(A ↦ B) ⟹ ¬(B ↦ A)" was my counter-claim. No, my counter claim is simply that because "B != A" therefore "A != B"
The statement "¬(A ↦ B) ⟹ ¬(B ↦ A)" is quite different, and I'm not 100% sure whether you are mixing set and logic notations? [0][1] It does seem you are bringing in number theory concepts, or we are misunderstanding one another?
I assume "↦" is the set mapping operator and "⟹" is logical implies, and "¬" is logical not. "¬(A ↦ B) ⟹ ¬(B ↦ A)" could potentially be phrased as: "If the element A cannot be mapped to B, then the element B cannot be mapped to A". I would agree that statement is not 'generally' true. Could you please clarify so we are not talking past each other.
[0] Per google: The symbol "↦" is a mapping symbol, typically used to indicate a function or relationship where one element maps to another. It's not a standard logical operator in the same way that symbols like &&, ||, or ¬ are. Instead, it represents a directional relationship between sets or elements, often seen in set theory and mathematical notations
As I said elsewhere (here for others), I'm not going to play this game of *willful misinterpretation.* Your very words do not hold up to the same bar your are attempting to hold mine to. I know you are upset you got caught in a lie, but fuck around and find out. ¯\_(ツ)_/¯ Domain experts can recognize other domain experts pretty easily
> No, my counter claim is simply that because "B != A" therefore "A != B"
We all know that this is not always true. It may be true in certain cases, in certain fields (pun intended), but you know that this is such a basic logical fallacy that it is taught to children.
Here's a counterexample so we can lay this to rest.
Let "A" be "a square"
Let "B" be "a rectangle"
B != A -> "A rectangle is not a square" (True)
A != B -> "A square is not a rectangle" (False)
Stop cosplaying, stop trolling, stop acting in bad faith.
Did you even look at my name? There's 2 people that should come to mind. Certainly any logician would notice.
> That's only in your head. Inventing claims so that you can pretend other people are wrong isn't a good move.
I'm not the only one that interprets a statement as "Math is Language" to be of the form "A = B" where "math" is A, "language" is B, and "is" is the equals operator, see: https://news.ycombinator.com/item?id=43874322
Seems kinda simple.. You're engaging in a almost pure personal attacks. Care to address the substance of how the 3 word long statement is not of the form "A = B", but is of some other different form? In which case, perhaps you can guide the conversation for why it is an interesting statement or not? If you want to change all the givens and use your own definitions, please provide those. Without common ground, this remains uninteresting.
> I'm not the only one that interprets a statement as "Math is Language" to be of the form "A = B" where "math" is A, "language" is B, and "is" is the equals operator, see: https://news.ycombinator.com/item?id=43874322
This appears to be a link that contains zero support for your sentence. Neither the comment you linked nor the response below it features such an interpretation.
> Seems kinda simple.. You're engaging in a almost pure personal attacks. Care to address the substance of how the 3 word long statement is not of the form "A = B", but is of some other different form?
You're in luck! I've already provided that material, and you responded to it. For further discussion, you'd need to have a better working understanding of English.
Very cool how the last sentence you leave off with always has to be a personal attack. It's breaking the rules of the discussion, irrelevant. Stop trying to win points. Let's focus on the substance.
> This appears to be a link that contains zero support for your sentence. Neither the comment you linked nor the response below it features such an interpretation.
It's interesting, because there is a disagreement about whether there is a paradox or not. There is a paradox if you take my perspective (that an equals relationship is being expressed), but none if an implication relationship is assumed.
These two sentences:
- "By that same logic you could also say that language is math"
- "Not quite, but the inverse is true."
The "not quite" says that "language is math" is not true. So we have "A = B", but "B != A", which is a paradox. OTOH it's not a paradox if what is actually being said is "A => B" but "B !=> A"
> You're in luck! I've already provided that material, and you responded to it.
Hello bad faith! I hope you are doing well today. Let's end the conversation here.
You're not making it look good. A common use of be is to express set membership rather than identity. For example, "two is an even number" or "tigers are cats".
We may hope that one day you'll come to realize that set membership is not reflexive, and - more to the point - also not symmetric.
> A common use of be is to express set membership rather than identity
Google for "logical "is a" vs logical "is"".
Google AI answers this:
> "is" typically represents an equality relation
Rest is from the AI response:
In logic, "is" typically represents an equality relation, while "is a" (or "is of the type") represents an inclusion relation. "Is" indicates that two things are the same or identical, while "is a" indicates that one thing is a member of a larger class or set of things.
Logical "is" (equality):
Meaning:
"A is B" means that A and B are the same thing, or have the same properties.
Example:
"The Eiffel Tower is in Paris" (the Eiffel Tower and the thing in Paris are the same thing).
Logical "is a" (inclusion or type):
Meaning: "A is a B" means that A belongs to the category or class of things that are B.
Example: "A dog is an animal" (dogs are a type of animal).
You're saying that "Math" and "Language" are 'sets'? And that the phrase "Math is Language" should be interpreted as expressing the relationship between sets?
If we do want to talk sets, that seems far more interesting.
The statements like "Math is a language", or "Math has equivalence classes within languages", or "Mathematics are a Language" are slightly more interesting to consider IMO.
>> Math major here.
> You say that like you think it's a qualification?
Agree, appeal to authority fallacy. I take that over mis-framing any day though.
Any old-timers might appreciate that we're arguing over the meaning of the word "is" =D
Finland tried teaching maths to children using Group Theory in the 70s [1], but the results weren't that good; it proved to be too abstract for young kids.
Ultimately, I believe basic algebra and geometry are the most important takeaways from math classes for most people.
I am no expert in pedagogy of mathematics, and I am sure someone will correct me if I am wrong, but I think there was/is a Russian academic program in which students were/are basically only taught algebra in an iteratively increasing manner.
As it was explained to me, one wouldn't take a "Calculus I" class as a prerequisite for say an entry-level engineering course. One typically had such a strong foundation of algebra, that when encountering a problem that required calculus, the student would just learn the necessary calculus at that point in time. In other words, with such a strong algebraic background, other aspects of math, within reason, were much easier to grok.
> the results weren't that good; it proved to be too abstract for young kids
You cannot make that conclusion as a result of the evidence. Yes, the evidence might support that conclusion, but there are many others that also could. For example, they could have just been really bad at teaching. This even seems like a likely one as it is difficult to perform such a reformulation and to do so broadly and quickly.
The other reason I'm willing to accept alternative conclusions is that France and the USSR had far more success than Finland (or even America). Their success contradicts a claim that "[it is] too abstract for young kids". You'd need to constrain it to something like "[it is] too abstract for Finish kids" which I think both of us would doubt such a claim.
I think you might be missing the point of the article: the study being cited isn't trying to establish the existence of a "language brain" or a "math brain", that's just the way the headline editorialized it to help people understand the conclusions.
The conclusion of the study was that linguistic aptitude seemed to be more correlated with programming aptitude than mathematical aptitude, which seems fairly interesting, and also fairly unconcerned with which specific physical regions in the brain might happen to be involved.
> The conclusion of the study was that linguistic aptitude seemed to be more correlated with programming aptitude than mathematical aptitude
And this is what I'm pushing back against and where I think you've misinterpreted.
> They found that how well students learned Python was mostly explained by general cognitive abilities (problem solving and working memory), while how quickly they learned was explained by both general cognitive skills and language aptitude.
I made the claim that these are in fact math skills, but most people confuse with arithmetic. Math is a language. It is a language we created to help with abstraction. Code is math. There's no question about this. Go look into lambda calculus and the Church-Turing Thesis. There is much more in this direction too. And of course, we should have a clear connection to connect it all if you're able to see some abstraction.
The word "is", maps to the logical "equals" operator. I agree with the example, but I don't agree it is relevant. There is no implies operator.
The statement "Math is Language", where A is Math and B is Language, maps to the logical assertion: "A = B".
If we are going to really be kinda twisty and non-standard, we could interpret the english "is" to be "is an equivalence class of". Which would map to your example pretty well: language is indeed an equivalence class of math, but math is not an equivalence class of language. Though, nobody is talking about implies operator or equivalence class here.. It's a "is" relationship, logical *equals*
You point out the "is a" relationship, not the "is" relationship, they are different. [0]
Find examples with two singular nouns and just the word 'is'.
The phrase in question: 'Math is language' is an example, or something like 'food is love' is too. I concede you could interpret those last few sentences with poetic license to be read more like: "A is a form of B", or "A is a B" - though that is not what was written and this is not a place to expect that much poetic license.
*edit*: a minute later, thought of a good example. "ice is water". True that "ice is a form of water", but strictly speaking no, "ice is not water". I'll concede there could exist an implied "is a", or an implied "is a form of", but that is poetic license IMO.
[0] Google AI summarized it pretty well: google "logical "is a" vs logical "is"
> In logic, "is" typically represents an equality relation, while "is a" (or "is of the type") represents an inclusion relation. "Is" indicates that two things are the same or identical, while "is a" indicates that one thing is a member of a larger class or set of things
> You point out the "is a" relationship, not the "is" relationship, they are different.
Well, what you reacted to was, let me copy'n'paste, "Math is a language". It was you who insisted that "is" in this sentence maps to "equals" relation, so thanks for agreeing that you were wrong.
Honestly I don't think his point even stands. We were using English to communicate and English doesn't have the strict rules of mathematics. That's literally why we created math (which I'll gladly call "a class of languages"). He's right, "is" maps to "equivalent" but he's also wrong because "is" also maps to "subset" and several other things. "Is" is a surjection.
The problem here all comes down to seadan83 acting in bad faith and using an intentional misinterpretation of my words in order to fit them to their conclusion. I'm not going to entertain them more because I won't play such a pointless game. The ambiguity of written and spoken language always allows for such abuse. So either they are a bad faith actor "having fun" (trolling) finding intentional misinterpretations to frustrate those who wish to act in good faith or they are dumb. Personally, I don't think they're dumb.
It's such an absurd thing to argue about that I just assumed that some massive brainfart happened there. It happens to everyone, not everyone doubles down on it though.
But then again, isn't a good portion of this thread non-mathematicians arguing about what math is? I really thought ndriscoll put it succinctly[0]
> It's like trying to argue about the distinction between U(1), the complex numbers with magnitude 1, and the unit circle, and getting upset when the mathematicians say "those are 3 names for the same thing".
I fear the day some of these people learn about Topology.
The point is that linguistic aptitude _is_ math aptitude, and vice versa.
From my experience, my ability to articulate myself well is bound up with my ability to abstract and detect patterns. It is the same thing I apply to crafting software, the same thing I apply to creating visual art.
I think high-cognitive-ability people segregating themselves into artsy vs mathy people has more to do with their experiences in their formative years.
Unfortunately I don't know any direct resources. I really do hope they are out there and some will share.
But if you're willing to hunt, I know that this idea was attempted before[0]. France and USSR had better success than the US. I'm sure there are still people working in this direction. I don't have children, but fwiw I've taught my nieces and nephews algebra and even some of calculus before they were 10 just in visiting time during vacations. They were bored and it seemed more fun than talking about the drama, politics, and religion that the rest of my family likes to spend most of their time on. Kids were similarly disinterested in that stuff lol. I've also seen my god{son,daughter} be able to learn these types of skills, so I'm highly confident it is doable.
> I'll claim that there's not real thing such as a "language brain" or "math brain.
Did you even read beyond the silly headline?
The article itself is about pre-testing subjects on a range of capabilities from problem solving ability to second (foreign) language learning ability, and then seeing how these correlated to the ability of the test subjects to learn to code.
The results were basically exactly what might be expected - people who learned Python the quickest were those who scored the best at learning a second language, and those who learned to wield it the best were those who had scored the best at problem solving.
Not surprisingly math ability wasn't much of a predictor since programming has little to nothing to do with math.
I think the above comment stands. The point is, what do they consider maths ability? High school maths has very little to do with programming but university-level maths certainly feels very similar to it in structure. Many of my (good) classmates in my maths degree were very bad at things like mental arithmetic. So maybe "maths ability" (defined some way) isn't very important for being good at "maths" (proper).
Yes. I'll also refer you to the HN guideline on this manner. You're welcome to disagree with me but you must communicate in good faith and unless you have a very specific reason for thinking I didn't "RTFM" then don't make the accusation.
I'm happy to continue discussing, but only on those terms. In fact, I think we're in far more agreement than your tone suggests. But I think you missed the crux of my point: math isn't number crunching
Your response opened with addressing the headline, which anyone who had "RTFM" (RTFA) would have realized was unrelated to the body of the article. You then veered off into a tangent about the nature of math which again was not addressing the content of the article.
The underlying nature article, linked from the posted story, makes it even more clear what is being discussed, with the abstract stating:
> This experiment employed an individual differences approach to test the hypothesis that learning modern programming languages resembles second “natural” language learning in adulthood.
Thank you. I’ve long held that calculus, formal logic, combinatorics, etc. should be started in elementary school, separate from arithmetic. Even after stripping away the numbers and procedural knowledge, there is tremendous real world value in simply understanding many of the concepts. Plus, (and I say this half-jokingly) won’t we all be using llms to vibe-math everything that needs math-ing anyway?
First red flag is here. The title rewrote this to be language only. That problem solving skills are relevant is pretty obvious, but language less so.
I’ve been programming for most of my life and I don’t consider myself a very good speaker. My language skills are passable. And learning new languages? Forget it. So I’m skeptical. Let’s look at the study.
First of all, “math” becomes “numeracy”. But I think programming is probably closer to algebra, but even then it’s less strict and easier to debug.
> Assessed using a Rasch-Based Nuemracy Scale which was created by evaluating 18 numeracy questions across multiple measures and determining the 8 most predictive items.
> That problem solving skills are relevant is pretty obvious, but language less so.
To me, problem solving ability is precisely the same as the ability to articulate the problem and a solution. I don't see a major difference.
If you can solve a problem but you can't articulate what the problem is or why the solution will address it, I wouldn't call you a good problem solver. If you can articulate the problem well but not come up with a solution, you're already doing better than a lot of programmers in the world, and I'd probably prefer working with you over someone who presents the solution without "showing their work".
In fact, what is problem solving without such articulation? It's hard to even grasp what the skill means in a raw sense. Arguably creativity in this context is just the ability to reframe a problem in a more approachable manner. Many times, if not most times, such framing implies some obvious solution or sets of solutions with clear tradeoffs.
There are different ways of solving a problem though, some which require more critical thinking than others. Trial and error (in industry you can sound fancy by calling it “choosing parameters empirically”) requires no understanding of the underlying process, only the ability to measure the outcome.
If you’re debugging, you can get by for a long time by trying things until the compiler shuts up. It’s not efficient or good but people do it.
That's fair. I agree that there's more to problem solving than just linguistic ability, so I rescind my claim that they're indistinguishable, but I still think there's a deep relationship between the two.
I have a very difficult time trying to extract the difference between "linguistic ability" and "critical thinking", though:
1. The core difference between "critical thinking" and "uncritical thinking" is the ability to discern incoherency from coherency.
2. Coherency is evaluated at the linguistic level: do the terms bind in meaningful ways to the problem? Do any statements contradict each other?
3. The remaining aspect is "creativity": can you come up with novel ways of approaching the problem? This is the hardest to tie to linguistic ability because it sort of exists outside our ability to operate within an agreed context.
So while I agree these are distinct skills, I still have difficulty identifying what remains in "critical thinking" after linguistic ability is addressed.
I have found a fairly interesting correlation between people who are good at learning programming and people who are good at English spelling bees. Something about holding a lot of anecdotes and esoteric rule exceptions when performing an otherwise algorithmic process.
Are you a teacher and/or someone that runs spelling bees or otherwise someone that has done the proper statistics to distinguish this correlation with generalised intelligence. Just wondering where you get this anecdata from?
Makes sense to me; I've always been a naturally good English speller. My brain just knows how to store and retrieve this type of data, which has a high correlation with e.g. CLI interfaces and their idiosyncratic command structures.
I can’t decide if you’ve cheekily included the classic homophonic there-error. This isn’t technically an issue with English spelling, but I think you intended to communicate that CLI interfaces have idiosyncratic command structures, not that they are idiosyncratic structures themselves. If I’ve been taken in by pedant-bait, I apologize.
Lol, nope, just a case of my fingers deciding they know what I'm typing, or possibly autocorrect "fixing" something that was already correct. (I don't know why but the Android GBoard has been introducing crazy numbers of wrong homophones recently.)
I'm a terrible speller; it's taken me ten years of typing "ammend" to learn its proper spelling. It also sort of goes against the "programmers are lazy" meme: why memorize what a computer can detect and correct?
As others have commented, I believe strongly that to define “language brain” and “math brain” is a false dichotomy and bunk neuroscience. Mathematics IS a language in itself; a collection of symbols created to express concepts. This study works really hard to make a statement that most people already know: anyone can be a good programmer, mathematician, “whatever” if they have the interest, drive, and opportunity to do it.
An interesting book that illustrates the evolving societal perception of mathematicians (and by extension, computer scientists) over time is “Duel at Dawn”. The modern (sic recent) notion is the reclusive genius (maybe even suggestion of autism) who possesses the alien, superhuman, unattainable, too-cool-for-you ability to process numbers and information that you can only be born with. (Those familiar with the TV show “the big bang theory” would recognize the trope of the “Sheldon Cooper” character.) This is False.
The reality is that no one is born with the super-human ability to do anything - anyone who is very good at something has worked very hard to get good at that thing regardless of the perception.
edit: my initial criticism of “the study” was based upon the article. On a skim of the actual cited paper, I revised my specific criticism, but the actual paper still comes off as no more than a mild eugenics argument dressed in psychology and statistics jargon.
The article is extremely misleading, I dare even say almost malignant.
The study itself claims:
- fluid reasoning and working-memory capacity explained 34%
- language aptitude (17%)
- resting-state EEG power in beta and low-gamma bands (10%)
- numeracy (2%)
They take math skills to equal numeracy. The study itself implies this too. I disagree on a fundamental level with that. Math skills align much more closely to fluid reasoning than to numeracy.
At the beginning, the article mentions correlation with language skills AND problem-solving. Focusing only on language skills in the second half is misleading. According to the abstract of the original paper, problem solving and working memory capacity were FAR MORE important.
Also, the article doesn't mention "math skills". It talks about numeracy, which is defined in a cited paper as "the ability to understand, manipulate, and use numerical information, including probabilities". This is only a very small part of mathematics. I would even argue that mathematics involves a lot of problem solving and since problem solving is a good predictor, math skills are good predictor.
Going further, it seems like Language Aptitude was primarily significant in explaining variance in learning rate, measured by how many Codecademy lessons they completed in the allotted time, but wasn't explanatory for learning outcomes based on writing code or answering multiple-choice questions.
Seeing as Codecademy lessons are written in English, I would think this may just be a result of participants with higher Language Aptitude being faster readers.
I do think that language skills are undervalued for programming, if only for their impact on your ability to read and write documentations or specifications, but I'm not sure this study is demonstrating that link in a meaningful way.
100% agree, I've been saying this for years. I'm terrible with arithmetic but great with symbols and relations. Recursion is also fundamentally linguistic, and although our internal "stacks" for processing it naturally are quite small, language remains the easiest demonstration of recursion in our daily lives.
Oddly, I also use spatial intuition when thinking about stuff like stacks and the shape of data structures.
This is just a guess on my part, but I'd also bet that writing inductive proofs (or proofs in general) require more of the language brain than just doing math problems.
Language has an inherently recursive structure: I saw the man who saw the man who saw the man who saw the man who saw the man who... While our brains have practical limits to how deeply such things can actually be nested, language has a recursive tree-like aspect to it.
Yes, but “language is fundamentally recursive” doesn’t mean the same thing as “recursion is fundamentally linguistic”. Language is just one example of a recursive structure.
I am also unsure whether recursion is fundamentally linguistic, but I thought that “language remains the easiest demonstration of recursion in our daily lives” to be useful. If I ever write another essay about recursion, I'll now consider starting with a linguistic example before diving into recursive functions or data structures.
Language has nothing that corresponds to a recursive function, so that is a bad example. You can write a sentence that could correspond to a call to a recursive function, but its not the same thing as a recursive function.
If recursion was just writing the function 10 times like you did in language then people wouldn't struggle with it.
Recursive functions are just a subset of all possibly recursive concepts. In the case of human spoken language, the recursion exists in our characterization of the grammar. You could just as easily frame this in "iterative" terms just like you can make any recursive function iterative, but that's less convenient for analysis.
So in this case, "recursive function" would be "clause" or something like that; I'm no linguist. But clauses can embed clauses which can embed further clauses, etc.
I think your usage of recursive functions is just high-level logic—you're describing an inductive proof. We also frame a lot of our social games as recursive processes. But these are conscious processes that we can evaluate consciously; the recursion in spoken language is largely unconscious and very shallow.
> In the case of human spoken language, the recursion exists in our characterization of the grammar
But people are constructing sentences, not grammars. When you construct a grammar you can add a recursive part to it, that is true, just like in a programming language, but constructing grammars is not what people mean with language skills.
A sentence can't be recursive since languages in themselves has no concept of applying a concept, for that you need an interpretation of the language references. For example, you can have a recursive function written in a programming language that doesn't have a recursive grammar, the concepts are different things.
There are two ways that recursion intersects with language that are relevant here:
1. Our spoken and especially written grammar is recursive. We do handle this unconsciously. This is not related to our ability to reason about recursion at a high level, and recursive grammars are not necessary to do so. This is not a skill in the normal sense and we have only (very) limited ability to improve our capacity to interpret deeply nested grammars. However, this is still a useful illustration of what recursion IS, which is why I brought it up.
2. Language also introduces the ability to semantically reason about recursiveness. This is still a linguistic thing—you need a symbol and relations among symbols in order for recursion to be meaningful—but this is a skill and is likely very related to linguistic skill. This is the part that really helps you to program: ultimately, you're just reasoning about symbols and looking for incoherency.
Can you come up with some conception of recursion that doesn't involve symbols referring to themselves, directly or indirectly? Ie what is left of recursion when you remove the linguistic component?
Recursion itself is simply a conjecture. Nothing fundamental about it unless you believe Chomsky, but his is a speculative claim, not empirical per se.
I don't really know what you mean by "conjecture", but I thought apriori was implied by positing it as a linguistic construct. "Fundamental" doesn't imply empiricism at all. All of apriori knowledge for a language is a set of all sets of coherent statements: the outer set represents a set of implied axioms required to make the statements cohere. Recursion just broadens the complexity of the statements you can express, but it's fundamentally a concept that arises from language and can be evaluated for coherency (like all other apriori concepts).
Edit: added a definition of apriori knowledge.
Edit2: to put this another way, nobody is arguing that recursion doesn't exist. Or that it is empirically-derived. No, it's a useful construct to show certain relations.
Edit3: added a sentence
Edit4: The extent to which our own grammars are inherently recursive vs this being culture or technology is irrelevant to identifying the concept of recursion as an apriori, linguistic concept.
Edit5: i suppose you might also be referring to the idea that we naturally process recursion. I mean, we clearly, evidently do; whether or not that's inherent to being human is a separate question entirely. Hell in the free software world there's a whole recursive acronym meme that taps into some part of our brain and tickles it.
It kinda is empirically true that human language is recursive. Every human language ever discovered is recursive, except, supposedly, for one: Pirahã. And Pirahã has mainly been described by one researcher whose results are controversial.
If you define recursion as a symbol referencing itself, either directly or indirectly, and if you define language as a system of relating symbols to each other, recursion is a linguistic concept, it is a concept that describes a relationship between symbols. There are good reasons to define each concept differently, but if you identify recursion empirically, recursion won't "actually" exist outside of the description of the process. It's our characterization of the process that reveals the recursive structure, even if that characterization doesn't actually exist outside of language.
> If you define recursion as a symbol referencing itself, either directly or indirectly, and if you define language as a system of relating symbols to each other, recursion is a linguistic concept
But that isn't what we mean with recursive function. We don't call this recursive:
> We don't call this recursive... it's just incrementing x
That's not a recursive function as it's written, but you could certainly consider it a form of symbolic recursion. This just isn't a very useful characterization in an iterative/imperative context. You could frame incrementing as recursive, though—this is just peano axioms/church encoding.
I agree that there's enormous value in carving out mathematics from other linguistic reasoning, but I don't see defining as something as mathematic rather than linguistic is generally useful. You use the same skills to look for incoherency in both situations, but human language is generally expected to be incoherent on some level.
Besides, a lot of what people mean when they say they're bad at math is that they're bad at arithmetic, which is honestly understandable.
This makes sense for my path to math. In high school I was bad at math and good at learning languages. Then I started learning Python and realized that it was just like learning a language. Then at some point I realized math notation was just another language for expressing the kinds of things you could express in Python. Now I'm in a job where I do math every day and I read math textbooks for fun.
It helped that Python was meant to resemble natural language. I had learned C++ and Perl before but they never stuck, because I never made the connection to language. Ironically, since Perl was designed by a linguist!
I've noticed that the people who initially grok programming language syntax are often better at learning new symbolic systems, but I think this is actually due to a persistent type I error their brains make that is OK for early learning.
People who end up being the best programmers have a deeper appreciation for semantics and information flow, but tend to commit more type II errors early on, making them inferior intro CS students.
Much of the CS curriculum (and typically also the required maths curriculum) in universities still favors the first type of student over the second, driving out the most capable and creative minds.
I'm strong in the belief that kids having lower reading scores is directly related to lower math scores. Learning to decode words is directly translatable to learning to decode equations. Down to some of my favorite math passages would include how to read something as it introduces a symbol or other construct.
