As I’ve said before, I’m planning to post over the coming weeks some thoughts on the essays (re)published in Charles Parsons’s Philosophy of Mathematics in the Twentieth Century (Harvard UP). I’m going to be reviewing the collection of Mind, and promising to comment here is a good way of making myself read through the book reasonably carefully. Whether this is actually going to be a rewarding exercise — for me as writer and/or you as reader — is as yet an open question: here’s hoping!
This kind of book must be a dying form. More and more, we put pre-prints, or at least late versions, of published papers on our websites (and with the drive to make research open-access, this will surely quickly become the almost universal norm). So in the future, what will be added by printing the readily-available papers together in book form? Perhaps, as in the present collection, the author can add a few afterthoughts in the form of postscripts, and write a preface drawing together some themes. But publishers will probably, and very reasonably, think that that’s very rarely going to be enough to make it worth printing the selected papers. Still, I am glad to have this collection. For Parsons doesn’t have pre-prints on his web page, the original essays were published in very widely scattered places, and a number of them are unfamiliar to me so it is good to have the spur to (re)read them. And I should add the volume is rather beautifully produced. So let’s make a start. For reasons I’ll explain in the next set of comments, I’ll begin with the second essay in the book, on predicativity.
Famously, Parsons and Feferman have disagreed about whether there is a sense in which there is an element of impredicativity even in arithmetic. Thus Parsons has argued for “The Impredicativity of Induction” in his well-known 1992 paper. And Feferman (writing with Hellman) explores what he calls “Predicative Foundations of Arithmetic”, in a couple of equally well-known papers, the second of which is in fact in the festschrift for Parsons published in 200o, edited by Sher and Tiezsen. Now the second essay in the present collection is in turn reprinted from Parsons’s contribution to the Feferman festschrift published a couple of years later, edited by Sieg and others. He again returns to issues about predicativity; but perhaps rather regrettably he doesn’t continue the substantive debate with Feferman but instead offers an historical piece “Realism and the Debate on Impredicativity, 1917-1944.” What can we get out of this?
As a preliminary warm-up, let’s remind ourselves of a familiar line about predicativism (the kind of thing we tell — or at least, which I used to tell — students by way of a first introduction). We start by noting the Russellian term of art: a definition is said to be impredicative if it defines an entity E by means of a quantication over a domain of entities which includes E itself. Frege’s definition of the property natural number is, for example, plainly impredicative in this sense. But so too it seems are more workaday mathematical definitions, as e.g. when we define a supremum by quantifying over some objects including that very one.
Now: Poincaré, and Russell following him, famously thought that impredicative definitions are as bad as more straightforwardly circular definitions. Such definitions, they suppose, offend against a principle banning viciously circular definitions. But are they right? Or are impredicative denitions harmless?
Well, Ramsey (and Gödel after him) famously noted that some impredicative definitions are surely quite unproblematic. Ramsey’s example: picking out someone, by a Russellian definite description, as the tallest man in the room is picking him out by means of a quantification over the people in the room who include that very man, the tallest man. And where on earth is the harm in that? And the definition of a supremum, say, seems to be exactly on a par.
Surely, there no lurking problem at all in the case of the tallest man. In this case, the men in the room are there anyway, independently of our picking any one of them out. So what’s to stop us identifying one of them by appealing to his special status in the plurality of them? There is nothing logically or ontologically weird going on. Likewise, if we think that, say, the real numbers are there anyway, picking out one of them by its special status as the supremum of a set of numbers is surely again harmless.
It is similar — to continue the familiar story — for other contexts where we take a realist stance, at least to the extent of supposing that reality already in some sense supplies us with a fixed totality of the entities to quantify over. If the entities in question are (as I put it before) ‘there anyway’, what harm can there be in picking out one of them by using a description that quantifies over some domain which includes that very thing?
Things are surely otherwise, however, if we are dealing with some domain with respect to which we take a less realist attitude. For example, there’s a line of thought which runs through Poincaré, through the French analysts (especially Borel, Baire, and Lebesgue), and is particularly developed by Weyl in his Das Kontinuum: the thought, at its most radical, is that mathematics should concern itself only with objects which can be defined. As the constructivist mathematician Errett Bishop later puts it
A set [for example] is not an entity which has an ideal existence. A set exists only when it has been defined.
On this line of thought, defining e.g. a set (giving a predicate for which it is the extension) is — so to speak — defining it into existence. And from this point of view, impredicative definitions involving set quantification can indeed be problematic. For the definitist thought suggests a hierarchical picture. We define some things; we can then define more things in terms of those; and then define more things in terms of those; and we can keep on going on (though how far?). But what we can’t do is define something into existence by impredicatively invoking a whole domain of things already including the very thing we are trying to define. That indeed would be going round in a vicious circle. [Strictly speaking, that’s not a reason to ban impredicative definitions entirely: but we will have to restrict ourselves to using such definitions to pick out in a new way something from among things that we have already harmlessly defined predicatively at an earlier stage.]
So much then for at least part of a familiar story (part of the conventional wisdom?). On the one side, the idea is that worries about impredicativity were/are generated by a constructivist/definitist view of some mathematical domain (and are indeed quite reasonable on such a view); on the other side, we have some view which takes the things in the relevant domain to be suitably ‘there anyway’ and so can insist that it is harmless to pin down one them by means of a quantification over all of them including that one.
