Year: 2009

Leiter Report

I notice that the Leiter Report pre-publication headlines about UK departments has Cambridge ranked third, after Oxford, and the Stirling/St Andrews show. A happier outcome than in the RAE — and the PGR rankings are rather a better indicator for prospective grad students, given that the RAE carves up the Cambridge philosophers into their separate institutional units, and the PGR rankings clump us together.

Not that we place too much store by such things. Oh, not at all. Perish the thought.

Can Smiley be Carnapped?

The Second Cambridge Graduate Conference on the Philosophy of Logic and Mathematics took place over the weekend. You can see what you missed here. It would be nice to drum up just a bit more support next time — for I’m sure there will be a third in the series. The most substantial talk was, perhaps unsurprisingly, by Tim Williamson, who was running through some of the arguments of his piece Barcan Formulas in Second-Order Modal Logic.

I was responding to a talk by Julien Murzi and Ole Hjortland, based on their ‘Inferentialism and the Categoricity Problem: Reply to Raatikainen‘ (which is coming out in Analysis). One part of their talk was about Timothy Smiley’s bilateralist treatment of the Carnap problem, and that’s what my comments focussed on. Here’s a slightly expanded version of my comments, defending the local hero, rewritten though to be more stand-alone.

Parsons Mathematical Thought: Sec. 51, Predicativity and inductive definitions

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.

Travel broadens the mind …

When I was editing Analysis, I went to quite a few conferences in the line of fairly pleasurable duty, to find out what the bright young things were up to, what the hot topics were. But since then I’ve become a stay-at-home, going to a few conferences here in Cambridge, but otherwise not venturing out much. Philosophical globe-trotting for the sake of it has never much appealed. So, it’s going a bit against type to have just agreed to spend a couple of months in New Zealand next year as a Visiting Erskine Fellow at the University of Canterbury. But by all accounts, the place is wonderfully welcoming to visitors, a gentle-paced sojourn in one place attracts me much more than the kind of whistle-stop tours some people delight in, and New Zealand is spectacularly beautiful. I’m beginning to look forward to it a lot.

Parsons Mathematical Thought: Sec. 50, Induction and impredicativity, continued

Suppose we help ourselves to the notion of a finite set, and say x is a number if (i) there is at least one finite set which contains x and if it contains Sy contains y, and (ii) every such finite set contains 0. This definition isn’t impredicative in the strict Russellian sense (as Alexander George points out in his ‘The imprecision of impredicativity’). Nor is it overtly impredicative in the extended sense covering the Nelson/Dummett/Parsons cases. We might argue that it is still covertly impredicative in the latter sense, if we think that elucidating the very notion of a finite set — e.g. as one for which there is a natural which counts its members — must in turn involve quantification over naturals. But is that right? This is where Feferman and Hellman enter the story. For, as Parsons remarks, they aim to offer in their theory EFSC a grounding for arithmetic in a theory of finite sets that is predicatively acceptable and that also explains the relevant idea of finiteness in a way that does not presuppose the notion of natural number. Though now things get a bit murky (and I think it would take us too far afield to pursue the discussion and further here). But Parsons’s verdict is that

EFSC admits the existence of sets that are specified by quantification over all sets, and this assumption is used in proving the existence of an N-structure [i.e. a natural number structure]. For this reason, I don’t think that … EFSC can pass muster as strictly predicative.

This seems right, if I am following. It would seem, then, Parsons would still endorse the view that no explanation of the property natural number is in sight that is not impredicative in a broad sense — where an explanation counts as impredicative in the broad sense if it is impredicative in Russell’s sense, or in the Parsons sense, or invokes concepts whose explanation is in turn impredicative in one of those senses. But the question remains: what exactly is the significance of that broad claim if I am right that even e.g. a constructivist needn’t always have a complaint about definitions which are impredicative in a non-Russellian way? It would have been good to have been told.

