Logic Matters

The final(!) section of Parsons’s book is one of the briefest, and its official topic is about the biggest — the question of the justification of set-theoretic axioms. But, reasonably enough, Parsons just offers here some remarks on how the case of justifying set theory fits with his remarks in the preceding sections.

First, on “rational intuition” again. We can work ourselves into sufficient familiarity with ZFC for its axioms to come to seem intrinsically plausible — but such rational intuitions (given the questions than have been raised, by mathematicians and philosophers) “fall short of intrinsic evidence“. Which isn’t very helpful.

And what about Parsons’s modified holism? In the case of set theory, is there “a dialectical relation of axioms and their consequences such as our general discussion of Reason would suggest”? We might suppose not, given that (equivalents of) the standard axioms were already “essentially in place in Skolem’s address of 1922″. Nonetheless, Parsons suggests, we do find such a dialectical relation, historically in the reception of the axiom of choice, and perhaps now in continuing debates about large cardinal axioms, etc., where “the role of intrinsic plausibility” is much diminished, and having the right (or at least desirable) consequences are an essential part of their justification. But, Parsons concludes — the final sentence of his book — “apart from the purely mathematical difficulties, many problems of methodology and interpretation remain in this area”. Which is, to say the least, a rather disappointing note of anti-climax!

Afterword Later this week, I’ll post (a link to) a single document wrapping up all the blog-posts here into a just slightly more polished whole, and then I must cut down 30K words to a short critical notice for Analysis Reviews. I feel I’ve learnt a lot from working through (occasionally, battling with) Parsons — but in the end I suppose my verdict has to be a bit lukewarm. I’m unconvinced about his key claims on structuralism, on intuition, on the impredicativity of the notion of number, in each case in part because, after 340 pages, I’m still not really clear enough what the claims amount to.

How does arithmetic fit into the sort of picture of the role of reason and so-called “rational intuition” drawn in Secs. 52 and 53?

The bald claim that some basic principles of arithmetic are “self-evident” is, Parsons thinks, decidedly unhelpful. Rather, “in mathematical thought and practice, the axioms of arithmetic are embedded in a rather dense network … [which] serves to buttress [their] evident character … so that in that respect their evident character does not just come from their intrinsic plausibility.” Moreover, there is a subtle interplay between general principles and elementary arithmetical claims — a dialectic “between attitudes towards mathematical axioms and rules and methodological or philosophical attitudes having to do with constructivity, predicativity, feasibility, and the like”. Which, as Parsons notes, is all beginning to sound rather Quinean. How is his position distinctive?

Not by making any more play with talk of “rational intuition”, which made its temporary appearance in Sec. 53 just as a way of talking about what is intrinsically plausible: indeed, the idea that the axioms of arithmetic derive a special status from being grounded in rational intuition is said to be “in an important way misleading”. Where Parsons does depart from Quine — and it is no surprise to be told, at this stage in the book! — is in holding that some elementary arithmetic principles can be intuitively known in the Hilbertian sense he discussed in earlier chapters. And the main point he seems to want to make in this chapter is that, as we move to more sophisticated areas of arithmetic which cannot directly be so grounded, so “the conceptual or rational element in arithmetical knowledge becomes much more prominent”, the web of arithmetic isn’t thereby totally severed from intuitive knowledge grounded in intuitions of stroke strings and the like. It is still the case that “an intuitive domain witnesses the possibility of the structure of numbers”.

Of course, how impressed we are by that claim will depend on how well we think Parsons defended his conception of intuitive knowledge in earlier chapters (and I’m not going to go over that ground again now, and nor indeed does Parsons). And what grounds the parts of arithmetic that don’t get rooted in Hilbertian intuition? To be sure, those more advanced parts can get tied to other bits of mathematics, notably set theory, so there is that much rational constraint. But that just shifts the question: what grounds those theories? (There are some remarks in the next chapter, but as we’ll see they are not very unhelpful.)

So where have we got to? Parsons’s picture of arithmetic retains a role for Hilbertian intuition. And unlike an “all-in” holism, he wants to emphasize the epistemic stratification of mathematics (though his remarks on that stratification really do little more than point to the phenomenon). But still, “our view does not differ toto caelo from holism”. And I’m left really pretty unclear what, in the end, the status of the whole web of arithmetical belief is supposed to be.

Back to Parsons, to look at the final chapter of his book, called simply ‘Reason’. And after the particularly bumpy ride in the previous chapter, this one starts in a very gentle low-key way.

In Sec. 52, ‘Reason and “rational intuition”‘, Parsons rehearses some features of our practice of supporting our claims by giving reasons (occasionally, he talks of ‘features of Reason’ with a capital ‘R’: but this seems just to be Kantian verbal tic without particular significance). He mentions five. (a) Reasoning involves logical inference (and “because of their high degree of obviousness and apparently maximal generality, we do not seem to be able to give a justification of the most elementary logical principles that is not in some degree circular, in that inferences codified by logic will be used in the justification”). (b) In a given local argumentative context, “some statements … play the role of principles which are regarded as plausible (and possibly even evident) without themselves being the conclusion of arguments (or at least, their plausibility or evidence does not rest on the availability of such arguments).” (c) There is there is a drive towards systematization in our reason-giving — “manifested in a very particular way [in the case of mathematics], though the axiomatic method”. (d) Further, within a systematization, there is a to-and-fro dialectical process of reaching a reflective equilibrium, as we play off seemingly plausible local principles against more over-arching generalizing claims. (e) Relatedly, “In the end we have to decide, on the basis of the whole of our knowledge and the mutual connections of its parts whether to credit a given instance of apparent self-evidence or a given case of what appears to be perception”.

Now, that final Quinean anti-foundationalism is little more than baldly asserted. And how does Parsons want us to divide up principles of logical inference from other parts of a systematized body of knowledge? His remarks about treating the law of excluded middle “simply as an assumption of classical mathematics” suggest that he might want to restrict logic proper to some undisputed core — though he doesn’t tell us what that is. Still, quibbles apart, the drift of Parsons’s remarks here will strike most readers nowadays as unexceptionable.

Sec. 52, ‘Rational inuition and perception’, says a bit more to compare and contrast intuitions in the sense of statements found in a given context of reasoning to be intrinsically plausible — call these “rational intuitions” — and intuitions in the more Kantian sense that has occupied Parsons in earlier chapters. As he says, “intrinsic plausibility is not strongly analogous to perception [of objects]“, in the way that Kantian intuition is supposed to be. But perhaps analogies with perception remain. For one thing, there is the Gödelian view that intrinsic plausibility for some mathematical propositions involves something like perception of concepts. And there is perhaps is another analogy too, suggested by George Bealer: reason is subject to illusions that, like perceptual illusions, persist even after they have been exposed. But Parsons only briefly floats those ideas here, and the section concludes with a different thought, namely there is a kind of epistemic stratification to mathematics, with propositions at the lowest level seeming indisputably self-evident, and as we get more general and more abstract, self-evidence decreases. Which is anodyne indeed.

The final section of Ch. 8 sits rather uneasily with what’s gone before. The preceding sections are about arithmetic and ordinary arithmetic induction, while this one briskly touches on issues arising from Feferman’s work on predicative analysis, and iterating reflection into the transfinite. It also considers whether there is a sense in which a rather different (and stronger) theory given by Paul Lorenzen some fifty years ago can also be called ‘predicative’. There is a page here reminding us of something of the historical genesis of the notion of predicativity: but there is nothing, I think, in this section which helps us get any clearer about the situation with arithmetic, the main concern of the chapter. So I’ll say no more about it.