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Of all the constructions of the reals, the construction of the surreals seems the most elegant to me.

It seems to immediately capture the total ordering and precision of Dedekind cuts at a fundamental level since the definition of a number is based entirely on how things are ordered. It avoids, or at least simplifies, the convergence question of Cauchy sequences. And it naturally transcends finiteness without sacrificing awareness of it.

The one "rumor" I've consistently heard is that it is hard to naturally define integrals and derivatives in the surreals, although I have yet to see a solid technical justification of that.

Are there known results that suggest we should avoid further study of this construction, or that show limitations of it?

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Regarding the difficulties wtih integrals and derivatives, see the appendix to the second edition of On Numbers and Games. I don't think that this is really relevant to your question, though. Once you have the reals you can just develop analysis as usual; you don't have to carry the surreals around as excess baggage everywhere if you don't want to. –  Timothy Chow Jun 24 '10 at 1:58
    
In asking a big-picture question I was looking for an equally vague-but-intuitive answer, although of course technical evidence is extremely useful! You're reference to the end of On Numbers and Games is probably the single best answer so far, or rather, Conway's epilogue itself. Thanks, Timothy. (I think I'll rephrase this as another answer). –  user2498 Jun 24 '10 at 22:06

5 Answers 5

up vote 29 down vote accepted

At a recent conference in Paris on Philosophy and Model Theory (at which I also spoke), Philip Ehrlich gave a fascinating talk on the surreal numbers and new developments, showcasing it as unifying many disparate paths in mathematics. The abstract is available here, on page 8, and here his draft article on the Absolute Arithmetic Continuum. The principal new technical development is a focus on the underlying tree.

Philip expressed his frustration that Conway often treated his creation of surreal numbers as a kind of game or just-for-fun project---an attitude reinforced by the excellent Knuth book---whereas they are in fact a profound mathematical development unifying disparate threads of mathematical investigation into a single unifying structure. And he made a very strong case for this position at the conference.

Meanwhile, perhaps exhibiting Philip's point, at a conference on logic and games here at CUNY, I once heard Conway describe the surreal numbers as one of the great disappointments of his life, that they did not seem after all to have the profound unifying nature that he (and many others) thought they might. Philip Ehrlich strove to make the case that Conway was his own worst enemy in promoting the surreals, and that they actually do have the unifying nature Conway thought they did, but that Conway scared people away from this perspective by treating them as a toy. I encourage you to read Philip's articles.

So my answer, supporting Philip, is that nothing is wrong with the surreals---please have at them! Of course they have their own issues, which will need to be surmounted, but we shall all benefit from a greater investigation of them.

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Unifying nature in what sense? That is, what are examples of mathematical (as opposed to philosophical) concepts from outside of pure set theory were once seen as unconnected but now are seen as unified by means of this work? And what does the tree allow one to do which couldn't be done before? Or is the interest internal to logic and set theory? A skim of the draft copy did not clarify this for me. –  Boyarsky Jun 24 '10 at 4:45
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Apparently the Continental philosopher Alain Badiou regards the surreals as being of substantial philosophical importance, since they supply a way of talking about the infinite which still satisfy the ordered field axioms (in contrast with, say ordinal arithmetic). (I can't honestly claim to understand Badiou, but I do enjoy reading him: it's like reading about the foundations of mathematics from some parallel universe where Russell became a disciple of Husserl instead of Frege..!) –  Neel Krishnaswami Jun 24 '10 at 8:29
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It seems the main thing that would be needed to broaden the interest of surreals would be to provide some tools from transferring results back from the surreals back to more classical number systems (analogous to the transfer principle in nonstandard analysis). The situation here reminds me of that of generalised functions in analysis. There are many, many ways to generalise the concept of a function, but only distributions have really been successful, because there are ways to get from distributions back to classical functions, e.g. by convolving with a test function. –  Terry Tao Jun 25 '10 at 0:04
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For the first 50 years after Hensel, $p$-adic analysis was championed by a few but viewed as esoteric by most; intrinsic beauty wasn't enough. Then Dwork used it to prove a Weil Conjecture and Tate invented rigid-analytic spaces to make analytic continuation possible over totally disconnected fields (with applications in number theory), and now it's a huge industry. Surreals are in need of a Dwork and Tate. (Although the set theory they require is elementary to set theorists, I conjecture it is a real obstacle for many without such expertise. Look at the history of non-standard analysis.) –  Boyarsky Jun 25 '10 at 1:21
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A Boyarsky turning to Dwork for an example. Am I the only one amused by this? –  KConrad Jun 25 '10 at 3:52

To me one of the more fascinating aspects of the surreals is that application by Kruskal and others to construct higher order asymptotic expansions. For example, if you want to understand the asymptotics of the function $$f(x)= {1\over 1-x}+e^{-1/x}$$ on $(0,\epsilon)$ and differentiate it from $g(x)={1\over 1-x}$ you look at the ``series" $$1+x+x^2+x^3+\dots+e^{-1/x}.$$ Kruskal and his co-authors have used surreal numbers to give an approach to these expansions and applications.

This type of expansion can also be be dealt with using the transseries of Ecalle or the logarithmic-exponential series developed in model theory.

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Conway himself lists a few disadvantages in On Numbers and Games, Chapter 2.

