Monday, January 26, 2009

Parsons's Mathematical Thought: Sec 55, Set theory

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.

MacBook Air, six months on

Anyone out there who is still wavering about getting a MacBook Air might be interested in some comments from a delighted owner who has now had one for six months (this updates my "after one month" post from last August). Everyone else can, of course, just cheerfully ignore this again!

As background, I make heavy academic use of a computer (particularly using LaTeX, and reading a lot of papers, even books, onscreen, as well of course as the usual surfing, emailing etc.) but don't really use one as a media centre except for light iPhoto use, and occasionally ripping CDs into iTunes for transfer to an iPod.

  1. The portability is fantastic. No question. Just to compare: I've had in the past a 15" Titanium PowerBook, a 15" G4 PowerBook, and a 17" MacBook Pro before; and they've of course been portable in the sense I could heave them from home to my office and back. But all of them were honestly just too heavy/bulky to make that particularly convenient. I very rarely bothered to take them elsewhere, e.g. to a coffee shop, or even a library. (You might well ask why on earth, in that case, I had portables at all! Answer: Partly because our Cambridge house is very small, my "study" is the size of a large cupboard, and I very much like to be able to work on the kitchen table for a change of scene, or answer emails with a computer on my knees in the living room in the evening. So I certainly want "local" portability. And partly I need to be able to drive data projectors when lecturing.) Anyway, by contrast with the earlier portables, I can and do cheerfully tote the MBA (in its snug protective sleeve) anywhere, without really thinking about it, whether or not I'm definitely planning to use it. It just is so light and convenient. Much lighter than the new aluminium 13" regular MacBook which I've tried out in the Apple Store too. Yet the MBA always feels remarkably sturdy. There's not a sign really of six months of constant use.
  2. Some early reviews complained about the MBA's footprint, saying that it isn't a genuine ultraportable. People still complain about that. Well, true, I can imagine e.g. it wouldn't be that easy to use in the cramped conditions of an airline seat. But that sort of issue just doesn't arise for me. It's not the footprint but the lightness and thinness which means that you can carry it so very comfortably in one hand, and of course the larger-than-ultra footprint goes with the stunningly good, uncramped, screen and the generous keyboard. In my kind of usage now, I've never found the footprint an issue.
  3. I rarely use the MBA to do anything very processor-intensive for a prolonged period of time, and it normally remains cool -- though the fans can sometimes kick in a bit enthusiastically e.g. when backing up. And the battery life seems just fine: well over three hours for writing, text-browsing, reading. Recharging though is pretty slow: but if you need to take it with you, then the MBA's charger is very small and portable (though I've bought a second one for my office in the faculty, and so don't find in practice I need to carry it around).
  4. One main reason I traded up a couple of years ago from the 15" G4 machine was that LaTeX ran pretty slowly: nearly 30 seconds to typeset my Gödel book on the G4, about 4 seconds on the new intel MacBook Pro. The MBA, despite its slower chip, seems almost as fast running LaTeX , and indeed in most other ways: occasionally, e.g. when opening an application, the MBA is noticeably slower -- but it has never been a particularly irritating issue. So this is plenty fast enough.
  5. And the reason, when I traded up the previous time, I chose the 17" MBP model was to have enough "real estate" to have a TeXShop editing window and the PDF output window side-by-side and comfortably readable. Obviously, I'm now looking at 1280 x 800 pixels, rather than 1680 x 1050 (so that's just 58% as much). But I've surprised myself by getting very used to working with overlapping windows again, and the screen quality is really terrific. The best I've ever had by far. Of course it is nicer e.g. for extended on-screen reading to plug in an external monitor as well. But not any sort of necessity -- and indeed I seem these days pretty often not to bother even if I'm sitting next to the external monitor.
  6. What about the paucity of ports, mentioned critically by all the reviewers, or the absence of an onboard CD drive? Really not an issue. I've a couple of times wished there were two USB ports, I bought a little one-to-two-port splitter, for very occasional home use, but in fact even when I don't have it with me, I've never been seriously annoyed. Of course, if I had one of the new version MBA's I'd be tempted with one of the new displays that also acts as a USB hub: but that would be an indulgence. I've latterly bought an external CD/DVD drive built for the MBA, for when I occasionally need one. (The one caveat concerns day one, long before I got the external drive. Since there is no firewire port, you can't migrate files from your old computer to your new MBA using the usual firewire connection. And using a wireless connection to migrate is both painfully slow and seems flaky. Is that a problem? I didn't find really it so. I installed new versions of necessary additional software, like the LaTeX installation, over the web, and then copied my documents folder and other bits and pieces from a SuperDuper! clone of the old hard disk on an external drive. Quick to do, and resulting in a clean and tidy MBA.)
  7. So that's all very, very positive. Are there any negatives? The flat keyboard is surprisingly nice to use (much better than I imagined it would be). But, unlike the almost silent similarly flat new iMac keyboards, I find the MBA version to does seem a bit noisier (and a bit more so than the MBP keyboard). But that's a very marginal disappointment.
  8. I thought, when I bought the MBA a month ago, I'd be using it very much as a second machine, carrying on using the 17" MPB (with external monitor) as a main, quasi-desktop, set-up. In fact I find I now almost never use the MPB.
  9. So, assuming a three year life cycle (and it seems very well built so should last longer with a battery refresh after a while), the MBA after education discount costs much less than half a pint of beer a day. Put like that, how can you resist?
  10. And then, of course, there is the "Wow!"-factor ...

Sunday, January 25, 2009

Parsons's Mathematical Objects: Sec. 54, Arithmetic

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.

Saturday, January 24, 2009

Parsons's Mathematical Objects: Secs 52-53, Reason, "rational intuition" and perception

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.

Wednesday, January 21, 2009

Wittgenstein's Notes on Logic

Just to say that my colleague Michael Potter's intriguing new book Wittgenstein's Notes on Logic is published tomorrow.

Tuesday, January 20, 2009

Nerdy stuff

Just for fellow Macaholics ...

  1. I've just noticed that a new version of TeXShop has been released in the last couple of weeks, with a couple of useful little tweaks.
  2. I had a pre-release trial copy of Things for a while, and now the first proper release is out. Very neat and very simple: so it is, for once, "task management" software -- ok, a fancy way of keeping To Do lists -- that I actually do use.
  3. Oh, and I've just got an Iomega eGo Helium external drive. Very small and no power block, so easy to tote, and no fan so very quiet. It's a bit sad to get even mildly pleased by a hard drive. But still, it is rather pretty ...

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.

Monday, January 19, 2009

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.

Tuesday, January 13, 2009

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!)

Sunday, January 11, 2009

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

Monday, January 05, 2009

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.

Saturday, January 03, 2009

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!