I'm reading and liking his book just called "Trees" (some bites), it is about group actions on trees, written in the '80. It talks about the modular group among others, the tree action also being used in Shai Haran work on the "real" prime (if the AIs let us dream the old way about classic problems). Serre et al graph of groups idea is very categorical. He could well have been present in the very seminars when Grothendiek was breweing what is now called the Grothendiek construction (absurdly, since G was a serial constructor). That wraps that graph of groups thing.
> I did not like, and did not understand, epsilons and deltas.
It's nice to have this perspective validated by someone like Serre! I felt like I was missing something when I first encountered that formalism. In fact, all of my introductory calculus classes sucked and turned me off of math for a few years.
Euler was a master manipulator of formal expressions; don't think it bothered him very much whether, e.g., an infinite series converged or not. (Unfortunately don't remember a source for this offhand -- maybe a leftover impression from having read ET Bell's book long ago? I also don't know how reliable Bell is.)
Nonstandard analysis [0] [1] uses infinitesimals but is still completely rigorous. I haven't ever really used nonstandard analysis myself, but there's a fairly well-regarded textbook available online [2].
Also conceptually it feels just right to use nilpotents to probe the smooth structure. In a way nilpotents are violently smaller than even non standard analysis infinitesimals, as the laters’ powers are incredibly small but never vanishing.
Another way to see this is that it makes Taylor expansion exact by killing terms above a bound so it works naturally with the ecosystem surrounding it
Finally duals are very similar to complex in a way. i can be defined as root of X^2 + 1 = 0 even if it felt impossible initially, the dual number as a non nul solution of X^2 = 0 even if it is as counterintuitive.
There's no alternative that's significantly easier to understand and to use. The so-called "nonstandard analysis" hasn't caught on, because it's mostly the exact same arguments wrapped in slightly different language, not making them any simpler or shorter.
The language used by mathematicians is subject to constant evolution. 18th and 19th century results in analysis are not expressed and taught in the same way their original authors did it. Newton, Leibniz, Euler, Lagrange, Fourier, Riemann -- none of them expressed their results in terms of epsilons and deltas. These only caught on in the second half of 19th century, and they did so, because they were a better tool to rigorously prove the ideas.
New terminology inventions that make the subjects easier to understand take the field by storm. Some of the relatively recent examples are category theory, homological algebra, or, for that matter, the notion of sheafs, popularized by J.P. Serre himself. Mathematicians are very open to innovation, and intransigence is not the reason why we're stuck with epsilon-delta.
The reason is that nobody has yet come up with a better way of talking about these concepts. I repeatedly observe many people who seem to believe that their difficulty in understanding math stems from mathematicians gatekeeping their results. I think that this belief is just a coping mechanism. Mathematics is genuinely hard, and when people have trouble understanding something, it's easier to think that it's someone else's fault, rather than accepting one's own deficiencies.
> The so-called "nonstandard analysis" hasn't caught on, because it's mostly the exact same arguments wrapped in slightly different language
No. Let's take a nonstandard proof of the intermediate value theorem on [0,1] by Nelson.
By the transfer principle it is enough to prove this for a standard continuous function f on [0,1] with f(0)<0<f(1).
Take a finite subset of [0,1] containing every standard point. Colour its points blue, green, or red according to whether f is negative, zero, or positive at tha point.
The first point of the interval is blue and the last red. Hence either awe can find some green point, or we can find two neighbouring points that have different colours, the first blue and the second red.
In the first case there is a zero, so we are done. In the second, let the neighbouring points be p and q. By the completeness of the real numbers, every nonstandard real in [0,1] is infinitesimally close to exactly one standard real. So p and q are infinitesimally close to some standard real number, let's call it z.
Standard continuous functions send infinitesimally close points to infinitesimally close points. So f(p) and f(q) are both infinitesimally close to f(z). But f(p) is negative and f(q) is positive. The only standard number infinitesimally close to both positive and negative numbers is zero. Thus f(z) is zero. This proves the theorem.
You tell me, which standard proof is this? It's certainly not the nested interval proof. Not the supremum proof. Not the bisection proof in disguise. Which argument does it wrap in slightly different language? Can you point to a single textbook, course note or lecture that gives such an argument?
No. One could of course argue that this is not simpler/shorter than the usual arguments. But it is very different from them. Saying that it's the same arguments repackaged in a different language is just wrong, and detracts from an otherwise valid point.
A professor of mine had an anecdote of meeting Serre and complaining to him about Bourbaki style and how hard it is for students.
Serre's reply was "But we never wrote those books for students! We wrote them for researchers to have a handy reference for all proofs of basic results."
I like this site of Mathematicians' biographies with some occasional quoted segments for extra flavor.
I also like how Serre wrote a book on linear representations of symmetry groups, because his wife needed a good exposition of the subject for her work on quantum chemistry, and that Serre described that as "fullfiling his duty as a husband" :-P
I'm reading and liking his book just called "Trees" (some bites), it is about group actions on trees, written in the '80. It talks about the modular group among others, the tree action also being used in Shai Haran work on the "real" prime (if the AIs let us dream the old way about classic problems). Serre et al graph of groups idea is very categorical. He could well have been present in the very seminars when Grothendiek was breweing what is now called the Grothendiek construction (absurdly, since G was a serial constructor). That wraps that graph of groups thing.
I especially liked the proof in that book that looks at the action of a group on its Cayley graph to show a subgroup of a free group is free.
The European Mathematical Society just published an interview with Serre on the occasion of his birthday: https://euromathsoc.org/news/ems-magazine-no.-141:-celebrati...