> There’s a lot of people out there who “aren’t math people,” but they just might be computer science people
This is beyond silly from my perspective. I know the field of CS is vast, but this seems to conflate programming with CS. My school was more theory heavy but there definitely came a point in certain paths of study where I didnt touch a line of code for a year, just pure math. I struggle to even understand how someone can think of this sentence - computer science at its core is underpinned by mathematics.
I just started reading this. Only 42 people in this study and only 36 completed the activity. I am a bit skeptical about the findings with such a small sample size.
I wonder if this finding hold across programming languages. I suspect the conclusions would be different for people programming in APL instead of Python, for example.
It surely helps. Take for example History, if I space out for whatever reason in the class and botch my examination on the Aztecs, that doesn't preclude me doing great in the classes about the Roman empire. History is like a sparse suburb, where a single house catching fire does not necessarily propagate and destroy the others. On the other side Math its like a skyscrapper, one floor build upon the other. If someone living in a lower floor botches a home renovation in his unit and damages a pillar, it risks taking down the whole building. In math if you botch a prerequisite, you'll ending messing your learning of subsequent concepts 95% of the time. So it is more unforgiving to people with problems like ADHD, that struggle to keep unbroken focus and discipline.
I'm very curious what this math pretest looked like, whether it was "proper" high level math or, like, computing some trig problems. Folks who have aptitude with algebra or number theory or topology, I'd expect that to be correlated, but not to rote computational math.
My take on skills related to programming is that it's a dance between writing skills and working memory. I've measurably bad working memory, which had led to compensation with writing skills, which led to a programming style that's fairly disciplined and focused on "wholesome" readability. A downside is that I still struggle interacting with code that's written by the working-memory-oriented programmers.
I've recently taken up a daily practice to improve my cognitive skills. Currently I'm using NeuroNation, and it is quite great. The stimulus to start the practice was getting my memory tested (clinically) and learning that it's below norm.
My experience is that they spit out reasonably looking solutions but then they don't even parse/compile.
They are OK to create small spinets of code and completion.
Anything past that they suck.
It's actually hilarious that AI "solved" bullshiting and and artistic fields much better and faster than say reasoning fields like math or programming.
It's the supreme irony. Even 5 years ago the status quo was saying artistic fields were completely safe from the AI apocalypse.
I disagree that the current generation of AI has "solved" artistic fields any more than it's solved math or programming.
Just as an LLM may be good at spitting out code that looks plausible but fails to work, diffusion models are good at spitting out art that looks shiny but is lacking in any real creativity or artistic expression.
> "looks shiny but is lacking in any real creativity or artistic expression."
My experience with that is that artistic milieus now sometimes even explicitly admit that the difference is who created the art.
"Human that suffered and created something" => high quality art
"The exact same thing but by a machine" => soulless claptrap
It's not about the end result.
A lot could be written about this but it's completely socially unacceptable.
Whether an analogous thing will happen with beautiful mathematical proofs or physical theories remains to be seen. I for one am curious, but as far as art is concerned, in my view it's done.
Truly great art, the kind that expands the field of artistry and makes people think, requires creativity; if you make something that's just a rehashing of existing art, that's not truly creative, it's boring and derivative.
This has nothing to do with whether a human or AI created the art, and I don't think it's controversial to say that AI-generated art is derivative; the models are literally trained to mimic existing artwork.
Creativity in AI art production is a fancy term for temperature that adds no semantic value.
Your "creativity" is just "high temperature" novel art done by the right person/entity.
This was something already obvious to anyone paying attention. Innovation from the "wrong people" was just "sophomoric", derivative or another euphemism, but the same thing from the right person would be a work of genius.
I would say the fact they do better than anyone could have imagined using just language points in the same direction, okay is very much up for debate imo.
It's like people enjoy extrapolating their surprise when it comes to LLMs, and I don't think it's very helpful.
I tell people all the time that the single greatest tool an aspiring lawyer can have is a background with programming, as the analytical and algorithmic mindset is FAR more important than being "good at public speaking" or any of the other base skills often cited as desirable for lawyers. I've also said the second greatest tool an aspiring lawyer can have (in my personal opinion) is a significant background in foreign language learning, as that is a skill that is closely related to programming, though a bit abstracted and coming from a different angle. I'm going to see if I can use this article to support that.
How much math is in a typical CS program these days? Calc 1-3 (maybe 3, varied by school), Linear Algebra, Statistics (inconsistent across programs), Discrete were pretty much it when I was in school 25+ years or so ago. That's only 4-6 courses depending on the university, though some where the CS dept was more strongly associated with an engineering college might have added Diff Eq and others. (I got interested in CS education and reviewed a lot of curricula in the US at the time, this is from memory.)
Some schools like MIT might have required more, but on average what I wrote was about it. Has it increased since then? Based on the new hires I've seen the last decade I'd have guessed the math requirements were mostly the same.
I'd argue none of that math is really necessary. While I have used most of my classes at least once, it was never a barrier to advancement in my career. Hell you could say the same thing about any of the theory. Like yea it's cool I know what a "merkle tree" is but it ultimately is a distraction from most of the skills you need to work with git.
Anyway, both computation and math are grouped under "apriori" knowledge. Any semantic distinction is ultimately silly. But we could just as easily be teaching programming as a craft in the context of the real world—I think this is closer to how it's done outside the US. I am not at all convinced the American style is what people ought to be paying for.
When I did my CS degree in New Zealand there were just two mandatory maths papers - statistics and discrete mathematics. Would be wrong to say I didn't get anything from them - but I'm not fumbling around truth tables or poisson distributions all that often either. Everything else was pretty standard: intro to programming, DSA, low level programming, compilers and networks. What I do find kind of mind blowing is comparing my lectures with the ones from MIT and CM (on YouTube) where they can't go more than a few seconds without jumping into math. Ultimately I'm left unconvinced I was deprived of anything important as a typical software engineer.
It was my grades in math that ultimately failed me out of my undergraduate CS program. My university had: Calculus 1, Calculus 2, Linear Algebra, Vector Geometry, Multi-Variable Calculus, Applied Combinatorics, Discrete Math, Differential Equations and maybe more that I don't remember. So many that CS majors could take one more math class and get a minor.
Yeah, I never thought this made sense, but so many people did; and, I always hear people on Slashdot talking about how programming IS math. None of that has been my personal experience, and I'm coming up on 21 years as software engineer. Discrete was the ONLY math course that I really enjoyed and did well in the first time around. For me, this always made sense.
I can count the times I've ever applied math past approximately high school algebra 1, on one hand. Period, in private life, in hobbies, at work. I'm not sure I've ever used any "college level" math, for anything at all.
I've, and other programmers I've known, gotten excited on the very few occasions anything even slightly mathematically-tricky came up, precisely because it almost never happens.
I certainly don’t dispute the empirical validity of the findings from the study - but there are important nuances to consider as well. I am certainly more naturally-attuned to languages as-far as language-learning and reading than mathematics, but I have also found myself understanding more mathematical and theoretical linguistics as well. I also love programming.
It wasn’t until high school, when I tested-into the highest math class that the school offered, that I began to unlock (with some initial struggle) more logical and procedural reasoning specific to mathematics that I had always done well in, but never explicitly went above-and-beyond in, despite hints of such in arithmetic competitions that my school would hold and that sort of thing. I just think my brain works well for both the linguistic aspects of programming (more naturally) and the computational problem-solving aspects of programming. Certainly there are individuals who have strengths in both cognitive aspects, despite being more naturally-attuned to one versus the other, at least presumably.
Perhaps this shows a cognitive profile that has natural strengths in both "brains", or maybe this highlights limitations of the article's potentially narrow definitions of "language" and "math", implying a more complex intellectual landscape.
Okay, yeah - fair point. I admittedly didn't look too closely at the article before posting this - and I'm not too statistically-minded in many respects. But upon further investigation, yeah, you seem to be right. I think this article is really just promotional material for something.
Abstraction is a huge factor in this too. Language may be dominant because we are at a stage in which the vast majority of programming is little more than hooking together tidy abstractions that are already highly optimized and predominantly organized linguistically.
It you ever need to get into the guts of a system or need to solve bleeding edge problems for which good abstractions don't yet exist, the "math brain" becomes significantly more relevant.
I say this as someone who studied literature and philosophy. The majority of what I know about programming and software engineering I either taught myself or learned from the tutelage of others on the job. Early on in my career, a solid mathematics background was, indeed, not that relevant. These days, though, I'd be lost without it. Whether you like it or not, when it comes to doing real engineering you necessarily need to establish bounds and prove things about those bounds and typically you'll need to do this numerically or at the very least using inductive structures. Linguistic aptitude is still relevant, but it helps less in these cases.
"Coding largely involves the 'logical part' of your brain. It tends to not include the 'language part' of your brain.
This is one reason why comments you add to code are so useful: they force you to engage both parts of your brain and therefore get better results.
This is also why when you go to explain a problem to a colleague, you often have a flash of brilliance and solve the problem: you are reframing the problem using a different part of your brain and the different part of your brain helps solve the problem."
I'm sure some readers here will say this is preposterous and there is no such thing as having "two parts of your brain".
This tracks with my experience perfectly. Hated the math at my Computer Science program in university, barely passed the math courses while excelling at the programming courses, loved taking linguistics, foreign languages, and literature as electives. To this day, programming, foreign languages, and literature is what I do all day every day.
Couldn't this difference be explained by the fact that the lessons were in English? If your language skills are poor, it's going to be hard to read the directions. You'll definitely be slower, which was the primary finding around language.
That side, I wonder if early programming was much more math heavy, and higher level languages have successively reduced that need over time.
If we distinguish between form and content: Mathematics = form; language = content. However, mathematics is only one type of form, and subject-predicate-object grammar also belongs to form, yet grammar falls under language.
For example: "Matter determines consciousness." If we apply first-principles thinking—where does matter come from? This statement then becomes: "XXX created matter, and matter determines consciousness." At this point, our interest shifts to the first-principle subject "XXX," focusing our attention on it. Who is XXX? God?
In this thought process, we use the subject-predicate-object grammatical structure to trace back the original subject "matter" in "matter determines consciousness"—where does matter come from? Although this formal reasoning does not involve specific mathematical formulas, it indeed employs formal logic to uncover a flaw and opens the door to deeper rabbit-hole
There's only a handful of key concepts people need to learn to understand the basics of 'coding'. If you understand what a variable is, how they're scoped, how step-wise transformations happen, and what loops are, you know 90% of "coding". I knew it at age 13 in 1981 doing "Basic" programming. Sure mastery of the art of coding does take decades, but that's just just adding polish, rather than capability.
Math is VASTLY different with VASTLY more concepts that are all much more abstract in nature and harder to understand the infinite numbers of different ways one mathematical construct can be applied to another. A person can "master" coding, but no one ever masters math.
So comparing math to language or to coding is silly. They're completely separate domains. Just because each of the three can encode and represent the other two doesn't make them similar in any way whatsoever.
Notwithstanding Puritanism like "language is maths" or "logic is maths" I definitely agree.
I learned to program when I was a kid and my maths skills were super basic. Programming can almost be distilled to something as basic as "if this, then do that", plus "do this x times". Then read API documentation and call functions that do what the docs say.
With just this basic understanding you can create a lot of stuff. The maths is obviously the foundation of computation, but to be a programming language user and build stuff, you don't actually need to understand most of it.
In university I did eventually do some math-y stuff (econ degree so prerequisites in stats, maths and even CS) and it helps with certain stuff (understanding graphics programming, ML and LLMs, plus knowing maths on its own is useful), but I still don't feel it was strictly necessary. Language and basic logic is enough IMO.
Started programming over half a century ago. The insurance company I was working for gave me a joint life actuarial evaluation problem for which I wrote a Fortran program, picked up from a book. My uni student buddy let me use his ID so I could drop decks into their IBSYS. Turnaround was about a day. My career as a professional manual reader began.
Well yes, my high school maths were in the high 90s - more than my language scores in French, German and Latin with some off curricular Russian. I guess being a polymath helps.
Unless you are doing an engineering or mathematical application you don't need much math, especially as you can just call a function in the vast majority of the time.
I did a number of software products and operating system modifications without using any math beyond arithmetic operations.
I was a resource for other programmers including the odd math PhD.
Of course it does. Math is not programming - it's almost entirely irrelevant to programming and certainly to software and architecture development. Most modern programming only involves math for trivial things like counting.
This classic article explains the real issue - like Mike Gancarz' classic on the Unix Philosophy, this is something all younger hackers should read, but few have, since these are the fundamental ideas that have created our modern world of computers: https://web.archive.org/web/20000529125023/http://www.wenet....
I’m old enough to have caught the tail end of CS being an offshoot of the Math department and I had a gaggle of fellow undergrads who were happy ours had been moved to the Engineering college and upset to know other students at other schools who were still in Math.
I don’t know about for learning but definitely for collaborating and mentoring. And it’s difficult to make a definition of mastery that excludes both of those, so I suppose after a fashion it’s right.
Despite being a professed lover of math, I scored higher on the verbal than the math SAT. There’s a lot of persuasive and descriptive writing in software, particularly if you’re trying to build a team that works smarter instead of finding more corners to cut.
I did not read any mentioning of testing non-verbal IQ. Many math skills also depend on verbal IQ. Verbal IQ correlates strongly with academic performance and probably also with learning new programming languages. I personally have a much stronger non-verbal IQ than a verbal IQ. Often when I think about algorithms, it is all about abstract blocks moving around even before I have written a single line of code. I usually see multiple solutions in my head and often find myself being stuck because I cannot make a choice. I always takes some effort to code the solutions and weed out stupid bugs due to spelling errors mixing up varisbles, but then things work correctly.
In my experience, language skill may or may not matter for programming, but it certainly matters for understanding a problem space. You can be a great programmer, building the wrong thing because you didn’t understand the requirements or see the gap in them and know to ask the right clarifying question, etc.
Lately I’ve also felt language skills matter when writing concise, specific AI prompts. This has become a useful part of my programming workflow in, I suppose, the last year or so. Before that it was knowing “how to Google” well, but that’s less language-dependent in my opinion.
The most obvious way in which I think "math" and "programming" are related is the way they're taught; by repetitive performance of problem sets. I got frustrated in math for this reason; it takes a lot of energy to sit and focus on a whole bunch of hard problems that all "look the same". It was too easy to do 1 or 2 and think "okay, I got it" and then move on, not realizing the repetition was the key to memorization.
I feel the same way about starting learning programming. Repetition, repetition, repetition, until you "get good".
As a self taught programmer with a Communication Studies degree, this definitely resonates. The ability to articulate the problem in code is kind of the starting point for most productive development work.
Any experienced programmer will be able to tell what kind of person another programmer is just by the way they write their code.
Code written by programmers with humanities backgrounds is easily identifiable as being of bad quality.
Kind of like vibe coding meets programming by accident, before vibe coding was really a term.
It's a faux pas to even mention this IRL, but good coders know what's up.
Those "coders" usually get promoted to people managers, which is usually what they want anyway because their self-worth relies on abusing others to mitigate the correct self-perception they have of being "inferior".
The problem is, things need to be solved and vibe+accident programming can only go so far.
But fear not, they can always scapegoat whoever solves the problems, because if they were not to blame, how could they know what was up or even feel the need to correct it?
This is a high level of bias and generalization about an enormous group of people with varied backgrounds and experiences, not to mention selection bias as you don't know many good coders backgrounds who may in fact be from humanities.
Even if many of the bad coders are those who were in humanities and don't have coding experience because they just entered the field (because once you get it you are no longer a humanities background)
Personally, I find ruby one of the most readable languages because of the language and its DSLs. To be fair, the DSLs are very readable, but the lines can blur between what is ruby and what is a DSL.
I have really come to enjoy using LINQ queries (specifically with lambda syntax) in the C# language. Although I’m a little biased because I work mostly in C#.
My own definition of mathematics for the sake of this comparison:
- If you can write a formal proof, starting from some assumptions and proving some result, it is mathematics. The assumptions are the axioms. Then you use logic, which is the same programming logic, and then you get to some result.
It can be an invariant in a programming function, it can be a more general result, if you can write a proof, it is mathematics. Most algorithms involve proofs, so they are mathematics.
You can make any subject "mathy", because there are relations, ratios, correlation and patterns in everything. My favorite example, is how Chomsky mathified linguistics, and made it esoteric, undecipherable, uncomputable and thereby linguists unemployable, even in research labs.
But as a matter of practice, teaching programming to engineers/scientists, even to mathematicians, is an order of magnitude easier than teaching math to CS folks. Simply quiz job candidates on fp arithmetics, and see how many fail miserably.
Interestingly, I became better at learning human languages after learning a couple of programming languages. I was good at math at the beginning. I guess programming kinda bridged the gap.
The article is from 2020. Recent LLM developments show that a model trained on math and code is better at coding than one that's trained on language and code without math.
I don't find this too surprising. The study itself was primarily just testing a students ability to identify syntax and remember what various functions do. I wouldn't expect math proficiency to help much in this area vs I would very much expect language to.
It'd be interesting to see correlations (language brain vs math brain) for how easy or hard it is for people to solve new problems with language after they already know the basics.
As an accomplished programmer who is not great at language nor math, I feel like my "modeling brain" is the key to success. I can envision all the aspects of an architecture, its data shapes and program flow, making it obvious which data structures and algorithms to use, or where to look for a bug. I did spend a lot of time naming things though... until ChatGPT replaced the need for a thesaurus.
I had to scroll through most of the paper to find this.
> All participants were right-handed native English speakers with no exposure to a second natural language before the age of 6 years
Which removes a confounder that Python mimics English syntax.
Still if this is a typical study recruiting thirty-some undergrads as subjects it's probably not generalizable, or even replicable given the same experimental setup.
> the requirements for advanced math classes for every computer science major seem unnecessary
Computer science is much more than programming - and I think that most of the value derived is from being able to think about problems, which largely require the abstract type of thinking encouraged by more advanced math. Code is just a tool.
Makes sense to me. "Language" is an oft-forgotten part of "programming language." I'm sure math skills probably would better predict, say, ability to do well in advanced theory classes, but programming doesn't require that. It DOES require learning a specialized language.
I feel like I'm using language part of the brain for almost everything. My performance at any task (except purely manual and automatic ones) drops to 0-15% when I can hear someone talking. Alternatively if I manage to focus on the task I immediately stop understanding what's being said.
I'd posit that this is part of why Pascal lost. Making and/or the same precedence as */+ only appeals to mathematicians and gets in the way of ordinary uses for Boolean expressions where you want them lower than the sub-expressions they concatenate in a language-like manner.
I would agree. Linguistics and literature have always been a strong interest of mine, and many computer scientists. Moreso, I find weaker programmers struggle with spelling, grammar, various dyslexias and torrent-of-thought code organisation.
Counter example: I can pick up any new programming language within 3-6 months and be proficient but it takes a year to pick up some minor fluency in a human language and I still have yet to obtain full fluency in any other language as an adult.
I suspect, however, that a programming language is a much more limited domain than a human language.
I often tell people it's not that learning a language is hard, is learning that language's software library... and learning a software library doesn't feel like learning a language. More like learning a set of tools.
As a new computer science professor at a community college, this is a timely article for me that may end up influencing how I teach, especially my introductory programming course as well as my discrete mathematics course.
but really it just depends:
- web or app information software: logic and language matters more here
- doing what Donald Knuth does: math matters more here
I guess its the ability to tell long, forking stories, traversing several abstraction layers. Dante Alighieri would be loving this! Its the ability to walk through a mental palace, room for room again.
I can't speak for other forms of FP, but symbol operators make communicating about haskell very annoying. Outside of that FP seems to be doing fine, IMO.
To be clear, the symbols themselves don't bother me so much as trying to refer to them in spoken english. I have no particular beef with the use of symbols in code, which can be quite readable.
Hold your horses. The summary of that article does not offer any proof of what the title says.
* It's a small sample, and they did not analyze the people who didn't complete the course. That's dubious. Those 6 could have had a massive influence on the outcome.
* The summary does not present the actual numbers. These are: "fluid reasoning and working-memory capacity explained 34% of the variance, followed by language aptitude (17%), resting-state EEG power in beta and low-gamma bands (10%), and numeracy (2%)". Note: numeracy, not math.
* The test result was only partially programming related. 50% consisted of the results of a multiple choice test with questions such as What does the “str()” method do?. Linguistic knowledge indeed.
* It's about completing a 7.5 hour Python course. That's learning indeed, but only the very beginning, where abstraction is not in play. The early phase is about welding bits of syntax into working order.
* The numeracy skills required are very low for such tasks, as the tasks are simple, and mainly require thinking in steps and loops, whereas numeracy aptitude is generally measured on rather problems involving fractions.
Edit: the paper uses the Rasch-Based Numeracy Scale for this, which seems to involve estimation and probabilities.
* 17% explained variance is a rather minimal result, and you cannot easily compare factors in such a small design, even if the other one is only 2%. That's a rather hairy statistical undertaking.
* Linguistic expedience might be explain the speed with which the course was followed, since the instruction is, obviously, linguistic. Hence, this factor is not necessarily related to the actual learning or programming.
* The argument from beta waves is clutching at straws.
* The argument that "perhaps women should have more of a reputation for being “good” at programming" because they score better on tests, is --however well meant-- utterly ridiculous. It reverses correlation to causation and then turns that into a fact.
* That linguistic skills are useful for programmers is widely understood. However, this is not because of the actual coding, but because the coder needs to understand the environment, the specs, the users, etc., all of which is transferred via language.
* And of course, the statistical result relies on Null Hypothesis Test Significance, which is rotten in its very foundations.
* Note that the CodeAcademy course "Learn Python 3" is 23 hours in 14 lessons.
Yes, nobody doubt about that. The article is about programming, which, even if it requires some math skills, require more an ability to deal with a language system.
Not sure why the title left out the essential term: it matters more for learning programming
I believe the goal is to encourage those (young people) allergic to math but good in languages to realize they could be good at programming. That's worthy and important (though ironic to use a scientific study to do so).
As for the larger question commenters are raising, I notice often programmers reducing programming to problem-solving, and that to modeling, thence to math writ large; then they prove their point by noting that the most significant algorithms have strong and deep roots in math. The argument is: if the pinnacle of programming is math, then the more math-like programming is (in people and style), the better. (Hence: functional programming and provable algorithms are the epitome)
This argument is invariably made only by math experts, and does have the effect of reducing competition and selecting for like-minded persons, so there's at least the possibility of infection with self-interest and bias to the familiar.
I think the real distinction lies in this: with math, once you have a model, you fit things into that, and exclude other things. You do perfectly, but only in your domain. With language, any differance (any difference that makes a difference) starts to be named and tracked and dealt with somehow. It may grow in confusing ways, but it turns out (like architecture) it actually makes much more sense to adapt the structure to the use and relevance than vice-versa.
Sure, some subset of architecture is engineering by necessity, and it's probably even the hardest subset. But that's also the most portable knowledge and practice, and I would argue, thus easier to find and deploy once you've isolated such a need, particularly since mathy stuff fits nicely in a reusable library.
So math is an important but relatively isolated part of programming, and thinking it's ideal for everything is missing the point that programming matters for some purpose.
"attention is all you need" is but one example of the priority of relevance over structure in systems.
It's funny that this popsci article instantly succeeded in dividing the HN commenters into math/not-math camps. Porque no los dos?
Programming is the manifestation of thought through the medium of a keyboard and screen. If you are a clear thinker, if you can hold multiple things in your head at once, if you can reason about things and their relations, well, you can be a strong programmer.
It seems wholly unremarkable to me that someone new to Python would not be fazed by it, given it's fundamental basis in words (print, if, etc.) Someone with a background in languages, who can think well enough to explicitly or implicitly pick up the structure of the language, is gonna do just fine. "Oh, so when I see a while, I need to end with a colon" isnt so different from "when I shout, I need to add a ! at the end"
(Java gets a special place in hell for forcing "public static void main" on every beginner.)
Math only really comes into it when you want to reason about things that have a history of being manipulated mathematically, typically for reasons of convenience. You could probably invert a matrix in SNOBOL, but its a lot easier to pull out lists and arrays and linear algebra.
In other words, lets see the follow up paper to this where Python newbies are asked to model the trajectory of a bead on a wire and see how they do.
Probably true. Math used to be written in prose style before we invented symbols to compress the prose. Most of the symbols are very recent just in the last 200 years or so.
Geometry only benefits from visual reasoning in 3d and lower, and there are a lot of dimensions above that!
You can see visual reasoning as a little cheat computation, you can run math problems through your sense-determining brain, which is what brains are really good at (robots struggle with our levels of dexterity). But the fact remains that you can only visualize in low dimensions, and there are infinitely many dimensions.
Note: You can reduce many problems to 3d, but also many problems in 3d have configuration spaces with much higher dimension, so there's some nuance.
This is largely true, but also, the dynamic shifts depending on how learners engage with peers as they move from understanding to synthesis.
In small peer groups (“pods”) that debug and learn together, communication becomes a core skill—and that can actually change how math skills are applied and developed. Language doesn’t just support learning; it reshapes the process.
Honestly, all this "math brain" vs "language brain" stuff just feels off to me - I've always been more of a pattern and abstraction person anyway. Makes me wonderif schools focused more on teaching real abstract thinking early, would way more kids just "get" programming without even stressing over math class?
Interesting. I grew up most of my life overseas learning various different languages. Anecdotal, but most of the best coders I know are talented writers and often are multi-lingual.
The language brain is more important than the math brain for reading the first lesson on how to program in Python. Fixed that for you.
