The Autonomy of Mathematical Knowledge, §§2.3-2.5

To return for a moment the question we left hanging: what is the shape of Hilbert’s “naturalism” according to Franks? Well, Franks in §2.3 thinks that Hilbert’s position can be contrasted with a “Wittgensteinian” naturalism that forecloses global questions of the justification of a framework by rejecting them as meaningless. “According to Hilbert … mathematics is justified in application” (p. 44), and for him “the skeptic’s path leads to the death of all science”. Really? But, to repeat, if that is someone’s basic stance, then you’d expect him to very much want to know which mathematics is actually needed in applications, and to be challenged by Weyl’s work towards showing that a “sceptical” line on impredicative constructions in fact doesn’t lead to the death of applicable maths. Yet Hilbert seems not to show much interest in that.

At other points, however, Franks makes Hilbert’s basic philosophical thought sound less than a claim about security-through-successful-applicability and more like the Moorean point that the philosophical arguments for e.g. a skepticism about excluded middle or about impredicative constructions will always be much less secure than our tried-and-tested methods inside mathematics. But in that case, we might wonder, if the working mathematician can dismiss such skepticism, why engage in “Hilbert’s program” and look for consistency proofs?

Franks’ headline answer is “The consistency proof … is a methodological tool designed to get everyone, unambiguously, to see [that mathematical methods are in good order].” (p. 36). The idea is this. Regimenting an area of mathematics by formalisation keeps us honest (moves have to be justified by reference to explicit axioms and rules of inference, not by more intuitive but risky moves apparently warranted by intended meanings). And then we can aim to use other parts of mathematics that aren’t under suspicion — meaning, open to mathematical doubts about their probity — to check the consistency of our formalized systems. Given that formalized proofs are finite objects, and that finitistic reasoning about finite objects is agreed on all sides to be beyond suspicion, the hope would be to give, in particular, finitistic consistency proofs of mathematical theories. And thus, working inside mathematics, we mathematically convince ourselves that our theories are in good order — and hence we won’t be seduced into thinking that our theories need bolstering from outside by being given supposedly firmer “foundations”.

In sum, we might put it this way: a consistency proof — rather than being part of a foundationalist project — is supposed to be a tool to convince mathematicians by mathematical means that they don’t need to engage in such a project. Franks gives a very nice quotation from Bernays in 1922: “The great advantage of Hilbert’s procedure rests precisely on the fact that the problems and difficulties that present themselves in the grounding of mathematics are transformed from the epistemological-philosophical domain into the domain of what is properly mathematical.”

Well, is Franks construing Hilbert right here? You might momentarily think there must be a deep disagreement between Franks with his anti-foundationalist reading and (say) Richard Zach, who talks of “Hilbert’s … project for the foundation of mathematics”. But that would be superficial. Compare: those who call Wittgenstein an anti-philosopher are not disagreeing with those who rate him as a great philosopher! — they are rather saying something about the kind of philosopher he is. Likewise, Franks is emphasizing the kind of reflective project on the business of mathematics that Hilbert thought the appropriate response to the “crisis in foundations”. And the outline story he tells is, I think, plausible as far as it goes.

It isn’t the whole story, of course. But fair enough, we’re only in Ch.2 of Franks’ book! — and in any case I doubt that there is a whole story to be told that gives Hilbert a stably worked out position. It would, however, have been good to hear something about how the nineteenth century concerns about the nature and use of ideal elements in mathematics played through into Hilbert’s thinking. And I do want to hear more about the relation between consistency and conservativeness which has as yet hardly been mentioned. But still, I did find Franks’ emphases in giving his preliminary orientation on Hilbert’s mindset helpful. To be continued

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