What's wrong with the surreals?

Question

Of all the constructions of the reals, the construction via 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 1

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 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 2

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 3

This is more of an extended footnote to Nombre’s answer than an answer itself. As Nombre’s observations would suggest, I heartily agree that the algebraico-tree-theoretic simplicity hierarchy is critical to the surreals. $\mathbf{No}$ is not just a monster ordered field containing the reals and the ordinals.

The following is a list of some recent papers on the surreals that make critical use of the simplicity hierarchy, and thereby lend credence to Nombre's observations. It is only the beginning of a new wave of work presently being done by model theorist, order algebrists and analysts that take advantage of $\mathbf{No}$’s simplicity-hierarchical structure.

Berarducci, A. and Mantova, V. (2018): Surreal numbers, derivations and transseries, Journal of the European Mathematical Society 20, pp. 339-390. arixv:1503.00315.

Berarducci, A. and Mantova, V. (forthcoming): Transseries as germs of surreal functions, Transactions of the American Mathematical Society, arXiv:1703.01995.

Aschenbrenner, M., van den Dries, L. and van der Hoeven, J. (2018): Numbers, germs and transseries, Proceedings of the International Congress of Mathematicians, Rio De Janeiro, 2018, arXiv:1711.06936.

Aschenbrenner, M., van den Dries, L. and van der Hoeven, J. (forthcoming): Surreal numbers as a universal $H$-field, Journal of the European Mathematical Society arXiv:1512.02267.

Ehrlich, P. and Kaplan, E.: Number systems with simplicity hierarchies: a generalization of Conway's theory of surreal numbers II, The Journal of Symbolic Logic 83 (2018), No. 2, pp. 617-633, arXiv:1512.04001.

Kuhlmann, S. and Matusinski, M. The exponential-logarithmic equivalence classes of surreal numbers, Order 32 (2015), no. 1, 53–68. arXiv:1203.4538.

Costin, O., Ehrlich, P. and Friedman, H. (24 Aug 2015): Integration on the surreals: a conjecture of Conway, Kruskal and Norton, preprint, arXiv:1505.02478.

The last paper is a rather old version of a paper now in the process of being revised and will eventually be two separate papers.

Edit. May 17, 2020.

The following recent paper by Elliot Kaplan and myself adds further credence to the idea that the algebraico-tree-theoretic simplicity hierarchy is of critical importance to the surreals.

Surreal ordered exponential fields: (https://arxiv.org/abs/2002.07739)

Abstract: In (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field $\mathbf{No}$ of surreal numbers was brought to the fore and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$-vector space) to be isomorphic to an initial subfield ($K$-subspace) of $\mathbf{No}$, i.e. a subfield ($K$-subspace) of $\mathbf{No}$ that is an initial subtree of $\mathbf{No}$. In this sequel to (Ehrlich, J Symb Log, 66, 2001: pp. 1231-1266), piggybacking on the just-said results, analogous results are established for ordered exponential fields. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $(\mathbf{No}, \exp)$. These include all models of $T(\mathbb{R}_W, e^x)$, where $\mathbb{R}_W$ is the reals expanded by a convergent Weierstrass system $W$. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of $\mathbf{No}$, which includes $\mathbf{No}$ itself, extend to canonical exponential functions on their surcomplex counterparts. This uses the precursory result that trigonometric-exponential initial subfields of $\mathbf{No}$ and trigonometric ordered initial subfields of $\mathbf{No}$, more generally, admit canonical sine and cosine functions. This is shown to apply to the members of a distinguished family of initial exponential subfields of $\mathbf{No}$, to the image of the canonical map of the ordered exponential field $\mathbb{T}$ of transseries into $\mathbf{No}$, which is shown to be initial, and to the ordered exponential fields $\mathbb{R}((\omega))^{EL}$ and $\mathbb{R}\langle\langle\omega\rangle \rangle$, which are likewise shown to be initial.

Answer 4

I take advantage of this question to propose an answer in the vein of that of Philip Ehrlich, and to advertise a little bit for $\mathbf{No}$.

I feel like the surreal numbers should be attractive for a several reasons, besides the recent interesting results that have been proven recently regarding them. My sense is that they fail to be in some measure because $\mathbf{No}$ is mostly known as a monster model for the theory of real closed fields, albeit with a nice inductive definition.

Let me recall a fundamental property (FP) of $\mathbf{No}$.

The linear order $(\mathbf{No},\leq)$ is equipped with a well-founded partial order $\leq_s$ of simplicity, where for numbers $x,y$, we have $x\leq_s y$ if as sign sequences, we have $x\subseteq y$.

