Wednesday, December 31, 2008

Parsons's Mathematical Thought: Sec. 49, Uniqueness and communication, continued

In sum, then, we might put things like this. Parsons has defended an 'internalist' argument -- an argument from "within mathematics'' -- for the uniqueness of the numbers we are talking about in our arithmetic, whilst arguing against the need for (or perhaps indeed, the possibility of) an 'externalist' justification for our intuition of uniqueness.

Can we rest content with that? Some philosophers would say we can get more -- and Parsons briefly discusses two, Hartry Field and Shaughan Lavine, though he gives fairly short shrift to both. Field has argued that we can appeal to a 'cosmological hypothesis' together with an assumption of the determinateness of our physical vocabulary to rule out non-standard models of our applicable arithmetic. Parsons reasonably enough worries: "If our powers of mathematical concept formation are not sufficient [to rule out nonstandard models], then why should our powers of physical concept formation do any better?'' Lavine supposes that our arithmetic can be regimented as a "full schematic theory'' which is in fact stronger than the sort of theory with open-ended induction that we've been considering, and for which a categoricity theorem can be proved. But Parsons finds some difficulty in locating a clear conception of exactly what counts as a full schematic theory -- a difficulty on which, indeed, I've commented elsewhere on this blog.

In both cases, I think Parsons's points are well taken: but his discussions of Field and Lavine are brief, and more probably needs to be said (though not here).

Monday, December 29, 2008

Parsons's Mathematical Thought: Sec. 49, Uniqueness and communication

Parsons now takes another pass at the question whether the natural numbers form a unique structure. And this time, he offers something like the broadly Wittgensteinian line which we mooted above as a riposte to skeptical worries -- though I'm not sure that I have grasped all the twists and turns of Parsons's intricate discussion.

We'll start by following Parsons in considering the following scenario. Michael uses a first-order language for arithmetic with primitives 0, S, N, and Kurt uses a similar language with primitives 0', S', N'. Each accepts the basic Peano axioms, and each also stands ready to accept any instances of the first-order induction schema for predicates formulable in his respective language (or in an extension of that language which he can come to understand). And we now ask: how could Michael determine that his 'numbers' are isomorphic to Kurt's?

We'll assume that Michael is a charitable interpreter, and so he thinks that what Kurt says about his numbers is in fact true. And we can imagine that Michael recursively defines a function f from his numbers to Kurt's in the obvious way, putting f(0) = 0', and f(Sn) = S'f(n) (of course, to do this, Michael has to add Kurt's vocabulary to his own, while shelving detailed questions of interpretation -- but suppose that's been done). Then trivially, each f(n) is an N' by Kurt's explicit principles which Michael is charitably adopting. And Michael can also show that f is one-one using his own induction principle.

In sum, then, Michael can show that f is an injection from the Ns into the N's, whatever exactly the latter are. But, at least prescinding from the considerations in the previous section, that so far leaves it open whether -- from Michael's point of view -- Kurt's numbers are non-standard (i.e. it doesn't settle for Michael whether there are also Kurt-numbers which aren't f-images of Michael-numbers). How could Michael rule that out? Well, he could show that f is onto, and hence prove it a bijection, if he could borrow Kurt's induction principle -- which he is charitably assuming is sound in Kurt's use -- applied to the predicate ∃m(Nm & fm = ξ). But now, asks Parsons, what entitles Michael to suppose that that is indeed one of the predicates Kurt stands prepared to apply induction to? Why presume, for a start, that Kurt can get to understand Michael's predicate N so as to bring it under the induction principle?

It would seem that, so long as Michael regards Kurt 'from the outside', trying to 'radically interpret' him as if an alien, then he has no obvious good reason to presume that. But on the other hand, that's just not a natural way to regard a fellow human being. The natural presumption is that Kurt could learn to use N as Michael does, and so -- since grasping meaning is grasping use -- could come to understand that predicate, and likewise grasp Michael's f, and hence come to understand the predicate ∃m(Nm & fm = ξ). Hence, taking for granted Kurt's common humanity and his willingingness to extend the use of induction to new predicates, Michael can then complete the argument that his and Kurt's numbers are isomorphic. Parsons puts it like this. If Michael just takes Kurt as a fellow speaker who can come to share a language, then

We now have a situation that was lacking when we viewed Michael's understanding of Kurt as a case of radical interpretation; namely, he will take his own number predicate as a well-defined predicate according to Kurt, and so he will allow himself to use it in induction on Kurt's numbers. That will enable him to complete the proof that his own numbers are isomorphic to Kurt's.
And note, the availability of the proof here ''does not depend on any global agreement between them as to what counts as a well-defined predicate'', nor on Michael's deploying a background set theory.

So far, then, so good. But how far does this take us? You might say: if Michael and Kurt in effect can come to belong to the same speech community, then indeed they might then reasonably take each other to be talking of the same numbers (up to isomorphism) -- but that doesn't settle whether what they share is a grasp of a standard model. But again, that is to look at them together 'from the outside', as aliens. If we converse with them as fellow humans, presume that they stand ready to use induction on our predicates which they can learn, then we can use the same argument as Michael to argue that they share our conception of the numbers. You might riposte that this still leaves it open whether we've all grasped a nonstandard model. But that is surely confused: as Dummett for one has stressed, in order to formulate the very idea of models of arithmetic -- whether standard or nonstandard -- we must already be making use of our notion of 'natural number' (or notions that swim in the same conceptual orbit like 'finite', or stronger notions like 'set'). To cast put that notion into doubt is to saw off the branch we are sitting on in describing the models. Or as Parsons says, commenting on Dummett,
[I]n the end, we have to come down to mathematical language as used, and this cannot be made to depend on semantic reflection on that same language. We can see that two purported number sequences are isomorphic without strong set-theoretic premisses, but we cannot in the end get away from the fact that the result obtained is one ''within mathematics" (in Wittgenstein's phrase). We can avoid the dogmatic view about the uniqueness of the natural numbers by showing that the principles of arithmetic lead to the Uniqueness Thesis ...
So, there is indeed basic agreement here with the Wittgensteinian observation that in the end there has to be understanding without further interpretation. But Parsons continues,
... but this does not protect the language of arithmetic from an interpretation completely from the outside, that takes quantifiers over numbers as ranging over a non-standard model. One might imagine a God who constructs such an interpretation, and with whom dialogue is impossible, and with whom dialogue is impossible. But so far the interpretation is, in the Kantian phrase, ''nothing to us". If we came to understand it (which would be an essential extension of our own linguistic resources) we would recognize it as unintended, as we would have formulated a predicate for which, on the interpretation, induction fails.
Well, yes and no. True, if we come to understand someone as interpreting us as thinking of the natural numbers as outstripping zero and its successors, then we would indeed recognize him as getting us wrong -- for we could then formulate a predicate 'is-zero-or-one-of-its-successors' for which induction would have to fail (according to the interpretation), contrary to our open-ended commitment to induction. And further dialogue will reveal the mistake to the interpreter who gets us wrong. However, contra Parsons, we surely don't have to pretend to be able to make any sense of the idea of a God who constructs such an interpretation and 'with whom dialogue is impossible': Davidson and Dummett, for example, would both surely reject that idea.

