What's wrong with the surreals? - MathOverflow

What's wrong with the surreals?

2010-06-24T01:00:12Z

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?

Answer by Timothy Chow for What's wrong with the surreals?

2010-06-24T01:53:07Z

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.]

Answer by Joel David Hamkins for What's wrong with the surreals?

2010-06-24T03:40:09Z

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.

Answer by tylern for What's wrong with the surreals?

2010-06-24T22:25:09Z

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:

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?
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 e^x instead of e^x-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

Answer by Dave Marker for What's wrong with the surreals?

2010-06-28T13:36:05Z

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.

Answer by Serge R. for What's wrong with the surreals?

2010-07-01T11:04:58Z

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.