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.
There are several perfectly rigorous ways to formalize this kind of reasoning, none of which require any nonstandard analysis (which you should be quite suspicious of as it relies on a weak choice principle to even get off the ground).
One of them is, as Robert Israel says, interpreting statements about infinitesimals as statements about limiting behavior as some parameter tends to zero. For example, you can define what it means for a function $f(x)$ to be differentiable at a point: it means there is some real number $f'(x)$ such that (in little-o notation)
$$f(x + \epsilon) = f(x) + f'(x) \epsilon + o(|\epsilon|)$$
as $\epsilon \to 0$. After you prove some basic lemmas about how little-o notation works, you get some very clean and intuitive proofs of basic facts in calculus this way. For example, here's the product rule:
$$\begin{eqnarray*} f(x + \epsilon) g(x + \epsilon) &=& \left( f(x) + f'(x) \epsilon + o(|\epsilon|) \right) \left( g(x) + g'(x) \epsilon + o(|\epsilon|) \right) \\
&=& f(x) g(x) + (f'(x) g(x) + f(x) g'(x)) \epsilon + o(|\epsilon|). \end{eqnarray*}$$
After writing down a bunch of arguments like this, if you're familiar with elementary ring theory it becomes very tempting to think of expressions that are $o(|\epsilon|)$ (meaning they grow more slowly than $|\epsilon|$ as $\epsilon \to 0$) as an ideal that you can quotient out by, and this intuition can also be formalized.
More precisely, in the ring $R = C^{\infty}(\mathbb{R})$ of smooth functions on $\mathbb{R}$, for any $r \in \mathbb{R}$ there's an ideal $(x - r)$ generated by the function $x$, consisting of all functions vanishing at $r$. Working in the quotient ring $R/(x - r)$ amounts to only working with the value at $r$ of a function. Working in the quotient ring $R/(x - r)^2$, though, amounts to working with both the value at $r$ and the first derivative at $r$, with multiplication given by the product rule. Similarly, working in $R/(x - r)^{n+1}$ amounts to working with the value at $r$ and the first $n$ derivatives at $r$.
Taking ideas like this seriously leads to things like formal power series, germs of functions, stalks of sheaves, jet bundles, etc. etc. It is all perfectly rigorous mathematics, and nonstandard analysis is a huge distraction from the real issues.
Best Answer
I believe this has to do with Jesuit opposition to atomism, rather than their position on infinitesimals. Of course, the two are linked and evolved together in the early 17th century. Today we consider atomism a physical theory, but at the time there was no distinction between a mathematical continuum and the physical continuum, just as there was no distinction between Euclidean geometry and the geometry of the space around us.
Aristotelian physics maintained that time, space, and matter were infinitely divisible, and the Jesuits had sided with this idea. They kept records over various ideas which they had debated and found to be flawed, and atomistic ideas appear here several times throughout the first half of the 17th century.
The idea that the continuum consisted of finitely many indivisible particles, each with some physical extension, was considered to be contrary to dogmas about the Holy Communion, and hence particularly offensive. It could be taken to imply that Christ was present in the bread and wine only to a limited degree, corresponding to the number of indivisibles present. This idea was explicitly forbidden in 1608, and in the following years the Jesuit doctrine was refined to forbid atomism also in the case when there were considered to be infinitely many indivisibles.
Galileo used some atomistic ideas to explain his new physics. When his Dialogue was published in February 1632, it would be natural to examine these ideas again, and presumably this is what happened in the meeting in August 1632 mentioned by Alexander.
(For some more details, see the chapter by Palmerino in The New Science and Jesuit Science: Seventeenth Century Perspectives. She does not mention the meeting in 1632, though.)