Back, though, to the question of induction. Dummett, to repeat, says that “the totality of natural numbers is characterised as one for which induction is valid with respect to any well-defined property” including ones whose definitions “may contain quantifiers whose variables range over the totality characterised”. Likewise Nelson. Now, as a gloss on what happens in various formalized systems of arithmetic, that is perhaps unexceptionable. But does the totality of natural numbers have to be so characterized? Return to what I called the simplest explanation of the notion of the natural numbers, which says that (i) zero is a natural number, (ii) if n is a natural number, so is Sn, and (iii) whatever is a natural number is so in virtue of clauses (i) and (ii). This explanation, Parsons argued, sustains induction for any well-defined property. But as we noted before, that argument leaves it wide open which are the well-definined properties. So it seems a further thought, going beyond what is given in the simplest explanation, to claim that any predicate involving first-order quantifications over the numbers is in fact well-defined. There are surely arithmeticians of finitist or constructivist inclinations, who fully understand the idea that the natural numbers are zero and its successors and nothing else, and understand (at least some) primitive recursive functions, but who resist the thought that we can understand predicates involving arbitrarily complex quantifications over the totality of numbers, since we are in general bereft of a way of determining in a finitistically/constructively acceptable way whether such a predicate applies to a given number. To put it in headline terms: it is a significant conceptual move to get from grasping PRA to grasping (first-order) PA — we might say that it involves moving from treating the numbers as a potential infinity to treating them as a completed infinity — and it wants an argument that someone who balks at the move has not grasped the property natural number.

How much arithmetic can we get it we do balk at the extra move and restrict induction to those predicates we have the resources to grasp in virtue of grasping what it is to be a natural number (plus at least addition and multiplication, say)? Well, arguably we can get at least as far as IΔ0, and Parsons talks a bit about this at the end of the present section. He says, incidentally, that such a theory is ‘strictly predicative’ — but I take it that this is meant in a sense consistent with saying an explanation ‘from outside’ of what the theory is supposed to be about, i.e. the natural numbers, is necessarily impredicative in the broad sense. I won’t pursue the details of the compressed discussion of IΔ0 here.

So where does all this get us? Crispin Wright has written

Ever since the concern first surfaced in the wake of the paradoxes, discussion of the issues surrounding impredicativity — when, and under what assumptions, are what specific forms of impredicative characterizations and explanations acceptable — has been signally tangled and inconclusive.

Indeed so! Given that tangled background, any discussion really ought to go more slowly and more explicitly than Parsons does. And I think we need to distinguish here grades of impredicativity in a way that Parsons doesn’t do. Agree that in the broadest sense an explanation of the natural numbers is impredicative: but this doesn’t mean that finitists or constructivists need get upset. Induction over predicates involving arbitrarily embedded quantifications over the numbers involves another grade of impredicativity, this time something the finitist or constructivist will indeed refuse to countenance. (I perhaps will return to these matters later: but for now, we must press on!)

Parsons’s Mathematical Thought: Sec. 50, Induction and impredicativity

Here’s the first half of an improved(?!?) discussion of this section: sorry about the delay!

Parsons now takes up another topic that he has written about influentially before, namely impredicativity. He describes his own earlier claim like this: “no explanation [of the predicate `is a natural number’] is in sight that is not impredicative”. That claim has been challenged by Feferman and Hellman in a couple of joint papers, and Parsons takes the present opportunity to respond. As the title of this section indicates, Parsons links claims about impredicativity to thoughts about the scope of induction: but as we’ll see, the link takes some teasing out.

What, though, does Parson mean by impredicativity? Oddly, he doesn’t come out with a straight definition of the notion. Nor does he really explain why it might matter whether definitions of the natural numbers have to be impredicative. So before tackling his discussion, we’d better pause for some preliminary clarifications and reflections.

The usual sort of account of impredicativity, in the same vein as Russell’s original (or rather, as one of Russell’s originals), runs roughly like this: ‘a definition … is impredicative if it defines an object which is one of the values of a bound variable occurring in the defining expression’, i.e. an impredicative specification of an entity is one ‘involving quantification over all entities of the same kind as itself’. (Here,the first quotation is from Fraenkel, Bar-Hillel and Levy, Foundations of Set Theory, p. 38, one of a number of very similiar Russellian definitions quoted by Alexander George in his ‘The imprecision of impredicavity; the second much more recent quotation is from John Burgess Fixing Frege, p. 40.) Thus Weyl, famously, argued against the cogency of some standard constructions in classical analysis on the grounds of their impredicativity in this sense. (And because ACA0 bans impredicative specifications of sets of numbers, it provides one possible framework for developing those portions of analysis which should be acceptable to someone with Weyl’s scruples. Now, as Parsons in effect notes, a theory like ACA0 which lacks an impredicative comprehension principle is often described as being, unqualifiedly, a predicative theory of arithmetic: but that description takes it for granted that its first-order core — usually first order Peano Arithmetic — isn’t impredicative in some other respect.)