But where exactly does all this leave us on the uniqueness question? To be continued ...

Saturday, December 27, 2008

Advances in education

Stuff the "Season of Goodwill". The only decent reaction to this kind of thing remains anger: "Female education is against Islamic teachings".

Wednesday, December 24, 2008

RAE 2008 again

Discussions of the RAE 2008 results for philosophy rumble on inconclusively.

One thing I'd be rather interested to know is how much the need to make a show in RAE returns (and so get promotion) constrains -- an even distorts -- the intellectual life of younger philosophers. Here's a scenario. Dr A writes a superb PhD thesis on topic X, gets a junior research fellowship, and turns the thesis after a few more years into a very impressive book on X, getting a permanent job at a good department on the basis of it. Understandably, after seven years intensive work, Dr A now wants to move on to thinking about something else. "Ah, no ..." say the newly appointing department, "your growing reputation is as a star thinker about X, so for the next RAE we do really need you to keep writing another few papers about that, because those papers are bound to be very well ranked. If you start working in another area, you might well not have publications as good in the needed time-frame." And so, not entirely happily, Dr A knuckles down to grinding out the needed papers ...

Just how frequently does this sort of scenario occur, I wonder? (This isn't a fanciful question, for I do have reason to suspect that this sort of thing happens.) The ever-increasing professionalization and specialization of philosophers does seem to be deeply at odds with that kind of wide-ranging intellectual curiosity, that liking for making connections and seeing "how things in the broadest possible sense of the term hang together in the broadest possible sense of the term", which gets people into philosophy in the first place.

But enough already. It is time to stop for a couple of days. Cambridge's winter speciality is grey dank days which are bony-chillingly damp without ever quite getting round to raining. But today has been bright and clear, and the evening sky is now streaked pink. After days when town has been swarming with distracted votaries of the gods of consumerism, it was back to a pleasantly human level of bustle. So a few presents are bought, the pheasants are in the fridge, the Barolo and Brunello under the stairs ...

Happy Christmas!

Tuesday, December 23, 2008

Another great day for obscurantism and stupidity

Well, we expect no other from the crazed geriatric Pope. But we might have hoped for better from science teachers than from some dingbat Muslim loony.

Meanwhile, in Saudi Arabia ...

Monday, December 22, 2008

Parsons's Mathematical Thought: Sec. 48, The problem of the uniqueness of the number structure: Nonstandard models

''There is a strongly held intuition that the natural numbers are a unique structure.'' Parsons now begins to discuss whether this intuition -- using 'intuition', of course, in the common-or-garden non-Kantian sense! -- is warranted. He sets aside until the long Sec. 49 issues arising from arguments of Dummett's: here he makes some initial points on the uniqueness question, arising from the consideration of nonstandard models of arithmetic.

It's worth commenting first, however, on a certain 'disconnect' between the previous section and this one. For recall, Parsons has just been discussing how we might introduce a predicate 'N' ('... is a natural number') governed by the rules (i) N0, and (ii) from Nx infer N(Sx), plus the extremal clause (iii) that nothing is a number that can't be shown to be so by rules (i) and (ii). Together with the rules for the successor function, the extremal clause -- interpreted as intended -- ensures that the numbers will be unique up to isomorphism. Conversely, our naive intuition that the numbers form a unique structure is surely most naturally sustained by appeal to that very clause. The thought is that any structure for interpreting arithmetic as informally understood must take numbers to comprise a zero element, its successors (all different, by the successor rules), and nothing else. And of course the numbers in each structure will then have a natural isomorphism between them (which matches zeros with zeros, and n-th successors with n-th successors). So the obvious issue to take up at this point is: what does it take to grasp the intended content of the extremal clause? Prescinding from general worries about rule-following, is that any special problem about understanding that clause which might suggest that, after all, different arithmeticians who deploy that clause could still be talking of different, non-isomorphic, structures? However, obvious though these questions are given what has gone before, Parsons doesn't raise them.

Given the ready availability of the informal argument just sketched, why should we doubt uniqueness? Ah, the skeptical response will go, regiment arithmetic however we like, there can still be rival interpretations (thanks to the Löwenheim/Skolem theorem). Even if we dress up the uniqueness argument -- by putting our arithmetic into a set-theoretic setting and giving a formal treatment of the content of the extremal clause, and then running a full-dress version of the informal Dedekind categoricity theorem -- that still can't be used settle the uniqueness question. For the requisite background set theory itself, presented in the usual first-order way, can itself have nonstandard models: and we can construct cases where the unique-up-to-isomorphism structure formed by 'the natural numbers' inside such a nonstandard model won't be isomorphic to the 'real' natural numbers. And going second-order doesn't help either: we can still have non-isomorphic ''general models'' of second-order theories, and the question still arises how we are to exclude {those}. In sum, the skeptical line runs, someone who starts off with worries about the uniqueness of the natural-number structure because of the possibilities of non-standard models of arithmetic, won't be mollified by an argument that presupposes uniqueness elsewhere, e.g. in our background set theory.

