Item — HN Reader

> And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities?

-- George Berkeley, namesake of UC Berkeley, in 1734, critiquing infitesimal approaches to calculus.

Math uses limits because "dx" as a concept is hard to define and relies on faith that such an object can exist. It behaves as zero when convenient yet is non-zero when that breaks math. Limits have a more rigorous footing.

3 comments

Infinitesimals don't rely on faith any more than any other mathematical idea. The "normal" calculus sequence being based on limits is solely due to the fact that limits were the first discovered method for rigorously formalizing the subject. Infinitesimals do a better job representing how most people intuitively think about calculus though, as evidenced by being the vehicle through which the entire field was discovered in the first place, and so now that they too rest on a formally rigorous foundation we should probably consider rebuilding calculus education around them.

Your comment was correct 100 years ago. But today it is highly inaccurate. Initially Calculus was developed using the ideas of infinitesimals throughout, although this was not yet fully formalized. The first to ground with mathematical rigor was O believe Cauchy with the epsilon-delta definition of limits. For historical reasons this caught on and is the standard way we introduce students to the subject till today. But since then we have already discovered fully rigorous and zero faith ways to define and work with limits: Robinson's non standard analysis and using nilpotent infinitesimals a la synthetic differential geometry. These provide completely rigorous way to view all the classical intuitions that initially develop the subject and are much easier to work with than the current standard epsilon-delta gymnastics. Unfortunately, mathematicians are very conservative and we tend to stick with the conventional way of doing things way more than we should. In fact a huge part of mathematical community have not fully engaged with the beautiful way of defining and using infinitesimals for calculus, even though it would greatly aid the students learning and intuition and solve the disconnect when working with physics using infinitesimals.

I wrote an article about Berkeley's critique here:

https://mindmatters.ai/2021/03/the-needless-complexity-of-mo...

I actually quite enjoy Berkeley. I wish he had framed his critique slightly differently, but the past is the past :)