Comment by pjlegato | Hacker News Reader

Comment by pjlegato | Hacker News Reader

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There was another step: Leibniz' pre-limits formulation of calculus used hyperreal numbers: infinite and infinitesimals (small-delta) numbers. This version is conceptually simpler and for many people more intuitive than the limits-based explanation. It was the standard approach for many years.

Infinitesimal calculus was discarded in late 19th century pedagogy in favor of limits as no rigorous proofs yet existed of its general correctness, whereas such proofs had been formulated for the limits version. It was suspected that while infinitesimals were a nice intuition, they lacked rigor which might lead to unknown errors.

It was eventually rigorously proven in the 1960s that the infinitesimals approach is indeed as complete and correct as the limits version, but by then standard pedagogy had become entrenched in limits.

More on this at https://en.wikipedia.org/wiki/Elementary_Calculus:_An_Infini... , the modern textbook that re-introduced infinitesimals.

For the formal underpinnings of the infinitesimal approach, the phrase to look for is "Nonstandard Analysis":

https://en.wikipedia.org/wiki/Nonstandard_analysis

Fun stuff.

I understand that nonstandard analysis is a firm footing on which to build calculus, but is it totally equivalent to real analysis?

I loved real analysis in college, for example I found cantors set kind of crazy. Does that "exist" in nonstandard analysis? Or are there other mind-bending implications?

Standard math is a subset of nonstandard, and you can prove the existence of NSA within ZFC, so they are exaxtly equivalent.

> I understand that nonstandard analysis is a firm footing on which to build calculus, but is it totally equivalent to real analysis?

This is a bit of a philosophical question which hinges on what the word "equivalent" means. I disagree with the other comment. It is not totally equivalent, or even at all equivalent.

I think about math in terms of set theory. You create sets, like the real numbers, the integers, or otherwise. You can add other elements to those sets. For example:

* You can add a variety of infinities (countable, uncountable, positive, negative, etc.). You have a self-consistent system.

* You can add imaginary numbers, think of the whole thing as a ball, with a zero on one end and infinity on the other. That gives you complex analysis, which is self-consistent as well.

All of those are helpful, useful, and powerful, and generally lead to the same place where they overlap, they're not very compatible with each other, either formally or intuitively.

I view the two formulations of calculus as similar. Even if they lead to the same results, that's not the same as saying they're equivalent. They're different formalisms.

Edward Nelson's Internal Set Theory is a somewhat cleaner and more intuitive foundation that's practically equivalent to non-standard analysis.

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