Does answering a quiz on the contents of the first lesson on how to program in Python really encapsulate anything concrete about who will and will not be able to actually program in Python?
I've always been disturbed by the disconnect between "first lessons" on programming languages and how I personally actually learn programming languages. I can't help thinking that the researchers have measured something else other than whether people have learned to program.
This mirrors my personal experience. My writing skills have been above average and my parents' presumption was that I would pursue some humanistic studies. I ended up studying software engineering and made a career out of it of almost three decades and counting. Meanwhile my wife who always had a "math brain" has struggled to learn to program at a decent competency level.
I don't think it's fair to attribute anything to anything. Stuff comes from all over the place. In other words, attributing programming prowess to math was a mistake, and we are making the same mistake again attributing it to language.
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Just one more:
Kurt Vonnegut: --consider himself in competition with a world's champion. And this is one reason good writers are unlikely to come from an English Department. It's because the English Department teaches you good taste too early.
I think his main point is when we put something on a pedestal, we actually limit people, whether that be math or language.
That is now obvious, any LLM is able to code, and is usually very bad at math.
This alone proves that a good part of programming is linked to language not math.
After I started my PhD program in CS/AI (late 90's), one of the faculty told me that the GRE verbal score was thing on the application most predictive of student success in the program. Of course, most applicants had perfect or near perfect GRE math scores, so there was no predictive power there. But the application also included the GRE subject test, transcripts, and letters of recommendation.
I also seem to recall from The Making of the Atomic Bomb that the theoretical physicists of the era were noted as having the highest verbal intelligence of any group of scientists. That struck me since I would not have expected it.
EDIT: I found the quote, from chapter 6:
Psychometricians have closely questioned American scientists of this first modern generation, curious to know what kind of men they were—there were few women among them—and from what backgrounds they emerged. Small liberal arts colleges in the Middle West and on the Pacific coast, one study found, were most productive of scientists then (by contrast, New England in the same period excelled at the manufacture of lawyers).
...
Theoretical physicists averaged the highest verbal IQs among all scientists studied, clustering around 170, almost 20 percent higher than the experimentalists. Theoreticians also averaged the highest spatial IQs, experimentalists ranking second.
Perhaps survivorship bias explain part of it. Many of those scientists arrived at US fleeing their home countries in WWII. Migrating under those circumstances is tricky, lacking the necessary support network or failing to successfully negotiate an encounter with border patrol may be the difference between reaching a safe place and being dragged to rot in the trenches. Good verbal intelligence is very helpful in this process. Perhaps the same way good verbal intelligence is over-represented in those who escaped the war, the trait of being taciturn was over-represented in all the scientists that got stuck and died in WWII[1].
In order to solve what are essentially "word problems" -- i.e. reading a verbal description of a problem and translating it to math -- you have to have very good reading comprehension.
The GRE math bar for excellence was quite low and so a perfect score was only the 92nd percentile or so. It’s not surprising that other metrics correlated better with success
Makes sense that verbal ability would line up more with success in CS, especially when math scores are already high across the board. A lot of programming leans on language-type skills: reading and understanding code, navigating docs, naming things clearly, writing maintainable logic etc.
The field probably does itself a disservice by overemphasising math. That framing can push people away who might actually do really well, especially those strong in reasoning, abstraction, or communication. Linked study is a good reminder to rethink how we present programming imo.
It does. The majority of what's popular and gets attention in programming is things like languages and frameworks which do the best smooth talking. The whole point of CS is to focus on the aspects of programming that aren't just vogue trends. With any field or aspect of modern life, language skills are the gateway to privilege and software engineering is no exception. But here it's respect for the math that stops it all from spiraling into a bunch of bs. You can't fool the math people.
Absolutely. That's why people can be perfectly fluent in a programming language and a high performer at work, but fail an interview algorithms/leetcode problem in that language, as algorithms are math problems that programmers almost never see at work.
The recruiter labels an algorithm problem as a "coding" test, but it's a math test, and concludes that most applicants who claim to be fluent in a programming language can't code and must have lied on their resume.
For context, I don't mind algorithm tests, but I strongly disagree with recruiters presenting it as a coding assessment.
Has anyone here actually read the paper? Heavily clickbait headline, completely exaggerating the study results, which themselves are not very rigorous (to be generous)
Tiny sample size - 36 people completed
Numeracy has R^2 = .27 Language has R^2 = .31
They then run stepwise regression to determine variance contributions, seemingly ignoring their earlier results, and this leads to almost no contribution from numeracy. Why? Because they have ~10% shared variance and stepwise regression is greedy - will just take whatever you give it first.
I can't mention this part enough. If you got a second, very similar language test, and added it to the model, you would _also_ find it has almost no unique variance added.
Every thing they measure is incredibly noisy and they do not once attempt to deal with this. Human based reviewers, time-to-completion, etc.
p-value for "language learning is more significant than numeracy" on the values they give (Steiger test) gives 0.772. Utterly insignificant.
Also, to beat the point home, just think about the argument here: * Numeracy contributes 27% of variance
* Language skills contribute 31% of variance
* After regression step, numeracy contributes only 2% of unique variance. Because you added a correlated variable!
To be somewhat fair, their numeracy scatter plot visually looks more like a dart board than any real process. The fact that you got a positive-sloping line out at all from the regression has to do more with the positions of the outliers than anything else. You're at least able to visually examine the language aptitude plot and see an up-and-to-the right connection between the two variables.
I agree, but also via some napkin math, the chance of getting those different results is something like 40%.
(Ie, if you sample the same signal twice for the numbers they did in the study, there’s a 40% chance it’ll be >0.04 away from the original sample, as numeracy was from language)
It reads like an ad for something.
Good code doesn’t just solve a problem, it solves it in a way that’s readable and modular.
I think the problem-solving part of coding requires math skills, while the organization part requires writing skills. The organization part affects the problem-solving part, because if you write messy code (that you can’t reread once you forget or extend without rewriting) you’ll quickly get overwhelmed.
Writing large math proofs also requires organization skills, since you’ll refer to earlier sections of your proof and may have to modify it when you encounter issues. But to me, math seems to have more “big steps”: sudden insights that can’t be derived from writing (“how did you discover this?”), and concepts that are intrinsically complicated so one can’t really explain them no matter how well they can write. Whereas programming has more “small steps”: even someone who’s not smart (but has grit) can write an impressive program, if they write one component at a time and there aren’t too many components that rely on each other.
Most coding doesn't need much of any math past boolean logic and very basic set operations. I'm much more likely to spend my time studying DB and interface schemas to understand how something works than doing a lot of mathy fiddling. Sure, some people write game engines and such, but even much of 3D graphics doesn't need anything more complicated than the first half of a linear algebra course.
Not "coding uses math", I mean it is math.
I don't know how you code, but I don't think I'm aware of code that can't be reasonably explained as forming relationships between objects. The face we can trace a program seems to necessitate this.[0] https://philosophy.stackexchange.com/questions/22440/what-di...
> Coding IS math.
Programming is an expression of logic, which is absolutely mathematics.
But then we also have to think about naming variables and classes, structuring our code so that it is more readable by other developers, and so on. That's less about formal reasoning and more about communication.
There is an engineering aspect to programming (prototyping, architecture, optimization, etc). It's a combination of mathematics and engineering. Software Engineering.
> Coding IS math.
The murky world of software patents would like a word about that.
For me, coding also feels artistic at times. But we see artistic beauty in mathematics all the time, so no wonder. But often I look at ugly code, and maybe that is subjective, but fixing that code makes it also feel prettier.
It is clear all developers have their owm artistic style, and probably that is why there is so much disagreement in the industry. Maybe we are lacking the pure mathemetical language to describe our intent in a more beutiful and precise way that is more clearly the-right-way. As in how we find beauty in simple physics equations.
Do you think mathematicians don’t have to think about naming variables and structuring proofs?
The presumably-mathematicians who wrote my Algorithms textbook (CLRS) didn't seem to think giving the variables in their algorithms useful names. They just use i, x, etc. all over the place and don't even consider giving things actual names. This makes the book much less accessible than it would be if they treated it more like most people write code.
In general, lower case letters are used as constants, excluding x,y,z,t,i,j,k (sometimes u,v,w). It isn't a hard rule, but strong preference to begin at the beginning of the alphabet for these. Capital letters usually are held for things like Variable Sets (like random variables). Greek letters need context for constants or variables. BB and Cal typefaces for sets (e.g. Real Numbers, Integers). And much more.
I think a lot of the difficulty in it is that these "rules" or patterns are generally learned through usage and often not explicitly stated. But learning them can really help read unfamiliar topics and is why "notation abuse" leads to confusion. But after all, math is all about abstraction so technically any symbol will do, but no doubt some are (significantly) better than others for communicating.
Picking what letters to use for what things can still be a struggle.
Mathematicians do have to deal with difficulties in naming things.
They just invent new characters.
What do you mean? Most letters used in math are just Latin and Greek letters in different fonts or stylizations.
That's how it is in applied mathematics. That's a hardcore computer science textbook. If your job is computer science research, inventing new algorithms, that is the kind of book you will get used to. There are better options for practical learning.
Of course they do, but that part is not "mathematics". It is communication. If they use the English language to write the proof that doesn't mean English is also mathematics.
Writing proofs is just as important to mathematics as writing code is to programming.
Yes it is important but that's not the point.
Structuring sentences and naming variables so that it is easier for other people to understand is less about formal mathematical reasoning, and more about communication.
You could name a variable x, y, or Banana, but it doesn't change the logic.
Neither is it "the point" in programming. You should be concerned with communication and have every right to get upset when someone is being needlessly convoluted but that's as much of a point in programming as it is in math, physics, or any domain.
I mean the reason we get mad at this is because it is someone destroying "society" in some sense. Even if that society is your team or just the set of programmers. It would be a pretty dick move were I to just use a word that significantly diverged from conventional meaning and expected you to mull it over. Similarly if I drop a unknown out of context math equation. It would be meaningless.
And I'm on your side, really! I strongly advocate for documenting. And let's be real, the conclusion of your argument more strongly argues for documentation than good variable names. Because variable names are much more constrained and much more easily misinterpreted considering how any word has multiple definitions. Surrounding code is often insufficient to derive necessary contextualization.
https://news.ycombinator.com/item?id=43874738
>Programming is an expression of logic, which is absolutely mathematics.
and also philosophy.
As in GNU vs. Microsoft? Or something more foundational in logic?
Classes are a platonic ideal representation of reality?
That's another gap: mathematical classes are ideal representation of fantasy, programming classes are leaky representation of reality.
I don't think that quote really supports coding and math being equivalent. To me, the quote provides an abstraction of math through a structuralist perspective. Language can also be viewed through this abstraction. I think coding could share the abstraction, but that doesn't make the three of these fields equivalent.
This seems like the setup to a joke about how mathematicians don't know how to communicate to ordinary folks
Well a lot of people did wildly misunderstand the Poincare quote. To me is is obviously about abstraction and I think this is true for any mathematician. I thought it would also be natural for programmers considering we use "object" quite similarly, if not identically. So... maybe it is or maybe this is the joke.
"There are those who tell us that any choice from among theoretically-equivalent alternatives is merely a question of taste. These are the people who bring up the Strong Church-Turing Thesis in discussions of programming languages meant for use by humans. They are malicious idiots. The only punishment which could stand a chance at reforming these miscreants into decent people would be a year or two at hard labor. And not just any kind of hard labor: specifically, carrying out long division using Roman numerals." — Stanislav Datskovskiy
No one is suggesting programming with lambda calculus. But it would be naïve to think lambda calculus isn't important. They serve different purposes.
We didn't:
> Not "coding uses math", I mean it is math
> I mean equivalent in the way mathematicians do
That sounds like you're backing off from your original claim, probably because it is impossible to defend.
That you can use mathematics to describe code doesn't seem very different from using math to describe gravity, or the projected winner in an election, or how sound waves propagate.
Isn't the primary purpose of math to describe the world around us?
Then it shouldn't be surprising that it can also be used to describe programming.
In the real world, however, software engineering has nothing to do with mathematical abstractions 99% of the time
A programmer constructs a function from some data type to another while a mathematician constructs a function from witnesses of some proposition to another?
Though interpreting a CRUD app as a theorem (or collection of theorems) doesn’t result in an interesting theorem, and interpreting a typical theorem as a program… well, sometimes the result would be a useful program, but often it wouldn’t be.
Interpreting a CRUD apps (or fragments of them) as theorems is interesting (given a programming language and culture that doesn't suck)! e.g. if you have a function `A => ZIO[Any,Nothing,B]`, then you have reasonable certainty that barring catastrophic events like the machine going OOM or encountering a hardware failure (essentially, things that happen outside of the programming model), that given an A, you can run some IO operation that will produce a B and will not throw an exception or return an error. If you have an `A => B`, then you know that given an A, you can make a B. Sounds simple enough but in practice this is extremely useful!
It's not the type of thing that gets mathematicians excited, but from an engineering perspective, such theorems are great. You can often blindly code your way through things by just following the type signatures and having a vague sense of what you want to accomplish.
It's actually the halting problem that I find is not relevant to practical programming; in practice, CRUD apps are basically a trivial loop around a dispatcher into a bunch of simple functions operating on bounded data. The hard parts have been neatly tidied away into databases and operating systems (which for practical purposes, you can usually import as "axioms").
I think we're in agreement. My comment about the halting problem was meant to refer to Rice's theorem (the name was slipping my mind), which I occasionally see people use to justify the idea that you can't prove interesting facts about real-world programs. In practice, real-world programming involves constantly proving small, useful theorems. Your useful theorem (e.g. `Map[User,Seq[Account]] => Map[User,NetWorth]`) is probably not that interesting to even the category theorists, but that's fine, and there's plenty you can learn from the theorists about how to factor the proof well (e.g. as _.map(_.map(_.balance).sum)).
Physicists use math, but that doesn't mean it is math. It's not the only language at their disposal nor do they use all of math.
I'd argue that 100% of the time it has to do with mathematical abstractions. Please read the Poincare quote again. Take a moment to digest his meaning. Determine what an "object" means. What he means by "[content] is irrelevant" and why only form matters. I'll give you a lead: a class object isn't the only type of object in programming, nor is a type object. :)[0] Technically a specific (class of) physics, but the physics that any reasonable reader knows I'm referencing. But hey, I'll be a tad pedantic.
[1] String Theory is untestable, therefore doesn't really reflect the observable world. Even if all observable consequences could be explained through this theory it would still be indistinguishable from any other alternative theory which could do so. But we're getting too meta and this joke is rarely enjoyed outside mathematician and physicist communities.
> No, that's Physics
Going on a total tangent, if you'll forgive me, and I ask purely as a curious outsider: do you think math could have ever come into being if it weren't to fill the human need of describing and categorizing the world?
What would have been the very beginning of math, the first human thought, or word or action, that could be called "math"? Are you able to picture this?
But my confusion is because I'm having a difficult time thinking where such things aren't also necessary consequences of just being a living being in general. I cannot think of a single creature that does not also have some world model, even if that model is very poor. My cat understands physics and math, even though her understandings are quite naive (also Wittgenstein[1] is quite wrong. I can understand my cat, even if not completely and even though she has a much harder time understanding me). More naive than say the Greeks, but they were also significantly more naive than your average math undergrad and I wouldn't say the Greeks "didn't do math".
It necessitates a threshold value and I'm not sure that this is useful framing. At least until we have a mutual understanding of what threshold we're concerned with. Frankly, we often place these contrived thresholds/barriers in continuous processes. They can be helpful but they also lead to a lot of confusion.
This too is hard to describe. Mull over the Poincare quote a bit. There's many thresholds we could pick from.I could say when the some of the Greeks got tired of arguing with people who were just pulling shit out of their asses, but that'd ignore many times other civilizations independently did the same.
I could say when the first conscious creature arose (I don't know when this was). It needed to understand itself (an object) and its relationship to others. Other creatures, other things, other... objects.
I could also say the first living creature. As I said above, even a bad world model has some understanding that there are objects and relationships between them.
I could also say it always was. But then we get into a "tree falls in a forest and no one is around to hear it" type of thing (also with the prior one). Acoustic vibrations is a fine definition, but so is "what one hears".
I'd more put the line closer to "Greeks" (and probably conscious). The reason for this is formalization, and I think this is a sufficient point where there's near universal agreement. In quotes because I'll accept any point in time that can qualify with the intended distinction, which is really hard to pin-point. I'm certainly not a historian nor remotely qualified to point to a reasonable time lol. But this also seems to be a point in history often referenced as being near "the birth" and frankly I'm more interested in other questions/topics than really getting to the bottom of this one. It also seems unprovable, and I'm okay with that. I'm not so certain it matters when that happened.
To clarify, I do not think life itself necessitates this type of formalization though. I'm unsure what conditions are necessary for this to happen (as an ML researcher I am concerned with this question though), but it does seem the be a natural consequence of a sufficient level of intelligence.
I'll put it this way, if we meet an alien creature I would be astonished if they did not have math. I have no reason to believe that their math would look remotely similar to ours, and I do think there would be difficulties in communicating, but if we both understand Poincare's meaning then it'll surely make that process easier.
Sorry, I know that was long and probably confusing. I just don't have a great answer. Certainly I don't know the answer either. So all I can give are some of my thoughts.
[0] https://www.inference.vc/causal-inference-3-counterfactuals/
[1] https://existentialcomics.com/comic/245
>Even if all observable consequences could be explained through this theory it would still be indistinguishable from any other alternative theory which could do so.
That's not unique, all quantitative theories allow small modifications. Then you select parsimonious theory.
Coding is math in the sense that coding is a subset of math.
Mathematics is a very extensive field, and covers a vast amount of subjects.
For the same reason it can't be said that mathematics is equivalent to coding, as there are many things in mathematics that are not relevant for coding.
However, by far the most interesting parts of coding are definitely related to mathematics.
Sure, lambda calculus is math. To call assembly or typical imperative C math, at least in the same sense, is a bit of a stretch.
Working with different objects doesn't make it any less of math. Just because you can derive calculus from set theory (analogous to assembly or even binary here) doesn't make calculus "not math".
Math is about abstractions and relations. See the Poincare quote again.
Plus, the Programming Languages people would like to have a word with you. Two actually: Category Theory. But really, if you get them started they won't shut up. That's either a great time or a terrible time, but I think for most it is the latter.
Wheeler: It from bit. Otherwise put, every it—every particle, every field of force, even the space-time continuum itself—derives its function, its meaning, its very existence entirely—even if in some contexts indirectly—from the apparatus-elicited answers to yes-or-no questions, binary choices, bits. It from bit symbolizes the idea that every item of the physical world has at bottom—at a very deep bottom, in most instances—an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes–no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and that this is a participatory universe.
That is not in contention with what I said, now is it?
Wheeler is arguing that if a tree falls in the forest, and there is nobody to hear it, that it still makes a sound because there are things that interact with the sound. But if the tree fell in a forest and there was nothing else in the universe then there is no sound because there is no observation.
It helps to read the whole thing[0] and to understand the context of the discussion. This is meta-physics and a deep discussion into what the nature of reality is. Ian Hacker has a good introduction to the subject but I find develop grave misunderstandings when they also do not have the strong math and physics background necessary to parse the words. Even people who understand the silliness of "The Secret" and that an observer need not be human often believe that this necessitates a multi-verse. A wildly convoluted solution to the problem of entropy not being invertible. Or closer to computer terms, a solution that insists that P = NP. There is information lost.
If you wanna argue that there's no difference between the word cup and a cup itself because there is no word without the observer who has the language, then yeah.
[0] https://historyofinformation.com/detail.php?id=5041
If nature is information theoretic, then physics is mathematics.
But that doesnt necessarily mean successful programmers are good at conventional math. This is why certain people in the department are identified as "math people".
I'm not sure why you'd think I disagree. It seems you understood I argued that it's unhelpful to make the distinction between math and "conventional" math
But I'll refer you to a longer conversation if it helps https://news.ycombinator.com/item?id=43872687
> Coding IS math.
> Not "coding uses math", I mean it is math.
Arguably writing a novel is math, if you use the right definition of math. But sometimes its more helpful to use more informal definitions that capture what people mean then what is technically accurate.
By that same logic you could also say that language is math. In fact I think your quote kind of disproves your point because the content/state of a program is super important in coding more than the form.
Coding used to be very close to pure math (many early computer science classes were taught in the Math Department in universities) but it has been so far abstracted from that to the point that it is its own thing and is as close to math as any other subject is.
>I can state incomprehensible sentences or words.
since you can express paradoxes with match, perhaps not that different.
I think you misunderstand what "paradox" means. While it can mean "self-contradictory" it can also mean "contrary to one's expectation." Math uses both, but in very different contexts.
The contradiction is used in proof formulation, specifically to invalidate some claim. I don't think this is what you're implying.
The latter is what it contextually sounds like you're stating; things like the Banach-Tarksi Paradox. There's no self-contradiction in that, but it is an unexpected result and points to the need to refine certain things like the ZFC set theory.
I'd also stress that there are true statements which cannot be proven through axiomatic systems. The Halting Problem is an example of what Godel proved. But that's not contradictory, even if unexpected or frustrating.
forming relationships between objects
This is vague and doesn't mean anything. People can't even agree what 'objects' are and people did a lot of programming before the term 'object' was invented.
Programming is about is fundamentally about instructions and data. Yin and yang of two things that are completely different, even if people can occasionally mix them and conflate them.
Category theory is often jokingly called the study of dots and arrows. What's a dot? Anything. What's an arrow? A relationship.
I'm surprised I hit a nerve with so many people. I'm quoting someone who's considered one of the greatest mathematicians. Obviously I don't know the backgrounds of people but it seems like programmers have strong opinions on what math is that disagrees with what mathematicians say they do.
https://en.wikipedia.org/wiki/Category_theory
Now it's not math it's 'category theory' and that's dots and arrows and dots are anything.
At what point does the abstract description just not offer any useful insight anymore?
I'm surprised I hit a nerve with so many people.
A lot of people had very good explanations for why you're pretty far off in trying to say two things are the same.
Programming is much more like building a machine than any sort of straight math. There is state, there is interactivity, there is performance, there is IO and there are real world implications to all of it.
Saying they are the same is like saying gardening is plumbing just because you sometimes use a hose.
That's a wild claim considering Category Theory is a branch of mathematics
It's not a wild claim since you misquoted me.
A lot of non-mathematicians disagreed with mathematicians.
Mathematicians can claim whatever they want, when it comes to programming, programmers understand it better and they're trying to explain to you why this is nonsense. Vanderbilt claims to be "the harvard of the south" but wouldn't you know it, harvard doesn't claim to be "the vanderbilt of the north".
Show me programming languages designed by mathematicians to be 'mathematically pure' and I'll show you a language that hasn't been used to ship software that people want to use.
It's math the way everything is physics, as well as the way physics is philosophy (as the natural sciences started out as branches of philosophy).
In other words: It's math in a sense that for most of us with a computer science background is often not very relevant to how we work.
>Coding IS math.
No, not always. Quite a lot of high-level code doesn't require any math at all. It doesn't take math to perform CRUD operations, which account for a lot of programming work. Sure, the underlying machine code is all based on math, but the higher level programming doesn't need to involve a single math equation for it to be useful. Let's see where the goalposts move now...
Mathematics is not "equations". Most of mathematics is not related with calculus either.
For anything CRUD, you put the data into a database. The underlying database is relational, or a key->store.
If it's relational, that's one branch of mathematics. If it's another kind of database, it's another branch of mathematics. Mathematics is extensive, and covers more things you can imagine at a glance.
The main difference between writing mathematics and programming, and this applies to any form of programming, is that in mathematics writing a formal proof is amazingly close to: you write the program, and you are also the compiler, and you are the CPU, performing all operations yourself. With programming you only have to write the program.
Source: On one hand I have studied pure mathematics (not the simplified applied mathematics that are taught in engineering, which is mostly equations), on the other hand I have been working as a software developer for over 15 years.
Sorry but you are wrong. "CREATE, READ, UPDATE, DELETE" operations are not mathematical in nature. It doesn't matter what the data is or how esoteric you want to get with it - a programmer doesn't need any math at all for simple CRUD. You're trying to move the goalposts all the way back to the 1800s to win a pointless internet argument.
A while back I made this[0] as a quick demo of generic batching for CRUD requests. It's very much practically oriented (it's for performance optimization while making the code nicely reusable/not muddying up the business logic), but also felt quite a bit like the same sort of things I did in my math degree.
Actually I'm starting to wonder whether the thing that made university math easy for me was I quickly developed a good internal "type checker"/"type inference engine" in my head, and that helped make the next steps of proofs seem straightforward.
[0] https://gist.github.com/ndriscoll/881c4f5f0398039a3a74543450...
I understand you have strong opinions, I just don't understand why.
In math are highly concerned with structures and abstraction. We have things called operators. They aren't just addition and multiplication. We also use those words to describe different operations. They have things like groups, rings, fields, and algebras. Yes, plural.
The purpose of these things is to create logical frameworks. It matters not what the operators are. Nor does it matter what objects we operate on. Poincaré is explicitly saying this.
The part you're not understanding is the abstraction. This is what math is about. It is also why the Programming Language people are deeper in the math side and love Category Theory (I have a few friends who's PL dissertations are more math heavy than some math dissertations I've seen). It's no surprise. What's a function? How do you abstract functions? How do you define structure? These are shared critical questions. PL people are more concerned with types but I wouldn't say that makes them any less of a mathematician than a set theorist.
We can perfectly describe the CRUD operations with set theory. Do most programmers need concern themselves with this? Absolutely not. But is it something people designing those operations and systems is thinking about? Yes.