FP: If $L,R$ are sets of surreal numbers with $L<R$ (i.e. $\forall l \in L,\forall r \in R, l<r$), then there is a unique $\leq_s$-minimal surreal number $x$ satisfying $L<x<R$.

This, along with the fact that for $x\in \mathbf{No}$, the class of numbers $z$ with $z<_s x$ is a set, characterizes the class $(\mathbf{No},\leq,<_s)$ up to unique isomorphism. Note that this fundamental property is a very nice and rare combination: order saturation + canonicity. That alone is a desirable quality in my opinion.

The FP allows one to define easily many operations on $\mathbf{No}$, as Conway did with the field operations and the normal form, as Kruskal-Gonshor did with exponentiation, and those are only the tip of the iceberg. So $\mathbf{No}$ is simply fun to play with, all the more so that easily defined operations can be very challenging to study. Basically, $\mathbf{No}$ is an immense playing field for curious mathematicians interested in order-related notions, as many definitions that are only order-consistent can be thrown into existence in $\mathbf{No}$.

However, the notion of simplicity has been somewhat downplayed by Conway who rather focused on the notion of earliness (and even Harry Gonshor who used it a lot didn't dedicate a symbol to it). The fundamental property is often taken as a tool designed to define the field operations, rather than a marvelous mathematical spell with various uses. This picture, in which $\mathbf{No}$ is a large ordered field containing the ordinals and the real numbers, hides the truth that surreal numbers can tell many other tales.

Answer 5

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 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 6

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 7

One thing that might boost the stock of surreal numbers is if the genetic approach gave us a good way to define functions that mathematicians need. Here I'll fantasize a bit: What if some sort of genetic procedure were the right way to construct an admissible function (in the sense of "New upper bounds on sphere packings I" by Cohn and Elkies) that would enable one to use their Theorem 3.2 to prove optimality of the best known sphere packing in dimension 8? I hasten to add that I know of no reason why a genetic construction procedure should be helpful here. The "missing function" of Cohn and Elkies is just one example of a mathematical function that ought to exist but which we don't know how to construct. At some point, someone will probably find a construction that "comes out of left field". If surreal analysis served as "left field", more people would be interested in it.

Answer 8

A quick search indicates that Peano axioms are not mentioned on this page. It seems reasonable to mention that there does not seem to be a good notion of natural number in the surreals that would satisfy Peano arithmetic. On the other hand, a maximal class-size surreal field is isomorphic to a suitable maximal class-size hyperreal field, so the hyperintegers can be imported to the other side, and with them all the required first-order properties.

Answer 9

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.

Answer 10

It seems, the main difficulty in defining integration on surreals stems from the set of axioms the authors decided to choose for their definition of integration to satisfy.

Particularly, the most problematic is the property of linearity against an infinitely-large factor. For instance, it is the property $(b)$ in Preposition 14 in this paper on surreal integration:

$$\int_a^b (\alpha f+\beta g)=\alpha \int_a^b f + \beta \int_a^b g,$$

where $\alpha, \beta \in \text{No}$.

In my view, this property is unnatural. For instance, assuming total order, you definitely want a surreal number, such big that it to correspond to the derivative of a function that makes a jump at a point, like $\operatorname{sgn}(x)$ makes at zero. You definitely want a surreal number such big that integration of a function over a range where the function takes this number as a value only at one point is non-zero.

Otherwise, you would be able to extend your order of surreal numbers by new numbers who would also be totally ordered. But surreal numbers are a maximal ordered field.

Once this axiom is dropped and replaced by the linearity against finite (real) values, the integration can be easily defined based on this natural definition:

$$\int_S u dt=\pi N(S) \partial(u),$$

where $u$ is a surreal constant, $N(S)$ is numerosity and $\partial(u)$ is derivation. The numerosity plays here the same role as measure in Lebesgue integral. Using this base formula, one can derive a formula for integrating arbitrary surreal-valued functions:

$$\int_a^b F(t) dt=\pi \omega_1\tilde{\int}_a^b \partial{(F(t))}dt+\int_a^b \operatorname{fin}F(t)dt,$$

where $b>a; a,b\in\text{No}$, $F$ is a surreal-valued function of surreal argument, the symbol $\tilde{\int}$ should mean that the integral is taken formally, without awareness of the surreal character of the expression under the integral, and $\operatorname{fin}F(t)$ means taking finite part of $F(t)$. The second integral is evaluated using Newto-Leibnitz formula after finding the indefinite integral symbolically.

