soft question - Which branch of mathematics rigorously defines infinitesimals?

I have some trouble doing standard computations in calculus because of the notion of a differential, otherwise known as an infinitesimal, being rather ill defined, in my experience.

Are there any fields of mathematics that someone can recommend for trying to come to a more rigorous grip on the notion of infinitesimals? I've heard and read up about non-standard analysis, but from what I can tell, even the rigour of non-standard analysis isn't as firm as that of more established branches of mathematics. How accurate is this perception?

Any help is appreciated, thank you.

Answer

Most objections to non-standard analysis seem to be about the use of the axiom of choice in the construction of the field of hyperreals. Non-standard analysis is completely rigorous, but if you're a hardcore constructivist then you may be a bit squeamish about it. Then again, there's always some things you need to take on faith in any branch of maths:

• If you're a hardcore finitist then you have to be really careful about analysis in general, since the conventional $\mathbb{R}$ as an object doesn't exist at all.


• If you don't accept the axiom of dependent choices then you're pretty limited in what you can do in real analysis, because many arguments rely on taking a sequence chosen arbitrarily.


• If you don't believe there is a nonprincipal ultrafilter on $\mathbb{N}$ then you can't construct the ultrapower required to create the hyperreals.

If you choose to allow more axioms ("there is an infinite set", "dependent choices", "there is a nonprincipal ultrafilter on $\mathbb{N}$") then you get access to correspondingly more interesting things you can do, but it's all still rigorous.

Note, however, that if you accept Choice then in a certain sense "anything you can do in non-standard analysis, you can also do without the hyperreals" (see https://math.stackexchange.com/a/51480/259262). It's an extra proof technique to make things easier by hiding many of the $\forall \exists$ quantifiers, rather than allowing you to prove genuinely new things that you couldn't prove before.