I'd encourage you to learn some set theory, abstract algebra, and maybe a bit of cat theory. It'll make these things pretty clear. But I'd caution about having strong opinions on topics you aren't intimately familiar with. Especially when arguing with those that are. Frankly, CRUD is a terrible example. I'm confident a quick google search (or asking a GPT) would quickly point you to relational algebra. It's discussed in most database classes. It certainly was in the one I was an assistant for.
> I understand you have strong opinions, I just don't understand why.
As someone that is "on the other side of the fence", and getting flamed for it, maybe I can shed some light since I have the other perspective as well. IMO, The reason for not seeing eye to eye is (for example) akin to saying "word problems are math". (Thinking of a grade school word problems for common reference). Yes, they are readily mapped to mathematical models that can solve the word problem & perhaps almost indistinguishably so. Though no - word problems are not math. Word problems are a series of phrases and words. That's where the talking past each other comes in... Different interpretations of "word problems are math", or "code is math". It's seemingly not clear whether we are talking about logical 'implies', 'element of', or 'equals'.
Which goes to "We can perfectly describe the CRUD operations with set theory.", we all agree there. That is not readily conveyed though when writing things like "code is math".
> word problems
That is a good analogy, except programming languages are formal languages, not natural languages.
> they are readily mapped to mathematical models
With code we are not talking about something that is mapped to mathematical models. Code is not modelled by mathematics, it is defined by mathematics (denotational semantics and operational semantics).
"Code is math" is true in a theoretical sense. Practically, coding doesn't always feel like what is commonly thought of as "doing mathematics", but it is essentially mathematical.
https://en.wikipedia.org/wiki/Formal_language
https://en.wikipedia.org/wiki/Semantics_(computer_science)
Yes, I agree we're misunderstanding each other because we're using different definitions but "you've" rejected "mine" without defining "yours" and expecting everyone to understand. Of course that'll lead to confusion. You can reject the mathematicians definition of math, but you sure gotta say more than "trust me" and it's a pretty wild thing to do, especially as non mathematicians.
The problem here is one side who's dabbled in cooking says "a chef makes desserts" and chefs are responding "we do a lot more than that". Maybe there's a few chefs that go "yeah all I do is dessert" and everyone points to that while ignoring the second part of their sentence is "but that's just my speciality." Wouldn't you think that conversation is insane? The only reason it's obviously so is because we all know what a chef is and agree on the definition. But who is better qualified to define the chef's job? The chef or consumer?
The way I'm responding I'd more characterize as: "wait, if what you are saying is true, then this other thing should be true too, but it does not seem to be. That would indicate what you are saying is not true."
In another thread, you characterized my response as stating: " ¬(A ↦ B) ⟹ ¬(B ↦ A)" (and this is a great example of language not being math, but math being language!). That was not at all my claim.
My claim is "I believe you are saying 'A = B'. It appears that 'B != A', therefore 'A != B'." My only claims are
(1) I believe you are writing to convey that you mean Math IS Language in the sense they are equal, identical, interchangeable, and fully equivalent, and bi-directionally so
(2) that: B != A
The only results can either be:
- "yeah, because B != A, the statement A = B is not true"
- Your claim (1) is false, I'm not actually saying "A = B"
- Your claim (2) is false, "B = A" is in fact true. I would find that to be an interesting assertion and would have loved to explore more why you think that.
You could say that math is behind the structure of a leaf, but the farmer doesn't care about that. Keep moving the goalpost all you want, I don't have time to argue these pointless things, and I stopped reading after the first sentence of your comment. I'm done here.
You seem really angry at the field of Computer Science for some reason.
We are talking about textbook CS fundamentals.
https://en.wikipedia.org/wiki/Semantics_(computer_science)
CS isn't programming, it doesn't write code.
CREATE, READ, UPDATE, DELETE are fundamentally mathematical in nature.
In a typical implementation these are database operations. That involves relational algebra operations, state transitions, boolean logic.
The READ part can be a very complex SQL query (composed of many algebraic operations) but even the simplest query (SELECT * from t where ID = 1) is filtering a set based on a predicate. That is mathematics.
No one is moving goalposts. Set theory and logic are at the foundations of mathematics.
It's almost the argument of programming vs computer science coming out here.
This is math:
{x | x.id = 1}
OTOH, a SQL query is a SQL query.
This thread is hilarious though. It's like
- Cashier: here is your change.
- Customer: you did math!
- Cashier, no, I gave you change.
- Customer: that IS math!
- Cashier: You mean, I used math to give you change?
- Customer: No, giving change doesn't use math, it IS math!!!!" [2]
= D
Moving along.. FWIW, indeed SQL was created to model set theory (relational algebra and tuple calculus), the close relationship is no accident of course [0][1])
> No one is moving goalposts
I feel too they are.
First goal post:
> Coding IS math. >> No, not always. Quite a lot of high-level code doesn't require any math at all. It doesn't take math to perform CRUD operations
Second goal post:
> CREATE, READ, UPDATE, DELETE are fundamentally mathematical in nature
CRUD is closely related to SQL, and SQL is closely related to various mathematics. Are they identical and therefore equivalent? No - because your database is not going to like it when you write "{x | x.id = 1}", and the Oracle DB might not like something that you can write for your Postgres DB.
[0] https://simpleprogrammer.com/mastering-sql/
[1] https://en.wikipedia.org/wiki/SQL
[2] To quote: """Not "coding uses math", I mean it is math""" @ https://news.ycombinator.com/item?id=43872771
The problem you're running into is that people who have some "mathematical maturity" don't get bogged down in notation, so it's difficult for them to see the distinction you're trying to draw between e.g. `{ x∈S | x.id = 1}` and `select x from S where x.id = 1`[0]. You say "a SQL query is a SQL query" and they just think "yes, which is also obviously a mathematical expression".
Computer programs are proofs[1]. This is intuitively and also formally true. You would agree writing proofs is doing math, yeah? Then obviously writing a computer program is also doing math.
Like I have a degree in math and have been a software engineer for over a decade. I do not know what distinction people are trying to get at. It's like trying to argue about the distinction between U(1), the complex numbers with magnitude 1, and the unit circle, and getting upset when the mathematicians say "those are 3 names for the same thing". Or saying that writing C is programming but writing in a functional language like Scala or Haskell (or Lean) is not.
[0] Modulo details like NULLs and multi-set semantics, but surely that's not the distinction?
[1] Up to isomorphism
Code is logical in nature and is defined by mathematics.
I'd agree code is usually governed by mathematics, not defined by it though.
Goes back to this ridiculous proposition:
- Cashier: You mean, I used math to give you change?
- Customer: No, giving change doesn't use math, it IS math!!!!" [2]
The proposition is that "code IS math", not defined by, not uses, not inspired by, not relies on, not modeled after, but IS.
Just because what you're working on specifically is an equivalent of middle-school math doesn't tell us much about the field as a whole.
Though middle-school or not, it's still math.
All of that code is a series of logical statements and expressions. Mathematical logic.
But the CRUD logic is so basic and boring, so obvious, that it doesn't require any thought.
>All of that code is a series of logical statements and expressions. Mathematical logic.
Which code? The machine code that underlies everything? Or the lines of simple high-level CRUD that don't even need a single number or logic statement to function? Not all programming has to be mathematical or even logical at a high enough level, and I say this as someone who's been coding assembly language for 40 years.
> Or the lines of simple high-level CRUD that don't even need a single number or logic statement to function?
Those lines ARE mathematical logical statements.
Each line defines logical operations that are executed by the computer.
Same for high level or low level programming. It's all logic.
Storing a string in some abstraction is not a mathematical operation. I'm done with this thread, it's going way too far down too many rabbit holes. The quarks that make up the matter that the computers are made of are pure "math". There, now I've moved the goalposts.
That is a narrow perspective of mathematics and computer science.
Assigning a variable is a mathematical operation. It is a logical state transition. This is formalized in Programming Language Theory.
We are not talking about quarks. We are talking about lines of code which are objectively mathematical statements, no matter how non-mathematical they seem to you subjectively.
You start with a practical problem to solve, and a computing machine that can perform logical operations on data. Your job is to figure out a correct sequence of logical operations that will solve the problem. The program that you have written is a mathematical structure. It is mathematics.
https://en.wikipedia.org/wiki/Semantics_(computer_science)
You confuse map with the territory. By the same logic you can say quarks are mathematics, because they are modeled by a theory that has some mathematics in it.
Our mathematical models of reality (i.e. Physics) is not the same as reality. So yes, by the same logic "quarks" are mathematics, but only in the same way that a "cup" is English. The quark was a great example, considering we can't observe it and purely rely on mathematical models, but the same argument would still hold true for a bowling ball. Physics is our description of reality, not reality itself. Our description of reality highly leverages the language of math, as that language was developed to enforce rules of consistency. Much the same way we design our programming languages, which is why programming language people study so much more math than your typical CS person.
If you're going to accuse someone of confusing the map with the territory, you really should make sure you aren't making the same error.
How math helps with programming languages? What math says about zero based indexes? How do you prevent off by one errors? How do you prevent buffer overflows? It's ergonomics problems.
It is hard to answer because of exactly what ndriscoll said[0]
As ndriscoll suggests, it is tautological. I mean look at what I said. I really need you to hear it. I said that coding is math. So what I hear is "How programming languages helps with programming languages?" Why are you expecting me to hear anything different? Start at 0? Start at 1? Who cares, it is the same thing. The natural numbers, non-negative integers, integers, even integers, who cares? They're the same thing. And who cares about indexing at 0 or 1 in programming? That's always been a silly argument that's inconsequential. By not being off by one? What's the question? Like being confused about if you start at 0 or start at 1 and how to get the right bound? It is a shift from one to the other, but they are isomorphic. We can perfectly map. But I really don't get the question. You can formalize these relationships with equations you know. I know it isn't "cool" but you can grab a pen and paper (or a whiteboard) and write down your program structure if you are often falling for these mistakes. This seems more about the difficulties of keeping track of a lot of things in your head all at once. By not going over your bounds? I'm so confused. I mean you are asking something like "if f(x) = inf when x > 10, how does math help you prevent the output of the function from being infinite?"Maybe what will help is seeing what some of the Programming Languages people do and why they like Haskell[1].
Or maybe check out Bartosz Milewski[2,3]. His blog[2], is titled "Bartosz Milewski's Programming Cafe: Category Theory, Haskell, Concurrency, C++". It may look very mathy, and you'd be right(!), but it is all about programming! Go check out his Category Theory Course[3], it is _for programmers_.
Don't trust me, go look at papers published in programming language conferences [4]. You'll find plenty of papers that are VERY mathy as well as plenty that are not. It really depends on the topic and what is the best language for the problems they're solving. But you'll certainly find some of the answers you're looking for.
Seriously, don't trust me, verify these things yourself. Google them. Ask an LLM. I don't know what to tell you because these are verifiable things (i.e. my claims are falsifiable!). The only thing you need to do is look.
[0] https://news.ycombinator.com/item?id=43882197
[1] https://excessivelyadequate.com/posts/isomorphisms.html
[2] https://bartoszmilewski.com/
[3] https://www.youtube.com/watch?v=I8LbkfSSR58&list=PLbgaMIhjbm...
[4] https://www.sigplan.org/Conferences/
Nonmathematical problems are difficult to answer when you try to find mathematical answers. It should be obvious, why it doesn't work.
Mentioning Scala is ironic, it's very light on math spik and to begin with was created to unify object oriented with functional programming, which is mathematically meaningless, because both are Turing complete and thus equivalent, tautological.
>So what I hear is "How programming languages helps with programming languages?"
Oh, right, in mathematics axiom is argument, but in reality it isn't. In programming you should assert what you assume, otherwise your assumptions can be wrong due to divergence from reality, but there no reality in mathematics, only fantasy, so you can't understand this with mathematics alone.
> but there no reality in mathematics, only fantasy, so you can't understand this with mathematics alone
No. Code is an abstraction. It exists as a logical structure, grounded in mathematical logic.
https://news.ycombinator.com/item?id=43888917
That may be true to some extent, but I think you are missing the point. Quarks are physical in nature, and code is logical in nature. Programs themselves are formal systems. Code isn’t just modelled by mathematics, it is defined by mathematics.
In the case of code, we can argue that the map is the territory.
Code is logical if you define logic as reasoning in general, broader than mathematics, and since it runs in physical environment, it now interacts with a messy world. Code is defined by business requirements, and there's no mathematics for that.
Now you’re talking about the human activity of writing code, not the code itself.
Those business requirements are inputs to the process of writing the code.
Once the code is actually written, that exists as a formal logical system, defined by mathematics, not business requirements.
The discussion started about whether human needs math skills to write code. That's what I mean when I say programming isn't mathematics. Meaning of code is defined by human, how do you intend code to be defined by mathematics? The human first imagines mathematical formulas, then by some unknown process they become code? I don't think anybody does it like that. You start with requirements, intuitively guess architecture, then decompose it, it works more like Cauchy problem (but less numeric, more conceptual, you have an owl, now draw it), but I don't think anybody models it like that.
>Once the code is actually written, that exists as a formal logical system, defined by mathematics
I still think that's not code, but your favorite model of code. For spellchecker language is defined by mathematics too: it splits text into words by whitespace, then for each word not found in dictionary it selects best matches and sorts them by relevance. Oh and characters are stored as numbers.
> The discussion started about whether human needs math skills to write code.
Writing code IS a math skill. When writing code you are writing logic in a formal system. Logic is mathematics.
You may be thinking that mathematics is just like doing arithmetic or solving equations. It is way deeper and broader than that.
> I still think that's not code, but your favourite model of code
Code is not just modelled through mathematics, it is actually defined by mathematics. It is fundamentally a mathematical construct, grounded in formal semantics.
https://en.wikipedia.org/wiki/Semantics_(computer_science)
That because mathematics doesn't need to match reality, so it's happy being ignorant about modelling? Anything you think is automatically true.
You are missing the point.
Code is not modelled mathematically, it is defined mathematically.
It exists as an abstraction which is fully defined by operational semantics and denotational semantics, not modelled or approximated.
In the counter example of a quark, that exists in nature and is modelled by mathematics, but not defined by mathematics.
https://en.wikipedia.org/wiki/Formal_language
https://en.wikipedia.org/wiki/Semantics_(computer_science)
It's definitely a gray area. Is a DAG traversal algo "math", or is it more computer-sciencey? What if you do it in SQL? Certainly there's a mix of more or less concentrated logic/math vs glue code, and most of that is very dependent on the domain you're working in.
I find this distinction useful in the abstract, that one can engage different parts of the brain for different components of development. This probably explains why a well-written DSL can be so powerful in the right context.
> Is a DAG traversal algo "math", or is it more computer-sciencey?
Firstly, computer science is math.
Secondly, I remember covering graphs in a discrete math course back when I was in college.
> What if you do it in SQL?
SQL is more-or-less a poor implementation of relational algebra. Ie, math.
Computer science is math in the same way that physics is philosophy, in that computer science certainly started as math, just like the natural sciences used to be subdisciplines of philosophy.
But it's hardly a useful grouping any more. You can study and do well in computer science with minimal knowledge of most of the core mathematical subjects.
While graph theory certainly crosses over into math, you can cover most of the parts of it relevant to most computer science as a discussion of algorithms that would not be the natural way of dealing with them for most mathematicians.
> You can study and do well in computer science with minimal knowledge of most of the core mathematical subjects.
You will fail at Theoretical Computer Science without mathematical proficiency. Go read some textbooks and papers in theoretical CS. It is a subfield of mathematics. Theorems and proofs. Rigorous and difficult mathematics.
https://en.wikipedia.org/wiki/Theoretical_computer_science
I've read plenty of theoretical computer science papers over the last 30+ years, and while some of it requires "rigorous and difficult mathematics" that is by no means universal.
I wrote my MSc thesis on the use of statistical methods for reducing error rates for OCR, and most of the papers in my literature review hardly required more than basic arithmetic and algebra.
So I stand by my statement.
Sure, there are subsets of computer science where you need more maths, just like in any field there are sub fields where you will need to understand other subjects as well, but that does not alter what I claimed.
EDIT:
Some authors are quicker to pull out the maths than others, and frankly in a lot of CS papers maths is used to obscure lack of rigor rather than to provide it. E.g the problem I ran across when writing my thesis was that once you unpacked the limited math into code you'd often reveal unstated assumptions that were less than obvious if you just read their formulas.
Is it possible your mental model of what CS is more aligned with software engineering rather than actual CS? Could you share some examples of what you consider to be CS but lacks any mathematical relation?
I agree is not a useful grouping in practice. I'm just interested in what makes you think like you do.
I did categorically not claim, nor even suggest, that any CS "lacks any mathematical relation".
What I claimed was that in computer science we often discuss things in terms that would not be the natural way of dealing with it in maths. We do that because our focus is different, and our abstractions are different.
It doesn't mean it's not math. It means it's not useful to insist that it isn't a different field, and its obtuse when people insist it's all the same.
Got it, thanks for the reply
Now we're getting into the "define maths" part of the discussion which is always where these discussions die. It can be argued that turning a kettle on and boiling some water is "maths" or it can be as narrow as "everything above basic arithmetic is logic, not maths."
So how much of programming is maths? Before we answer that, let's answer: How much of maths is actually maths? Because first we define maths, and then we define programming based on whatever that is, but until we have that first concrete definition this discussion cannot occur.
I will add that "it is taught by the maths department in college" is a flimsy argument, and frankly one the Physics department in particular would mock.
I think for this discussion, "math is stuff you'd learn in a math department" is a pretty useful definition, even if it's not a very good one. There's a lot of math involved in the design and manufacture of the kettle, electrical grid, and water utilities, but a person's ability to put a kettle on isn't going to be improved by math classes. In that way, programming probably is a bit mathy, but good programming is more like good technical writing than it is like math.
> I think for this discussion, "math is stuff you'd learn in a math department" is a pretty useful definition
That means that definition shifts over time. For example, courses on numerical analysis, graph algorithms, programming, and on compilers used to be part of “what you’d learn in a math department”.
It likely also even today will show geographical variation.
Overlap between programming and math is quite small, e.g. halting problem is solvable in programming (by flow analysis) but isn't solvable in math. Programming deals only with practical problems and can be completely guided by practical considerations, while math requires abstract outlandish skills - exact opposite. Why talk about math at all if it's already well known that programming is engineering?
Probability is math and there you have caching, hashing, etc. Then there are permutations, combinations, etc. with a lot of usecases in software. Distributing graphs across nodes? More math.
State machines too
Boolean logic and set theory are very important to computer logic, and graph theory is not far behind. But you can also learn Boolean logic in the philosophy dept. Which I unfortunately learned the hard way by taking Philosophy 101 after already having the CS class on logic. Took a semester to go over what we did for the midterm. Got a lot of naps that semester.
The difficulty is in how many relationships you need to keep in mind, not in how hard each of them are.
Just like in math.
BTW relational DBs are math.
> BTW relational DBs are math.
It's funny, reading the post you're replying to, I basically read it as
> I don't need math, I need <math, but by another name>
My teenage daughter used to complain about math, and I spent some time trying to explain to her that we use math every day... EVERY day. Now, when I see her do something that was math (even if it's not obvious it was math), I say "Math... every day". I say that a lot.
Also, yes, my daughter finds me annoying. But also funny; but likely not for the math thing.
https://news.ycombinator.com/item?id=43872687
DBs aren't math. In math it's perfectly ok for computation to take infinite time, you can just assume it already completed, but DBs don't work like that.
> Good code doesn’t just solve a problem, it solves it in a way that’s readable and modular.
Strongly disagree.
There are plenty of cases where non-modular code is more readable and faster than modular code (littered, presumably, with invocations of modular logic).
There are also countless cases, particularly in low-level languages or languages with manual memory management, where the best solution -- or the correct solution -- is far from readable.
Readability is anyways in the eye of the beholder (My code is readable. Isn't yours?) and takes a back seat to formal requirements.
Just as a matter of personal style, I prefer long, well-organized functions that are minimally modular. In my experience, context changes -- file boundaries, new function contexts, etc -- are the single greatest source of complexity and bugs, once one has shipped a piece of code and needs to maintain it or build on it. Modular code, by multiplying those, tends to obscure complexity and augment organizational problems by making them harder to reason about and fix. Longer functions certainty feel less readable initially but I'd wager they produce better, clearer mental models, leading to better solutions.
I think the important concept that "readable and modular" is trying to get at is how easy is it to continue working on the code in future. There's definitely codebases that are easier to work on than others, even when the domain is the same.
I'd say that readability, which often boils down to consistency, and modularity are ways to do this, but they aren't the only ways. And as you say, sometimes there's a need for "unreadable" code, so not everything can be easy.
Regarding mathematical
> concepts that are intrinsically complicated,
I'm not a mathematician, but I figure mathematicians aim for clean, composable abstractions the same way programmers do. Something complicated, not just complex in its interactions with other things, seems more useful as a bespoke tool (e.g. in a proof) than as a general purpose object?
> Whereas programming has more “small steps”: even someone who’s not smart (but has grit) can write an impressive program, if they write one component at a time and there aren’t too many components that rely on each other.
This is well put. I often wonder if a merely average working memory might be a benefit to (or at least may place a lower bound on the output quality of) a programmer tasked with writing maintainable code. You cannot possibly deliver working spaghetti if you can't recall what you wrote three minutes ago.
This is a baldly self-serving hypothesis.
> You cannot possibly write or deliver spaghetti, working or otherwise, if you're not capable of remembering what you wrote three minutes ago.
Forth programmers make a similar point. Forth is stack based; you typically use stack operations rather than local variables. This is ok when your 'words' (analogous to functions/procedures) have short and simple definitions, but code can quickly become unreadable if they don't. In this way, the language strongly nudges the programmer toward developing composable words with simple definitions.
(Of course, Forth sees little use today, and it hasn't won over the masses with its approach, but the broader point stands.)
> Whereas programming has more “small steps”: even someone who’s not smart (but has grit) can write an impressive program, if they write one component at a time and there aren’t too many components that rely on each other.
... In my experience, learning to write one component at a time (and try the code, and make sure it works before proceeding) is itself a skill that many struggle to develop. Similarly for avoiding unnecessary dependencies between components. Oh, and also being able to analyze the problem and identify separable components.
One of the most frustrating things about teaching programming, for me, is the constant insistence from other teachers that you have to maintain an "absolutely everyone can learn to program" attitude at all times. Many people who start to learn programming have misguided or confused reasons for doing so and - to say the least - could make much more effective use of their time developing other skills. (It's not a question of elitism; I'd surely flounder at some tasks that others find natural.)
I dislike "everyone can learn to program" because it conflates many, many levels of skills and expertise.
I very much think many people could learn the more advanced Excel Formulas, Power Automate and even simple Bash/PowerShell scripting to make their work more effective. I've met quite a few folks who had been intimidated out of trying who could do it.
On the other hand, how many people on this site could bootstrap a linux kernel on either very new or very old hardware? I know there are some, but they are certainly not the majority. I certainly won't be the first person to get linux and doom to run on a quantum computer.
But that is similar to other professions. Everyone with a largely functioning body can learn to turn a few planks and some metal parts into a functional shed door with some basic tools or to put up a decent brick wall that won't topple over in a month.
That doesn't mean everyone is qualified to pour concrete for a dam or a bridge foundation, or to re-do some historical work in original style.
> Everyone with a largely functioning body can learn to turn a few planks and some metal parts into a functional shed door
It's shocking how little physical and spatial ability some people have - that is definitely not true. Sometimes it might be a personal discount or lack of confidence, but this remains true regardless of the cause.
> "everyone can learn to program"
> That doesn't mean everyone is qualified to pour concrete for a dam or a bridge foundation, or to re-do some historical work in original style.
Exactly!
I think statements like that are more concerned with philosophy than reality. Any discussion surrounding topics like this typically ends up being a discussion around definitions.
I believe the vast majority of human beings are capable of learning how to program in the most extreme elementary sense of the word. As in, outside of severe disabilities or complete and utter inaccessibility to circumstances in which one could learn program, then I think the remaining population of people could learn to program to some degree. Obviously, not everyone will learn to program due to a near infinite number of reasons.
I would argue it's like music. Anyone can make 'music.' Just make a sound -- any sound. The difference between noise and music is subjective. I would not argue that everyone could be the next Lovelace, Turning, Ritchie, Thompson, Torvalds, etc..
Now, for my jaded opinion, I think a lot of the "everyone can learn to program" talk does not come from a place of desire to share the gift of knowledge and joy of programming. I think it's more of a subtle way to encourage people to go into programming so that they may be hired by mega corps. In order to keep the Capitalist machine running. It's like the National Hockey League's slogan, "Hockey is for everyone." That is just a fancy way of saying, "everyone's money can be spent on the NHL."
I recall reading about a professor at uni which carefully designed his tests throughout the year so he could determine if a student had grasped a certain programming topic. He also tracked effort, through handins and such.
After collecting data for a few semesters he concluded his students could be clearly divided into three categories: those who just "got" programming, those who understood it after working hard, and a small group that just didn't grasp regardless of effort.
I guess it depends on what is meant by "everyone can learn to program". It's a bit like saying everyone can learn to write, or to do math, or play tennis, etc.
I'm sure everyone is capable of learning some basic level of programming, just as they are able to learn a basic (high school) level of any subject. However, not everyone is going to have the aptitude to take that to an advanced professional level, however hard they try. We're not all cut out to be artists, or writers, or doctors, or scientists, or developers, etc.
