The Autonomy of Mathematical Knowledge, §3.3

Back, after a longer-than-intended break, to Curtis Franks’s The Autonomy of Mathematical Knowledge. I did complain a bit at the outset about Franks’s prose. You can certainly tell he hasn’t been through the rigours of tripos, suffering a one-to-one supervision for an hour every week with someone repeatedly asking “And what exactly do you mean by that?”, which after sixty or seventy sessions does tend to have an effect. Here, there’s just too much arm-waving loose talk for me.  And some of it isn’t even in English. For example, what are we to make of a sentence starting “If the purpose of foundations is not to vouchsafe questionable mathematics by reducing it to some privileged theory …”? I can only assume that Franks thinks that “vouchsafe” means something like “vouch for the safety of”: which it doesn’t. (As I know from my own experiences, CUP copy-editing standards these days certainly can leave something to be desired.) Sorry to bang on about this: but it is bad news when the prose gets in the way of transparently clear philosophy. Enough said.

§3.3 discusses the Foundations of Geometry. Hilbert’s originality is to show how we can use model-theoretic means to prove consistency and independence results. In particular, he shows that a certain axiomatized theory of three-dimensional geometry is consistent by constructing an intepretation in the domain Ω (the domain containing all those numbers you can get by starting from 1, and applying addition, subtraction, multiplication, division and the operation √(1 + x²) a finite number of times). Treat points as ordered pairs, lines as certain ratios, and so on. The geometric axioms then all come out true on this interpretation when spelt out. Hence, says Hilbert, “From these considerations, it follows that every contradiction resulting from our system of axioms must also appear in the arithmetic related to the domain Ω.”

The way Franks reports this is pretty odd though. For a start, in fn. 1 on p. 69, he says “Hilbert’s model Ω had a finite domain.” Not so: it’s countably infinite. Again, “Hilbert constructed models out of the positive integers” (pp. 68–69). Not so: they are constructed from a subset of the algebraic reals. And “… under this interpretation, the axioms of Euclidean geometry, the axioms of Euclidean geometry expressed arithmetical truths”. Again, not so: or at least not in the sense of “arithmetical truths” in modern philosophical writing, which means truths about the natural numbers (a reading naturally triggered by talking a couple of lines earlier about positive integers, and also suggested by drawing a contrast between this “arithmetical” consistency proof and a “proof relative to the theory of real numbers”). If he is using “arithmetic” in the broader sense that Hilbert used when he talked, old-style, of the arithmetic of Ω, Franks should have explicitly said so.

Is this last comment just captious? The trouble is that in a few pages Franks is talking about Poincaré on the consistency of arithmetic, apparently meaning  the consistency of arithmetic narrowly conceived. Franks gives no sign that there’s a change of subject here.

So presentationally, this all leaves something to be desired. But what is the point that Franks wants to make in this section? A key theme is this:

Hilbert’s choice of an arithmetical interpretation of the Euclidean axioms … has nothing to do with the epistemological status of arithmetic. … The value of his consistency proof, rather, was simply that it showed for the first time that the consistency of geometry could be proved mathematically and was therefore not dependent on grounding in Kantian intuition and the like. (p. 70, p. 72, my emphases)

I worry, though, about the “nothing” here. And I worry too about the “therefore”.

Take those worries in turn. First, if Hilbert had thought that the consistency of the arithmetic of Ω was problematic, then would his consistency proof have done what it is supposed to do, i.e. to demonstrate that “it is not possible to deduce from [his] axioms, by any logical process of reasoning, a proposition which is contradictory to any of them” (Hilbert, Foundations, p. 27.)? He clearly announces a consistency proof: and giving a relative consistency proof for geometry only counts as delivering the desired goods if the background “arithmetic” can indeed be taken to be consistent. So his choice of background theory has everything to do with its assumed security!

Second, suppose that we in fact hold that a proper confidence in “arithmetic” is in fact grounded in Kantian intuition, then the fact that we’d mathematically proved geometry consistent relative to arithmetic would not deliver us a proof that geometry is consistent (which is, to repeat, what Hilbert wants) without in addition invoking something grounded in Kantian intuition. So Franks’s “therefore” is also misjudged.

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