In Conway's "On Numbers And Games," page 44, he writes:
NON-STANDARD ANALYSIS
We can of course use the Field of all numbers, or rather various small
subfields of it, as a vehicle for the techniques of non-standard
analysis developed by Abraham Robinson. Thus for instance for any
reasonable function $f$, we can define the derivative of $f$ at the
real number $x$ to be the closest real number to the quotient$$\frac{f[x + (1/\omega)] – f(x)}{1/\omega}$$
The reason is that any totally ordered real-closed field is a model
for the elementary states about the real numbers. But for precisely
this reason, there is little point in using subfields of $\mathbf{No}$
when so many more visible fields will do. So we can say in fact the
field $\mathbf{No}$ is really irrelevant to non-standard analysis.
Conway here makes clear that you could, if you wanted, use the surreal numbers for non-standard analysis, because they are a real-closed field and thus a model for the theory of all elementary (first-order) statements about the reals. Conway does also make the point, in the last two sentences, that he doesn't view nonstandard analysis as the ultimate application of the surreals. But, for the sake of curiosity, I'm quite interested in understanding how you could do what Conway is hinting at above.
However, the statement that you can use the surreals for nonstandard analysis is really quite strong, and much stronger than just the field being real-closed. The real meat of the claim being made is the expression $f(x + 1/\omega)$ even exists at all. This would demand some kind of "transfer principle" for $f(x)$ to the surreals. Just being real-closed wouldn't be enough for this: the real algebraic numbers are real closed, but that doesn't mean we can use the real algebraic numbers for nonstandard analysis. But Conway says this is "of course" possible with the surreals.
So the main question is: how would such a transfer principle work?
Or an even stronger question: do we need anything like an ultrafilter lemma for Conway's claim to be true? Unlike the hyperreals, the surreals don't require any ultrafilter at all — which is a pretty significant achievement, really, since we are at least certain something like $1/\omega$ exists even in ZF, and that the resulting field is real-closed. It would seem to be possible to just go through the motions with taking the nonstandard derivative of, for instance, $\frac{\exp(x + 1/\omega) – \exp(x)}{1/\omega} = \exp{x}\left(\frac{\exp^{1/\omega} – 1}{1/\omega}\right) \approx \exp(x)$, treating $1/\omega$ as a formal expression satisfying the first-order properties of the reals and deriving $\exp(x)$ as the closest real number to the result, none of which seems to have required any kind of ultrafilter. (Or has it, implicitly?)
I've tried to keep this short but there is an enormous amount of subtlety to this question, so I will go into some of that below.
Some later results have clarified the relationship between the surreals and hyperreals, so some additional detail regarding what is being asked is probably necessary.
There has been a little bit of prior discussion about this, for instance in this post, where it is talked about the much more modern result that the surreals are isomorphic to the proper-class sized ultrapower of the reals. These isomorphisms can be thought of as various ways to transfer real functions to the surreal numbers. So in one sense, the answer is yes, a transfer principle exists in theory. But the pitfall with this approach is that everything requires ultrafilters, and is non-constructive, and there is no canonical choice of isomorphism. This is very different from the way that the surreals are built, which do not require ultrafilters.
On the other hand, Conway's book was written before any of the above results were published (with possibly an exception regarding one paper of Keisler). So partly the question is informal – what did Conway have in mind? But the other part of it is to formally ask if there is some other way to do this that doesn't involve this very particular method of using these isomorphisms, or even to use ultrafilters at all. For instance, what if we don't have the ultrafilter lemma? Then the hyperreals don't necessarily exist at all, but we can still build the surreals, which don't even require choice. Even if we don't have the ultrafilter lemma, can we still just go ahead anyway and say that $\frac{f(x + 1/\omega) – f(x)}{1/\omega}$ is a well-defined expression, and look for the closest real number to it, using some other way to derive a transfer principle?
The other part of the question is admittedly a soft question, but still well worth answering. The ultrafilter construction makes it very easy to see how such a transfer principle would work. Every hyperreal is a set of reals (or an equivalence class thereof), and to transfer any first-order predicate to some hyperreal, you simply ask the predicate of every real in the set and see if it's true of "most" of them (where "most" means "in the ultrafilter"). Thus you have a real-closed field, a "transfer principle," and all of that. Conway, on the other hand, has a very interesting way of building up the surreals in his book which is somewhat agnostic to the choice of set theory, using "birthdays," "left and right sets," etc. I am curious if there is some way to interpret Conway's assertion regarding the existence of $f(x+1/\omega)$ using his own machinery for the surreals, perhaps doing something clever and inductive with the left and right sets, rather than using these later developments involving isomorphisms with the ultrapower.