>"absolutely everyone can learn to program"
Personally I've always considered a solid grasp of algebra to be the minimum bar for being able to program, at least for anything that isn't utterly trivial. Being able to take a word problem and turn it into a system of equations and solve it is a pretty close analog to being able to take some sort of problem or business requirement and turn it into code.
And the sad truth is that a huge percentage of the population struggle with just arithmetic, let alone algebra.
I mean - not "absolutely everyone" can learn to do just about anything. There's a wide distribution of intelligence, aptitude/ability and desire amongst folks amongst all of the various things that you might learn (not to mention learning disabilities, impairments, etc).
I might be capable of learning advanced accounting, but that sounds like torture to me and I'll be damned if I'll ever take that on! I'm sure programming feels like that to a wide variety of people, and I don't see any need for us to try to pretend otherwise - outside of a bizarre ideological desire for equivalent outcomes from disparate groups.
What do you mean by math? Abstractions or calculations? The effort it takes me to do long division in my head matches the effort it takes me to follow some obtuse spaghetti code. See, I can get good and fast at long division in my head, but I may never understand the fundamental theorem of calculus. Some people are really good at mucking around garbage code (they have no choice, they get paid to), but what part of programming did they get good at? Obviously, some part of it, but nothing to write home about. Whenever I sense that I'm just getting practice at doing the equivalent of mental long division at work, that's when I always seek a new job. No amount of money is worth falling behind like that.
I'm thinking of "computation", "intuition", and "organization".
Computation is following an algorithm. e.g. long division or computing a derivative.
Intuition, AKA brilliance, is finding a non-obvious solution. Think "solving an NP problem without brute force"*. e.g. solving an integral (in a form that hasn't already been memorized) or discovering an interesting proof.
Organization is recording information in a way that a) is easy for you to recall later on (and get insights from) and b) is digestible by others**. e.g. explaining how to compute a derivative, solve an integral, or anything else.
Math, programming, and writing each require all skills. The kind of math taught in school (e.g. long division) and your boring jobs are primarily computation. I believe advanced math (e.g. calculus) is primarily intuition; it requires some organization because big theories are broken into smaller steps, but seems to mostly involve smart people "banging their head against the wall" to solve problems that are still quite unclear***. Programming is primarily organization. It requires some intuition (I think this is why some people seemingly can't learn to code), but in contrast to math, most programs can be broken into many relatively-simple features. IMO implementing all the features and interactions between them without creating a buggy, verbose, and unmaintainable codebase is programming's real challenge. Writing is also primarily organization, but finding interesting ideas requires intuition, and worldbuilding requires computation (even in fiction, there must be some coherence or people won't like your work).
> Some people are really good at mucking around garbage code (they have no choice, they get paid to), but what part of programming did they get good at? Obviously, some part of it, but nothing to write home about.
I agree that work you find boring should be avoided, and I also try to avoid working with it. But some people really seem to like working on esoteric code, and I think there are some skills (beyond computation) developed from it, that even apply when working with good code. Building a mental model of a spaghetti codebase involves organization, and if the codebase uses "genius hacks", intuition. Moreover, the same techniques to discern that two code segments in completely different locations are tightly coupled, may also discern that two seemingly-separate ideas have some connection, leading to an "intuitive" discovery. There's an MIT lecture somewhere that describes how a smart student found interesting work in a factory, and I think ended up optimizing the factory; the lesson was that you can gain some amount of knowledge and growth from pretty much any experience, and sometimes there's a lot of opportunity where you'd least expect it.
* Or maybe it is just brute force but people with this skill ("geniuses") do it very fast.
** These are kind of two separate skills but they're similar. Moreover, b) is more important because it's necessary for problems too large for one person to solve, and it implies a).
*** And whatever method solves these problems doesn't seem to be simplification, because many theories and proofs were initially written down very obtuse, then simplified later.
A lot of people in this type of threads always makes the same mistake: confusing what math is with branches of math, or rather, ways in which math is used. The way the education system is built certainly contributes to this.
I've always found the car metaphor to work very good to understand this: A car is a machine that can transport itself to point A to B (some other rules apply). There are different types of cars, but you certainly haven't understood the definition of you say that something is not a car because is not a Volvo, or because it doesn't look like a Ford, when it's clearly able to transport itself.
Math is the study of ideal objects and the way they behave or are related to each others. We have many branches of mathematics because people have invented so many objects and rules to play with them. Programming is nothing if not this very definition. The fact that you don't have to "use math" when programming is not really addressing the point, it's like saying a car is not a car because it has no discernible brand.
Declarative construct is made of relations, but imperative execution isn't, rather it's a process in time, but time is not a thing in math.
Another misconception I'd say.
"Time is not a thing in math" is not understanding what math is. Time is another ideal object following certain rules under a given domain. Programming is coming up with objects of different size, with different characteristics, with interact at different points in time, i.e. following certain rules.
Its very rare imo that computational problems emerge fully formed & ready to be tackled like proofs.
Usually even deciding what the problem is is in part an art, requires an act of narrativization, to shape and form concepts of origin, movement, and destination.
A good problem solver has a very wide range of abstract ideas and concepts and concrete tools they can use to model and explain problem, solution, & destination. Sometimes raw computational intellect can arrive at stunningly good proposals, can see brilliant paths through. But more often, my gut tells me it's about having a breadth of exposure, to different techniques and tools, and being someone who can both see a vast number of ways to tackle a situation, and being able to see tradeoffs in approaches, being able to weight long and short term impacts.
It has been reimplemented in a wide variety of languages today. I look forward to the next iteration: WolfgangSort or FatimaSort or XiaomiSort or whatever.
The a capital example of an exception that prove the rule.
I absolutely value & have huge respect for the deeply computational works that advance us all along!
But this is an exceedingly rare event. Most development is more glue work than advancing computional fundamentals. Very very very little of the industry is paid to work on honing data structures so generally.
+1. Excellent description of how the skills relate. I often read, listen to poetry when stuck in a math or programming problem. ... or just talk to the rubber duck. :-)
>Good code
"Good code" is very subjective. Even readability and modularity can be taken too far.
The problem with doing things right the first time is some people look at it and just say, “well of course it should work that way.” Yes but did you think of doing it that way?
Organization may also benefit from spacial skills.
Although the study by Prat et al. (2020) suggests that linguistic aptitude is a better predictor than numeracy for learning programming in Python, it should be read carefully, as it can easily be oversimplified.
To begin with, the study measures functional numeracy: the ability to solve everyday numerical problems. This is quite different from the kind of advanced mathematics often associated with programming, such as formal logic, symbolic abstraction, or the use of formal languages (as found in denotational semantics or type theory).
These more abstract skills—not basic arithmetic—are essential for understanding recursion, type inference, or algorithm design. That functional numeracy has low predictive power in this study does not imply that deep mathematical reasoning is irrelevant to programming.
Moreover, the language used in the study is Python, which was explicitly designed to be readable and semantically close to natural language. This may give an advantage to individuals with strong verbal skills, but the results don’t necessarily generalize to languages like C, Lisp, or Haskell, where symbolic and logical density is much higher.
Finally, language and mathematics are not opposing domains. They share cognitive underpinnings, such as working memory, executive attention, and hierarchical structure processing. The key is not which one "wins," but how they interact and complement each other in different programming contexts.
> These more abstract skills—not basic arithmetic—are essential for understanding recursion, type inference, or algorithm design.
No they're not. Academia has spent decades trying to formalize many aspects of programming and continues to be confused by the lack of correlation between comp sci grads and innovative programmers. Why is it that the drop-outs are succeeding so wildly?
Recursion, for example, is learned by most of us real world achievers when we hit a brick wall in programming that other methods won't solve, and we have that aha moment of "this is why this exists". Not because we studied advanced math with symbolic abstraction, denotational semantics and type theory.
The uncomfortable truth is that almost all of professional programming and innovative programming (creating useful stuff never before seen) never uses any of the advanced math skills that are prerequisites in every degree program. I think much of the sadism around teaching this is perpetuated by "I did it so you have to" and academic gatekeeping.
When you get really really good at programming and hit the most productive zone in your life, it feels like language. That you have the ability to just say it.
The drop-outs are "succeeding wildly" because people like Turing, Church, and Hilbert invented the sand pit they're playing in.
Knuth created LaTeX. Pandoc is written in Haskell, famous for being a completely useless academic language with no real purpose beyond torturing undergraduates (it says here.) Efficient search and data compression algorithms aren't hacked together in late night hobby coding sessions.
Cryptography, digital signal processing for images, sound, and video, and ML core algorithms are all mathematical inventions. The digital world literally runs on them.
"Real world achievers" might want to try being a little less parochial and a little more educated about the originators of the concepts and environments they take for granted.
Vibe coding "Social AI chatbot network with ads = $$profit$$" or "Cat videos as a service" is only possible because the entire field stands on the shoulders of mathematical giants.
Well said. I guess such ignorance is not that surprising given that most "programming" today is glueing code together with scripting languages. This is really just a form of configuration. It certainly requires domain knowledge and a tolerance for digital bureaucracy, but not always maths. I do wish schools would do a better job of teaching that the entire fabric of western civilization rests upon mathematics.
It sounds like these advanced skills might be needed to build the platforms that the dropouts build on top of. But I agree that, as a professional programmer of 20 years with a 4-year business degree from a state school with a weak math background (and also as someone with a talent for writing and language), I have very little need for advanced math in my day-to-day.
>mathematical inventions
Like Rijndael?
You are making a rather wild assumption that dropouts don't have the abstract math skills that correlate with programming skill.
I'd argue that if they can figure out recursion after hitting a brick wall like you describe, then that's a good indication they did have abstract math aptitude to begin with.
In my personal experience the hardest part of mathematics is it's grammar and language. It's very different from natural language, whereas programming is much closer. You can take nearly any math problem and convert it to pseudo code and it'll be much more understandable for those programmers who never studied (or struggled with) advanced math.
Programming requires a base level of natural language aptitude that nearly all adults have, there's diminishing returns for anything approaching the levels of a poet or novelist for example.
When you get good at math, it also feels like language. Mathematicians see expressions like `|x-y|` and the read it as synonymous with "distance" just as they would "distancia". Physicists pick out fragments of large equations and say "this term acts sort of like an 'effective' mass", etc. There is a sort of vocabulary to mathematical patterns.
Picking up a skill without intentional study is great, but you still learned the skill. Programming languages are formal languages. Most mathematicians don't study foundations either.
Professional programming doesn't often make use of specific advanced mathematical knowledge, but I find it makes everyday use of the skills.
My own definition of mathematics for the sake of this comparison: - If you can write a formal proof, starting from some assumptions and proving some result, it is mathematics. The assumptions are the axioms. Then you use logic, which is the same programming logic, and then you get to some result.
It can be an invariant in a programming function, it can be a more general result, if you can write a proof, it is mathematics. Most algorithms involve proofs, so they are mathematics.
It has nothing to do with it being "sadism" or academic gatekeeping.
These people are doing mathematics without knowing it is mathematics. That's all.
> Why is it that the drop-outs are succeeding so wildly?
Here is where you can learn about confirmation bias and educate yourself.
This reads like an LLM wrote it.
It does! Who structures their posts like a platonic rhetorical piece??
There is even the em dash in it
This totally makes sense to me. I've always been a very good / fast reader, which has been incredibly useful in my programming career. I had a good SAT math score (710) but got a perfect score (800) on the SAT verbal (in the late 90s).
I remember when I first started working on my Master's project on wireless sensor networks, my advisor sat me down and said "I think I know a good project for you. I want you to print out the source code for TinyOS, study it for a week, and come back to me when you think you know enough to make these changes." This was a sort of formative experience for me, and ever since when joining a new project I made sure to take the time to read through the code to understand how things fit together.
Also checking in as someone with strong language skills. When I was a kid in school a million years ago I was actually rather weak in math and crapped out around trig, but excelled in English, Spanish and C. Now I'm a senior web dev, and I still can't do much math beyond the basics required for everyday living (though 18 years in food service means I'm really quick at estimating percentages to within delta)
When I tell you that I was a server and bartender it should all become very clear. Though your guess isn't far off, it's a skill that's served me well since I've picked up breadmaking as a hobby
> I've always been a very good / fast reader, which has been incredibly useful in my programming career. I had a good SAT math score (710) but got a perfect score (800) on the SAT verbal (in the late 90s).
This is me and also many of the CS students in my cohort, and AFAIK something that universities actively selected for, and students also self-selected around in an era of RTFM/MUDs/IRC before LLMs or youtube. The best programmers I've worked with are still always very linguistically brained.. polyglots even when they didn't have to be, or with a long track record of engaging with difficult literature. If nothing else.. just very witty in that certain way that's meta-cognitive, meta-linguistic.
This is still true I think but it's much harder see out in the wild due to the degree/career trajectory popularity. Plus as long we're optimizing for leet-coding even though built-from-scratch algorithms are a very rare need compared to skills with good design/exposition.. naturally the math-brain is favored.
Remember people whining about weak ecosystems? That's a lot of opportunity to build things from scratch instead of making another - for real this time - IRC clone with Electron.
Early, strong reader here. Off-the-charts spatial reasoning, as measured by tests.
Terrible at math, I hate it and feel dyslexic trying to read most mathematical writing. I excelled at it in elementary school, then quickly came to feel frustratingly stupid at it as it became less about algorithms (more on that in a bit...) and all about equations and abstract stuff with unknown applications.
However, programming was natural and easy to pick up. I've repeatedly had to take more time convincing myself I actually understand some supposedly "hard" thing, like pointers or recursion, than it took to learn them in the first place, because they were in fact very easy to understand so I kept second-guessing myself—"I must not get it, because that was easy". I've been the go-to guy for "hard" and "low-level" problems basically everywhere I've worked.
What I've noticed is that when I must read math, the only way I can get any headway is to turn everything into steps, through which some example values may pass and affect one another. I have to turn it all into algorithms. Algorithms, I can get. Attempts to express meaning through equations and proofs, no, I have to painstakingly turn every single boundary between every symbol into a step and "walk through" it to have any hope of understanding it, and once I do, this new understanding only barely illuminates the original representation.
I think programming clicked for me because, as typically encountered and taught, it's very heavy on algorithms and very light on other varieties of mathematical presentation. Plus, having so very much more context available about what variables represent and what routines do, than a jumble of letters and symbols. FFS, if we say Perl is line noise, what's mathematical writing? Straight gibberish from a brain-wrecked Cthulhu cultist? Perl's the clearest thing in the world by comparison!
... where I do run into trouble is languages with "mathy" syntax, where the idiomatic style favors single-letter variables and accomplishing-things-by-asserting-equality. I can't read Haskell to save my life. Put the same supposedly-tricky concepts (monads, type classes) in any more-ordinary language, and it's easy, but when I tried to learn them using Haskell, I couldn't get anywhere at all. Shit, it takes me forever just to understand fizzbuzz-level programs written in Haskell.
Very interesting, as I am nearly the opposite. Bad spatial reasoning though pretty good at most other things. In school, geometry was fun and interesting - I enjoyed proofs. And I majored in English in college with a CS minor, so I think there must be a variety of in-roads for programming.
I think one "verbal" skill that has served me well is fast reading. When I had to read a 300 page novel once a week you learn how to skim for key elements, which is immensively useful for getting up to speed in a new codebase/language or locating a bug.
Same but I am literally math dyslexic (formal diagnosis of dyscalculia).
It prevented me from having a CS degree, I was unable to complete the math courses, but as far as actual programming and "software engineering" goes (design, etc) it's never hindered me. I can work out the logic and I let the computer do the math.
Edit: I'm downvoted below zero for this comment. I don't know what people are so offended by?
Yeah, to be clear, I'm not diagnosed with anything like that, and am only likening the experience to what I imagine it's like for dyslexics of humans language as a ready metaphor—meanwhile, the concepts aren't hard for me, like, a lot of people with no diagnosis of anything fall off the math-wagon right around the time operations on fractions are introduced, but the concepts have always posed no trouble to me and I breezed through early potential trip-ups like that. The style of presentation, specifically, is what gives me such a hard time and is what makes trying to approach even fairly easy "real" mathematics so hard for me.
> It prevented me from having a CS degree, I was unable to complete the math courses, but as far as actual programming and "software engineering" goes (design, etc) it's never hindered me. I can work out the logic and I let the computer do the math.
This is what's wild to me: I have a long, successful career in a "STEM" field that's allegedly math-heavy, while being practically incapable of working with math. Like, it's never even been slightly a problem. I can't relate at all to characterizations of programming as heavy on math. It's never been my experience of it, and at this rate, probably never will be. If it were, I'd for-sure be in a different job.
They probably suspect you to be a slow coder who writes code with bad abstractions because they percieve the maths skills as how one does things in software engineering.
Oh wow, are you me - I had an almost identical SAT experience. Oddly enough compared to another poster in this thread, I love functional programming and Haskell and studying things like dependent types in a programming context has helped me patch up my crappy understanding of "actual math", proofs, etc.
I came here to comment the same -- also ready to share my SAT scores from the 1980s. I was also much highter on the verbal side and I have had an awesome software engineering career using many, many languages (often together) on large, professional projects at large companies anybody here would recognize.
There is no such thing as a "language brain" or a "math brain" unless you show experimentally that those bunch of neurons can be grouped into two non-overlapping regions dedicated to "language" and "mathematics".
Mathematics itself is a human-made formal language that can be bootstrapped from definitions and axioms of logics and set theory, which have to be given in human language first.
Experienced mathematicians read formal theorems written in Greek letters off their blackboards as if it was normal English, suggesting they think about it like it was just normal English. This is not to say they cannot also view in front of their mental eye visual representations isomorphic with that language if they chose to.
Dyscalculia, dysgraphia and dyslexia can all present independent of one another, heavily implying that different processes govern math, reading and writing. To call them "language brain" and "math brain" is simply a useful fiction, whether they're actually separate piles of meat is irrelevant.
math != calculation.
It's an actively stupid fiction for people who don't understand what math is.
> It's an actively stupid fiction for people who don't understand what math is.
This comes off quite judgmental, and doesn't help me understand your actual point. Could you elaborate on the differences, as you see them?
In higher math (typically 2nd year and onwards at US universities) there is fairly little explicit calculation, and math is more about studying the (formal) relationships between mathematical objects. For example, we can abstractly define topological spaces and abstractly define algebraic “groups,” and algebraic topology studies the relationships between topological spaces and algebraic groups. This is somewhat like having a “data type” defined by some library, examining ways it can be transformed into data types defined by some other library, and studying what invariants hold under those transformations.
In some sense, an undergraduate math education is akin to learning the “standard library” (in the software engineering sense) of higher mathematics. Most courses start with basic abstractions of some mathematical object and repeatedly construct more and more abstractions on top of those. The structure of those abstractions is similar to how you might build a library. A professional mathematician is expected to be fluent in the mathematical standard library, just like how you might expect an experienced software engineer to be fluent in Python’s standard library.
If this analogy is true, people who can learn Python relatively quickly might be able to also learn higher mathematics relatively quickly under the right pedagogical environment.
One personal project I'd love to find the time for is to try to write out a sequence of math notes that essentially builds up a Lean repository, intentionally "refactoring" as it goes along. So you do some explicit coordinate vector or system-of-equation calculations with both real and complex numbers, eventually define interfaces for fields and vectors, do your linear algebra proofs, realize that linear endomorphisms are almost-but-not-quite a field, so define rings and go back and refactor your field proofs to use them, realize vector spaces are "K[T] not-quite vector spaces", define modules, refactor your vector space code, etc.
This was kind of how math classes worked, but without that explicit phrasing. It would certainly make the analogy between the two activities more obvious. I also wonder whether people would have less trouble with quantifiers if they were phrased in programming terms: a proof of "forall x, p(x)" is a function x=>p(x), and a proof of "there exists x such that p(x)" is a pair (x, p(x)). e.g.
Continuity: (epsilon: R, h: epsilon>0, x_0: R) => (delta: R, h2: (x: R, h3: d(x,x_0) < delta) => d(f(x),f(x_0)) < epsilon)
Uniform continuity: (epsilon: R, h: epsilon>0) => (delta: R, h2: (x: R, x_0: R, h3: d(x,x_0) < delta => d(f(x),f(x_0)) < epsilon))
proof of UC => C = (epsilon, h, x_0) => let delta, h2 = UC(epsilon,h) in (delta, h2(_,x_0,_))
So when you're trying to figure out how to do the proof, it's clear what kind of type you need to return and your IDE could help you with autocomplete based on type inference.
It is hugely judgmental of people making claims and doing studies who don't understand that elementary arithmetic is a tiny part of math. I judge them. It's ok to be ignorant of a topic, but they shouldn't discuss it or make suggestions.
The comment below/above sort of explains it, except I'd argue covering it starts/can start/should start much earlier than at university. Basically, the vast majority of most branches of math, especially pure math, involves very little in the way of calculation. And the way math gets taught (and stupid claims about "math brain") means that many kids who aren't in the top few % of "doing calculation" never get to do the classes where it's less important.
LLMs, trained on words/tokens and symbols and logical combinations of them, are proving to be good at math, and bad at calculation/arithmetic. If an LLM went to school we'd never let it train on real math tokens. It would get shoved in the corner as a "model bad at math" because it was a "model bad at arithmetic".
Most math is more about definition, proof and structuring/relationships than about numbers. It is a bit of a joke among math students that numbers above 5 are effectively infinity, they crop up that rarely.
This. And we do a huge disservice to tons of people by basically instilling the opposite idea. So many of the people I know that either claim they are "not math people" or who claim they "dislike math" say this because they were subject to nothing but tedious calculation in school and were never show the logical, conceptual, and actually interesting parts of mathematics. I despised math until I read Frege's foundations of arithmetic in college since it was the first time I ever had someone actually show the point and make it interesting.
"language brain" or "math brain" are obviously non-scientific terms.
I think most people would agree that problem solving, expressing ideas verbally, and expressing ideas in math are very different skills.
The other thing I don't like about this framing is the implicit false dichotomy. The best programmers I've met often have strong language and math skills.
Poetry is the art of giving different names to the same thing. Mathematics is the art of giving the same name to different things. (Henri Poincaré)
It's basically being good at "naming things", in particular abstractions.
> Poetry is the art of giving different names to the same thing.
Did Robert Frost say something like, "all language is poetry?" A chair is the term I learned to describe the object I am currently sitting on while typing this comment. Germans might call the same object a 'Stuhl', the French might call it a 'chaise', etc.. My point being that the object isn't technically any of those words, but rather those words symbolically map to that object.
"naming things" is really about "recognizing things" and/or "understanding things". It much deeper than just language.
Yes, exactly, just watch how little babies recognize and communicate about hundreds of objects, emotions and situations before they're able to utter momma or dadda.
I wanted to say something like this, but in the opposite direction. I think perhaps more so there’s a difference between abstract and concrete thinking? But tests like the SAT in my experience (which is now 20 years out of date) treat the verbal section as more of a set of abstractions (analogies, for instance) and the math the same. I got almost equal marks in each (660 verbal/650 math) on my only try.
> unless you show experimentally that those bunch of neurons can be grouped into two non-overlapping regions
A good place to look is after brain trauma. Where people end up with weird missing functionality.
Oliver Sacks writes about some of the weird side effects of brain trauma.
We could experiment on mathematicians or writers by causing trauma to different parts of the brain...
Or neural stimulation/anaesthetic of brain regions might show something.
Yeah, the brain is a sparse MoE. There is a lot of overlap in the hardware of the "language brain" and the "math brain". That being said, I can discuss software concepts in a foreign language, but struggle with basic arithmetic in anything but English. So while the hardware might be the same, the virtualization layer that sits on top might have some kind of compartmentalization.
Yes, but there is a visual/spatial ability and a verbal/mathematical, and you can do maths with the visual/spatial ability or with the verbal/mathematical ability.
It leads to different styles of thinking and problem solving.
I figure the programming brain matters the most for programming.
> There is no such thing as a "language brain" or a "math brain"
Tell that to the people who designed the study, the people that approved the study, and the people that funded the study.
At the end of the day, the study set out certain criteria for assessing certain skill types/competencies and divided the people by those well defined criteria. I think it’s pretty hard to argue against the idea that people might have at least some level of aptitude for different types of activity and skill.
I think its worth noting the study did not actually make this claim, the news article did. I suspect the study authors would agree the term "language brain" or "math brain" is misleading.
The types of people who do these types of experiments almost never have a real understanding of math. They equate arithmetic with math.
What about Wernicke's area and Broca's area? If damaged they produce causal changes to our use and processing of language. I agree it's not as simple "language brain" but there are physical structures in the brain that are directly related to language.
It think this is silly on multiple accounts. I'll claim that there's not real thing such as a "language brain" or "math brain." I'll also claim that most people don't know what math is, and that their evidence supports a "math brain".
Math isn't about calculations/computations, it is about patterns. You get to algebra and think "what are these letters doing in my math" but once you get further you think "what are these numbers doing in my math?"
A great tragedy we have in math education is that we focus so much on calculation. There's tons of useful subjects that are only taught once people get to an undergraduate math degree or grad school despite being understandable by children. The basics of things like group theory, combinatorics, graphs, set theory, category theory, etc. All of these also have herculean levels of depth, but there's plenty of things that formalize our way of thinking yet are easily understandable by children. If you want to see an example, I recommend Visual Group Theory[0]. Math is all about abstraction and for some reason we reserve that till "late in the game". But I can certainly say that getting this stuff accelerates learning and has a profound effect on the way I think. Though an important component of that is ensuring that you really take to heart the abstraction, not getting in your own way by thinking these tools only apply in very specific applications. A lot of people struggle with word problems, but even though they might involve silly situations like having a cousin named Throckmorton or him wanting to buy 500 watermelons, they really are part of that connection from math to reality.
