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> why infinitesimals might be useful

Ok, I'll ask. Why might they be useful? Is there any situation that you know of where infinitesimals can better attack a problem than the old-fashioned Calculus?

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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.

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?

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.

I think that's why I'm redoing my old Physics problem sets - but using the infinitesimal approach this time. To see is it more useful. So far the gains are modest but I find it easier to 'reason' about some of the calculations.

The author Seth Braver has two nice examples of reasoning with infinitesimals in the book intro - the first few chapters are available for free: https://www.bravernewmath.com/

Time will tell if the study will pay off. In later years of the Physics degree I ended up doing lots of algebraic manipulation without much understanding. Maybe because I had no intuitive 'feel' for the Calculus and it all felt like symbol manipulation ... As another commenter said, somehow infinitesimals allowed the giants like Newton & Lebnitz to work their way to some amazing results (especially about motion...)

> somehow infinitesimals allowed the giants like Newton & Lebnitz to work their way to some amazing results

I'm not that familiar with Leibnitz's work, but Newton understood calculus from many different angles. I heard this (probably apocryphal) saying by Feynman that you truly understand something if you understand it in three different ways.

Newton was like that with calculus, and he probably understood it in more than three ways. In particular he presented the world the theory of gravity using only geometry. Just take a look at [1], and see if you find anything that looks like limits, derivatives or integrals. You only see geometrical figures.

Newton was great at manipulating polynomials. He introduced what we call nowadays the "Newton-Raphson" method via an example of finding a root of a cubic polynomial. He never mentioned derivatives or tangents or slopes, or anything that we would now associate with calculus.

Of course, we know that Newton knew the binomial formula, some people wrongly think he invented it. What he did was that he generalized it to non-integer powers, so he could calculate the infinite series of things like sqrt(1+x) or sqrt(1-x^2). From here it doesn't take that long to derive the series for sine and cosine, especially if your name is Newton, and he did the arcsine and arctan for good measure too. (And from here he calculated many more digits of pi than anyone before him, by a good margin).

And Newton was intimately familiar with interpolation. Even today we have the concept of Newton interpolating polynomial [2]. Interpolation was indispensable in those times, even Briggs used it in his logarithmic tables which he published in 1617. Here's a quote from [3]: "Briggs’ quinquisection is actually a special case of Newton’s formula seen from a different vantage point". But "Newton was apparently unaware of Briggs’ work on finite differences and subtabulation".

[1] https://www.gutenberg.org/cache/epub/28233/pg28233-images.ht...

[2] https://en.wikipedia.org/wiki/Newton_polynomial

[3] https://inria.hal.science/inria-00543939/PDF/briggs1624doc.p...

Totally agree about Newton and his 3 ways. I remember reading in Burton's History of Mathematics:

"Newton developed 3 different versions of his calculus, apparently searching for the best approach; or maybe each version served a different purpose.

- 'Infinitesimals': largely a geometric approach, - 'Fluxions': a kinematic approach, - 'Prime and ultimate ratios': his most rigorous, "algebraic" approach.

The 3 methods weren't always kept apart when solving problems. See: DT Whiteside, Mathematical Papers Isaac Newton."

You might enjoy Tristan Needham's book on Visual Differential Geometry where he really dives into Newton's geometric approach.

Thanks for the other links... must go through them. Lots of gold there.

Solving an ODE by separation of variables. With the limit definition this is just a notational trick, requires additional proof to justify, and confuses students. With infinitesimals, the separation dy = v * dx is a rigorous statement, making the logic of the method immediately obvious.

I think Gian-Carlo Rota would disagree with you. See Section 7 of

https://web.williams.edu/Mathematics/lg5/Rota.pdf

Rota is complaining that differentials are introduced as an ad hoc technique, seemingly breaking the rules. If differentials are taught from the beginning, i.e. y + dy = f(x + dx) where f'(x) is a convenience function, this is not an issue. Of course, there are other issues with teaching differentials, namely why products of differentials vanish. On the other hand, limits aren't rigorously justified in an introductory course either.

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