The Autonomy of Mathematical Knowledge, §§2.1-2.2

Hilbert in the 1920s seems pretty confident that classical analysis is in good order. “Mathematicians have pursued to the uttermost the modes of inference that rest on the concept of sets of numbers, and not even the shadow of an inconsistency has appeared …. [D]espite the application of the boldest and most manifold combinations of the subtlest techiniques, a complete security of inference and a clear unanimity of results reigns in analysis.” (p. 41 — as before, references are to passages or quotations in Franks’ book.) These don’t sound like the words of a man who thinks that the paradoxes cause trouble for ‘ordinary’ mathematics itself — compare Weyl’s talk of the “inner instability of the foundations on which the empire is constructed” (p. 38). And they don’t sound like the words of someone who thinks that analysis either has to be revised (as an intuitionist or a predicativist supposes) or else stands in need of buttressing “from outside” (as the authors of Principia might suppose).

Franks urges that we take Hilbert at his word here: in fact, “the question inspiring [Hilbert] to foundational research is not whether mathematics is consistent, but rather whether or not mathematics can stand on its own — no more in need of philosophically loaded defense than endangered by philosophically loaded skepticism” (p. 31). So, on Franks’ reading, Hilbert in some sense wants to be an anti-foundationalist, not another player in the foundations game standing alongside Russell, Brouwer and Weyl, with a rival foundationalist programme of his own. “[Hilbert’s] considered philosophical position is that the validity of mathematical methods is immune to all philosophical skepticism, and therefore not even up for debate on such grounds” (p. 36). Our mathematical practice doesn’t need grounding on a priori principles external to mathematics (p. 38). Thus, according to Franks, Hilbert has a “naturalistic epistemology. The security of a way of knowing is born out, not in its responsibility to first principles, but in its successful functioning” (p. 40).

Functioning in what sense, however? About this, Franks is (at least here in his Ch. 2) hazy, to say the least. “The successful functioning of a science … is determined by a variety of factors — freedom from contradiction is but one of them — including ease of use, range of application, elegance, and amount of information (or systematization of the world) thereby attainable. For Hilbert mathematics is the most completely secure of our sciences because of its unmatched success.” Well, ease of use and elegance are nice if you can get them, but hardly in themselves signs of success for theories in general (there are just too many successful but ugly theories, and too many elegant failures). So that seemingly leaves (successful) application as the key to the “success”. But this is very puzzling. Hilbert, after all, wants us never to be driven out of Cantor’s paradise where — as Franks himself stresses in Ch. 1 — “mathematics is entirely free in its development”, meaning free because longer tethered to practical application. Odd then now to stress application as what essentially legitimises the free play of the mathematical imagination! (Could the idea be that some analysis proves its worth in application, and hence the worth of the mathematical methods by which we pursue it, and then other bits of mathematics pursued using the same methods get reflected glory? But someone who takes that line could hardly e.g. be as quickly dismissive of the predicative programme as Hilbert was or Franks seems to be at this point — for Weyl, recall, is arguing that actually applicable analysis can in fact all be done predicatively, and so no reflected glory will accrue to classical mathematics pursued with impredicative methods since those methods are not validated by essentially featuring in applicable maths.)

So what does Hilbert’s alleged “naturalism” amount to? To be continued.

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