This is why "advanced" math accelerating my learning, because that "level" of math is about teaching you abstractions. Ways to think. These are tremendously helpful even if you do not end up writing down equations. Because, math isn't really about writing down equations. But we do it because it sure helps, especially when shit gets complicated.
[0] https://www.youtube.com/watch?v=UwTQdOop-nU&list=PLwV-9DG53N...
It took me a long time to realize math consisted of abstractions.
I was never told, for example, that matrices are a useful abstraction (shorthand?) for representing linear equations.
Or that imaginary numbers were an invented abstraction that made certain calculations easier.
It would have been nice to learn it from the perspective of how those abstractions came into being and why they're useful.
> I was never told, for example, that matrices are a useful abstraction (shorthand?) for representing linear equations.
Congratulations on overcoming the (very weird) choice of whoever taught your intro to linear algebra class.
Or just your regular 8/9th grade Algebra course where you should have learned systems of equations. That's where I first encountered using matrices for systems of equations at least. (maybe it was Algebra 2+ but it was definitely before taking Linear Algebra in college).
I'm not sure how to interpret this comment, it could easily be read in multiple ways. (Depends on priors)
Regardless, it's a shockingly common occurrence. I'd agree it's not the right way, but it is also a common way. That perspective might define those priors
When growing up, I'm sure my fellow students and I were not the only ones get responses to questions like "Why can't one divide by zero?" or "Why is Pi necessary to compute to area of a circle?" answered like "because that's just how things work" or "because it just is."
I'm not going to sit here and act like I was a star student or anything. I was more of a class clown type. I absolutely hate math and all things math. That was, until I went to college. A switch flipped when I was in a Calculus II class.
Our professor asked, "What is a 100 divided by 0?" People in the class responded with, "You can't divide by 0 because it's undefined." To which our professor responded with, "Why?" Then a student took out a calculator and showed the professor that the answer as indeed undefined. To which he responded with, "Ok, how do you that calculator is correct? Do you just believe it because people told you that dividing by zero is undefined? Ok, the answer is undefined... but why?"
Right then and there, a switch flipped in my brain. I realized that I was basically institutionalized to just not question math and to just accept that things work a certain way "just because." It was at that point I actually started to become interested in math, and it completely changed my outlook on math in a positive manner. I love math now (despite being horrible at it).
This reminds me of a lot of what I see in grad school and academia. Something akin to what Tom Wolf talks about here[0]
I don't think this part is isolated to math, but there's a lot of acceptance for "because" being an answer. Being in an authoritative position and being challenged can be frustrating, but I think a lot of that frustration is self-generated. We stifle creativity when young and it should be no surprise that frequently when people challenge, they don't have the tools to do so (or be receptive) effectively. Truthfully, "I don't know"[1] still shuts down the conversation.But I think people themselves are uncomfortable with not knowing. I know I am! But that isn't a feeling of shame, it is a feeling that creates drive.
[0] https://thomwolf.io/blog/scientific-ai.html
[1] Alternatively, variations like: "That's a good question, I don't know" or "Great question, but we don't have the tools to address that yet". The last one need not undermine your authority either. Truthfully, the common responses resulted in me becoming strongly anti-authoritarian. Which, also has greatly benefited my career as a researcher. So, thanks lol
I remember my high school math teacher responding "that's an interesting question, I don't know" and then coming back with an answer the next lesson after researching the subject.
I guess I've been lucky with my math teachers.
Yeah, I was worried it might come off as sarcastic or something. Hopefully the original commenter doesn’t take it that way, or reads this follow up. Unfortunately, adding stuff like “honestly” often makes a comment look even more sarcastic. It is a real pain.
But yes, I am expressing honest admiration—it was a bad move on the part of the teacher I think, which the poster seems to have overcome!
Well I appreciate clarifying! I originally took it as sarcastic but second guessed myself. I figured math is pedantic, so maybe being a little pedantic here could help haha.
I went through the same thing, it really isn't easy. But it is also why I don't blame others for not seeing it. It would be hypocritical to do so. I'm just not sure what's the best way to share, I'm open to ideas. Unfortunately we have to contend with priors that we see here, though I'm happy if my efforts even make one more person able to share in this beauty.
No worries I did not take it that way!
No, it is "very weird". Without knowing that a matrix represents a system of equations, Gaussian elimination would be pointless.
I may not have been clear, but it was given to us as a definition in class. As in we saw the matrix and the corresponding equation.
But nobody even told us why! And at that time I never thought to ask.
I think we're all in agreement. No need to fight.
Numbers are an abstraction, not just "imaginary" numbers, all numbers are imaginary they don't exist in reality.
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Even if we narrow math to numbers, what many people thing of as "math", we quickly move from arithmetic to patterns.
The patterns of repetitions of single things, repetitions over different things, repetitions between different things, repetitions over repetitions, ...
Natural/counting numbers are just the simplest patterns of repetition.
If we taught music the way we teach math. No wonder so many people are "bad at math".
“Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.”
https://en.m.wikipedia.org/wiki/A_Mathematician%27s_Lament
Of course Brett Victor has the essay on his website: https://worrydream.com/refs/Lockhart_2002_-_A_Mathematician'...
It is a great read!
Frankly, with all the mess that music notation is and the way it's often being taught, this may actually be a slight upgrade over status quo. I have never experienced more "that's just how it is" than in music.
> It think this is silly on multiple accounts. I'll claim that there's not real thing such as a "language brain" or "math brain."
It seems plainly obvious that this language just means “areas of brain that activate when dealing with math problems” vs “areas of brain that activate when dealing with language problems” and yes there is hard evidence that there is a difference between them.
Please reread my comment in full. I'm willing to bet we disagree on the definition of math. I'll strongly insist that the one I'm using is common among mathematicians. If you'd like to retort by saying it's about semantics then congrats, we're on the same page (and can be verified by reading you sibling comments and/or my replies to some of them)
It has nothing to do with the definition of math. If you put people in an MRI and give them random problems then you’ll notice a distribution in the patterns of brain activity that are distinct. Some of these groupings we’ll call “math like” because the problems are more like mathematics than linguistics. And in the other distribution we’ll call them “word like” because the problems fall into the category we associate with word problems.
The fact that we see separate patterns of brain activity is the interesting part not any semantic argument about what you or mathematicians feel should be defined as being math or not.
I am willing to bet good money that teaching pure mathematics like group theory, abstract algebra, or stuff like big Oh and combinatronics will do our kids a lot more good and advance them far more than calculus or geometry as taught in schools will.
I have the same experience as you did, in that studying by myself abstract algebra accelerated my learning and reasoning skills.
I think differential equations was where the switch from "I'm good at math" to "I think I want to actually know more of this" happened. I changed my major from engineering to physics. But agreed, abstract algebra was another big change and after that I took every class my university would let me and the chair of the department even made some new ones for me and some others.
I don't think this experience is uncommon (among people who get to these levels). Which is why it is really sad. Especially given how linear algebra and abstract algebra are in a lot of ways easier than calculus. I also think they should be taught earlier purely due to the fact that they teach abstraction and reasoning.
Just anecdotally based on my 2000something kid high school, the copious amount of tiering of math classes seemed to indicate to me that either we're really bad at teaching math to 80% of people, or only 20% of people will be able to handle precalculus.
We had basically 4 tracks: one ended with you doing algebra 1 in senior year, another ended with you doing trig in sr year, yet another that ended with trig (no precalculus), and then one that ended with you doing trig and precalc. That final class then had further subdivisions that were too small to have their own full classes: some kids just did precalc, some did calc 1 and took the AP Calculus AB exam and/or IB Math SL, while even even smaller group took AP Calculus BC and/or IB Math HL. The total number of kids who took the AP Calc AB exam in my year was 20ish, out of a graduating class of 500-600.
This system sounds extremely complicated.
> the copious amount of tiering of math classes seemed to indicate to me that either we're really bad at teaching math to 80% of people, or only 20% of people will be able to handle precalculus
Or maybe it indicates that the people designing this system should be fired? Job security through complexity?
(Or maybe I’m just biased by the system I know… I’m just asking questions)
The classic example is conspiracy theorists. But lots of bad faith actors also use it to create leading questions that generally ignore or lead away from important context. Again, I do not think you're doing this. The rest of your comment makes me think you're acting in good faith.
I went to a small high school; my graduating class was around 30 people.
Math classes weren't separated by grade. So, I took Algebra 2 in 8th grade alongside a cross section of the school; there was one other 8th grader, two seniors, and a selection of people in between.
There was no path to take the AP Calculus AB. Trig / Calc A was offered as two semesters, and then Calc B / Calc C was offered as two more semesters, after which you'd take the BC test. There was also no such thing as "precalculus". Trig followed Algebra II.
In my Calc C class, there were probably 8ish people, of which one or two (besides me) would have been in my grade.
In either case it would seem to suggest we should radically change the system
To be fair, the original study uses "numeracy", and the correlation numbers are almost exactly like the ones for "language aptitude".
At the same time, the study excluded "five participants ... due to attrition (not completing the training sessions), and one participant ... because he was an extreme outlier in learning rate (>3 sd away from the mean)." I mean, if you are to exclude 15% of your subjects without looking at their aptitude (maybe they didn't do it because it was too hard to pass the training tests to move to the next lesson, yet their language aptitude is high?), with only 36 subjects of which 21 are female (it's obvious programming is male dominated, so they only had 15 males: maybe it doesn't matter, but maybe it does), how can you claim any statistical significance with such small numbers?
100%. I have 2 kids and what they learn at school is absurd... I have to put them in an extra class outside school where they learn a lot of the more abstract stuff, albeit as you say at a level which makes sense (not too deep).
what is the name / curriculum of such a class?
"russian school of math". Putin may be a bad dude, but those russians seem to know a thing or two about math...
Yeah, certainly we'd need to adapt classes to match their level, but we've tried before[0]. While it "failed" it also seemed pretty successful, especially in France and the USSR.
I mean is it any surprise kids get bored in math? They spend years learning the same thing. You spend years learning addition and multiplication (subtraction and division are the same operators). I sure as hell hated math as a kid. Especially doesn't help that it is always taught by someone who's also dispassionate about the subject. You really can get a process going where most middle schoolers are doing calculus and linear algebra (and advance ones are doing this in elementary). It isn't as far fetched as many would believe.
[0] https://en.wikipedia.org/wiki/New_Math
> I mean is it any surprise kids get bored in math?
Indeed. Kids did not evolve to learn things because they would be useful years down the road. But they did evolve to inhale knowledge at very high rates wherever it extends their capabilities in the moment.
My take on this is that math should be tied to kids crafting, experimenting, and learned directly in the context of its fun and creative uses.
Geometry screams out to be developed within a context of design, crafting, art and physical puzzles. Algebra, trigonometry, calculus ... they all have direct (and fun) uses, especially at the introduction stage.
The availability for graphics software, sim worlds, 3D printing, etc. should make math, from the simplest to most advanced, more fun and immediately applicable than ever.
Boredom with math is a failure to design education for human beings.
(Then there is the worst crime of all - moving kids through math in cohorts, pushing individuals both faster and slower than they are able to absorb it. Profound failure by design.)
In general I agree but
I strongly disagree with this. Children play. Most animals play. It is not hard to see how the skills learned through play lead to future utility despite potentially none at the time. We invent new games, new rules, and constantly imagine new worlds. The skills gained from these often have no utility beyond that that is self-constructed. Though many do have future rewards. I think we often miss those though because that path is through generalization. Learning how to throw a ball can help with learning physics, writing, driving, and much more. They don't perfectly transfer but it'd be naive to conclude that there aren't overlaps.I think the truth is that as long as we are unable to predict the future, we really can't predict what skills will be useful and what specific knowledge should be pursued. We can do this in a more abstract sense as we can better predict the near future than far but that also means we should learn creativity and abstraction, as these allow us to adapt to changes. And that is why I believe we evolved these methods, and why you see things like play in most creatures.
> I strongly disagree with this. Children play. Most animals play.
I am not arguing against this. On the contrary, I agree with this completely.
Pairing the opportunity to learn and use highly general long term knowledge, toward progression in their own intrinsic interests (crafting, whatever) is the point I am making.
The alternative, a long line of math for math's sake, which neglects to harness and support each child's need and motivation to continually advance their own natural interests in all kinds of idiosyncratic directions.
Interesting to me how Wernicke’s Aphasia works then (which I have occasionally; my epilepsy stems from there + broca’s area). Could you explain that?
What is the question asking? Why do you have Aphasia? I'm not sure what the relationship here is. Or are you suggesting you're good at math but not good at language?
I feel weird answering even if I infer the right question because it feels tautological. If you have Wernicke's Aphasia does it not create the possibility that I already have but your condition resulted in misunderstanding. Given the condition does it not create a high probability that a response will similarly be misunderstood? Is not Anosognosia quite common?
Maybe I'm really misunderstanding but honestly I'm not sure what you're asking
By the very expansive definition that you advocate for, what exactly do we do that isn't math?
One could argue that all we do is turn thoughts, senses, and memories into further thoughts and appropriate actions, which is applying a pattern, which is math. But at that point the definition is too broad to be helpful for anything but playing word games.
There are no rules. On the other hand, code is very formal. It is a set of rules. It has structures. There is wrongness and you can do verification that is not the result of opinion nor is it subjective.
But yes, math is very broad. But it doesn't cover everything. At least not at this time.
> most people don't know what math is
It's surprisingly common. Case in point: "The unreasonable effectiveness of mathematics in the natural sciences".
A normal person wouldn't be surprised that describing how something works is a good way to understand it.
Math is about form and traditionally the science of quantity. It is called a formal science for this reason.
Language also concerns form. Grammar has form. Concepts are forms.
Math is language. 'Everything' is language. Language is the image of reality.
In the beginning was the Logos...
[0] https://ncatlab.org/nlab/show/image
Math major here. 'is' is a reflexive relationship. If math is language, therefore language is math. I believe clearly language is not math, therefore math is not language. Math is described with language, Math itself is not language. It is a 'has-a' relationship vs a 'is-a' relationship.
The reason for doubt is a failure in fairly basic logic. Your claim is:
I'd expect anyone willing to claim the title of "math major" is aware of structures other than bijections and isomorphisms. The mapping operator doesn't form an abelian group.[0] https://news.ycombinator.com/item?id=43873351
The word "is", typically refers to the logical equals operator (written as "="). Let A = Math, and let B = Language, the phrase "Math is Language" is therefore "A = B". My claim is that "B != A", which implies "A != B" because "!=" is reflexive. "A != B" then contradicts the claim "A = B". In words: because "language is not math", therefore it cannot be true that "math is language".
This is not number theory, not set theory, just logic. I don't see how the properties of Abelian groups applies. I do suspect that non-abelian groups is a case where "¬(A ↦ B) ⟹ ¬(B ↦ A)" is false, and would be a contadiction to my counter claim if "¬(A ↦ B) ⟹ ¬(B ↦ A)" was my counter-claim. No, my counter claim is simply that because "B != A" therefore "A != B"
The statement "¬(A ↦ B) ⟹ ¬(B ↦ A)" is quite different, and I'm not 100% sure whether you are mixing set and logic notations? [0][1] It does seem you are bringing in number theory concepts, or we are misunderstanding one another?
I assume "↦" is the set mapping operator and "⟹" is logical implies, and "¬" is logical not. "¬(A ↦ B) ⟹ ¬(B ↦ A)" could potentially be phrased as: "If the element A cannot be mapped to B, then the element B cannot be mapped to A". I would agree that statement is not 'generally' true. Could you please clarify so we are not talking past each other.
[0] Per google: The symbol "↦" is a mapping symbol, typically used to indicate a function or relationship where one element maps to another. It's not a standard logical operator in the same way that symbols like &&, ||, or ¬ are. Instead, it represents a directional relationship between sets or elements, often seen in set theory and mathematical notations
[1] https://simple.wikipedia.org/wiki/Implication_(logic)
As I said elsewhere (here for others), I'm not going to play this game of *willful misinterpretation.* Your very words do not hold up to the same bar your are attempting to hold mine to. I know you are upset you got caught in a lie, but fuck around and find out. ¯\_(ツ)_/¯ Domain experts can recognize other domain experts pretty easily
We all know that this is not always true. It may be true in certain cases, in certain fields (pun intended), but you know that this is such a basic logical fallacy that it is taught to children.Here's a counterexample so we can lay this to rest.
Stop cosplaying, stop trolling, stop acting in bad faith.Did you even look at my name? There's 2 people that should come to mind. Certainly any logician would notice.
> This is not number theory, not set theory, just logic.
Try that in a philosophy class and you can expect an F.
A math class too.
The proposition was "A = B" (logical equals, A and B are entities)
The simple refutation was: because B != A, therefore A != B
That WAS the math exam!! No group or set theory needed. It's simple and this is not interesting.
> The proposition was "A = B" (logical equals, A and B are entities)
That's only in your head. Inventing claims so that you can pretend other people are wrong isn't a good move.
> That WAS the math exam!! No group or set theory needed. It's simple and this is not interesting.
I'm surprised you can pass a math class.
> That's only in your head. Inventing claims so that you can pretend other people are wrong isn't a good move.
I'm not the only one that interprets a statement as "Math is Language" to be of the form "A = B" where "math" is A, "language" is B, and "is" is the equals operator, see: https://news.ycombinator.com/item?id=43874322
Seems kinda simple.. You're engaging in a almost pure personal attacks. Care to address the substance of how the 3 word long statement is not of the form "A = B", but is of some other different form? In which case, perhaps you can guide the conversation for why it is an interesting statement or not? If you want to change all the givens and use your own definitions, please provide those. Without common ground, this remains uninteresting.
> I'm not the only one that interprets a statement as "Math is Language" to be of the form "A = B" where "math" is A, "language" is B, and "is" is the equals operator, see: https://news.ycombinator.com/item?id=43874322
This appears to be a link that contains zero support for your sentence. Neither the comment you linked nor the response below it features such an interpretation.
> Seems kinda simple.. You're engaging in a almost pure personal attacks. Care to address the substance of how the 3 word long statement is not of the form "A = B", but is of some other different form?
You're in luck! I've already provided that material, and you responded to it. For further discussion, you'd need to have a better working understanding of English.
Very cool how the last sentence you leave off with always has to be a personal attack. It's breaking the rules of the discussion, irrelevant. Stop trying to win points. Let's focus on the substance.
> This appears to be a link that contains zero support for your sentence. Neither the comment you linked nor the response below it features such an interpretation.
It's interesting, because there is a disagreement about whether there is a paradox or not. There is a paradox if you take my perspective (that an equals relationship is being expressed), but none if an implication relationship is assumed.
These two sentences:
- "By that same logic you could also say that language is math"
- "Not quite, but the inverse is true."
The "not quite" says that "language is math" is not true. So we have "A = B", but "B != A", which is a paradox. OTOH it's not a paradox if what is actually being said is "A => B" but "B !=> A"
> You're in luck! I've already provided that material, and you responded to it.
Hello bad faith! I hope you are doing well today. Let's end the conversation here.
> Math major here.
You say that like you think it's a qualification?
You're not making it look good. A common use of be is to express set membership rather than identity. For example, "two is an even number" or "tigers are cats".
We may hope that one day you'll come to realize that set membership is not reflexive, and - more to the point - also not symmetric.
> A common use of be is to express set membership rather than identity
Google for "logical "is a" vs logical "is"".
Google AI answers this:
> "is" typically represents an equality relation
Rest is from the AI response:
In logic, "is" typically represents an equality relation, while "is a" (or "is of the type") represents an inclusion relation. "Is" indicates that two things are the same or identical, while "is a" indicates that one thing is a member of a larger class or set of things.
Logical "is" (equality):
Logical "is a" (inclusion or type):Try applying some thought. You asked a stupid question and you got a stupid answer. What conclusions do you think that supports?
What does "silk is cloth" mean?
You're saying that "Math" and "Language" are 'sets'? And that the phrase "Math is Language" should be interpreted as expressing the relationship between sets?
If we do want to talk sets, that seems far more interesting. The statements like "Math is a language", or "Math has equivalence classes within languages", or "Mathematics are a Language" are slightly more interesting to consider IMO.
>> Math major here.
> You say that like you think it's a qualification?
Agree, appeal to authority fallacy. I take that over mis-framing any day though.
Any old-timers might appreciate that we're arguing over the meaning of the word "is" =D
Real
Finland tried teaching maths to children using Group Theory in the 70s [1], but the results weren't that good; it proved to be too abstract for young kids.
Ultimately, I believe basic algebra and geometry are the most important takeaways from math classes for most people.
[1]: https://www.hs.fi/tiede/art-2000004823594.html (sorry, it's in Finnish and behind a paywall)
I am no expert in pedagogy of mathematics, and I am sure someone will correct me if I am wrong, but I think there was/is a Russian academic program in which students were/are basically only taught algebra in an iteratively increasing manner.
As it was explained to me, one wouldn't take a "Calculus I" class as a prerequisite for say an entry-level engineering course. One typically had such a strong foundation of algebra, that when encountering a problem that required calculus, the student would just learn the necessary calculus at that point in time. In other words, with such a strong algebraic background, other aspects of math, within reason, were much easier to grok.
The other reason I'm willing to accept alternative conclusions is that France and the USSR had far more success than Finland (or even America). Their success contradicts a claim that "[it is] too abstract for young kids". You'd need to constrain it to something like "[it is] too abstract for Finish kids" which I think both of us would doubt such a claim.
Yeah fair point.
However, curious of what you base your claim "France and the USSR had far more success" on?
I think you might be missing the point of the article: the study being cited isn't trying to establish the existence of a "language brain" or a "math brain", that's just the way the headline editorialized it to help people understand the conclusions.
The conclusion of the study was that linguistic aptitude seemed to be more correlated with programming aptitude than mathematical aptitude, which seems fairly interesting, and also fairly unconcerned with which specific physical regions in the brain might happen to be involved.
I understood it.
And this is what I'm pushing back against and where I think you've misinterpreted. I made the claim that these are in fact math skills, but most people confuse with arithmetic. Math is a language. It is a language we created to help with abstraction. Code is math. There's no question about this. Go look into lambda calculus and the Church-Turing Thesis. There is much more in this direction too. And of course, we should have a clear connection to connect it all if you're able to see some abstraction.> Math is a language.
Language is not math, therefore math is not language.
Logic doesn't follow.
There is no problem with A -> B ∧ B -/-> A
Here's an example. "I live in San Francisco" would imply "I live in the US". But "I live in the US" does not mean "I live in San Francisco".
Here's a more formal representation of this: https://en.wikipedia.org/wiki/Bijection,_injection_and_surje...
The word "is", maps to the logical "equals" operator. I agree with the example, but I don't agree it is relevant. There is no implies operator.
The statement "Math is Language", where A is Math and B is Language, maps to the logical assertion: "A = B".
If we are going to really be kinda twisty and non-standard, we could interpret the english "is" to be "is an equivalence class of". Which would map to your example pretty well: language is indeed an equivalence class of math, but math is not an equivalence class of language. Though, nobody is talking about implies operator or equivalence class here.. It's a "is" relationship, logical *equals*
> The word "is", maps to the logical "equals" operator.
It very obviously doesn't. A square is a rectangle. seadan83 is (probably) a mammal. Math is a language.
You point out the "is a" relationship, not the "is" relationship, they are different. [0]
Find examples with two singular nouns and just the word 'is'.
The phrase in question: 'Math is language' is an example, or something like 'food is love' is too. I concede you could interpret those last few sentences with poetic license to be read more like: "A is a form of B", or "A is a B" - though that is not what was written and this is not a place to expect that much poetic license.
*edit*: a minute later, thought of a good example. "ice is water". True that "ice is a form of water", but strictly speaking no, "ice is not water". I'll concede there could exist an implied "is a", or an implied "is a form of", but that is poetic license IMO.
[0] Google AI summarized it pretty well: google "logical "is a" vs logical "is"
> In logic, "is" typically represents an equality relation, while "is a" (or "is of the type") represents an inclusion relation. "Is" indicates that two things are the same or identical, while "is a" indicates that one thing is a member of a larger class or set of things
> You point out the "is a" relationship, not the "is" relationship, they are different.
Well, what you reacted to was, let me copy'n'paste, "Math is a language". It was you who insisted that "is" in this sentence maps to "equals" relation, so thanks for agreeing that you were wrong.
Honestly I don't think his point even stands. We were using English to communicate and English doesn't have the strict rules of mathematics. That's literally why we created math (which I'll gladly call "a class of languages"). He's right, "is" maps to "equivalent" but he's also wrong because "is" also maps to "subset" and several other things. "Is" is a surjection.
The problem here all comes down to seadan83 acting in bad faith and using an intentional misinterpretation of my words in order to fit them to their conclusion. I'm not going to entertain them more because I won't play such a pointless game. The ambiguity of written and spoken language always allows for such abuse. So either they are a bad faith actor "having fun" (trolling) finding intentional misinterpretations to frustrate those who wish to act in good faith or they are dumb. Personally, I don't think they're dumb.
It's such an absurd thing to argue about that I just assumed that some massive brainfart happened there. It happens to everyone, not everyone doubles down on it though.
Oh, they did it much more than twice: https://news.ycombinator.com/item?id=43873381
But then again, isn't a good portion of this thread non-mathematicians arguing about what math is? I really thought ndriscoll put it succinctly[0]
I fear the day some of these people learn about Topology.[0] https://news.ycombinator.com/item?id=43882197
The point is that linguistic aptitude _is_ math aptitude, and vice versa.
