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Comment by yummyfajitas | Hacker News Reader

Calculus was described by them using the concept of 'infinitesimals', not through Limits. Limits was a clever abstraction that was evolved later to better explain Calculus and keep it consistent.

It also isn't even necessary to use limits. Deriving calculus using the hyperreal number system (which has both infinites and infinitesimals) has been proved to be equivalent to calculus with limits (i.e., any theorem true in one is true in the other).

So if we want to, we could go back to using infinitesimals.

http://en.wikipedia.org/wiki/Hyperreal_number

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Thanks for the link.

I wouldn't want to rigorously teach Calculus using *R unless the relevant proofs are reasonably accessible (which, given that we're talking Number Theory, seems like a remote possibility), but it's wonderful how much simpler they make the arguments if you work with them.

This is a good book. I found it preferable to regular epsilon delta when I was in highschool.

http://www.math.wisc.edu/~keisler/calc.html

The relevant proofs are easily accessible, probably more so than the limit based proofs of regular calculus.

The only thing that isn't easily accessible is the proof of transference, namely that what's true for star-R (don't know how to make asterisks format nicely) is also true for R.

I mean the equivalence / transference proofs. Otherwise you're pulling a theorem out of the air and telling students to just accept it. And by "accessible" I mean easy enough to understand that they don't become a huge distraction (or worse).

If one starts with star-R to begin with, then why does one need the transference theorems?

You're then discussing math in terms of a set whose properties don't obviously match anything the student has dealt with. It needs to be established, one way or another, that adding infinitesimals etc. is not introducing new behavior.

I don't see why that can't just be stated. Students need somewhere to start from, and there's plenty already that we say "trust us for now, we'll prove it later" - and it's not like the reals actually match anything the student has dealt with, at the corner cases, either...

No, it means you are starting with a different (but equivalent) set of axioms.

Our use of R rather than star-R is merely a quirk of history - we managed to justify R to ourselves first, hence it's the "standard".

You're working with strictly _more_ axioms, and the extra axioms don't seem justified unless you use the equivalence proof to show you haven't added undesired behavior.

(If you look at the other axioms, the hardest one to justify is the bounding axiom, and its necessity is reinforced by proof. Every other axiom fits in with the student's understanding of arithmetic.)

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