One that can be dealt with quickly is that it is quite tricky to make the process stop after constructing the reals! We can cure this by adding to the construction the proviso that if $L$ is non-empty but with no greatest member, then $R$ is non-empty with no least member, and vice versa. This happily restricts us exactly to the reals. The remaining disadvantages are that the dyadic rationals receive a curiously special treatment, and that the inductive definitions are of an unusual character. From a purely logical point of view these are unimportant quibbles (we discuss the induction problems later in more detail), but they would predispose me against teaching this to undergraduates as "the" theory of real numbers.

[Edit: I interpreted the question as, what is wrong with the construction of the real numbers via surreals? This might have been a misinterpretation. The question does, after all, literally ask what is wrong with the surreals. Obviously there is nothing wrong with the surreals. I thought this was obvious and so I assumed the question must have been something else.]

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I actually remember one more disadvantage mention by Conway: equality is a defined relation, and a pretty subtle one at that. It's not too easy to figure out when two expressions for surreal numbers are actually equal. –  Alon Amit Jun 24 '10 at 15:14
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Equality is also a "defined relation" if you define the real numbers as Cauchy sequences of rationals, or even if you define the rational numbers as ordered pairs of integers, although in most cases we sweep it under the rug by talking about "equivalence classes." So I don't really see why the fact that equality of surreals is a "defined relation" should be regarded as a disadvantage. –  Mike Shulman Jun 25 '10 at 5:19
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@Timothy: I don't want to blindly go ahead and pretend; I want clear definitions first. Notions like "set of all maps" and "set of all subsets" underlie many constructions. The way of thinking about many things is within a set-theoretic framework. How do you define and construct algebraic closures of fields and topological absolute Galois groups, or define what a "field" is, without set theory? There are reasons I think the analogy with category theory is not so apt, but this isn't the place to get into it. Saying "classes behave just like sets" sounds like word games; it can't be that easy. –  Boyarsky Jun 26 '10 at 7:08
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The ordinals and cardinals also form proper classes, but these number concepts have proved enormously fruitful. For that matter, the collection of all sets is also a proper class, but we still work with sets. So why not surreals? Of course, one must respect the set/class issues, but this is neither difficult nor mysterious. –  Joel David Hamkins Jun 28 '10 at 18:07
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There are mainly two main points to classes: the first is that classes, unlike sets, cannot be themselves members of other classes (they only contain elements, but they're not elements of anything). The second is that you need to be able to describe classes to use them, i.e. you need to consider definable classes only (if you're sticking to the usual set theory, ZFC, rather than someting like NBG or Morse-Kelley). Keeping those two in mind is all you need to clarify the difference between sets and classes, and to use classes correctly. –  David FernandezBreton Jun 24 '11 at 14:46

I just read the epilogue of On Numbers and Games as suggested by Timothy Chow, and here I see very concrete references to some technical difficulties that may demystify some research in this area, although it is apparent by other answers that there is still much optimism.

Some remarks based on the epilogue (written in 2000 by Conway):

There is a nice definition of the surreals which does not require equality as a defined relation. It is not formally given, but I'm guessing we can think of it as a mapping from an ordinal set to signs {-,+}, each sign being a direction we take in the surreal tree. Then identity is equality.

However, Conway remarks that this has two problems:

  1. It forsakes the "genetic" (his word, also in quotes at first) approach of the L,R definition. I don't fully understand this, but I'm guessing he means that we're building everything on the intuition of a total ordering (and maybe a "time-of-creation" idea), and the surreals will always be identifiable with L,R sets, so why not just define them that way?

  2. The sign-sequence definition requires that the ordinals are defined first.

Conway goes on to discuss work by Simon Norton (a proposed definition of an integral) and Martin Kruskal.

The general direction here is to define things in terms of (L,R) sets (classes??) in such a way that equal numbers (in the defined equality) give equal answers; and that classical analysis remains intact.

Conway gives Norton's integral definition, which has some good properties, but fails to integrate the surreal-exponential function in accordance with classical analysis (we get ex instead of ex-1 when integrating over [0,x]).

In summary, I'm choosing to interpret all of these comments and answers together (thanks to all) as: the surreals are indeed a worthwhile construction, although there is a noted lack of progress on extending calculus to work equally elegantly in a surreal-general setting.

In case others are curious, here are some references (I have read none of them, yet) Conway gives in this epilogue:

  • The Theory of Surreal Numbers by Harry Gonshor
  • Foundations of Analysis over Surreal Number Fields by Norman Alling
  • Real Numbers, Generalizations of the Reals, and Theories of Continua by Philip Ehrlich
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Gonshor's book is quite elegant and starts defining surreal numbers right away as sequences on {-,+}. Of course you need to define ordinals first, but for me that's not a disadvantage (maybe for Conway it is, since he seems to try to avoid to "restrict" himself to a particular axiomatic system such as ZFC, assuming that it's possible to even make sense of such a viewpoint). –  David FernandezBreton Jun 24 '11 at 1:40

This is not a major issue but there was a remark made in the master thesis that can be found at the following address http://www.mamane.lu/concoq/ that there is a small gap in the proof of the transitivity of the ordre relation in the original book of Conway. See the report, page 49-53.

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