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

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."

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.

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 European Mathematical Society just published an interview with Serre on the occasion of his birthday: https://euromathsoc.org/news/ems-magazine-no.-141:-celebrati...

Happy birthday “Tonton Serre”