But why should we care about avoiding impredicative definitions for Xs? Why should such definitions lack cogency? Well, suppose we think that Xss are in some sense (however tenuous) ‘constructed by us’ and not determined to exist prior to our mathematical activity. Then, very plausibly, it is illegitimate to give a recipe for constructing a particular X which requires us to take as already given a totality of Xs which includes the very one that is now being constructed. So at least any definition which is to play the role of a recipe-for-construction had better not be impredicative. Given Weyl’s constructivism about sets, then, it is no surprise that he rejects impredicative definitions of sets. I’ll not pause to assess this line of thought any further here: but I take it that it is a familiar one. (By the way, I don’t want to imply that constructivist thoughts are the only ones that might make us suspicious of impredicative definitions: though as Ramsey and Gödel pointed out, it is far from clear why a gung-ho realist should eschew impredicative definitions.)

Now, on the Russellian understanding of the idea, a definition of the set of natural numbers will count as ‘impredicative’ if it quantifies over some totality of sets including the set of natural of numbers. Modulated into property talk, we’d have: a definition of the property of being a natural number will count as impredicative if it quantifies over some totality of properties including the property of being a natural number. Some familiar definitions are indeed impredicative in this sense: take, for example, a Frege/Russell definition which says that x is a natural number iff x has all the hereditary properties of zero. Then, the quantification is over a totality which includes the property of being a natural number, and the definition is impredicative in a Russellian sense. But are all explanations we might give of what it is to be a natural number impredicative in the same way?

Take, for example, what I’ll call ‘the simplest explanation’: (i) zero is a natural number, (ii) if n is a natural number, so is Sn, and (iii) whatever is a natural number is so in virtue of clauses (i) and (ii) — and hence, almost immediately, (iv) the natural numbers are what we can do induction over. This characterization of the property of being a natural number which Parsons gives in Sec. 47 does not explicitly involve a quantification over a class of properties including that of a natural number. And though it might be claimed that an understanding of the extremal clause (iii) requires a grasp of second-order quantification, I’ve urged before that this view is contentious (and indeed the view doesn’t seem to be one that Parsons endorses — see again the discussion of his Sec. 47). So here we have, arguably, an explanation of the concept of number which isn’t impredicative in the Russellian sense. But does the quantification in (iii) make the explanation impredicative in some different, albeit closely related, sense?

Well here’s Edward Nelson, at the beginning of his book, Predicative Arithmetic. In induction we can use what he calls ‘inductive formulae’ which involve quantifiers over the numbers themselves. This, he supposes, entangles us with what he calls an ‘impredicative concept of number’:

A number is conceived to be an object satisfying every inductive formula; for a particular inductive formula, therefore, the bound variables are conceived to range over objects satisfying every inductive formula, including the one in question.

Dummett, quoted approvingly by Parsons, says much the same:

[T]he notion of `natural number’ … is impredicative. The totality of natural numbers is characterised as one for which induction is valid with respect to any well-defined property, … the impredicativity remains, since the definitions of the properties may
contain quantifiers whose variables range over the totality characterised.

So the thought seems to be that any definition of the numbers is more or less directly going to characterize them as what we can do induction over, and that `a characterization of the natural numbers
that includes induction as part of it will be impredicative’ (to quote Parsons’s gloss). But note, Dummett says that there is impredicativity here, not because the totality of natural numbers is being defined in terms of a quantification over some domain which has as a member the totality of natural numbers itself (which is what we’d expect on the Russellian definition), but because the totality is defined in terms of a quantification whose domain is (or includes) the same totality. To quote Parsons again:

Because the number concept is characterized as one for which induction holds for any well-defined predicate or property, there is impredicativity if those involving quantification over numbers are included, as they evidently are.

