[SyzygyRhythm]

Infinitesimal calculus is the old-fashioned calculus! It was what Newton and Leibniz invented. Limits only came into play later when mathematicians wanted a more robust foundation. But then Robinson proved that infinitesimals were perfectly rigorous. IMO, non-standard analysis is more intuitive than limit-based calculus.

[credit_guy]

Ok, it might be more intuitive. But in terms of applications, is there any example where there's any advantage of using infinitesimal calculus or non-standard analysis?

[numpy-thagoras]

Yes, any time you have to reduce something to a point for analysis in any geometric problem.

You can also vary infinitesimals and utilize them not just in nonstandard analysis, but in fractional calculus, such as for inferring stock market motions.

They have helpful applications in physics, especially field theory.

*

I can imagine, a long time from now, many elegant mathematical constructs simplified by the use of, e.g. infinitesimals, Clifford algebras, category theory, etc. There's a lot of complicated ideas that are nicely simplified, and are even more intuitive, easy to teach the fundamentals of, rather than the standard approach.

I think it's important to understand that the canonical calculus approach came from rather mechanical questions in analysis and proofs, and the math is layered with that, as well as the notational conveniences of forms of calculus commonly used for electromagnetism, classical mechanics, etc. There's a lot of legacy syntax there, and we just live with it, but it's not optimal. Infinitesimals are a way to go back to applications and to better syntax.