From my experience, my ability to articulate myself well is bound up with my ability to abstract and detect patterns. It is the same thing I apply to crafting software, the same thing I apply to creating visual art.
I think high-cognitive-ability people segregating themselves into artsy vs mathy people has more to do with their experiences in their formative years.
Do you recommend any other resources for children to learn the basics of those theories?
Unfortunately I don't know any direct resources. I really do hope they are out there and some will share.
But if you're willing to hunt, I know that this idea was attempted before[0]. France and USSR had better success than the US. I'm sure there are still people working in this direction. I don't have children, but fwiw I've taught my nieces and nephews algebra and even some of calculus before they were 10 just in visiting time during vacations. They were bored and it seemed more fun than talking about the drama, politics, and religion that the rest of my family likes to spend most of their time on. Kids were similarly disinterested in that stuff lol. I've also seen my god{son,daughter} be able to learn these types of skills, so I'm highly confident it is doable.
[0] https://en.wikipedia.org/wiki/New_Math
A book of abstract algebra is very approachable!
> I'll claim that there's not real thing such as a "language brain" or "math brain.
Did you even read beyond the silly headline?
The article itself is about pre-testing subjects on a range of capabilities from problem solving ability to second (foreign) language learning ability, and then seeing how these correlated to the ability of the test subjects to learn to code.
The results were basically exactly what might be expected - people who learned Python the quickest were those who scored the best at learning a second language, and those who learned to wield it the best were those who had scored the best at problem solving.
Not surprisingly math ability wasn't much of a predictor since programming has little to nothing to do with math.
I think the above comment stands. The point is, what do they consider maths ability? High school maths has very little to do with programming but university-level maths certainly feels very similar to it in structure. Many of my (good) classmates in my maths degree were very bad at things like mental arithmetic. So maybe "maths ability" (defined some way) isn't very important for being good at "maths" (proper).
I'm happy to continue discussing, but only on those terms. In fact, I think we're in far more agreement than your tone suggests. But I think you missed the crux of my point: math isn't number crunching
Sheesh - why so touchy?!
Your response opened with addressing the headline, which anyone who had "RTFM" (RTFA) would have realized was unrelated to the body of the article. You then veered off into a tangent about the nature of math which again was not addressing the content of the article.
The underlying nature article, linked from the posted story, makes it even more clear what is being discussed, with the abstract stating:
> This experiment employed an individual differences approach to test the hypothesis that learning modern programming languages resembles second “natural” language learning in adulthood.
https://www.nature.com/articles/s41598-020-60661-8
Thank you. I’ve long held that calculus, formal logic, combinatorics, etc. should be started in elementary school, separate from arithmetic. Even after stripping away the numbers and procedural knowledge, there is tremendous real world value in simply understanding many of the concepts. Plus, (and I say this half-jokingly) won’t we all be using llms to vibe-math everything that needs math-ing anyway?
> language ability and problem solving skills
First red flag is here. The title rewrote this to be language only. That problem solving skills are relevant is pretty obvious, but language less so.
I’ve been programming for most of my life and I don’t consider myself a very good speaker. My language skills are passable. And learning new languages? Forget it. So I’m skeptical. Let’s look at the study.
First of all, “math” becomes “numeracy”. But I think programming is probably closer to algebra, but even then it’s less strict and easier to debug.
> Assessed using a Rasch-Based Nuemracy Scale which was created by evaluating 18 numeracy questions across multiple measures and determining the 8 most predictive items.
Also, the whole thing is 5 years old now.
> That problem solving skills are relevant is pretty obvious, but language less so.
To me, problem solving ability is precisely the same as the ability to articulate the problem and a solution. I don't see a major difference.
If you can solve a problem but you can't articulate what the problem is or why the solution will address it, I wouldn't call you a good problem solver. If you can articulate the problem well but not come up with a solution, you're already doing better than a lot of programmers in the world, and I'd probably prefer working with you over someone who presents the solution without "showing their work".
In fact, what is problem solving without such articulation? It's hard to even grasp what the skill means in a raw sense. Arguably creativity in this context is just the ability to reframe a problem in a more approachable manner. Many times, if not most times, such framing implies some obvious solution or sets of solutions with clear tradeoffs.
There are different ways of solving a problem though, some which require more critical thinking than others. Trial and error (in industry you can sound fancy by calling it “choosing parameters empirically”) requires no understanding of the underlying process, only the ability to measure the outcome.
If you’re debugging, you can get by for a long time by trying things until the compiler shuts up. It’s not efficient or good but people do it.
That's fair. I agree that there's more to problem solving than just linguistic ability, so I rescind my claim that they're indistinguishable, but I still think there's a deep relationship between the two.
I have a very difficult time trying to extract the difference between "linguistic ability" and "critical thinking", though:
1. The core difference between "critical thinking" and "uncritical thinking" is the ability to discern incoherency from coherency.
2. Coherency is evaluated at the linguistic level: do the terms bind in meaningful ways to the problem? Do any statements contradict each other?
3. The remaining aspect is "creativity": can you come up with novel ways of approaching the problem? This is the hardest to tie to linguistic ability because it sort of exists outside our ability to operate within an agreed context.
So while I agree these are distinct skills, I still have difficulty identifying what remains in "critical thinking" after linguistic ability is addressed.
I have found a fairly interesting correlation between people who are good at learning programming and people who are good at English spelling bees. Something about holding a lot of anecdotes and esoteric rule exceptions when performing an otherwise algorithmic process.
Are you a teacher and/or someone that runs spelling bees or otherwise someone that has done the proper statistics to distinguish this correlation with generalised intelligence. Just wondering where you get this anecdata from?
They get it from being an engineer!
I had a CS prof who couldn't spell to save his life and caught a lot of good-natured flak about it from his students.
Makes sense to me; I've always been a naturally good English speller. My brain just knows how to store and retrieve this type of data, which has a high correlation with e.g. CLI interfaces and their idiosyncratic command structures.
I can’t decide if you’ve cheekily included the classic homophonic there-error. This isn’t technically an issue with English spelling, but I think you intended to communicate that CLI interfaces have idiosyncratic command structures, not that they are idiosyncratic structures themselves. If I’ve been taken in by pedant-bait, I apologize.
Lol, nope, just a case of my fingers deciding they know what I'm typing, or possibly autocorrect "fixing" something that was already correct. (I don't know why but the Android GBoard has been introducing crazy numbers of wrong homophones recently.)
I'm a terrible speller; it's taken me ten years of typing "ammend" to learn its proper spelling. It also sort of goes against the "programmers are lazy" meme: why memorize what a computer can detect and correct?
As others have commented, I believe strongly that to define “language brain” and “math brain” is a false dichotomy and bunk neuroscience. Mathematics IS a language in itself; a collection of symbols created to express concepts. This study works really hard to make a statement that most people already know: anyone can be a good programmer, mathematician, “whatever” if they have the interest, drive, and opportunity to do it.
An interesting book that illustrates the evolving societal perception of mathematicians (and by extension, computer scientists) over time is “Duel at Dawn”. The modern (sic recent) notion is the reclusive genius (maybe even suggestion of autism) who possesses the alien, superhuman, unattainable, too-cool-for-you ability to process numbers and information that you can only be born with. (Those familiar with the TV show “the big bang theory” would recognize the trope of the “Sheldon Cooper” character.) This is False.
The reality is that no one is born with the super-human ability to do anything - anyone who is very good at something has worked very hard to get good at that thing regardless of the perception.
edit: my initial criticism of “the study” was based upon the article. On a skim of the actual cited paper, I revised my specific criticism, but the actual paper still comes off as no more than a mild eugenics argument dressed in psychology and statistics jargon.
Can't see any eugenics there, can you quote?
The article is extremely misleading, I dare even say almost malignant.
The study itself claims:
- fluid reasoning and working-memory capacity explained 34%
- language aptitude (17%)
- resting-state EEG power in beta and low-gamma bands (10%)
- numeracy (2%)
They take math skills to equal numeracy. The study itself implies this too. I disagree on a fundamental level with that. Math skills align much more closely to fluid reasoning than to numeracy.
At the beginning, the article mentions correlation with language skills AND problem-solving. Focusing only on language skills in the second half is misleading. According to the abstract of the original paper, problem solving and working memory capacity were FAR MORE important.
Also, the article doesn't mention "math skills". It talks about numeracy, which is defined in a cited paper as "the ability to understand, manipulate, and use numerical information, including probabilities". This is only a very small part of mathematics. I would even argue that mathematics involves a lot of problem solving and since problem solving is a good predictor, math skills are good predictor.
Going further, it seems like Language Aptitude was primarily significant in explaining variance in learning rate, measured by how many Codecademy lessons they completed in the allotted time, but wasn't explanatory for learning outcomes based on writing code or answering multiple-choice questions.
Seeing as Codecademy lessons are written in English, I would think this may just be a result of participants with higher Language Aptitude being faster readers.
I do think that language skills are undervalued for programming, if only for their impact on your ability to read and write documentations or specifications, but I'm not sure this study is demonstrating that link in a meaningful way.
100% agree, I've been saying this for years. I'm terrible with arithmetic but great with symbols and relations. Recursion is also fundamentally linguistic, and although our internal "stacks" for processing it naturally are quite small, language remains the easiest demonstration of recursion in our daily lives.
Oddly, I also use spatial intuition when thinking about stuff like stacks and the shape of data structures.
> Recursion is also fundamentally linguistic
You sure about that? How about inductive proofs?
I would just say that language is more familiar to most. Mathematics are also languages, but more formal and foreign to most.
This is just a guess on my part, but I'd also bet that writing inductive proofs (or proofs in general) require more of the language brain than just doing math problems.
Language has an inherently recursive structure: I saw the man who saw the man who saw the man who saw the man who saw the man who... While our brains have practical limits to how deeply such things can actually be nested, language has a recursive tree-like aspect to it.
Yes, but “language is fundamentally recursive” doesn’t mean the same thing as “recursion is fundamentally linguistic”. Language is just one example of a recursive structure.
I am also unsure whether recursion is fundamentally linguistic, but I thought that “language remains the easiest demonstration of recursion in our daily lives” to be useful. If I ever write another essay about recursion, I'll now consider starting with a linguistic example before diving into recursive functions or data structures.
Language has nothing that corresponds to a recursive function, so that is a bad example. You can write a sentence that could correspond to a call to a recursive function, but its not the same thing as a recursive function.
If recursion was just writing the function 10 times like you did in language then people wouldn't struggle with it.
Recursive functions are just a subset of all possibly recursive concepts. In the case of human spoken language, the recursion exists in our characterization of the grammar. You could just as easily frame this in "iterative" terms just like you can make any recursive function iterative, but that's less convenient for analysis.
So in this case, "recursive function" would be "clause" or something like that; I'm no linguist. But clauses can embed clauses which can embed further clauses, etc.
I think your usage of recursive functions is just high-level logic—you're describing an inductive proof. We also frame a lot of our social games as recursive processes. But these are conscious processes that we can evaluate consciously; the recursion in spoken language is largely unconscious and very shallow.
> In the case of human spoken language, the recursion exists in our characterization of the grammar
But people are constructing sentences, not grammars. When you construct a grammar you can add a recursive part to it, that is true, just like in a programming language, but constructing grammars is not what people mean with language skills.
A sentence can't be recursive since languages in themselves has no concept of applying a concept, for that you need an interpretation of the language references. For example, you can have a recursive function written in a programming language that doesn't have a recursive grammar, the concepts are different things.
There are two ways that recursion intersects with language that are relevant here:
1. Our spoken and especially written grammar is recursive. We do handle this unconsciously. This is not related to our ability to reason about recursion at a high level, and recursive grammars are not necessary to do so. This is not a skill in the normal sense and we have only (very) limited ability to improve our capacity to interpret deeply nested grammars. However, this is still a useful illustration of what recursion IS, which is why I brought it up.
2. Language also introduces the ability to semantically reason about recursiveness. This is still a linguistic thing—you need a symbol and relations among symbols in order for recursion to be meaningful—but this is a skill and is likely very related to linguistic skill. This is the part that really helps you to program: ultimately, you're just reasoning about symbols and looking for incoherency.
Can you come up with some conception of recursion that doesn't involve symbols referring to themselves, directly or indirectly? Ie what is left of recursion when you remove the linguistic component?
Yes. But it's in my mind, I can't write it down for you.
How do you communicate an inductive proof without language? Even formal symbolic logic is fundamentally linguistic.
> How do you communicate an inductive proof without language?
With that argument everything is fundamentally linguistic since everything is communicated using a language.
Can you come up with a more reasonable argument?
Recursion itself is simply a conjecture. Nothing fundamental about it unless you believe Chomsky, but his is a speculative claim, not empirical per se.
I don't really know what you mean by "conjecture", but I thought apriori was implied by positing it as a linguistic construct. "Fundamental" doesn't imply empiricism at all. All of apriori knowledge for a language is a set of all sets of coherent statements: the outer set represents a set of implied axioms required to make the statements cohere. Recursion just broadens the complexity of the statements you can express, but it's fundamentally a concept that arises from language and can be evaluated for coherency (like all other apriori concepts).
Edit: added a definition of apriori knowledge.
Edit2: to put this another way, nobody is arguing that recursion doesn't exist. Or that it is empirically-derived. No, it's a useful construct to show certain relations.
Edit3: added a sentence
Edit4: The extent to which our own grammars are inherently recursive vs this being culture or technology is irrelevant to identifying the concept of recursion as an apriori, linguistic concept.
Edit5: i suppose you might also be referring to the idea that we naturally process recursion. I mean, we clearly, evidently do; whether or not that's inherent to being human is a separate question entirely. Hell in the free software world there's a whole recursive acronym meme that taps into some part of our brain and tickles it.
It kinda is empirically true that human language is recursive. Every human language ever discovered is recursive, except, supposedly, for one: Pirahã. And Pirahã has mainly been described by one researcher whose results are controversial.
> Recursion is also fundamentally linguistic
What does this mean exactly?
>> Recursion is also fundamentally linguistic
> What does this mean exactly?
What does this mean exactly?
Cute, but do you have a serious answer?
If you define recursion as a symbol referencing itself, either directly or indirectly, and if you define language as a system of relating symbols to each other, recursion is a linguistic concept, it is a concept that describes a relationship between symbols. There are good reasons to define each concept differently, but if you identify recursion empirically, recursion won't "actually" exist outside of the description of the process. It's our characterization of the process that reveals the recursive structure, even if that characterization doesn't actually exist outside of language.
> If you define recursion as a symbol referencing itself, either directly or indirectly, and if you define language as a system of relating symbols to each other, recursion is a linguistic concept
But that isn't what we mean with recursive function. We don't call this recursive:
Its just incrementing x.> We don't call this recursive... it's just incrementing x
That's not a recursive function as it's written, but you could certainly consider it a form of symbolic recursion. This just isn't a very useful characterization in an iterative/imperative context. You could frame incrementing as recursive, though—this is just peano axioms/church encoding.
I would argue that
- Defining recursion is linguistic
- Defining a function recursively is mathmatic
I agree that there's enormous value in carving out mathematics from other linguistic reasoning, but I don't see defining as something as mathematic rather than linguistic is generally useful. You use the same skills to look for incoherency in both situations, but human language is generally expected to be incoherent on some level.
Besides, a lot of what people mean when they say they're bad at math is that they're bad at arithmetic, which is honestly understandable.
This makes sense for my path to math. In high school I was bad at math and good at learning languages. Then I started learning Python and realized that it was just like learning a language. Then at some point I realized math notation was just another language for expressing the kinds of things you could express in Python. Now I'm in a job where I do math every day and I read math textbooks for fun.
It helped that Python was meant to resemble natural language. I had learned C++ and Perl before but they never stuck, because I never made the connection to language. Ironically, since Perl was designed by a linguist!
I've noticed that the people who initially grok programming language syntax are often better at learning new symbolic systems, but I think this is actually due to a persistent type I error their brains make that is OK for early learning.
People who end up being the best programmers have a deeper appreciation for semantics and information flow, but tend to commit more type II errors early on, making them inferior intro CS students.
Much of the CS curriculum (and typically also the required maths curriculum) in universities still favors the first type of student over the second, driving out the most capable and creative minds.
I'm strong in the belief that kids having lower reading scores is directly related to lower math scores. Learning to decode words is directly translatable to learning to decode equations. Down to some of my favorite math passages would include how to read something as it introduces a symbol or other construct.
> There’s a lot of people out there who “aren’t math people,” but they just might be computer science people
This is beyond silly from my perspective. I know the field of CS is vast, but this seems to conflate programming with CS. My school was more theory heavy but there definitely came a point in certain paths of study where I didnt touch a line of code for a year, just pure math. I struggle to even understand how someone can think of this sentence - computer science at its core is underpinned by mathematics.
They call "numeracy" math skills. From that perspective it seems not that weird to call programming CS.
I just started reading this. Only 42 people in this study and only 36 completed the activity. I am a bit skeptical about the findings with such a small sample size.
That's what I was going to say. Big conclusions based of tiny sample sizes are ridiculous.
I wonder if this finding hold across programming languages. I suspect the conclusions would be different for people programming in APL instead of Python, for example.
And/or application domain. Lower level programming might lean more on the math side? Wasn't the modulus operator a common complaint about fizz-buzz?
Low level requires more discipline, similar to math, that's about it.
Math doesn't require discipline though, many learn and do math without much discipline.
It surely helps. Take for example History, if I space out for whatever reason in the class and botch my examination on the Aztecs, that doesn't preclude me doing great in the classes about the Roman empire. History is like a sparse suburb, where a single house catching fire does not necessarily propagate and destroy the others. On the other side Math its like a skyscrapper, one floor build upon the other. If someone living in a lower floor botches a home renovation in his unit and damages a pillar, it risks taking down the whole building. In math if you botch a prerequisite, you'll ending messing your learning of subsequent concepts 95% of the time. So it is more unforgiving to people with problems like ADHD, that struggle to keep unbroken focus and discipline.
Sure does for me, a thousand rules to follow.
I was thinking about whether it would vary with task, but I think you have a good point and it would vary even more with the language.
I would take the other side of the bet.
I'm very curious what this math pretest looked like, whether it was "proper" high level math or, like, computing some trig problems. Folks who have aptitude with algebra or number theory or topology, I'd expect that to be correlated, but not to rote computational math.
It was arithmetic. The study is n=42 p-hacking nothingness, and then a highly misleading misinterpretation by the junk popsci website.
My take on skills related to programming is that it's a dance between writing skills and working memory. I've measurably bad working memory, which had led to compensation with writing skills, which led to a programming style that's fairly disciplined and focused on "wholesome" readability. A downside is that I still struggle interacting with code that's written by the working-memory-oriented programmers.
I've recently taken up a daily practice to improve my cognitive skills. Currently I'm using NeuroNation, and it is quite great. The stimulus to start the practice was getting my memory tested (clinically) and learning that it's below norm.
So maybe it makes sense that LLMs do okay at programming even though they lack the ability to reason?
They don't do okay. Quite on the contrary.
My experience is that they spit out reasonably looking solutions but then they don't even parse/compile.
They are OK to create small spinets of code and completion.
Anything past that they suck.
It's actually hilarious that AI "solved" bullshiting and and artistic fields much better and faster than say reasoning fields like math or programming.
It's the supreme irony. Even 5 years ago the status quo was saying artistic fields were completely safe from the AI apocalypse.
I disagree that the current generation of AI has "solved" artistic fields any more than it's solved math or programming.
Just as an LLM may be good at spitting out code that looks plausible but fails to work, diffusion models are good at spitting out art that looks shiny but is lacking in any real creativity or artistic expression.
> "looks shiny but is lacking in any real creativity or artistic expression."
My experience with that is that artistic milieus now sometimes even explicitly admit that the difference is who created the art.
"Human that suffered and created something" => high quality art
"The exact same thing but by a machine" => soulless claptrap
It's not about the end result.
A lot could be written about this but it's completely socially unacceptable.
Whether an analogous thing will happen with beautiful mathematical proofs or physical theories remains to be seen. I for one am curious, but as far as art is concerned, in my view it's done.
Truly great art, the kind that expands the field of artistry and makes people think, requires creativity; if you make something that's just a rehashing of existing art, that's not truly creative, it's boring and derivative.
This has nothing to do with whether a human or AI created the art, and I don't think it's controversial to say that AI-generated art is derivative; the models are literally trained to mimic existing artwork.
Creativity in AI art production is a fancy term for temperature that adds no semantic value.
Your "creativity" is just "high temperature" novel art done by the right person/entity.
This was something already obvious to anyone paying attention. Innovation from the "wrong people" was just "sophomoric", derivative or another euphemism, but the same thing from the right person would be a work of genius.
I would say the fact they do better than anyone could have imagined using just language points in the same direction, okay is very much up for debate imo.
It's like people enjoy extrapolating their surprise when it comes to LLMs, and I don't think it's very helpful.
I tell people all the time that the single greatest tool an aspiring lawyer can have is a background with programming, as the analytical and algorithmic mindset is FAR more important than being "good at public speaking" or any of the other base skills often cited as desirable for lawyers. I've also said the second greatest tool an aspiring lawyer can have (in my personal opinion) is a significant background in foreign language learning, as that is a skill that is closely related to programming, though a bit abstracted and coming from a different angle. I'm going to see if I can use this article to support that.
Programming education should have more humanities such as writing, sociology, epistemology and design, and not nearly as much maths.
How much math is in a typical CS program these days? Calc 1-3 (maybe 3, varied by school), Linear Algebra, Statistics (inconsistent across programs), Discrete were pretty much it when I was in school 25+ years or so ago. That's only 4-6 courses depending on the university, though some where the CS dept was more strongly associated with an engineering college might have added Diff Eq and others. (I got interested in CS education and reviewed a lot of curricula in the US at the time, this is from memory.)
Some schools like MIT might have required more, but on average what I wrote was about it. Has it increased since then? Based on the new hires I've seen the last decade I'd have guessed the math requirements were mostly the same.
I'd argue none of that math is really necessary. While I have used most of my classes at least once, it was never a barrier to advancement in my career. Hell you could say the same thing about any of the theory. Like yea it's cool I know what a "merkle tree" is but it ultimately is a distraction from most of the skills you need to work with git.
Anyway, both computation and math are grouped under "apriori" knowledge. Any semantic distinction is ultimately silly. But we could just as easily be teaching programming as a craft in the context of the real world—I think this is closer to how it's done outside the US. I am not at all convinced the American style is what people ought to be paying for.
When I did my CS degree in New Zealand there were just two mandatory maths papers - statistics and discrete mathematics. Would be wrong to say I didn't get anything from them - but I'm not fumbling around truth tables or poisson distributions all that often either. Everything else was pretty standard: intro to programming, DSA, low level programming, compilers and networks. What I do find kind of mind blowing is comparing my lectures with the ones from MIT and CM (on YouTube) where they can't go more than a few seconds without jumping into math. Ultimately I'm left unconvinced I was deprived of anything important as a typical software engineer.
It was my grades in math that ultimately failed me out of my undergraduate CS program. My university had: Calculus 1, Calculus 2, Linear Algebra, Vector Geometry, Multi-Variable Calculus, Applied Combinatorics, Discrete Math, Differential Equations and maybe more that I don't remember. So many that CS majors could take one more math class and get a minor.
Yeah, I never thought this made sense, but so many people did; and, I always hear people on Slashdot talking about how programming IS math. None of that has been my personal experience, and I'm coming up on 21 years as software engineer. Discrete was the ONLY math course that I really enjoyed and did well in the first time around. For me, this always made sense.
25-year career as a programmer here.
I can count the times I've ever applied math past approximately high school algebra 1, on one hand. Period, in private life, in hobbies, at work. I'm not sure I've ever used any "college level" math, for anything at all.
I've, and other programmers I've known, gotten excited on the very few occasions anything even slightly mathematically-tricky came up, precisely because it almost never happens.
That program makes sense if you want to be a Computer Scientist, or study machine learning / AI.
You chose CS but really wanted Software Engineering. Discrete Mathematics was all you needed for that.
I certainly don’t dispute the empirical validity of the findings from the study - but there are important nuances to consider as well. I am certainly more naturally-attuned to languages as-far as language-learning and reading than mathematics, but I have also found myself understanding more mathematical and theoretical linguistics as well. I also love programming.
It wasn’t until high school, when I tested-into the highest math class that the school offered, that I began to unlock (with some initial struggle) more logical and procedural reasoning specific to mathematics that I had always done well in, but never explicitly went above-and-beyond in, despite hints of such in arithmetic competitions that my school would hold and that sort of thing. I just think my brain works well for both the linguistic aspects of programming (more naturally) and the computational problem-solving aspects of programming. Certainly there are individuals who have strengths in both cognitive aspects, despite being more naturally-attuned to one versus the other, at least presumably.
Perhaps this shows a cognitive profile that has natural strengths in both "brains", or maybe this highlights limitations of the article's potentially narrow definitions of "language" and "math", implying a more complex intellectual landscape.
Interesting findings nonetheless.
> I certainly don’t dispute the empirical validity of the findings from the study
You absolutely should. Tiny sample size and poor statistical method. It is p-hacking plain and simple
Okay, yeah - fair point. I admittedly didn't look too closely at the article before posting this - and I'm not too statistically-minded in many respects. But upon further investigation, yeah, you seem to be right. I think this article is really just promotional material for something.
Abstraction is a huge factor in this too. Language may be dominant because we are at a stage in which the vast majority of programming is little more than hooking together tidy abstractions that are already highly optimized and predominantly organized linguistically.
It you ever need to get into the guts of a system or need to solve bleeding edge problems for which good abstractions don't yet exist, the "math brain" becomes significantly more relevant.