Now, that skeptical line of thought will, of course, be met with equally familiar responses (familiar, that is, from discussions of the philosophical significance of the existence of nonstandard models as assured us by the Löwenheim/Skolem theorem). For example, it will be countered that things go wrong at the outset. We can't keep squinting sideways at our own language -- the language in which we do arithmetic, express extremal clauses, and do informal set theory -- and then pretend that more and more of it might be open to different interpretations. At some point, as Wittgenstein insisted, there has to be understanding without further interpretation (and at that point, assuming we are still able to do informal arithmetical reasoning at all, we'll be able to run the informal argument for the uniqueness of the numbers).

How does Parsons stand with respect to this sort of dialectic? He outlines the skeptical take on the Dedekind argument at some length, explaining how to parlay a certain kind of nonstandard model of set theory into a nonstandard model of arithmetic. And his response isn't the very general one just mooted but rather he claims that the way the construction works ''witnesses the fact the model is nonstandard" -- and he means, in effect, that our grasp of the constructed model which provides a deviant interpretation of arithmetic piggy-backs on a prior grasp of the standard interpretation -- so the idea that we might have deviantly cottoned on to the nonstandard model from the outset is undermined. Yet a bit later he says he is not going to attempt to directly answer skeptical arguments based on the L-S theorem. And he finishes the section by saying the theorem ''seems still to cast doubt on whether we have really 'captured' the 'standard' model of arithmetic''. So I'm left puzzled.

Parsons does, however, touch on one interesting general point along the way, noting the difference between those cases where we get deviant interpretations that we can understand but which piggy-back on a prior understanding of the theory in question, and those cases where we know there are alternative models because of the countable elementary submodel version of the L-S theorem. Since the existence of such submodels is given to us by the axiom of choice, these resulting interpretations are, in a sense, unsurveyable by us, so -- for a different reason -- are also not available as alternative interpretations we might have cottoned on to from the outset. The point is worth further exploration which it doesn't receive here.

Thursday, December 18, 2008

RAE 2008

Well, the RAE results for UK philosophy departments are out (here's the Guardian's summary page: the two Cambridge entries are for HPS, ranked higher, and for the smaller Philosophy Faculty ranked at equal 12th).1 The results for us, and our relative placing in the scheme of things, were I think slightly disappointing but broadly predictable. What with one thing and another, the timing happened not to be great; and we'd perhaps rather too much disdained the game-playing.

Brian Leiter asks one of the right questions, though: What do the rankings actually mean for a student choosing graduate programs? After all, a department full of monomaniacal, autistic, world-class researchers would get a great score but give students a horrible time!

One of our grads put it this way this afternoon: "I'd much rather you guys continue running two or three good graduate seminars and reading groups in my area week in, week out, rather than sitting in your offices with the doors shut trying to improve your research ratings." Which gets to the nub of what matters as far as sensible choices for graduate study are concerned.

1. In the unlikely event of there being anyone interested enough to read this who doesn't know what the figures mean, the basic story is that each "research active" member of the department submitted four pieces of work for assessment, which were separately graded as 4* (the best), 3*, etc. Then this research output profile is combined with two other profiles for "research environment" and "esteem" to give the rounded profiles of the departments in each category. So the published figures at this stage aren't exactly transparent in their significance. More details are published later.

Wednesday, December 17, 2008

Parsons's Mathematical Thought: Sec. 47, Induction and the concept of natural number (continued)

To continue. Parsons now takes up three more issues about his self-styled “justification” of induction.

1. His first question is: “What is the range of the first-order variables?” over which we can apply the rules which ground his “justification” of induction? Some domain of entities, presumably, that can be given to us prior to our specifying its “numbers”, i.e. the zero and its successors. “However,” says Parsons, “this is . . . to assume that some infinite structure is given to us independently of our knowledge of the kind of structure the natural numbers instantiate.”

But I’m not sure why Parsons says this. Take any domain which contains a zero element 0 and for which a function S is defined. Then, whether the function S is injective or otherwise, whether the domain is finite or infinite, we’ll be able to similarly define N -- meaning ‘is 0 or one of its successors’ -- and the induction rule will hold for the Ns. We need, of course, further rules governing S to ensure that the Ns form an infinite progression: but Parsons’s “justification of induction” seems to work equally well whether they do and whether they don’t. If he thinks that there is something special about the infinite case, then he doesn’t bring the point out clearly here.

2. Second, Parsons comments on “the schematic character of the induction rule. . . . the applicability of the rule is not limited to predicates defined in some particular first-order language such as that of first-order arithmetic. But we must not take it as implying the unavoidability or even the legitimacy of full second-order logic.” The target here, I suppose, is Kreisel’s well known contrary claim that we accept instances of a schematic form of the induction rule because we already accept the full second-order induction axiom -- though Parsons doesn’t mention Kreisel here. I take it that the argument is that the reasoning that led us to accept the induction rule was silent on the particular character of the filling for φ – that, it seems, was left entirely open ended (the permitted fillings will be whatever we can make sense of, as wide or as narrow a class as that is): but silence doesn’t mean agreeing to the coherence of the full second-order notion of quantifying over arbitrary properties, where these are conceived of as being in effect arbitrary subsets of the domain of the first-order variables (when that domain is infinite). I agree.

3. “A third question is whether and in what sense is induction an analytic or conceptual rule or truth.” Parsons’s line is that “The explanation of the number concept by rules makes induction follow from an explanation of that concept: it is certainly in some sense ‘conceptual’.” But then what of someone who does not accept induction across the board -- say, a finitist who doesn’t countenance Σ1 induction? Is he then guilty of failing to acknowledge a conceptual truth? No, says Parsons, and surely rightly. We should take the finitist objection to be not to the schematic rule but rather to the admission of certain [say, Σ1] predicates as fully kosher.

Monday, December 15, 2008

Parsons's Mathematical Thought: Sec. 47, Induction and the concept of natural number

Why does the principle of mathematical induction hold for the natural numbers? Well, arguably, “induction falls out of an explanation of the meaning of the term ‘natural number’”.

How so? Well, the thought can of course be developed along Frege’s lines, by simply defining the natural numbers to be those objects which have all the properties of zero which are hereditary with respect to the successor function. But it seems that we don’t need to appeal to impredicative second-order reasoning in this way. Instead, and more simply, we can develop the idea as follows.

