Sunday, January 25, 2009

Mathematica materialis, or How not to be lured into Plato's cave

The great David Hilbert's* birthday (b. January 23, 1862) was this past Friday. Here are two Hilbert quotes that people still talk about:

1. No one shall expel us from the paradise that Cantor has created for us.

2. Mathematics is a game played according to certain simple rules with meaningless marks on paper.

Now the first is sourced, while the second is disputed. But I'll pretend he "said" them both.

These two statements, I think, sum up the two ways people (mathematicians?—are they people?) view the subject of mathematics: Platonism (mathematics is the exploration and discovery of truths in a world—beyond our feeble, material world—where there is Truth) and Formalism (mathematics is really just a game of symbol manipulation with established, agreed upon rules—like chess). Now this is a conflation of the bewildering zoo of philosophies of mathematics (everyone has their favorite animal), but I think it basically comes down to these two views. But when you get down to it, these are not necessarily opposing views: a Formalist can be (and probably is) a Platonist, and a Platonist can still do their mathematical work as if they were a Formalist. So Hilbert could have made both these quotes.

Some history: Cantor came along and "saw" not only an infinite number of "natural" numbers (1, 2, 3, ...) but that some infinities (all the real numbers, 3.14159..., etc.) are more infinite than others. And even that infinite one-upmanship goes on infinitely. Nice one, Cantor. (One the other hand, Hilbert got his comeuppance from Gödel on the first "quote": Hilbert's trial balloon [the so-called Hilbert's Program] that all mathematical "truth" could be found in a formal proof system was popped.)

However, thank Jan [Mycielski: Locally Finite Theories, Analysis Without Actual Infinity], there is a third view apart from Platonism and Formalism [The Meaning of Pure Mathematics]: Intentionalism (mathematics is "a description of finite structures consisting of finitely many individually imagined objects"). And, thanks to Shaughan [Lavine: Understanding the Infinite], there is a layman's book for laying out what Mycielski is up to.

Boiled down, this is what Intentionalism is: a re-reading of mathematical texts. And it all comes down to quantifiers.

In a mathematical text, one might read an axiom that basically says: "Between every two distinct rational numbers there is another rational number". In a Platonist reading, one can only "think" of rational numbers being infinitely divisible, somehow. And real numbers? There are "even more" of those! But in a "deconstructionist" Intententionalist view, this same axiom is read: "For-all[set-p] two distinct rational numbers x and y, there-exists[set-q] another rational number z between them". The quantifiers (for-all[set-p], there-exists[set-q]) are qualified: they range over finite sets that are of arbitrary size, but the qualification is that the more deeply nested a qualifier is in a statement, the larger (i.e., set-q is a superset of set-p) the set of its range. (There are more "rules" in play in Lavine/Mycileski, but I'll leave it at that.)

Since mathematical text consists of (first-order) statements of nested qualifiers, an Intentionalist** reading exorcizes the ghosts of infinity. And Platonists are left behind in Plato's cave, boxing at shadows.

* David Hilbert's legacy can probably be summed up in three bullets
  • Axiomization of Euclid's Geometry (its descendants found in high school geometry texts)
  • Hilbert Spaces (a mathematical basis for quantum theory)
  • Hilbert's Program (see above)

** I think the term Intentionalism is fine, but, with some liberty perhaps, I use the term (mathematical) Materialism to mean the same thing. If it is the case that nature is 'quantal', then there are no real numbers in the first place!