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
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.
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."
[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...
The formalism is very simple symbolically. But the mathematical machine behind it is very complex.
This is treated more rigorously and generically in the subject of synthetic differential geometry.
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.
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.
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.
It is not.
I'll be honest: your one sentence response tells me you did not read the proof above in any detail.
I chose Nelson's proof precisely because its construction is well-studied and well-understood. The same construction of a mesh containing all standard points, with the coloring forcing a tiny multicolored cell, extends from the interval to the triangle. In one dimension you get two adjacent differently colored points; in two dimensions you get an infinitesimal triangle whose three vertices have the three relevant colors. Taking their common standard part and applying continuity gives a short proof of Brouwer's fxied-point theorem on the triangle.
But it is well-understood (there's a whole field studying such questions [2]) that the nested interval proof of the Intermediate Value Theorem does not generalize to proving Brouwer's fixed point theorem on the triangle [1]. This fact can be derived from a computability argument as well [3].
Nelson's argument does generalize to prove Brouwer, so it's not the nested intervals argument. But really, nobody cares about these technical reasons. It's obvious to most math undergraduates that Nelson's proof is not the nested interval proof, the clear absence of any nested construction kinda gives it away. The only reason it was necessary to get technical is that you did not really inspect the proof before claiming it was nested intervals. The technical results cited above are just a formal way to show that any correspondence you might imagine between the two proofs is just not there.
[1] Shioji/Tanaka: "Fixed Point Theory in Weak Second-Order Arithmetic", Annals of Pure and Applied Logic v47, pp 167188 (1990).
[2] https://en.wikipedia.org/wiki/Reverse_mathematics
[3] Potgieter: "Computable counter-examples to the Brouwer fixed point theorem", https://arxiv.org/abs/0804.3199 (2008).
Edit: added semi-source
> 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!
Smartphones prevent most teenagers of this era to read a book. This may apply to lots of adults.
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.
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!