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?

Two quick observations: (i) one thing that caught my attention was that, in the postscript, Parsons mentions that even intuitionistic mathematics is, in a sense, impredicative; hence, it’s clear that the predicative/impredicative debate cuts across the traditional realist/anti-realist debate. (ii) In the same postscript, Parsons seems to concede McCarty’s point that what justifies one to allow impredicative definitions just is the enormous success of impredicative mathematics. He frames the debate as not being about whether impredicative definitions should be allowed, but rather to what extent we can do without them (which, as you noted, is essentially Feferman’s position). In other words, if one frames the question as you did, it seems that the justification for realism would be mainly pragmatic (or perhaps conservative), i.e. that’s what it takes to develop mathematics as we know it.

Yes indeed! Many thanks for this — I’d run out of steam, and was going to return briefly to the postscript in a further shorter post, and now I don’t need to as this sums things up beautifully. Though I’d like to say just a bit more about (i) and (ii) when I have a moment … watch this space!

To continue: Re (i) it is perhaps worth mentioning that it is a bit quick to say that “even inituitionistic mathematics is … impredicative”. Thus, Gödel in 1933 takes it that intuitionism should

reject“notions introduced by impredicatived definitions”. And later we get both predicative and impredicative versions of intuitionistic analysis developed. So a question is going to be: what broadly anti-realist conceptual motivation for intuitionism (qua refusal to accept excluded middle) is there which doesn’t also give us a reason to eschew impredicativity? I guess more needs to be said about that before we can be confident that “the predicative/impredicative debate cuts across the … realist/anti-realist debate”.Re (ii) if Parsons does end up happy with a merely pragmatic defence, with the appeal to the “enormous success of impredicative mathematics” — as indeed his Postscript suggests — I wonder how well this chimes with his other views (about arithmetic, about set theory, etc.) where he sees — doesn’t he? — a role for conceptual defences of some of our practices (and a cause for at least some initial concern when our practices outrun what we can say by way of conceptual motivation). And I’m not sure that going pragmatic puts him quite in the same camp as Feferman either, whose interest in predicativist versions of analysis is driven by an avowed anti-platonist philosophical position.

I’m in a bit of a hurry, so I’ll be quick.

Re (i): Well, this may be related to Parsons views about the natural number series, which, as you note, he doesn’t develop in the article (though there are some interesting remarks on the matter in his

Mathematical Thought and Its Objects). If one considers the natural number series as being impredicative, I can’t see a way out for the intuitionist not to admit impredicative definitions. Perhaps here an appeal tointuitionwould be called for.Re(ii): I don’t see how Parsons’s position here differs from Feferman; he too seems to be in the anti-platonist camp (I may be mistaken, though; most of what I’ve read from Parsons is his historical papers, so I may be missing something); and Feferman himself, as far as I remember, is also driven to recognize that there are important parts of mathematics which are impredicative (I may be misremembering his preface to

In Light of Logic, though). In any case, that’s the position I got from his Postscript. Perhaps he’s more explicit about this in the relevant sections ofMathematical Thought and Its Objects.Re (i), I really must one day get down to writing a piece on why I think Parsons is wrong about the “impredicativity of induction”. (Sure it takes a ‘new thought’ to get us from say PRA to arithmetic with full induction — but I’d argue that it is a different new thought than the one involved in standard cases of impredicativity. But that’s a story for another day!) But yes, true, IF you think that there’s impredicativity in first-order PA, then there’s equal impredicativity in first-order HA. And yes, the issue is then whether “intuition” or something else that is anti-realistically acceptable is enough to defend that degree of impredicativity, or whether (as indeed some have worried) standard intuitionism overshoots.

(ii) Yes, Parsons is in some sort of anti-platonist camp (or some readings of what counts as platonism). I wasn’t doubting that, but wondering about the hook-up between (a possible tension between?) his more substantive critical philosophical views and a willingness (if that’s what it is) to quietistically say “Hey, impredicative maths is a successful game, so that’s fine!”.

But thanks again for the comments!

May I suggest that it is difficult to come to grips with this topic and Parson’s treatment without discussion of the finite|infinite distinction which concerns Parsons both in this chapter and the preceding first chapter of the book?

It seems to me that the idea of defining entities into existence is, if anything, more problematic (and opaque) than the idea that they are “there anyway”; and if, as Bishop puts it, “a set exists only when it has been defined”, what sort of existence does it have, if not an ideal one? Or if we construct the entities, what are we constructing?

I supposed it could be argued that such talk of defining and constructing is merely a “so to speak” and not to be taken seriously. We “define” or “construct” a set, for instance, but that doesn’t mean the set then exists. But then it’s not clear what the objection to impredicative definitions amounts to. It can’t be that they “define something into existence by impredicatively invoking a whole domain of things already including the very thing we are trying to define”, because we’re no longer seeing definitions as defining anything into existence.

Anyway, since (as Parsons points out) the paper is “almost entirely historical”, shouldn’t it be evaluated primarily as history? Then the main questions would be ones such as “is he correct about what was said and thought?” and “does he offer new insights or help us understand the people it discusses?” Instead, the discussion has so far been primarily about what Parsons’s own views might be and how much sense they make, and what we should think now about impredicativity.

I guess I agree about the obscurity of talk of “defining entities into existence”, though I think I’d still want to say that the constructivist metaphor isn’t empty. But yes, it is a very good question what a careful, slow-motion version of the constructivist line of thought comes to.

Parsons paper is historical, but not

merely: as he puts it, “I want to argue that the conventional wisdom [about realism as a response to concerns about impredicativity] is to some degree oversimplified, both as a story about the history and as a substantive view”. He takes — doesn’t he? — the history to illuminate the substantial issues which are, I confess, more my concern.