The last subtlety involves a philosophical point that has sometimes been raised with the topic of surreals vs hyperreals, but it is also worth addressing. There is, for instance, some debate regarding how functions like $\sin$ and $\cos$ should be transferred to the surreals. In theory, you could say that since we have these isomorphisms to the hyperreals, which have a transfer principle, these guarantee the existence of some kind of function on the surreals with the required first-order properties. But the surreals are very tangible in a very constructive sort of way, whereas these isomorphisms are typically totally non-constructive, so there is no way to use them to see what $\sin(\omega)$ should be, if it's positive or negative, etc. On the other hand, you could raise the same philosophical issue with the hyperreals, because there also is no real answer regarding what $\sin((1,2,3,4,…))$ should be, where $(1,2,3,4…)$ is a particular hyperreal number. The answer depends on the ultrafilter, which determines what $(1,2,3,4…)$ even means to begin with, or what properties it has, or if you like, which hyperreal it's referring to.
But what you can do with the hyperreals, which is part of the appeal, is you can kind of get "part of the way there" in a totally constructive manner. You know, for instance, that whatever $\sin((1,2,3,4,…))$ is, in some sense it's $(\sin(1), \sin(2), \sin(3), \sin(4), …)$. Since we don't know what the ultrafilter is we don't know exactly what properties that has, but we do know something: we know that for any ultrafilter this value will not be an integer, for instance, or any rational number. So, we have some idea of what the transferred sin function would have to look like on the surreals as a result, at least given that $\omega$ is some hypernatural. So even though the ultrafilter is non-constructive, you can at least get "part of the way there" in an entirely constructive manner, which is part of what makes the entire thing interesting. And of course you don't really need to know much more than these few constructive things to actually do nonstandard analysis, just kind of happily plodding along formally doing nonstandard derivatives, with the understanding that the ultrafilter handles all of the various pathological, undefinable sets of indices in some logically consistent way or another.
So the last question is if there is some way for us to do something similar with the surreals, to get "part of the way there," in this sense. That is, to at least have enough constructive "transfer" for us to play around with all of this stuff, but using the framework of the surreals rather than the hyperreals, so that we can see that $f(1 + 1/\omega)$ even makes sense to begin with and play around with it. Something like Terry Tao's "cheap nonstandard analysis", perhaps.
Best Answer
Conway was of course correct in saying that NSA is irrelevant to the surreals, but like Mike I found Conway’s further remarks about NSA puzzling and I am not sure what he had in mind. What I think Conway might have said is: "$\mathbf{No}$ is really irrelevant to nonstandard analysis", and, vice versa. After all, whereas the transfer property of hyperreal number systems, a property not possessed by $\mathbf{No}$, is central to the development of nonstandard analysis, the s-hierarchical (i.e the algebraico-tree-theoretic) structure of $\mathbf{No}$, which is absent from hyperreal number systems, is central to the theory of surreal numbers and is responsible for its canonical nature. On the other hand, as I have noted on occasion, I do not rule out the possibility that down the road there might be cross-fertilization between the two theories. However, even if not, it seems to me that to not appreciate that both theories—which are used for quite different purposes--are remarkably powerful and beautiful is a demonstration of ignorance.
While surrealists have thus far shown little interest in applying surreal numbers to nonstandard analysis or in providing an infinitesimalist approach to classical analysis based on surreal numbers more generally, a number of surrealists beginning with Norton, Kruskal, and Conway have fostered the idea of extending analysis to the entire surreal domain. While this has been a comparatively small focal point of the theory thus far and progress has been slow, I am happy to say that, contrary to the assertions of Sam and Emanuele, there is now a reasonably strong theory of surreal integration. After laying dormant for quite some time and in need of much revision, work on surreal integration has recently made substantial progress and I expect that a paper on the subject by Ovidiu Costin and myself will be posted before too long. The theory extends integration from the reals to the surreals for a large portion of Écalle’s class of resurgent functions, in particular, a large subclass of the resurgent functions that are found in applied analysis. On the other hand, there are considerations in the foundation of mathematics that preclude the theory from being extended very much further. So, for example, the theory cannot be applied in general to the class of smooth functions.