Now, enter Parsons at the beginning of his paper:
There is a conventional wisdom, to which I myself have subscribed in some published remarks, that the defense of impredicativity in classical mathematics rests on a realist or platonist conception. Such a view is fostered by Gödel’s famous discussion of Russell’s vicious circle principle. Still, I want to argue that this conventional wisdom is to some degree oversimplified, both as a story about the history and as a substantive view. I don’t think it even entirely does justice to Gödel.
On reflection, at some level this got to be partly right. If the attack on impredicativity is generated by constructivist/definitist views, then the defence just needs to resist going down a constructivist or definitist road. And there is clear water between (A) resisting some form of constructivism strong enough to make predicativism compelling, and (B) defending a view that is realist or platonist in some interestingly strong sense.
That’s because one way of doing (A) without doing (B) would be just to resist the whole old-school game of looking for extra-mathematical, ‘foundational’, ideas against which mathematical practice needs to be judged. Mathematicians should just go about their business, without worrying whether it is warranted — from the outside, so to speak — by this or that conception of the enterprise. Just lay down clear axioms — e.g. for the reals, as it might be — and adopt a clear deductive framework and that’s enough: doing this “is logically completely free of objections, and it only remains undecided …whether the axioms don’t perhaps lead to contradiction”. To be sure, a (relative) consistency proof would be nice if we can get it, but that’s just mathematical cross-checking: we don’t need external validation by some philosophically motivated constructivist standards or by realist standards either. Thus, of course, the modern “naturalist” about mathematics. But we can perhaps read Hilbert, from whom the quote comes, in such a broadly naturalistic spirit — and Bernays too, who gave a very early lecture commenting on Weyl (which isn’t to say that there aren’t more ingredients to Hilbert’s evolving position at the time). So yes, as Parsons nicely explains, from the very outset one line of defence against predicativist attack was (not to substitute a realist for a constructivist philosophical underpinning of mathematics but) in effect to resist the pressure to play a certain foundationalist game. The paradoxes call us to do mathematics better, more rigorously, not to get bogged down in panicky revisionism. Or so a story goes.
However, Weyl in his predicativist phase, or other philosophically motivated mathematical revisionists like the intuitionists (including Weyl in a later phase) will presumably complain that this riposte is thumpingly point-missing. And again, those like Feferman who don’t go the whole hog, but still find considerable significance in the project of seeing how far we can get using predicative theories because they minimize ontological and proof-theoretic baggage, will presumably also want to resist a Hilbertian refusal (if that’s what it is) to engage with any reflections about the conceptual motivations of various proof-procedures. And given Parsons’s own philosophical temperament (as evinced e.g. by his well-known concern with the question of how far you can get one the basis of something like Kantian ‘intuition’), you would have thought that here at any rate he would side with Feferman.
So yes, in remarking that the Hilbertians opposed Weyl without being platonists, Parsons has a real point against the familiar story that sets up too easy an binary opposition between predicativists and realists. But does he want to occupy the further ground thus marked out and be a refusenik about the role of a certain kind of conceptual reflection in justifying proof procedures? Well, see the first comment from Daniel Nagase below, and my note in reply!
Let’s turn now briefly to Parsons’s remarks on Gödel. Gödel undoubtedly wanted to do (A) and he did, famously, come to endorse (B), taking a strongly platonist stance — or so it seems, though what this really comes too is difficult to get straight about.
Now, Parsons urges that we need to distinguish a platonism about objects (the objects of the relevant domain are ‘there anyway’ as I put it, or ‘independent of definitions and constructions’ as Parsons puts it) and a platonism about truth (truth concerning the domain in question is ‘independent of our knowledge, perhaps even of our possibilities of knowledge’ as Parsons puts it). Gödel at first blush accepted both strands in platonism, in some form. Parsons then remarks that only the first strand seems to be involved in the rejection of predicativism, which Gödel takes to be rooted in the opposing idea that the entities involved are “constructed by ourselves”. But I’m not sure how exciting or novel it is to point that out, at least if this isn’t accompanied with rather more discussion about what platonism about a domain of objects really comes to, once supposedly distinguished from platonism about truth. (I say “supposedly” because it isn’t so clear on further reflection that we can elucidate what it is for e.g. numbers to be ‘there anyway’ except via the claim that certain truths that purport to refer to and quantify over numbers are true, where their truth is sufficiently independent of knowledge).
I found Parsons’s discussion here, which is quite brief, a bit unclear. But then, to put it baldly, the realist idea of objects being ‘there anyway’ remains pretty opaque, and Parsons really doesn’t help us out much in this essay (fair enough, it is a relatively modest length historically focused piece). In fact, speaking for myself, rather than appeal to such an idea in order to try to ground accepting predicative definitions over various domains, I’d be tempted to put it the other way about. I’d rather say: accepting the legitimacy of impredicative definitions over a domain constitutes one kind of realism about that domain. Understood that way, ‘realism’ at least has a tolerably clear shape. But then, thus understood, realism can hardly be a ground for accepting impredicativity, as the conventional wisdom would have it. So then, where do we go for arguments?