Put ‘N’ for ‘. . . is a natural number’. Then we have the obvious ‘introduction’ rules, (i) N0, and (ii) from Nx infer N(Sx), together with the extremal clause (iii) that nothing is a number that can’t be shown to be so by rules (i) and (ii).

Now suppose that for some predicate φ we are given both φ(0) and φ(x) → φ(Sx). Then plainly, by repeated instances of modus ponens, φ is true of 0, S0, SS0, SSS0, . . .. Hence, by the extremal clause (iii), φ is true of all the natural numbers. So it is immediate that the induction principle holds for φ – e.g. in the form of this elimination rule for N:
Thus far, then, Parsons.

So: two initial issues about this, one of which Parsons himself touches on, the other of which he seems to ignore.

First, as an argument warranting induction doesn’t this go round in a circle? For doesn’t the observation that each and every instance φ(SS . . . S0) is derivable given φ(0) and φ(x) → φ(Sx) itself depend on an induction? Parsons says that, yes, “As a proof of induction, this is circular. . . . Nonetheless, . . . it is no worse than arguments for the validity of elementary logical rules.” This of course doesn’t count against the claim that “induction falls out of an explanation of the meaning of the term ‘natural number’” – it is just that the “falling out” is so immediate that we can’t count as fully grasping the idea of a natural number while not finding inductive arguments primitively compelling (in something like Peacocke’s sense). I’m minded to agree with Parsons here.

But, second, some will complain that Parsons’s preferred way of seeing induction as given to us in the very notion of ‘natural number’ is actually not significantly different from Frege’s way, because the extremal clause (iii) is essentially second order. It will be said: the idea in (iii) is that something is a natural number if belongs to all sets which contain 0 and are closed under applications of the successor function – which is just Frege’s second-order definition put in set terms. Now, Parsons doesn’t address this familiar line of thought. However, I in fact agree with his implicit assumption that his preferred line of thought does not presuppose second-order ideas. In headline terms, just because the notion of transitive closure can be defined defined in second-order terms, that doesn’t make it a second-order notion (compare: we can define identity in second-order terms, but that surely doesn’t make identity a second-order notion!). And it is arguable that the child who picks up the notion of an ancestor doesn’t thereby exhibit a grasp of second-order quantification. But more really needs to be said about this (for a little more, see my Introduction to Gödel’s Theorems, §23.5).

To be continued

Friday, December 12, 2008

Parsons's Mathematical Thought: Secs. 40-45, Intuitive arithmetic and its limits

Here, as promised, are some comments on Chapter 7 of Parsons's book. They are quite lengthy, and since in writing them I found myself going back to revise/improve some of my discussions of earlier sections, I'm just posting a single composite version of all my comments on the first seven chapters. I'm afraid that is already over 20K words and 36 single-spaced pages (start at p.31 for the substantially new stuff). So I am sounding off at some length: but it seems to me that the topics tackled in Mathematical Thought are so very central as to be well worth extended discussion.

I've still two more chapters to go: next up is a fifty page chapter on induction, which I think can be discussed fairly independently from what's gone before. So I'll revert to section-by-section blogging here.

Thursday, December 11, 2008

What have I missed?

It's around now that the reviews pages and the literary supplements carry lists of their Books of the Year. So what really worthwhile books on logic matters were first published in 2008? Alan Sokal's Beyond the Hoax is amusing in parts but added less than I'd hoped to Intellectual Impostures; John P. Burgess's collected papers are worth having together in Mathematics, Models and Modality (but content-wise, that doesn't really count as a new book); Graham Priest's second edition of Introduction to Non-classical Logic is a great textbook but isn't exactly full of news for old logic hands! So ok, what can I particularly commend that pushes ideas on in a novel and thought-provoking way?

Ermmm .... This is really rather embarrasing. I can't think of anything to suggest! The books that I have read and most enjoyed recently seem all to have been published in previous years.

Ah, Hartry Field's Saving Truth from Paradox was published this year, but it is still sitting on my shelves waiting to be read. But what have I missed? Perhaps I'm forgetting or have just not registered the publication of some terrific books over the last year in the areas of philosophy of science/philosophy of logic/philosophy of maths. What would you recommend from the class of '08?

Friday, December 05, 2008

Meanwhile, reading Giaquinto

I'm still working away on Parsons's book -- and I'm rather stumped by his claims about "intuitive knowledge". One worry is this: he introduces the notion in cases where we acquire propositional knowledge by, as it were, "just seeing" e.g. that "|||" is the successor of "||". But he fairly rapidly wants to extend the notion of intuitive knowledge so that it is preserved under some, but not all, logical inference, and some but not all applications of arithmetical induction. And I just can't see what the constraint on the notion is that rules some cases in and others out -- for that constraint certainly isn't implicit in the cases which introduce the notion in the first place.

Well, be all that as it may. For light relief -- and to see if any sideways light can be thrown on Parsons's on intuition more generally -- I'm reading Marcus Giaquinto's Visual Thinking in Mathematics (OUP, 2007). The first few chapters already show that, unsurprisingly, the book is written with Marcus's customary clarity and good sense. I'll report back in due course as I read more: but already I'm having to backtrack a bit and slightly rethink things that I earlier wrote on Parsons.

Wednesday, December 03, 2008

Darwin Day!

There has just been a national petition started, supporting a proposal to make Charles Darwin's birthday (12th February) a UK Bank Holiday. Well yes, let's celebrate the great man -- and just possibly send a signal marking some opposition to the noisy fringe of know-nothing, anti-science, religious loonies.

Why not (1) spend just a moment to sign the petition at (if you are a British citizen or resident)? (2) Warmly encourage/nag a couple of friends to sign too. And then (3) spread the word to other groups/e-lists that you belong to -- e.g. by distributing the text of this post.

(For here, perhaps, is a meme worth propagating!)

Monday, December 01, 2008

Fraser MacBride moving to Cambridge

Delighted to report that Fraser MacBride has accepted the offer of a job in the Cambridge philosophy faculty. A terrific outcome for our recent appointment process. And a real strengthening of philosophy of maths as well as metaphysics here (as the post replaces an ancient philosopher).