However, to repeat, that involves a non-Russellian notion of impredicativity. In fact it seems that Parsons would also say that an explanation of the concept P — whether or not couched as an explicit definition — is impredicative if it involves a quantification over the totality of things of which fall under P. It is perhaps in this extended sense, then, that our ‘simplest explanation’ of the property of being a natural number might be said to be impredicative.

But now note that it isn’t at all obvious why we should worry about about a property’s being impredicative if it is a non-Russellian case. Suppose, just for example, we want to be some kind of constructivist about the numbers: then how are our constructivist principles going to be offended by saying that the numbers are zero, its successors, and nothing else? Prescinding from worries about our limited capacities, the ‘simplest explanation’ of the numbers tells us, precisely, how each and every number can be ‘constructed’, at least in principle, and tells us not to worry about there being any ‘rogue cases’ which our construction rules can’t reach. What more can we sensibly want? We might add that, if we are swayed by the structuralist thought that in some sense we can only be given the natural numbers all together (whether by a general method of construction, or otherwise), then perhaps we ought to expect that any acceptable explanation of the property of being a natural number will — when properly articulated — involve us in talking of all the numbers, at least in that seemingly anodyne way that is involved in the extremal clause (iii) above.

These preliminary reflections, then, seem rather to diminish the interest of the claim that characterizations of the property natural number are inevitably impredicative, if that is meant in the in the Parsons sense. But be that as it may. Let’s next consider: is the claim actually true?

To be continued

Welcoming Aatu to the blogosphere

It is very good indeed to see that Aatu Koskensilta has started a blog. Ignore the self-deprecating ‘About Me’: as long time readers of the newsnet group sci.logic will know, Aatu is a fount of very considerable technical knowledge combined with philosophical good sense (the Nordic spirit of Torkel Franzén lives on).

Sci.logic seems to have gone into terminal decline, largely taken over by the maunderings of a few unteachable idiots (plus generous helpings of spam), and I suspect that Aatu’s wise words have been largely wasted there. So I look forward a lot to seeing Aatu put his considerable energies into the new blog, which ought to reach a different and much more discerning audience.

Sitting in the UL tearoom …

Actually, I fib: I’m not in the University Library tearoom right now — I meant to post from there, but I got talking to one of our grad students (about Parsons on impredicativity, what else?), and the opportunity passed. But I’d been musing a bit earlier, sitting in the book stacks, on why — with one of the greatest lending libraries in the English-speaking world more or less on my doorstep — I should still be so tempted to buy philosophy books. Of course it is good having a “working library” at home: but that only excuses buying books you know you need, rather than buying in a much more speculative way. There are no doubt some deep irrational motivations at work: but despite myself, I keep being tempted. Here’s what I’ve just bought, on the basis of browsing in Heffers:

  1. Penelope Maddy, Second Philosophy (OUP, 2007). Well, a student who’d in the past done some work on earlier Maddy wanted to write another paper on naturalism — so I suggested we both read this.
  2. C. S. Jenkins, Grounding Concepts: An Empirical Bass for Arithmetical Knowledge (OUP, 2008). Carrie gently chided me for not mentioning her book in my ‘What have missed?‘ post. Knowing a bit about ‘where she is coming from’ I’m not sure I’m going be persuaded; but this is written with entirely admirable directedness and clarity, so it will be a pleasure to read.
  3. E. J. Lowe and A. Rami (eds) Truth and Truth-making (Acumen, 2009). Well, I’ll read this just because I should get more on top of this stuff.
  4. W. V. Quine, Confessions of a Confirmed Extensionalist and Other Essays, ed D. Follesdal and D. B. Quine (Harvard U.P., 2008). I guess in lots of ways my instincts are still quite Quinean, such was his influence when I was a lad! So a must-have, for old times sake. And not that expensive for such a beautifully produced book.
  5. Alexander Bird, Nature’s Metaphysics: Laws and Properties (OUP, 2007). Like the Maddy book, another rather belated bit of catching up.
  6. Jonathan Barnes, Truth, etc.: Six Lectures on Ancient Logic (OUP, 2007). This was the serendipitous find from browsing. Just looks a fascinating and a fun read.

So now I simply need the time to read them …. I keep telling myself that I don’t really believe in too much of this philosophy stuff: but then I get drawn back in!

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