> 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.
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.
jey · · focus · HN ↗
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.
zahlman · · focus · HN ↗
sheafification · · focus · HN ↗
This is treated more rigorously and generically in the subject of synthetic differential geometry.
Nesco · · focus · HN ↗
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.