Without giving anything away, knowing the shortlist, things are looking good for the Knightbridge chair replacement too.

The way things are developing, at this rate I won't want to retire ...

Richter's Schubert

Ok, ok -- I know that, by now, it is a such a banal thing to say: but I'm still frequently bowled over by the quality of information that there is out there on the net, freely available, the result of the labours of amateur enthusiasts. I wanted to check whether a recording of Richter playing D960 duplicated one I already had. Within three minutes I had the answer: one Paul Geffen maintains a Richter discography, and here's the Schubert page. Terrific.

Sunday, November 30, 2008

Parsons again

There's now a version of my posts on the first five chapters of Parsons book: so the newly added pages are on Chapter 5 of his book, on "Intuition". I found these sections unconvincing (when I didn't find them baffling) -- a reaction that seemed to be shared by other members of the reading group here which is working through the book. So again, all comments and suggestions will be very gratefully received!

Thursday, November 27, 2008

Back to Parsons

Well, "blogging at a snail's pace'' is all well and good, but my posts about Parsons have recently ground to a complete halt. Sorry about that. Pressure of other things. But I'm back on the case, now with the pressure of a deadline, and so here is a significantly expanded/improved version of my posts on the first four chapters of his Mathematical Thought and Its Objects. I'll post on the next three chapters over the coming week. And then comment on the last two chapters the following week.

All comments will be very gratefully received as I'm going to be mining these long ruminations for a critical notice of the book.

Wednesday, November 26, 2008

Reading Russell's Introduction to Mathematical Logic

Gregory Landini is talking at our Logic Seminar next week about Frege and Russell on cardinal numbers. Since our students tend to know a lot more about Frege than Russell, we had a preparatory session on Russell last week, in which I got a chance to show off my stunning historical ignorance. But it was fun to re-read (after a long time) the opening chapters of the Introduction to Mathematical Philosophy. These were, as much as anything, the pages that got me interested in philosophy and the foundations of mathematics when I was a maths student.

Fun to re-read, but also oddly very disappointing. Chapter II starts with stirring words which I well remembered: 'The question "What is a number?" is one which has been often asked, but has only been correctly answered in our own time' (meaning, of course, in 1884 in the Grundlagen). But I'd quite forgotten this passage, later in the same chapter, where Russell writes

We naturally think that the class of all couples is different from the number 2. But there is no doubt about the class of all couples: it is indubitable and not difficult to define, whereas the number 2, in any other sense, is a metaphysical entity about which we can never feel sure that it exists or that we have tracked it down. It is therefore more prudent to content ourselves with the class of couples, which we are sure of, than to hunt for a problematic number 2 which must always remain elusive.
So Russell's stirring words are misleading: he isn't after all claiming to have located, thanks to Frege, the one true metaphysical story about numbers (as classes of classes). It's rather that here we have one way of replacing a problematic entity with something clear and sharply defined that can do the job. And then, of course, Russell is cheating. There isn't such a thing as the class of all couples. Far from there being no doubt about it, he doubts it himself: his official story has a type hierarchy, with classes of couples at each level of the hierarchy above the bottom two. The seductive clarity of the opening chapters IMP is sadly only superficial!

Wednesday, November 12, 2008

Why you should sometimes ask your local logicians ...

What are we to make of this passage?

Among the triumphs of set theory are Gödel's Incompleteness Theorems and Paul Cohen's proof of the independence of the Continuum Hypothesis. Gödel's theorems in particular had a dramatic effect on philosophical perceptions of mathematics, though now that it is understood that not every mathematical statement has a proof or disproof most mathematicians carry on much as before, since most statements they encounter do tend to be decidable. However, set theorists are a different breed. Since Gödel and Cohen, many further statements have been show to be undecidable, and many new axioms have been proposed that would make them decidable.
Well, we might complain that this is at least three ways misleading:
  1. Gödel's Incompleteness Theorems are a triumph, but not a triumph of set theory.
  2. Gödel's Incompleteness Theorems do not show that "not every mathematical statement has a proof or disproof".
  3. Cohen's and Gödel's results are significantly different in type and shouldn't be so swiftly bracketed together. While the Cohen proof leaves it open that we might yet find some new axiom for set theory which settles the Continuum Hypothesis (and other interesting propositions which can similarly be shown to be independent of ZFC), Gödel's First Theorem -- perhaps better called an Incompletability Theorem -- tells us that adding new axioms won't ever give as a negation-complete theory (unless we give up on recursive axiomatizability of our theory).
I'm slightly embarrassed to report, then, that the rather dodgy passage above is by a local Cambridge hero, Timothy Gowers, writing in the Princeton Companion (p. 6). Oops. I rather suspect he didn't run this bit past his local friendly logicians ...

Wednesday, November 05, 2008

Yes we can ... can't we?

In 1997 -- before I got the job back in Cambridge -- I was travelling up and down to lecture: and I happened to be here the morning after the Blair victory. And, a bit bleary-eyed from a late night in front of the election results show, I bumped into a friend in town on a lovely morning and we sat outside a café drinking coffee and thinking how much better the world seemed. We knew Blair was an unprincipled opportunist, but he was -- so to speak -- our unprincipled opportunist, and there were, we hoped, enough decent people around him to keep things on track. That was, it turned out, wildly over-optimistic.

It is easy to lose heart. And perhaps (I wish I could say "unbelievably") 56 million Americans did just vote to try to put an egregious cartoon character within a heartbeart of the presidency. But, at least for today, let's be a bit optimistic again.

Sunday, November 02, 2008

Logic keeps you sane

Non-philosophical events have really been rather stressful over the last six weeks or so. While I'm keeping my head above water as far as teaching and seminars are concerned, lots of other things -- like sounding off here! -- have been pushed right into the background. And I'm going to pressed for time this coming week too. For I'm down to respond to Luca at a Faculty Colloquium this Friday: we're talking about Graham Priest's animadversions against the iterative conception of sets, and I need to write a talk. How on earth this will go down with a general audience of non-logicians, heaven alone knows. But then, the whole idea of a general colloquium seems to me mildly daft. Still, the topic is a good one to think about.

