Sunday, January 18, 2004

The Axiom of Choice and Assumptions in Applied Mathematics

Via Jacques Distler, I came across this extremely informative discussion of the Axiom of Choice by Eric Schechter. The Axiom of Choice (or AoC, as I'll call it from here on) is justly notorious for being easy to state, seemingly obviously true, and yet leading to some very strange conclusions:

Let C be a collection of nonempty sets. Then we can choose a member from each set in that collection. In other words, there exists a function f defined on C with the property that, for each set S in the collection, f(S) is a member of S.

Straightforward enough, right? And there are even more seemingly obvious formulations, such as the following one:

Given any two sets, one set has cardinality less than or equal to that of the other set -- i.e., one set is in one-to-one correspondence with some subset of the other.

Or try the following equivalent statement:

Every vector space has a basis.

Now, what could seem more obvious than that? The problem is that buying into any of these equivalent formulations of the AoC means buying into results like the Banach-Tarski Decomposition Theorem, which basically says that it is possible to take a solid ball, cut it up into no more than five pieces, and then re-arrange these pieces to obtain two balls of the same size as the original! Either the old saying about being able to get something for nothing is false, and the conservation laws of physics do not hold, or something is very wrong in our mathematical assumptions.

The traditional reaction most mathematicians have had to the Banach-Tarski "Paradox" has been to either regard it as grounds to reject the AoC outright, or, finding it too useful to do without, to argue that theirs is simply a formalist enterprise, mere symbol manipulation without any consideration for the meanings of the symbols being studied; in fact, whatever they might say to the contrary, few mathematicians either believe or act like they believe in formalism while going about their business - as the old saying goes, "platonism on weekdays, formalism on weekends."

The unvarnished truth is that the AoC is simply too obviously true, and too fruitful to mathematics, for most mathematicians to spend serious time doubting its veracity, and a little consideration shows that the Banach-Tarski "Paradox" (quote-unquote) actually gives us no reason to do so; for there is another assumption on which Stefan Banach and Alfred Tarski's strange theorem relies, an assumption of a much more dubious nature where the world about us is concerned - that there exists such a thing as a physical manifestation of the continuum.1

The assumption that various phenomena are continuous in nature has been an extremely fruitful one in science and engineering, and it is only natural that engineers, economists and other users of mathematics should have come to take it for granted that any phenomena of interest to them can be treated as being continuous. The problem, however, is that nature apparently is not continuous, particularly not at the smallest scales. Both string theory and loop quantum gravity, the main competitors for a unification of quantum mechanics and general relativity, tell us that the universe is fundamentally discrete at the lowest levels - that spacetime is quantized, and cannot come in the arbitarily sized bundles demanded by the assumption of continuity.

For economists going about analyses of market behavior, or engineers working on aircraft design, such reconceptualizations of the nature of the universe are of no practical impact, but I think that they, in combination with the Banach-Tarski result, ought at least to make one a little more cautious in jumping to the conclusion that continuity is always a safe assumption to make. It is almost certainly the case that one other assumption that engineers and economists are fond of, namely that linear differential equations provide a good approximation of the phenomena they wish to study, has sometimes proven in practice to be catastrophically flawed.2

What does all of this mean for those for whom mathematics is merely a tool rather than an end in itself? Does it mean that engineering types should be expected to add courses on the foundations of mathematics to their curricula? Not at all. I'd just say that they would do well to be more conservative both in the assumptions they choose to make as well as in their estimates of the applicability of the models they create. Too much engineering mathematics is little more than a "plug and chug" application of formulas learned by rote, with little understanding evident of the domain of applicability of the techniques being used. The seductions of extrapolating from limited data, with little to rely on other than the conviction that continuity must hold, must also be resisted; even when continuity does hold, it is dangerous to imagine that changes can happen only in a constrained3 manner; finally, simplicity does not always imply predictability - a point well illustrated by the behavior of the logistic equation.

In summary, engineers, economists and other heavy users of mathematics would do well to take a bit of mathematical rigor on board themselves, rather than imagining it as only of importance for pure mathematicians working in their ivory castles.

1 - In plain English, the real line, which consists not just of the rationals, their nth roots and the rest of the algebraic numbers, but also transcendantal numbers like e and π. (In fact, nearly all numbers on the real line are transcendental, as the algebraic numbers, being countable, have zero measure.)

2 - McKenna, P.J. 1999. "Large torsional oscillations in suspension bridges revisited: Fixing an old approximation." American Mathematical Monthly 106(January):1. See this MathTrek article by Ivars Peterson for an overview of McKenna's paper.

3 - I.e., that functions must be uniformly continuous, or, even more optimistically, Lipschitz continuous.