My favorite part curled up in a footnote:
> AI told me that, among my books, this is the most difficult to read for students. I write for mathematicians, not for students.
Gépété raté.
EDIT:
Actually, I didn't know he bouldered!
> I did not like, and did not understand, epsilons and deltas.
It's nice to have this perspective validated by someone like Serre! I felt like I was missing something when I first encountered that formalism. In fact, all of my introductory calculus classes sucked and turned me off of math for a few years.
He describes the style he did like as "Euler's style, so to speak". What was Euler's style in this context? i.e. as opposed to epsilons and deltas?
Euler was a master manipulator of formal expressions; don't think it bothered him very much whether, e.g., an infinite series converged or not. (Unfortunately don't remember a source for this offhand -- maybe a leftover impression from having read ET Bell's book long ago? I also don't know how reliable Bell is.)
Edit: added semi-source
What's the alternative for explaining those concepts that's still reasonably rigorous?
Nonstandard analysis [0] [1] uses infinitesimals but is still completely rigorous. I haven't ever really used nonstandard analysis myself, but there's a fairly well-regarded textbook available online [2].
[0]: https://en.wikipedia.org/wiki/Nonstandard_analysis
[1]: https://math.stackexchange.com/questions/51453/is-non-standa...
[2]: https://people.math.wisc.edu/%7Ehkeisler/keislercalc-06-03-2...
When I first learned non-standard analysis, my reaction was that we don't need the axiom of choice to find the derivative of x^2.
The formalism is very simple symbolically. But the mathematical machine behind it is very complex.
Various algebras of dual numbers are used in most automatic derivative routines.
This is treated more rigorously and generically in the subject of synthetic differential geometry.
wanted to say this.
Also conceptually it feels just right to use nilpotents to probe the smooth structure. In a way nilpotents are violently smaller than even non standard analysis infinitesimals, as the laters’ powers are incredibly small but never vanishing.
Another way to see this is that it makes Taylor expansion exact by killing terms above a bound so it works naturally with the ecosystem surrounding it
Finally duals are very similar to complex in a way. i can be defined as root of X^2 + 1 = 0 even if it felt impossible initially, the dual number as a non nul solution of X^2 = 0 even if it is as counterintuitive.
There's no alternative that's significantly easier to understand and to use. The so-called "nonstandard analysis" hasn't caught on, because it's mostly the exact same arguments wrapped in slightly different language, not making them any simpler or shorter.
The language used by mathematicians is subject to constant evolution. 18th and 19th century results in analysis are not expressed and taught in the same way their original authors did it. Newton, Leibniz, Euler, Lagrange, Fourier, Riemann -- none of them expressed their results in terms of epsilons and deltas. These only caught on in the second half of 19th century, and they did so, because they were a better tool to rigorously prove the ideas.
New terminology inventions that make the subjects easier to understand take the field by storm. Some of the relatively recent examples are category theory, homological algebra, or, for that matter, the notion of sheafs, popularized by J.P. Serre himself. Mathematicians are very open to innovation, and intransigence is not the reason why we're stuck with epsilon-delta.
The reason is that nobody has yet come up with a better way of talking about these concepts. I repeatedly observe many people who seem to believe that their difficulty in understanding math stems from mathematicians gatekeeping their results. I think that this belief is just a coping mechanism. Mathematics is genuinely hard, and when people have trouble understanding something, it's easier to think that it's someone else's fault, rather than accepting one's own deficiencies.
> The so-called "nonstandard analysis" hasn't caught on, because it's mostly the exact same arguments wrapped in slightly different language
No. Let's take a nonstandard proof of the intermediate value theorem on [0,1] by Nelson.
By the transfer principle it is enough to prove this for a standard continuous function f on [0,1] with f(0)<0<f(1).
Take a finite subset of [0,1] containing every standard point. Colour its points blue, green, or red according to whether f is negative, zero, or positive at tha point.
The first point of the interval is blue and the last red. Hence either awe can find some green point, or we can find two neighbouring points that have different colours, the first blue and the second red.
In the first case there is a zero, so we are done. In the second, let the neighbouring points be p and q. By the completeness of the real numbers, every nonstandard real in [0,1] is infinitesimally close to exactly one standard real. So p and q are infinitesimally close to some standard real number, let's call it z.
Standard continuous functions send infinitesimally close points to infinitesimally close points. So f(p) and f(q) are both infinitesimally close to f(z). But f(p) is negative and f(q) is positive. The only standard number infinitesimally close to both positive and negative numbers is zero. Thus f(z) is zero. This proves the theorem.
You tell me, which standard proof is this? It's certainly not the nested interval proof. Not the supremum proof. Not the bisection proof in disguise. Which argument does it wrap in slightly different language? Can you point to a single textbook, course note or lecture that gives such an argument?
No. One could of course argue that this is not simpler/shorter than the usual arguments. But it is very different from them. Saying that it's the same arguments repackaged in a different language is just wrong, and detracts from an otherwise valid point.
He later participated in Bourbaki, who were known by their overly formal style, tough.
A professor of mine had an anecdote of meeting Serre and complaining to him about Bourbaki style and how hard it is for students.
Serre's reply was "But we never wrote those books for students! We wrote them for researchers to have a handy reference for all proofs of basic results."
Happy birthday “Tonton Serre”
I like this site of Mathematicians' biographies with some occasional quoted segments for extra flavor.
I also like how Serre wrote a book on linear representations of symmetry groups, because his wife needed a good exposition of the subject for her work on quantum chemistry, and that Serre described that as "fullfiling his duty as a husband" :-P
"When I was 14 or 15, I used to look at these books, and study them"
Smartphones prevent most teenagers of this era to read a book. This may apply to lots of adults.