Below is the code for Mathematica that realizes this formula. For simplicity, in input w corresponds to $\omega$ and W to $\omega_1$. The input accepts any surreal numbers composed of these values as integration limits and any surreal-valued function of integration variable x as the expression under integral. In output, derivation is denoted as ꝺ.

f[x_] = w^W;
a = 0;
b = w;
f[x_] = f[x] /. W -> W[w];
Unprotect[Power]; Power[0, 0] = 1; Protect[Power];
Fin[t_] := (PowerExpand[f[t]] /. Log[w] -> 0 /. w -> 0) + 
     Limit[Evaluate[LaplaceTransform[ D[f[t], w], t, x]] + 
        Evaluate[LaplaceTransform[ D[f[t], w], t, -x]], x -> 0]/
      2 /. \[Infinity] -> 0 /. -\[Infinity] -> 0 // 
  FullSimplify  (*Finding the finite part so to integrate separately*)
int = \[Pi]  W  Integrate[D[f[t], w], {t, a, b}] + 
    Integrate[Fin[t], {t, a, b}] // FullSimplify;
int = If[a == b, \[Pi]  D[f[t], w], int, int] /. 
     Derivative[1][W][w] -> ꝺ[W] /. W[w_] :> W // Normal;
Print[Inactivate[
  Integrate[
   f[x] /. W[w] -> W /. w -> \[Omega] /. 
    W -> Subscript[\[Omega], 1], {x, 
    a /. w -> \[Omega] /. W -> Subscript[\[Omega], 1], 
    b /. w -> \[Omega] /. W -> Subscript[\[Omega], 1]}], 
  Integrate], "=", 
 int /. w -> \[Omega] /. W -> Subscript[\[Omega], 1]]

The code correctly gives all the results from the other my answer, but also can find more complicated results, such as:

• $\int _0^{\omega }e^{c x \omega }dx=\frac{c \omega ^3+\omega +\pi \left(e^{c \omega ^2} \left(c \omega ^2-1\right)+1\right) \omega _1}{c \omega ^2}$

• $\int _0^{\omega }e^{c x^2 \omega }dx=\omega +\frac{1}{2} e^{c \omega ^3} \pi \omega _1-\frac{\pi ^{3/2} \text{erfi}\left(\sqrt{c} \omega ^{3/2}\right) \omega _1}{4 \sqrt{c} \omega ^{3/2}}$

• $\int _0^{\omega }\log (c x \omega )dx=\omega (\log (c)+\log (\omega )-1)+\pi \omega _1$

• $\int _0^{\omega }\omega ^xdx=\frac{\frac{\pi \left((\omega \log (\omega )-1) \omega ^{\omega }+1\right) \omega _1}{\omega }+1}{\log ^2(\omega )}$

• $\int _0^{\omega }\omega ^{\omega }dx=\pi (\log (\omega )+1) \omega _1 \omega ^{\omega +1}+\omega$

• $\int _0^{\omega }p \omega ^{q \omega }dx=p \omega \left(\pi q (\log (\omega )+1) \omega _1 \omega ^{q \omega }+1\right)$

• $\int _0^{\omega }\omega ^{\omega _1}dx=\pi \omega _1 \left(\omega \partial \left(\omega _1\right) \log (\omega )+\omega _1\right) \omega ^{\omega _1}$

• $\int _0^{\omega }\omega _1^{\omega }dx=\pi \omega \left(\frac{\omega \partial\left(\omega _1\right)}{\omega _1}+\log \left(\omega _1\right)\right) \omega _1^{\omega +1}+\omega$

• $\int _0^{\omega }\omega _1^xdx=\frac{\omega _1^{\omega }-1}{\log \left(\omega _1\right)}+\frac{\partial\left(\omega _1\right) \left(\pi \left(\omega \log \left(\omega _1\right)-1\right) \omega _1^{\omega +1}+\pi \omega _1+\omega \right)}{\log ^2\left(\omega _1\right) \omega _1}$

• $\int _0^{\omega _1}\omega ^xdx=\frac{\omega _1 \left(\pi \left(\log (\omega ) \omega _1-1\right) \omega ^{\omega _1}+\pi +1\right)}{\omega \log ^2(\omega )}$

• $\int _0^{\omega }\log \left(\omega _1\right)dx=\omega \left(\log \left(\omega _1\right)+\pi \partial\left(\omega _1\right) \omega _1\right)$

• $\int _0^{\omega }\log \left(\omega _1\right)dx=\omega \left(\log \left(\omega _1\right)+\frac{\pi \partial\left(\omega _1\right) \omega _1}{\omega _1(\omega )}\right)$

etc.