Indeed, logic matters have been a lot of fun this last week, and a very welcome distraction to keep me sane. Two of our MPhil students gave very nice and very helpful presentations at the two seminars -- one on Chap. 3 of Parsons's book in the reading group on that, and one on Big Typescript §113, the section on "Ramsey's Theory of Identity". None of us ended up very much clearer about what Parsons's noneliminative structuralism comes to exactly: but I'll return to that here shortly. Lectures are going pretty enjoyably too (for me, at least!). I'm probably going too slowly talking about Gödel's theorems to cover everything I really should, but at least people seem to still be on board. And I unthinkingly produced a frisson of mildly scandalized reaction in the first year logic course by saying en passant that, while Wittgenstein might have been a great philosopher -- he'd just come up in the context of talking about the idea of a tautology -- he seems to have been a bit of a shit as a human being. Still, we do need to encourage some healthy disrespect!

Thursday, October 30, 2008

Congratulations to Luca Incurvati

Champagne in the graduate room this afternoon (and more afterwards during the logic seminar) to celebrate Luca's election to a three-year Junior Research Fellowship at Magdalene from next October. Great news.

Friday, October 24, 2008

The Gowers' Companion/Davenport's Higher Arithmetic

Tim Gowers's Princeton Companion to Mathematics -- which has been available from Amazon USA for a discounted price for a little while -- is now available at a 25% discount from Amazon UK. Hooray! I've sent off for it and will no doubt be commenting here.

Meanwhile, to get in practice for reading a stack of mathematics (outside the logician's usual diet, for once), I'm devouring Davenport's The Higher Arithmetic which has just appeared in the C.U.P. bookshop in its new eighth edition. I first read chunks of this a long time ago as a schoolboy: and it's old-style mode of presentation -- mercifully without Bourbachiste over-formalization -- is a delight. I'd of course entirely forgotten most of this stuff: some of it is very pretty!


We've now had three seminars on Wittgenstein's remarks on the foundations of mathematics in the Big Typescript, taking things very slowly. The first week, I talked about Sec. 108 (the contrast between arithmetic and a game). One of our final year undergrads gave an admirable presentation in the second week on Secs 109-11. This week, we battled with Secs 112 and 114 (leaving the discussion on Ramsey and identity in Sec. 113 till next week).

Now, a few pages ago, in Sec. 108, it seemed that it is the use or application of arithmetic that is supposed to distinguish it as mathematics from a mere game. What kind of applicability is in question? "It is mathematics, I should think, when it is used for the transition from one proposition to another." (Sec. 108, p. 372e). So there, at any rate, Wittgenstein offers the beginnings of a story about applicability. But now, in Sec. 112, we have some distinctly odd remarks about applications. For example, "Arithmetic is its own application." (p. 382e, repeated p. 385). What does that mean? I think it's fair to report that we were left baffled.

To be sure, we presumably do want a story about the difference between using an empirical theory to take us from one proposition to another and using arithmetic. And Wittgenstein in effect remarks that if we use arithmetic and get empirically the wrong answer, we don't blame arithmetic. "It might look as though the mathematical computation entitled us to make a prediction [e.g. about how many apples each a group of people will have, if you divide the pile of twelve apples between four]. But that isn't so. What justifies us in making this prediction is a hypothesis of physics, which lies outside the calculation. The calculation is only an examination of logical forms, of structures, and of itself can't yield anything new." [p. 383e]. But just what does that last sentence mean? And let's suppose for the sake of argument that, as part of our overall practice, we have the rule that we do not revise arithmetical propositions in the light of empirical results. That doesn't make it any less the case that arithmetic is being applied to apples or whatever, or make it appropriate to say instead that "arithmetic is its own application".

Any pointers to helpful discussions in the literature that makes sense of what is going on here will be very gratefully received!

Tuesday, October 21, 2008

Peanuts on intuition ...

Thanks to David Auerbach for this! (Click to enlarge.)

Monday, October 20, 2008

Parsons's Mathematical Thought: Sec. 35, Intuition of finite sets

Suppose we accept that "it is not necessary to attribute to the agent perception or intuition of a set as a single object" in order to ground arithmetical beliefs. Still, we might wonder whether some such intuition of sets-as-objects might serve to "give an intuitive foundation to theories of finite sets".

But Parsons finds problems with this suggestion too. One difficulty can be introduced like this. Suppose I perceive the following array:

Then do I 'intuit' six dollar signs, a single set of six dollar signs, a set of three elements each a set of two signs, or even a set containing the empty set together with a set of six signs? Which way do I 'bracket things up'?
The possibilities are many -- indeed literally endless, if we are indeed allowed the empty set (and what is our intuition of that?). So it seems that the "intuition" here has to involve some representational ingredient to play the role of the brackets in the various possible bracketings. But then we are losing our grip on any putative analogy between intuition and perception (as Parsons puts it, "in a perceptual situation involving the application of certain concepts, we not expect that a linguistic of other embodiment of the concepts should be perceptually present in that very situation").

Secondly, note that we can in fact give a theory of those "bracket terms" -- putatively for hereditarily finite sets constructed from a given domain D of individuals -- which uses a relative substitutional semantics. That is to say, we can start with a first-order language for which D is the domain, add terms for hereditarily finite sets of elements from D, and variables and quantifiers for them, which we then interpret substitutionally relative to D. Parsons spells this out in an Appendix, but the general idea will be familiar to readers of his old paper on 'Sets and Classes'. And the upshot of this, Parsons says, "is that if we take the relative substitutional semantics as capturing a speaker's understanding of the language of hereditarily finite sets ... then we largely remove the motives for characterizing awareness of such sets as initution". That's a significant "if" of course: but we might indeed wonder why we should take elementary talk about finite sets (and sets of those, and so on) to be more committing than the substitutional interpretation allows.

Note that this isn't to say that we have entirely eliminated a role for intuition. For on the relative substitutional interpretation we still need the idea of sequences of individuals from D. And we might suppose that that notion is grounded in intuition. But even if true, that still falls well short of the original thought that we could need intuitions of sets-as-objects to give a foundation to theories of finite sets.