Thus far, a sizable portion of the literature on the surreals has dealt with the theory of ordered algebraic systems and (as Emanuele suggests) model-theoretic issues thereof—issues involving ordered abelian groups, ordered domains, ordered fields, ordered exponential fields and ordered differential fields. In the latter two cases, there has been considerable work on Hardy fields and Écalle’s ordered differential field of transseries and generalizations thereof. Some of the work on ordered exponential fields and ordered differential fields, however, go well beyond ordered algebraic and model-theoretic considerations and are concerned with developing asymptotic differential algebra--the subject that aims at understanding the asymptotics of solutions to differential equations from an algebraic point of view—for the surreals. In their 2017 ICM talk (On Numbers, Germs, and Transseries), Aschenbrenner, van den Dries and van der Hoeven outline the program they (along with Mantova, Berarducci, Bagayoko and Kaplan) are engaged in for developing an ambitious theory of asymptotic differential algebra for all of the surreals, though one that would require a derivation on $\mathbf{No}$ having compositional properties not enjoyed by the derivation introduced by Mantova and Berarducci in the paper cited by Emanuele. Such a program, if successful, would provide the most dramatic advance towards interpreting growth rates as numbers since the pioneering work of Paul du Bois-Reymond, G. H. Hardy and Felix Hausdorff on "orders of infinity" in the decades bracketing the turn of the 20th century. Unlike the surreals, the framework on NSA does not appear to be particularly well suited to this end, despite the fact that Robinson and Lightstone did make modest contributions to the theory of asymptotics in their nice monograph that applies NSA to the subject.
In his response, Sam noted that (as far as he knew) the surreals do not have a visible copy of surnatural numbers. Sam is, of course, correct; but given the recursive nature of the construction of the surreals no one should expect to find one, and for the purposes for which it has been used it has not proven to be a limitation. On the flip side, of course, unlike the surreals, the number systems employed in NSA do not have a canonical copy of the ordinals or significant initial segments thereof. However, for the purpose of developing an infinitesimalist approach to classical analysis, I don’t see this to be a problem either. Given all the areas and questions to which mathematicians apply finite, infinite and infinitesimal numbers, it is extremely unlikely there will ever be one theory ideally suited for all applications. An assertion like "For all its weaknesses" made by Sam about the surreals, fails to appreciate this. What would be a weakness in one context need not necessarily be a weakness in another.
All of the people I know who are working in the theory of surreal numbers are familiar with, and have great respect for, NSA. Sadly, however, there appears to be a small segment of the contemporary NSA community who, while repeatedly demonstrating their lack of knowledge of the subject and its applications, attack it time and again, one person (as is evident from the comments) even describing those who work on it as members of a cult. During the 19th century, Cantor repeatedly attacked the works of du Bois-Reymond, Stolz, and Veronese on their non-Cantorian theories of the infinite (and infinitesimal), theories designed to deal with issues not addressed by Cantor’s theory—non-Archimedean geometry, the rates of growth of real functions, and non-Archimedean ordered algebraic systems. Abraham Robinson, who was as gracious a person as he was a great and knowledgable mathematician, attempted to soften his well-deserved implicit critique of Cantor’s misguided and narrow-minded attacks by noting: "It may be recalled that, at that time, Cantor was fighting hard in order to obtain recognition for his own theory" (The Metaphysics of the calculus, p. 39). I wonder to what extent some of the aforementioned attacks are motivated by similar considerations, despite the fact that NSA is already widely, albeit not universally, regarded as a major contribution. However, whatever the motivation may be, I believe losing sight of the lesson of Cantor or the humanity of Robinson would be an unfortunate mistake indeed.
P.S. One of the longstanding bugaboos in the aforementioned attacks on the surreals has been that there is no natural sine and cosine functions for the surreals. For a proof that this contention is mistaken, see Section 11 of Kaplan and the author’s recent Surreal Ordered Exponential Fields, The Journal of Symbolic Logic, 86 (2021) pp.1066-1115.