Year: 2009

Hilbert writes

Just as the physicist investigates his apparatus … the mathematician has to secure his theorems by a critique of this proofs, and for this he needs proof theory. (p. 61)

Indeed: if you are a physicist getting surprising results (or worse) thrown up by your apparatus, then you will take a hard look at the apparatus — but that’s a matter of doing more physics, of course, and doesn’t involve going outside the methods of physics.  If you are mathematician getting surprising results (or worse, contradictions!) then you’ll similarly want to take a hard look at your purported proofs and the assumptions you’ve been using/smuggling in — and that’s a matter of doing more mathematics. In particular, formalization and “proof theory” is a mathematical tool to help the mathematician put his own house in order, removing any need to search for philosophical “foundations” to ensure consistency. Or at least, that’s the kind of line that we’ve seen Franks attributing to Hilbert.

Sometimes, however, Franks seems to claim more for the project of a Hilbertian proof theory. For example, he comments on the quote  above that

Hilbert wants to demonstrate that the reliability [of mathematical techniques] is self-witnessing.

Reliability for what? Mathematical truth? Self-witnessing in what sense? Franks doesn’t say.

I won’t dwell on this, but to see quickly that there is an issue here, suppose you are the kind of set theorist who thinks that mathematics aims to reveal the truth about a unique universe of pure sets which is there for us to explore. For such a theorist, proving consistency of our theory is not enough. We believe, for example, that ZFC + V = L is consistent, but our set theorist typically doesn’t think that that makes it the correct theory of the universe of sets. But then what are the techniques for theory-choice here? And why are they reliable? Proof theory is plainly not much help in answering that question for our set-theorist or for witnessing that his methods are in good order. I’m not saying that the question isn’t answerable from  a naturalist perspective, in a broad sense of “naturalism”– cf. Maddy’s discussions of such questions: I’m just noting that taking a Hilbertian line on the significance of consistency proofs — as explained by Franks — falls well short of completing the job.

That said, the first main point of Franks’s third chapter, which is called ‘Arithmetization’, is to claim that Gödel’s “technique for the arithmetization of syntax is perhaps the most significant positive contribution to Hilbert’s program” (p. 64). Which certainly, at first sight, seems a pretty extravagant claim.

Actually, as an aside, it is not the only strange claim in §3.2. Franks writes

Gödel’s two incompleteness theorems follow without much work from an application of the fixed point theorem of Rudolf Carnap [1934] and the traditional analysis of paradoxical sentences like the “liar sentence”. Thus the substantial analytical work preceded him. The innovation in Gödel’s paper was the technique … of the arithmetization of syntax … and is what allowed Gödel to apply the fixed point theorem and the analysis of paradoxical statements to the case of provability. (p. 67)

This is hopeless. (1) The fixed point theorem wasn’t proved by Carnap until after he learnt of Gödelian incompleteness — so that “substantial analytical work” did not in fact precede Gödel. (2) The fixed point theorem for a suitable theory T is of the form that for every open sentence φ(x) there is a sentence γ such that T proves γ ↔ φ(“γ”), where “γ” is the Gödel-number, in a given scheme, for γ. So the fixed point theorem cannot precede the arithmetization of syntax: it requires arithmetization for its very statement. (3) And what on earth is “the traditional analysis of paradoxical sentences like the ‘liar sentence'” and how is that supposed to have been applied in Gödel’s incompletness proof?

Back, though, to the extravagant claim that arithmetization is the most — yes, most — significant positive contribution to Hilbert’s program (so much for Gentzen, then, and work in proof-theory downstream from him; so much for what many regard as the direct descendent of Hilbert’s program in the project of reverse mathematics). What can Franks mean?

Well, the thought is that, even if Gödel’s idea of arithmetization did lead to the incompleteness theorems that sabotaged Hilbert’s more ambitious hopes for proof theory, still this was only possible because

the same discovery vindicated Hilbert’s principal philosophical conviction: that by being cast within mathematics itself important questions about mathematics could be investigated without favouring any philosophical tendencies over others. (p. 66)

But I don’t find that particularly convincing as a defence of the extravagant claim. After all, Hilbert already had emphasized the key point that when we go metatheoretical, and move from thinking about sets (for example) to thinking about formalized-theories-about-sets, we move from considering infinite sets to considering suites of finite formal objects (wffs, and finite sequences of wffs) — and we might then hope to bring to bear, at the metatheoretical level, merely finitary formal reasoning about these suites of finite formal objects in order to prove consistency, etc. It’s the formalization that gives us a new, finitary, subject matter apt for formal investigation — apt, in other words, for mathematical treatment. The additional fact that we can arithmetically code up finite objects by mapping them to numbers means we might be able to bring some more, already familiar, mathematics to bear on these objects (just as when we co-ordinatize a space we can bring the arithmetic of the reals to bear on geometrical objects). But just as we don’t need co-ordinatization to vindicate treating geometrical objects mathematically, we don’t need arithmetization to vindicate bringing formal methods to bear on the formal objects given us when we formalize a theory and go metatheoretical.

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

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.