Sunday, October 19, 2008

D960 for a desert island?

Sometimes, in an idle moment, I jot down -- be honest, don't we all? -- a list of the eight discs I would select as my Desert Island Discs. Impossibly difficult of course! But one constant choice is the last Schubert piano sonata, D960 (and if I had to save one of the eight from the waves, then this would probably be it). But which recording? .... Well, that's almost impossible too.

I've for a while had Schnabel, Brendel (1972, 1988, 2000), Richter (three recordings of his too -- extraordinary, Schubert stretched to the limit), Imogen Cooper (underrated, but very fine), Schiff, Mitsuko Uchida, Perahia, and Kovacevich. Surely I had Kempff too but that seems to have gone walkabout. And very recently I bought the recording by Paul Lewis, which has received much praise. Yes, I agree! -- I really should kick this buying habit. But the prospect of another possible great performance was irresistible. Still, Lewis doesn't quite work ideally for me: I'll listen more -- but it isn't the one for the desert island. I still think that that has to be one of the Brendel recordings: after listening to others, I always listen to him again with a sense of coming home. Perhaps I love his 1988 recording the most.

Monday, October 13, 2008

Parsons's Mathematical Thought: Secs 33, 34, Finite sets and intuitions of them

So where have we got to in talking about Parsons's book? Chapter 6, you'll recall, is titled "Numbers as objects". So our questions are: what are the natural numbers, how are they "given" to us, are they objects available to intuition in any good sense? I've already discussed Secs 31 and 32, the first two long sections of this chapter.

There then seems to be something of a grinding of the gears between those opening sections and the next one. As we saw, Sec. 32 outlines rather incompletely the (illuminating) project of describing a sequence of increasingly sophisticated but purely arithmetical language games, and considering just what we are committed to at each stage. But Sec. 33 turns to consider the theory of hereditarily finite sets, and considers how a theory of numbers could naturally be implemented as an adjunct to such a theory. I'm not sure just what the relation between these projects is (we get "another perspective on arithmetic", but what exactly does that mean? -- but, looking ahead, I think things will be brought together a bit more in Sec. 36).

Anyway, in Sec. 33 (and an Appendix to the Chapter) Parsons outlines a neat little theory of hereditary finite sets, taking a dyadic operation x + y (intuitively, x U {y}) as primitive alongside the membership relation. The theory proves the axioms of ZF without infinity and foundation. I won't reproduce it here. In such a theory, we can define a relation x ~ y that holds between the finite sets x and y when they are equinumerous. We can also define a "successor" relation between sets along the following lines: Syx iff (Ez)(z is not in x and y ~ x + z).

Now, as it stands, S is not a functional relation. But we can conservatively add (finite) "cardinal numbers" to our theory by introducing a functor C, using an abstraction axiom Cx = Cy iff x ~ y -- so here "numbers are types, where the tokens are sets and the relation ~ is that of being of the same type". And then we can define a successor function on cardinals in terms of S in the obvious way (and go on to define addition and multiplication too).

So far so good. But quite how far does this take us? We'd expect the next step to be a discussion of just how much arithmetic can be constructed like this. For example, can we cheerfully quantify over these defined cardinals? We don't get the answer here, however. Which is disappointing. Rather Parsons first considers a variant construction in which we start not with the hierarchical structure of hereditarily finite sets but with a "flatter" structure of finite sequences (I'm not too sure anything much is gained here). And then -- in Sec. 34 -- he turns to consider whether such a story about grounding an amount of arithmetic in the theory of finite sets/sequences might give us an account of an intuitive grounding for arithmetic, via a story about intuitions of sets.

Well, we can indeed wonder whether we "might reasonably speak of intuition of finite sets under somewhat restricted circumstances" (i.e. where we have the right kinds of objects, the objects are not too separated in space or time, etc.). And Penelope Maddy, for one, has at one stage argued that we can not only intuit but perceive some such sets -- see e.g. the set of three eggs left in the box.

But Parsons resists at least Maddy's one-time line, on familiar -- and surely correct -- kinds of grounds. For while it may be the case that we, so to speak, take in the eggs in the box as a threesome (as it might be) that fact in itself gives us no reason to suppose that this cognitive achievement involves "seeing" something other than the eggs (plural). As Parsons remarks, "it seems to me that the primary elements of a story [a rival to Maddy's] would be the capacity to classify what one sees ... and to recognize identities and differences" -- capacities that could underpin an ability to judge small numerical quantifications at a glance, and "it is not necessary to attribute to attribute to the agent perception or intuition of a set as a single object". I agree.

Sunday, October 12, 2008

Life is too short ...

After empty months over the summer, you are suddenly faced with hectic weeks when there is far too much going on by way of seminars and discussion groups. I assume it is much the same in most places; but perhaps the phenomenon is exaggerated here in Cambridge, where 'Full Term' is so short and intense.

My coping strategy is to be hyper-selective, on the policy 'if in doubt, miss it out'. But I do try to keep an eye on what is going on, just in case. I note, for example, that there is a philosophy of mind reading group, and wonder what it is getting up to. Reading next, apparently, Charles Travis's 'Reason's Reach' European Journal of Philosophy 2007, pp. 225-248. So I take a speculative look. Ye gods. Life is surely far too short to bother with stuff written in such a ghastly pretentious style (this sort of thing would have been straight to the instant reject pile in my editorial days!). If philosophy is worth doing at all -- and of course much of it isn't -- then let's have it in straight-talking prose with absolutely maximal clarity so we can see precisely what the arguments are. To the flames with the rest!

Friday, October 10, 2008


There's a piece on academic blogging in today's Times Higher Education. It doesn't contain any big surprises, but it is mildly interesting -- and I do get a few mentions. Fame at last.

I have an entirely enviable walk into the Faculty: through a couple of pleasant late Victorian/Edwardian backstreets, across Midsummer Common (yes, there really are cattle grazing near the centre of Cambridge) and Jesus Green, then perhaps through Trinity Great Court and Neville's Court and out over the river to the backs; along behind Clare and King's (here there are white cattle in the meadow, colour-coded to match the Gibbs Building) and then a few hundred yards further to the Raised Faculty Building. I couldn't wish for much better.