I say this as someone who studied literature and philosophy. The majority of what I know about programming and software engineering I either taught myself or learned from the tutelage of others on the job. Early on in my career, a solid mathematics background was, indeed, not that relevant. These days, though, I'd be lost without it. Whether you like it or not, when it comes to doing real engineering you necessarily need to establish bounds and prove things about those bounds and typically you'll need to do this numerically or at the very least using inductive structures. Linguistic aptitude is still relevant, but it helps less in these cases.
This reminds me of the following quote:
"Coding largely involves the 'logical part' of your brain. It tends to not include the 'language part' of your brain.
This is one reason why comments you add to code are so useful: they force you to engage both parts of your brain and therefore get better results.
This is also why when you go to explain a problem to a colleague, you often have a flash of brilliance and solve the problem: you are reframing the problem using a different part of your brain and the different part of your brain helps solve the problem."
I'm sure some readers here will say this is preposterous and there is no such thing as having "two parts of your brain".
To them I suggest watching:
1. "You are are two" (about people with their corpus callosum being severed) https://www.youtube.com/watch?v=wfYbgdo8e-8
2. "Conscious Ants and Human Hives" by Peter Watts https://www.youtube.com/watch?v=v4uwaw_5Q3I
This tracks with my experience perfectly. Hated the math at my Computer Science program in university, barely passed the math courses while excelling at the programming courses, loved taking linguistics, foreign languages, and literature as electives. To this day, programming, foreign languages, and literature is what I do all day every day.
Couldn't this difference be explained by the fact that the lessons were in English? If your language skills are poor, it's going to be hard to read the directions. You'll definitely be slower, which was the primary finding around language.
That side, I wonder if early programming was much more math heavy, and higher level languages have successively reduced that need over time.
If we distinguish between form and content: Mathematics = form; language = content. However, mathematics is only one type of form, and subject-predicate-object grammar also belongs to form, yet grammar falls under language.
For example: "Matter determines consciousness." If we apply first-principles thinking—where does matter come from? This statement then becomes: "XXX created matter, and matter determines consciousness." At this point, our interest shifts to the first-principle subject "XXX," focusing our attention on it. Who is XXX? God?
In this thought process, we use the subject-predicate-object grammatical structure to trace back the original subject "matter" in "matter determines consciousness"—where does matter come from? Although this formal reasoning does not involve specific mathematical formulas, it indeed employs formal logic to uncover a flaw and opens the door to deeper rabbit-hole
There's only a handful of key concepts people need to learn to understand the basics of 'coding'. If you understand what a variable is, how they're scoped, how step-wise transformations happen, and what loops are, you know 90% of "coding". I knew it at age 13 in 1981 doing "Basic" programming. Sure mastery of the art of coding does take decades, but that's just just adding polish, rather than capability.
Math is VASTLY different with VASTLY more concepts that are all much more abstract in nature and harder to understand the infinite numbers of different ways one mathematical construct can be applied to another. A person can "master" coding, but no one ever masters math.
So comparing math to language or to coding is silly. They're completely separate domains. Just because each of the three can encode and represent the other two doesn't make them similar in any way whatsoever.
Notwithstanding Puritanism like "language is maths" or "logic is maths" I definitely agree.
I learned to program when I was a kid and my maths skills were super basic. Programming can almost be distilled to something as basic as "if this, then do that", plus "do this x times". Then read API documentation and call functions that do what the docs say.
With just this basic understanding you can create a lot of stuff. The maths is obviously the foundation of computation, but to be a programming language user and build stuff, you don't actually need to understand most of it.
In university I did eventually do some math-y stuff (econ degree so prerequisites in stats, maths and even CS) and it helps with certain stuff (understanding graphics programming, ML and LLMs, plus knowing maths on its own is useful), but I still don't feel it was strictly necessary. Language and basic logic is enough IMO.
Started programming over half a century ago. The insurance company I was working for gave me a joint life actuarial evaluation problem for which I wrote a Fortran program, picked up from a book. My uni student buddy let me use his ID so I could drop decks into their IBSYS. Turnaround was about a day. My career as a professional manual reader began.
Well yes, my high school maths were in the high 90s - more than my language scores in French, German and Latin with some off curricular Russian. I guess being a polymath helps.
Unless you are doing an engineering or mathematical application you don't need much math, especially as you can just call a function in the vast majority of the time.
I did a number of software products and operating system modifications without using any math beyond arithmetic operations.
I was a resource for other programmers including the odd math PhD.
Of course it does. Math is not programming - it's almost entirely irrelevant to programming and certainly to software and architecture development. Most modern programming only involves math for trivial things like counting.
This classic article explains the real issue - like Mike Gancarz' classic on the Unix Philosophy, this is something all younger hackers should read, but few have, since these are the fundamental ideas that have created our modern world of computers: https://web.archive.org/web/20000529125023/http://www.wenet....
I’m old enough to have caught the tail end of CS being an offshoot of the Math department and I had a gaggle of fellow undergrads who were happy ours had been moved to the Engineering college and upset to know other students at other schools who were still in Math.
I don’t know about for learning but definitely for collaborating and mentoring. And it’s difficult to make a definition of mastery that excludes both of those, so I suppose after a fashion it’s right.
Despite being a professed lover of math, I scored higher on the verbal than the math SAT. There’s a lot of persuasive and descriptive writing in software, particularly if you’re trying to build a team that works smarter instead of finding more corners to cut.
I did not read any mentioning of testing non-verbal IQ. Many math skills also depend on verbal IQ. Verbal IQ correlates strongly with academic performance and probably also with learning new programming languages. I personally have a much stronger non-verbal IQ than a verbal IQ. Often when I think about algorithms, it is all about abstract blocks moving around even before I have written a single line of code. I usually see multiple solutions in my head and often find myself being stuck because I cannot make a choice. I always takes some effort to code the solutions and weed out stupid bugs due to spelling errors mixing up varisbles, but then things work correctly.
In my experience, language skill may or may not matter for programming, but it certainly matters for understanding a problem space. You can be a great programmer, building the wrong thing because you didn’t understand the requirements or see the gap in them and know to ask the right clarifying question, etc.
Lately I’ve also felt language skills matter when writing concise, specific AI prompts. This has become a useful part of my programming workflow in, I suppose, the last year or so. Before that it was knowing “how to Google” well, but that’s less language-dependent in my opinion.
The most obvious way in which I think "math" and "programming" are related is the way they're taught; by repetitive performance of problem sets. I got frustrated in math for this reason; it takes a lot of energy to sit and focus on a whole bunch of hard problems that all "look the same". It was too easy to do 1 or 2 and think "okay, I got it" and then move on, not realizing the repetition was the key to memorization.
I feel the same way about starting learning programming. Repetition, repetition, repetition, until you "get good".
As a self taught programmer with a Communication Studies degree, this definitely resonates. The ability to articulate the problem in code is kind of the starting point for most productive development work.
Any experienced programmer will be able to tell what kind of person another programmer is just by the way they write their code.
Code written by programmers with humanities backgrounds is easily identifiable as being of bad quality.
Kind of like vibe coding meets programming by accident, before vibe coding was really a term.
It's a faux pas to even mention this IRL, but good coders know what's up.
Those "coders" usually get promoted to people managers, which is usually what they want anyway because their self-worth relies on abusing others to mitigate the correct self-perception they have of being "inferior".
The problem is, things need to be solved and vibe+accident programming can only go so far.
But fear not, they can always scapegoat whoever solves the problems, because if they were not to blame, how could they know what was up or even feel the need to correct it?
This is a high level of bias and generalization about an enormous group of people with varied backgrounds and experiences, not to mention selection bias as you don't know many good coders backgrounds who may in fact be from humanities.
Even if many of the bad coders are those who were in humanities and don't have coding experience because they just entered the field (because once you get it you are no longer a humanities background)
Personally, I find ruby one of the most readable languages because of the language and its DSLs. To be fair, the DSLs are very readable, but the lines can blur between what is ruby and what is a DSL.
I have really come to enjoy using LINQ queries (specifically with lambda syntax) in the C# language. Although I’m a little biased because I work mostly in C#.
My own definition of mathematics for the sake of this comparison:
- If you can write a formal proof, starting from some assumptions and proving some result, it is mathematics. The assumptions are the axioms. Then you use logic, which is the same programming logic, and then you get to some result.
It can be an invariant in a programming function, it can be a more general result, if you can write a proof, it is mathematics. Most algorithms involve proofs, so they are mathematics.
You can make any subject "mathy", because there are relations, ratios, correlation and patterns in everything. My favorite example, is how Chomsky mathified linguistics, and made it esoteric, undecipherable, uncomputable and thereby linguists unemployable, even in research labs.
But as a matter of practice, teaching programming to engineers/scientists, even to mathematicians, is an order of magnitude easier than teaching math to CS folks. Simply quiz job candidates on fp arithmetics, and see how many fail miserably.
Interestingly, I became better at learning human languages after learning a couple of programming languages. I was good at math at the beginning. I guess programming kinda bridged the gap.
The article is from 2020. Recent LLM developments show that a model trained on math and code is better at coding than one that's trained on language and code without math.
I don't find this too surprising. The study itself was primarily just testing a students ability to identify syntax and remember what various functions do. I wouldn't expect math proficiency to help much in this area vs I would very much expect language to.
It'd be interesting to see correlations (language brain vs math brain) for how easy or hard it is for people to solve new problems with language after they already know the basics.
As an accomplished programmer who is not great at language nor math, I feel like my "modeling brain" is the key to success. I can envision all the aspects of an architecture, its data shapes and program flow, making it obvious which data structures and algorithms to use, or where to look for a bug. I did spend a lot of time naming things though... until ChatGPT replaced the need for a thesaurus.
I had to scroll through most of the paper to find this.
> All participants were right-handed native English speakers with no exposure to a second natural language before the age of 6 years
Which removes a confounder that Python mimics English syntax.
Still if this is a typical study recruiting thirty-some undergrads as subjects it's probably not generalizable, or even replicable given the same experimental setup.
> the requirements for advanced math classes for every computer science major seem unnecessary
Computer science is much more than programming - and I think that most of the value derived is from being able to think about problems, which largely require the abstract type of thinking encouraged by more advanced math. Code is just a tool.
Interesting tidbit:
>>> In this study, high levels of these beta oscillations were associated with faster learning and more programming knowledge
This makes me think those binaural beat programs attuned to beta wave frequency might help with heavy coding sessions?
Can anyone point to studies that confirm/reject this?
Makes sense to me. "Language" is an oft-forgotten part of "programming language." I'm sure math skills probably would better predict, say, ability to do well in advanced theory classes, but programming doesn't require that. It DOES require learning a specialized language.
> It DOES require learning a specialized language.
Which... is what math is, too.
I feel like I'm using language part of the brain for almost everything. My performance at any task (except purely manual and automatic ones) drops to 0-15% when I can hear someone talking. Alternatively if I manage to focus on the task I immediately stop understanding what's being said.
I'd posit that this is part of why Pascal lost. Making and/or the same precedence as */+ only appeals to mathematicians and gets in the way of ordinary uses for Boolean expressions where you want them lower than the sub-expressions they concatenate in a language-like manner.
For learning a programming environment*
This has the effect of making programming easier, but don't confuse it.
I would agree. Linguistics and literature have always been a strong interest of mine, and many computer scientists. Moreso, I find weaker programmers struggle with spelling, grammar, various dyslexias and torrent-of-thought code organisation.
Counter example: I can pick up any new programming language within 3-6 months and be proficient but it takes a year to pick up some minor fluency in a human language and I still have yet to obtain full fluency in any other language as an adult.
I suspect, however, that a programming language is a much more limited domain than a human language.
I often tell people it's not that learning a language is hard, is learning that language's software library... and learning a software library doesn't feel like learning a language. More like learning a set of tools.
That makes sense.
Probably the most valuable math classes for me, were ones that had me use algebra to solve word problems.
And, fundamentally, all languages describe the same stuff, using different tokens. That is pretty much in line with programming languages.
If you are bad at math, don't assume you will be bad at programming. Similarly, if good at math, don't assume you will be bad at programming.
If you try programming and you don't like it chances are you won't be very good at it.
Those long held assumptions were held by people who aren't professional programmers, right?
Because, if you do what we do, it's obvious that language > math for most of this stuff.
I'd go so far as to say that music > math for programming.
(2020)
As a new computer science professor at a community college, this is a timely article for me that may end up influencing how I teach, especially my introductory programming course as well as my discrete mathematics course.
I could have told you this for free :)
but really it just depends: - web or app information software: logic and language matters more here - doing what Donald Knuth does: math matters more here
I guess its the ability to tell long, forking stories, traversing several abstraction layers. Dante Alighieri would be loving this! Its the ability to walk through a mental palace, room for room again.
This says more about our programming languages than it does about the brain.
I've always wondered why FP isn't more popular. I concluded it's because most folks don't like thinking like abstract math.
I can't speak for other forms of FP, but symbol operators make communicating about haskell very annoying. Outside of that FP seems to be doing fine, IMO.
Try APL, you'll be begging for FP.
To be clear, the symbols themselves don't bother me so much as trying to refer to them in spoken english. I have no particular beef with the use of symbols in code, which can be quite readable.
Hold your horses. The summary of that article does not offer any proof of what the title says.
* It's a small sample, and they did not analyze the people who didn't complete the course. That's dubious. Those 6 could have had a massive influence on the outcome.
* The summary does not present the actual numbers. These are: "fluid reasoning and working-memory capacity explained 34% of the variance, followed by language aptitude (17%), resting-state EEG power in beta and low-gamma bands (10%), and numeracy (2%)". Note: numeracy, not math.
* The test result was only partially programming related. 50% consisted of the results of a multiple choice test with questions such as What does the “str()” method do?. Linguistic knowledge indeed.
* It's about completing a 7.5 hour Python course. That's learning indeed, but only the very beginning, where abstraction is not in play. The early phase is about welding bits of syntax into working order.
* The numeracy skills required are very low for such tasks, as the tasks are simple, and mainly require thinking in steps and loops, whereas numeracy aptitude is generally measured on rather problems involving fractions.
Edit: the paper uses the Rasch-Based Numeracy Scale for this, which seems to involve estimation and probabilities.
* 17% explained variance is a rather minimal result, and you cannot easily compare factors in such a small design, even if the other one is only 2%. That's a rather hairy statistical undertaking.
* Linguistic expedience might be explain the speed with which the course was followed, since the instruction is, obviously, linguistic. Hence, this factor is not necessarily related to the actual learning or programming.
* The argument from beta waves is clutching at straws.
* The argument that "perhaps women should have more of a reputation for being “good” at programming" because they score better on tests, is --however well meant-- utterly ridiculous. It reverses correlation to causation and then turns that into a fact.
* That linguistic skills are useful for programmers is widely understood. However, this is not because of the actual coding, but because the coder needs to understand the environment, the specs, the users, etc., all of which is transferred via language.
* And of course, the statistical result relies on Null Hypothesis Test Significance, which is rotten in its very foundations.
* Note that the CodeAcademy course "Learn Python 3" is 23 hours in 14 lessons.
Not treating math as just another language for expressing your thoughts is a weird position to take. I cannot take this position seriously.
Whenever encountering such sensationalist headline, it's good to remind ourselves of "[Replication needed]" tag.
Computer science definitely requires math brain
Yes, nobody doubt about that. The article is about programming, which, even if it requires some math skills, require more an ability to deal with a language system.
Not sure why the title left out the essential term: it matters more for learning programming
I believe the goal is to encourage those (young people) allergic to math but good in languages to realize they could be good at programming. That's worthy and important (though ironic to use a scientific study to do so).
As for the larger question commenters are raising, I notice often programmers reducing programming to problem-solving, and that to modeling, thence to math writ large; then they prove their point by noting that the most significant algorithms have strong and deep roots in math. The argument is: if the pinnacle of programming is math, then the more math-like programming is (in people and style), the better. (Hence: functional programming and provable algorithms are the epitome)
This argument is invariably made only by math experts, and does have the effect of reducing competition and selecting for like-minded persons, so there's at least the possibility of infection with self-interest and bias to the familiar.
I think the real distinction lies in this: with math, once you have a model, you fit things into that, and exclude other things. You do perfectly, but only in your domain. With language, any differance (any difference that makes a difference) starts to be named and tracked and dealt with somehow. It may grow in confusing ways, but it turns out (like architecture) it actually makes much more sense to adapt the structure to the use and relevance than vice-versa.
Sure, some subset of architecture is engineering by necessity, and it's probably even the hardest subset. But that's also the most portable knowledge and practice, and I would argue, thus easier to find and deploy once you've isolated such a need, particularly since mathy stuff fits nicely in a reusable library.
So math is an important but relatively isolated part of programming, and thinking it's ideal for everything is missing the point that programming matters for some purpose.
"attention is all you need" is but one example of the priority of relevance over structure in systems.
Learning mathematics is likely to benefit more from whatever a language brain is as opposed to whatever a math brain is.
Strangely (I was a math major as an undergrad), I never doubted this. It just feels more like writing.
It's funny that this popsci article instantly succeeded in dividing the HN commenters into math/not-math camps. Porque no los dos?
Programming is the manifestation of thought through the medium of a keyboard and screen. If you are a clear thinker, if you can hold multiple things in your head at once, if you can reason about things and their relations, well, you can be a strong programmer.
It seems wholly unremarkable to me that someone new to Python would not be fazed by it, given it's fundamental basis in words (print, if, etc.) Someone with a background in languages, who can think well enough to explicitly or implicitly pick up the structure of the language, is gonna do just fine. "Oh, so when I see a while, I need to end with a colon" isnt so different from "when I shout, I need to add a ! at the end"
(Java gets a special place in hell for forcing "public static void main" on every beginner.)
Math only really comes into it when you want to reason about things that have a history of being manipulated mathematically, typically for reasons of convenience. You could probably invert a matrix in SNOBOL, but its a lot easier to pull out lists and arrays and linear algebra.
In other words, lets see the follow up paper to this where Python newbies are asked to model the trajectory of a bead on a wire and see how they do.
You know where we got a lot of really solid programmers before we had formal computer science degrees?
English majors.
Math is a subset of language, surely.
Probably true. Math used to be written in prose style before we invented symbols to compress the prose. Most of the symbols are very recent just in the last 200 years or so.
Some types of math benefit a lot from visual reasoning. Like geometry.
Geometry only benefits from visual reasoning in 3d and lower, and there are a lot of dimensions above that!
You can see visual reasoning as a little cheat computation, you can run math problems through your sense-determining brain, which is what brains are really good at (robots struggle with our levels of dexterity). But the fact remains that you can only visualize in low dimensions, and there are infinitely many dimensions.
Note: You can reduce many problems to 3d, but also many problems in 3d have configuration spaces with much higher dimension, so there's some nuance.
It's easily argued that languages are subsets of math.
Easily? Not sure. But perhaps it's a linguistic version of the Skolem paradox. Looks like Hilary Putnam had some things to say on that..
Before Godel, yeah. Many in math departments would like to believe that claim but would not accept it when pressed.
Left as an exercise for the reader.
and both are instances / processes of symbolic composition.
This is largely true, but also, the dynamic shifts depending on how learners engage with peers as they move from understanding to synthesis.
In small peer groups (“pods”) that debug and learn together, communication becomes a core skill—and that can actually change how math skills are applied and developed. Language doesn’t just support learning; it reshapes the process.
You guys have more than one brain?
This sounds obvious. To me it seems the majority of developers hardly use any math.
For the 99% of programming I come across, I need about 0% math.
Honestly, all this "math brain" vs "language brain" stuff just feels off to me - I've always been more of a pattern and abstraction person anyway. Makes me wonderif schools focused more on teaching real abstract thinking early, would way more kids just "get" programming without even stressing over math class?
Interesting. I grew up most of my life overseas learning various different languages. Anecdotal, but most of the best coders I know are talented writers and often are multi-lingual.
The best programmers were good in Latin at school
If you grew up where it was in the curriculum ;)
Is 42 participants statically significant?
we'll eventually find that the language brain matters more for math than the math brain.
The language brain is more important than the math brain for reading the first lesson on how to program in Python. Fixed that for you.
Does answering a quiz on the contents of the first lesson on how to program in Python really encapsulate anything concrete about who will and will not be able to actually program in Python?
I've always been disturbed by the disconnect between "first lessons" on programming languages and how I personally actually learn programming languages. I can't help thinking that the researchers have measured something else other than whether people have learned to program.
This is why women are so dominant in the software development field - because women are known to have higher language skills than men.
This mirrors my personal experience. My writing skills have been above average and my parents' presumption was that I would pursue some humanistic studies. I ended up studying software engineering and made a career out of it of almost three decades and counting. Meanwhile my wife who always had a "math brain" has struggled to learn to program at a decent competency level.
the language brain matters for math much more than the math brain.
Isn't math somewhat a language in of itself? If not, a language has surely derived from math.
As a mathematician by trade, the math brain is a language brain. Mathematics is about abstract communication rather than mechanics.
Oh wait neuroscientists, explains it all. A statisticians favourite target for being unable to interpret data correctly.
tautological. math is language.
And yet CS grad programs seem to care about only the math section of the GRE
Societally we made that association, so it's not the programs fault, they simply live the same world we live in.
Kurt Vonnegut: See, I came up through a chemistry department.
Charlie Rose: Yeah, right.
Kurt Vonnegut: And so I wrote and there was nobody there to tell me whether it was any good or not. I was just making my soul grow, writing stories.
There's some stuff about his opinion on training for writing that could be relevant:
https://charlierose.com/videos/25437
I don't think it's fair to attribute anything to anything. Stuff comes from all over the place. In other words, attributing programming prowess to math was a mistake, and we are making the same mistake again attributing it to language.
---
Just one more:
Kurt Vonnegut: --consider himself in competition with a world's champion. And this is one reason good writers are unlikely to come from an English Department. It's because the English Department teaches you good taste too early.
I think his main point is when we put something on a pedestal, we actually limit people, whether that be math or language.
Video isn’t loading for me. But from the still frame of the preview, it seems to be this one on YouTube:
https://www.youtube.com/watch?v=ROyr83rib3Q
Maybe they didn’t invent the problem, but the admissions priorities are theirs to repair
That is now obvious, any LLM is able to code, and is usually very bad at math. This alone proves that a good part of programming is linked to language not math.
Even if CS is sort of applied mathematics.
After I started my PhD program in CS/AI (late 90's), one of the faculty told me that the GRE verbal score was thing on the application most predictive of student success in the program. Of course, most applicants had perfect or near perfect GRE math scores, so there was no predictive power there. But the application also included the GRE subject test, transcripts, and letters of recommendation.
I also seem to recall from The Making of the Atomic Bomb that the theoretical physicists of the era were noted as having the highest verbal intelligence of any group of scientists. That struck me since I would not have expected it.
EDIT: I found the quote, from chapter 6:
Psychometricians have closely questioned American scientists of this first modern generation, curious to know what kind of men they were—there were few women among them—and from what backgrounds they emerged. Small liberal arts colleges in the Middle West and on the Pacific coast, one study found, were most productive of scientists then (by contrast, New England in the same period excelled at the manufacture of lawyers).
...
Theoretical physicists averaged the highest verbal IQs among all scientists studied, clustering around 170, almost 20 percent higher than the experimentalists. Theoreticians also averaged the highest spatial IQs, experimentalists ranking second.
Perhaps survivorship bias explain part of it. Many of those scientists arrived at US fleeing their home countries in WWII. Migrating under those circumstances is tricky, lacking the necessary support network or failing to successfully negotiate an encounter with border patrol may be the difference between reaching a safe place and being dragged to rot in the trenches. Good verbal intelligence is very helpful in this process. Perhaps the same way good verbal intelligence is over-represented in those who escaped the war, the trait of being taciturn was over-represented in all the scientists that got stuck and died in WWII[1].
[1] https://worldscientific.com/worldscibooks/10.1142/q0436#t=ab...
I'd think strong verbal skills would make for better collaboration between scientists.
In order to solve what are essentially "word problems" -- i.e. reading a verbal description of a problem and translating it to math -- you have to have very good reading comprehension.
The GRE math bar for excellence was quite low and so a perfect score was only the 92nd percentile or so. It’s not surprising that other metrics correlated better with success
Sure, but also, if you can't read and write well, you're going to struggle in a PhD, because that's really a lot of what you have to do.
Putnam exam score might be more predictive?
Makes sense that verbal ability would line up more with success in CS, especially when math scores are already high across the board. A lot of programming leans on language-type skills: reading and understanding code, navigating docs, naming things clearly, writing maintainable logic etc.
The field probably does itself a disservice by overemphasising math. That framing can push people away who might actually do really well, especially those strong in reasoning, abstraction, or communication. Linked study is a good reminder to rethink how we present programming imo.
It does. The majority of what's popular and gets attention in programming is things like languages and frameworks which do the best smooth talking. The whole point of CS is to focus on the aspects of programming that aren't just vogue trends. With any field or aspect of modern life, language skills are the gateway to privilege and software engineering is no exception. But here it's respect for the math that stops it all from spiraling into a bunch of bs. You can't fool the math people.
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Absolutely. That's why people can be perfectly fluent in a programming language and a high performer at work, but fail an interview algorithms/leetcode problem in that language, as algorithms are math problems that programmers almost never see at work.
The recruiter labels an algorithm problem as a "coding" test, but it's a math test, and concludes that most applicants who claim to be fluent in a programming language can't code and must have lied on their resume.
For context, I don't mind algorithm tests, but I strongly disagree with recruiters presenting it as a coding assessment.