The question is the title of a 2013 publication in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics is based on the assumption of an inevitable evolution toward the real continuum-based framework as developed by Cantor, Dedekind, Weierstrass (referred to as the "great triumvirate" by Carl Boyer here) and others. Taking some seminal remarks by Felix Klein as their starting point, the authors argue that the traditional view is lopsided and empoverishes our understanding of mathematical history. Have the historians systematically underplayed the importance of the infinitesimal strand in the development of analysis? Editors are invited to submit reasoned responses based on factual historical knowledge, and refrain from answers based on opinion alone.
To be even more explicit, we ask for additional examples from history that support either Boyer's viewpoint or the NAMS article viewpoint. That is, limit the question to facts and not opinions (based on a comment by Willie Wong at meta).
Note 1. For a closely related MO thread see this.
Note 2. A reaction to the Notices article by Craig Fraser was published here.
Note 3. Another would-be victor Gray is analyzed in this MSE thread.
Note 4. The Notices article originally contained a longish section on Euler, which was eventually split off into a separate article. The article shows, using the writings of Ferraro as a case study, how an assumption of default Weierstrassian foundations deforms a scholar's vision of Euler's mathematics. The article was recently published in 2017 in Journal for General Philosophy of Science.
Note 5. A response to Craig Fraser's reaction was published in 2017 in Mat. Stud.; see this version with hyperlinks.
Best Answer
Certainly the victors write the history, generally. But when the victory is so complete that there is no further threat, the victors sometimes feel they can beneficently tolerate "docile" dissent. :)
Srsly, folks: having been on various sides of such questions, at least as an interested amateur, and having wanted new-and-wacky ideas to work, and having wanted a successful return to the intuition of some of Euler's arguments ... I'd have to say that at this moment the Schwartz-Grothendieck-Bochner-Sobolev-Hilbert-Schmidt-BeppoLevi (apologies to all those I left out...) enhancement of intuitive analysis is mostly far more cost-effective than various versions of "non-standard analysis".
In brief, the ultraproduct construction and "the rules", in A. Robinson's form, are a bit tricky (for people who have external motivation... maybe lack training in model theory or set theory or...) Fat books. Even the dubious "construction of the reals" after Dedekind or Cauchy is/are less burdensome, as Rube-Goldberg as they may seem.
Nelson's "Internal Set Theory" version, as illustrated very compellingly by Alain Robert in a little book on it, as well, achieves a remarkable simplification and increased utility, in my opinion. By now, having spent some decades learning modern analysis, I do hopefully look for advantages in non-standard ideas that are not available even in the best "standard" analysis, but I cannot vouch for any ... yet.
Of course, presumably much of the "bias" is that relatively few people have been working on analysis from a non-standard viewpoint, while many-many have from a "standard" viewpoint, so the relative skewing of demonstrated advantage is not necessarily indicative...
There was a 1986 article by C. Henson and J. Keisler "on the strength of non-standard analysis", in J. Symbolic Logic, 1986, maybe cited by A. Robert?... which follows up on the idea that a well-packaged (as in Nelson) version of the set-theoretic subtley of existence of an ultraproduct is (maybe not so-) subtly stronger than the usual set-theoretic riffs we use in "doing analysis", even with AxCh as usually invoked, ... which is mostly not very serious for any specific case. I have not personally investigated this situation... but...
Again, "winning" is certainly not a reliable sign of absolute virtue. Could be a PR triumph, luck, etc. In certain arenas "winning" would be a stigma...
And certainly the excesses of the "analysis is measure theory" juggernaut are unfortunate... For that matter, a more radical opinion would be that Cantor would have found no need to invent set theory and discover problems if he'd not had a "construction of the reals".
Bottom line for me, just as one vote, one anecdotal data point: I am entirely open to non-standard methods, if they can prove themselves more effective than "standard". Yes, I've invested considerable effort to learn "standard", which, indeed, are very often badly represented in the literature, as monuments-in-the-desert to long-dead kings rather than useful viewpoints, but, nevertheless, afford some reincarnation of Euler's ideas ... albeit in different language.
That is, as a willing-to-be-an-iconoclast student of many threads, I think that (noting the bias of number-of-people working to promote and prove the utility of various viewpoints!!!) a suitably modernized (= BeppoLevi, Sobolev, Friedrichs, Schwartz, Grothendieck, et al) epsilon-delta (=classical) viewpoint can accommodate Euler's intuition adequately. So far, although Nelson's IST is much better than alternatives, I've not (yet?) seen that viewpoint produce something that was not comparably visible from the "standard" "modern" viewpoint.