Looking for a comprehensive text that compares the development of calculus using limits vs infinitesimals

Question

I am looking for a book that covers the development of calculus using the ideas of limits working within the reals and also infinitesimals by extending the reals to hyperreals. I have seen people talk about how non-standard analysis provides more concise and intuitive proofs compare to the standard version using the idea of limits. I am looking for a self-contained text that covers the development of both simultaneously and provides a full comparison. Is there any such text available? Please help.

Answer 1

Having done my PhD in non-standard analysis, I've never seen such a book, and I doubt that you'll find one. The reason for that has nothing to do with the subject matter: mathematics books comparing multiple developments of the same subject in parallel just happen to be exceedingly rare, even in far more popular subfields.

You'd be hard-pressed to find a self-contained book-length treatment of sheaves on topological spaces that develops everything using both étalé spaces and functors, much less one that provides a deep comparison. You won't even find a book that combines and compares Linear Algebra Done Right with Linear Algebra Done Wrong in this manner!

There are books that teach (some) real analysis and (some) non-standard analysis at the same time [1] [2] [3], but they lack parallel developments of the same theorems. Instead, the authors will select what they consider the best approach to each theorem. Again, the same thing happens in other fields: a sheaf theory book might construct pullbacks using the étalé space perspective but pushforwards using the functorial perspective, and a linear algebra textbook might use determinants for some proofs but not others.

The closest thing I can suggest: reading some articles that give nonstandard proofs of specific theorems, such as Monotone subsequence via ultrapower by Błaszczyk, Kanovei, Katz and Nowik. Articles of this type tend to include at least some kind of comparison with the more standard proofs. You can find some article-length treatments that compare standard and nonstandard proofs of common theorems along some philosophical angle as well, such as On the plausibility of nonstandard proofs in analysis by Farkas and Szabo. Katz's bibliography on the history, mathematics, and philosophy of infinitesimals is a great resource for finding articles of both kinds. Tao's blog post about A cheap version of non-standard analysis might also provide you with some intuition, explaining why certain types of non-standard arguments feel simpler than their ordinary counterparts (Beware, though: I've met many people who took away the wrong message from this post! Tao discusses one specific situation where non-standard analysis comes in handy, which is not the only, nor necessarily the most common such situation!). Needless to say, these resources are not self-contained, and most of them assume that you have some familiarity with both standard and nonstandard real analysis.

[1] R. F. Hoskins. "Standard and Nonstandard Analysis: Fundamental Theory, Techniques, and Applications", Ellis Horwood Series on Mathematics, 1990. Comment: this is a mention, not necessarily a recommendation for learning analysis.

[2] H. J. Keisler. "Elementary Calculus: An Approach Using Infinitesimals", Prindle Weber & Schmidt, ISBN 978-0871509116, (online 3rd edition), 1976. Comment: The first, and still one of the few available, infinitesimals-first calculus textbooks; manages to teach a fair bit of standard first course real analysis material.

[3] N. Vakil: "Real Analysis through Modern Infinitesimals", Cambridge University Press, 2011. Comment: got a reminded of the existence of this book in personal correspondence about my answer; this one takes my preferred, Internal Set Theoretic approach, it does try to give self-contained $\varepsilon,\delta$ and infinitesimal characterizations of continuity etc. (again, without the parallel development of proofs that OP seeks)