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Category Archives: Surreal Numbers

1996: John Horgan declares THE END OF SCIENCE. 1997: first E. coli genome sequence published. 1998: expansion of Universe found to be accelerating.
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In the appendix to a paper I am currently co-authoring, I recently wrote the following within a parenthetical excursus:

When talking of dynamical systems, our probability assignments really carry two time indices: one for the time our betting odds are chosen, and the other for the time the bet concerns.

A parenthesis in an appendix is already a pretty superfluous thing. Treating this as the jumping-off point for further discussion merits the degree of obscurity which only a lengthy post on a low-traffic blog can afford.

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Among the fun math books I have on my overburdened bookshelves is John Conway and Richard Guy’s fascinating volume, The Book of Numbers (1996). In following up one of the topics discussed in its very last chapter, I discovered that Conway and Guy had made a bibliographic error, which in the interests of scholarship should be publicly noted. While I could give the correction in a line and be done with it, the topic and its background are curious enough to merit a few paragraphs. To wit:

Anybody who has had a brush with calculus is familiar with taking derivatives of a function. The derivative of a function is a whole new function which gives the rate of change of the original; plug a function into the machine, and out comes a new one, which is also just as “complicated” as the one you started with. If your initial function was something like

[tex]f(x) = x^2,[/tex]

which maps each real number to a real number, then the derivative will be something like

[tex]f^\prime(x) = 2x,[/tex]

which also maps elements of [tex]\mathbb{R}[/tex] to elements of [tex]\mathbb{R}[/tex]. That’s a whole lot of mappings! If we were so inclined, we could also represent the “growth rate” of functions by numbers, instead of by functions. The operation of “finding the growth rate” would then be a functional, mapping functions to numbers — though the sort of numbers we find ourselves using are a little out of the ordinary.
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I’ve been going through André Joyal’s category-theoretic construction of the surreal numbers, futzing around to see how restricting the construction to yield only the real numbers affects things. (See my earlier remarks on this and/or Mark Chu-Carroll’s discussion for background information.) If I were an actual mathematician, I’d probably be done by now. Instead, I have a headful of half-baked notions, which I’d like to spill out into the Intertubes (mixing my metaphors as necessary).

Why would a physics person even care about the surreal numbers? Well, ultimately, my friends and I were going through Baez and Dolan’s “From Finite Sets to Feynman Diagrams” (2000), which touches upon the issue that subtraction can be a real pain to interpret categorically. If you interpret the natural numbers as the decategorification of FinSet, then addition is easy: you’re just talking about a coproduct. But subtraction and negation — oh, la!
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Due to an illness among the teaching staff, today’s statistical-mechanics session will be postponed until later in the week. In the meantime, check out science writer Carl Zimmer and his experience being quote-mined by global warming denialists. To cheer up after that study in human folly, try Mark Chu-Carroll’s ongoing series on surreal numbers. Based on his prior habits, I’m curious to see if he takes a crack at explaining the relation between surreals and category theory (definitely one of the branches of mathematics you need to study if you want to be like the guy in Pi).

One point should be raised about the surreals which has not yet appeared in Mark Chu-Carroll’s exposition. You can get the reals — that ordinary, familiar number line — from the surreals by imposing an extra condition on the construction. It’s exercise 17 in the back of Knuth’s Surreal Numbers (a problem originally suggested to Knuth by John Conway). A number x is defined to be real if -n < x < n for some integer n, and if x falls in the same equivalence class as the surreal number

({x – 1, x – 1/2, x – 1/4, …}, {x + 1, x + 1/2, x + 1/4, …)}.

This topic is also discussed in chapter 2 of Conway’s On Numbers and Games. Theorem 13 proves that dyadic rationals are real numbers, and Conway then deduces that each real number not a dyadic rational is born on day ω (“Aleph Day” in Knuth’s book).

The practical upshot of all this is that surreals may provide a better pathway to understanding the real numbers than the standard way of teaching real analysis! Dealing with Dedekind cuts, for example, leads you to an explosion of special cases and general irritation. Conway says:

This discussion should convince the reader that the construction of the real numbers by any of the standard methods is really quite complicated. Of course the main advantage of an approach like that of the present work is that there is just one kind of number, so that one does not spend large amounts of time proving the associative law in several different guises. I think that this makes it the simplest so far, from a purely logical point of view.

Nevertheless there are certain disadvantages. One that can be dealt with quickly is that it is quite difficult 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.

There is another way out. If we adopt a classical approach as far as the rationals Q, and then define the reals as sections of Q with the definitions of addition and multiplication given in this book, then all the formal laws have 1-line proofs and there is no case-splitting. The definition of multiplication seems complicated, but is fairly easy to motivate. Altogether, this seems the easiest possible approach.