Serre wasn't there, but did a talk over Zoom from Switzerland where he now lives. I wasn't there either, but followed (and enjoyed) his talk over the YouTube stream. This was both a reminiscence of the mathematics of the 50s, and reflections on the role of counterexamples in mathematics.
Fun facts: Serre is still the youngest mathematician to be awarded the Fields medal, at 27. And Serre still publishes mathematics papers, the last one in 2025 (he was 99). A volume V of his Collected Papers (1998-2025) was published this summer by Springer. What a career!
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].
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.
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."
on slightly related note of longevity - Fred Haise, one of three Apollo 13 astronatuts celebrated 92nd birthday day before yesterday, Jim Lovell died at 97 and only one who lived short life was Jack Swigert who died at 51 (respiratory failure)
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
There was a conference yesterday in his honour in Paris: https://serre100.sciencesconf.org/
Serre wasn't there, but did a talk over Zoom from Switzerland where he now lives. I wasn't there either, but followed (and enjoyed) his talk over the YouTube stream. This was both a reminiscence of the mathematics of the 50s, and reflections on the role of counterexamples in mathematics.
Fun facts: Serre is still the youngest mathematician to be awarded the Fields medal, at 27. And Serre still publishes mathematics papers, the last one in 2025 (he was 99). A volume V of his Collected Papers (1998-2025) was published this summer by Springer. What a career!
Proving Hardy’s “A Mathematician’s Apology” wrong?
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
1 reply →
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...
1 reply →
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.
1 reply →
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.
1 reply →
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."
on slightly related note of longevity - Fred Haise, one of three Apollo 13 astronatuts celebrated 92nd birthday day before yesterday, Jim Lovell died at 97 and only one who lived short life was Jack Swigert who died at 51 (respiratory failure)
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.
Smartphones and now AI...
I'm looking forward to Dario's essay begging for regulating the use of AI on education.
Oh wait, that goes against his economic interest.