> 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.
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.
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
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.
[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.
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