That walk is mostly very quiet: so I can listen to music en route (the Lindsays playing Haydn works really well). And the weather these first few days of term has been stunning: perfect early autumn. Which has all put me in a good mood for the first couple of intro logic lectures which went by with only very minor hiccups -- though however many times I do this, the first lecture or so is still surprisingly nerve-wracking. I've also given the first class in my Gödel's Theorems course (depressingly few are taking the math. logic paper again this year: despite our best efforts, a Cambridge tradition seems to be in decline). And the first logic seminar went pretty well. Or at least, I enjoyed it. There were over twenty there, to battle with §108 of the Big Typescript (of course, the numbers won't last!). I gave the talk which I posted a draft of here, and Michael Potter added some very useful comments. When I've got a moment I'll put together a revised version to take account of some things said in the discussion.

So that's the first days of term survived. And now, I hope, back to Parsons!

Sunday, October 05, 2008

LaTeX for Logicians

Just to report that there's now an smarter, updated, port of the old LaTeX for Logicians pages now available at my composite website All corrections and further contributions very gratefully received. (Another pre-term task ticked off the list!)

Friday, October 03, 2008

Online logic texts resources

In case you missed this over at Richard Zach's blog, here's a link to an excellent page by Henri Galinon which in turn links to freely available logic texts and surveys of various kinds. Very well worth checking out. [Link updated!]

iTunes: why?

Life has been more than a bit distracting for the last few weeks. So apart from stuff that has had to be done -- revising some lectures for the beginning of term, and putting together some thoughts for the first logic seminar of term -- logical matters have been put on the back burner. But I hope to get back to normal service here a.s.a.p. (For a start, I need to continue working through Parsons's book. Also, I need to think a bit over the next few weeks about the relation between set theory and category theory prompted in part by Graham Priest's provoking remarks in Ch.2 of In Contradiction. So watch this space.)

Meanwhile, I've been cheering myself up with a bit of retail therapy in a very small way. In particular -- given the stunning reviews -- I want to get into Paul Lewis's Beethoven cycle. Fact: Vol. 2 is £24.99 on iTunes for pretty lossy compressed files (ok on an iPod with dodgy headphones, but that's about it); the original CDs are £9.28 including postage from Caiman via Amazon. It takes just a few minutes to import the CDs to your iPod in great quality files, and there's no fuss about backing up as you have the originals. So exactly why would I prefer to use iTunes?

Wednesday, October 01, 2008

Mathematics and games, again

OK, I've had a chance to get back to wrestling with Sec. 108 of the Big Typescript. So here's a draft handout for the first seminar of term -- mostly for third year undergrads and beginning post-grads (so this is neither very detailed nor very sophisticated; but I hope it is at least comprehensible and will provoke some discussion). Comments very welcome though!

Sunday, September 28, 2008

If you were a set

Amazon's algorithm for telling you about books they think you might find interesting (given your past purchases) can deliver some amusing results. "Greetings," they say today, "we've noticed that customers who have purchased Classical Recursion Theory: The Theory of Functions and Sets of Natural Numbers by P. Odifreddi have also purchased If You Were a Set by Evelyn M Aboff ...".

But maybe, on second thoughts, I should take a look at this book for small kids: those misconceptions about sets that the average first year philosophy student seems to have must come from somewhere!

Friday, September 26, 2008

Mathematics and games

Of course, the trouble with tackling Wittgenstein is you can bogged down so easily and distracted into various kinds of detective work (sometimes I wonder if half the attraction the sage has for some of his less critical fans is that he offers his readers the pleasure of puzzle-cracking as one tries to track down the sense of the more gnomic utterances). So, I'm still wrestling with Sec. 108 of the Big Typescript, the first section in the last third of the book which comprises remarks on the foundations of mathematics. But here, to be going on with, are my reconstructive efforts organizing and padding out the remarks of Sec. 108 into more continuous prose: commentary to follow. [For an updated link for the version with commentary, see posting for 1st Oct.]

Sunday, September 21, 2008

On the irritation of reading Wittgenstein

I've got to another sticky point in Parsons's book (some irritating obscurity), and am a bit stumped to know what to think. But I'll return to that in due course.

In the meantime, one other thing I've started doing in a very busy week is to look at the remarks on the foundations of mathematics in the last third of Wittgenstein's Big Typescript. Michael Potter and I are going to run a seminar on this material during the coming term. Why? Well, we both are interested in reading this relatively recently published text -- and in Wittgenstein's Cambridge it seems odd not to return occasionally to think again about his distinctly odd ruminations about mathematics as new generations of graduate students come through.

Yet, as ever, I can't but be irritated by Wittgenstein's affectation in refusing to write decent connected prose (albeit a different kind of irritation from that in reading Parsons). Oh yes, I know we are supposed to find deep significance in his choice of the aphoristic style. But most of what is written about that is pretentious bollocks, of course. (Wittgenstein's epigones like to intimate that if you don't appreciate the deep significance of the master's allusive style, you are an illiterate philistine. Which is both fatuous and offensive.) Anyway, just as an exercise, I'm having some amusement taking a section of the Big Typescript (which at least is divided into sections) and imagining embedding the fragmentary remarks into some connected prose in a sensible ordering and with the twists and turns of argument signalled. If something useful comes out of it, I'll post a version here!

Friday, September 19, 2008


A while back I posted about trying the OmniFocus 'task management' software which implements Getting Things Done type lists. As I said, it's not that I haven't tasks to do, and the GTD idea really does work. But, having played about a bit with it, I reckoned my life isn't so cluttered that carrying on using NoteBook and iCal wouldn't work well enough for me.

I've not changed my mind about Omnifocus. But now I've just discovered an alternative, lighter weight, more free-form task management OS X application simply called Things. Still in beta and free, but very well regarded (for a tour, see here): very clean and easy to learn, and even easier to use. I'm a convert. Well worth checking out.

Friday, September 12, 2008

Blackburn vs Polkinghorne

In my post about the LHC, I mentioned John Polkinghorne (that's the Reverend Professor Sir John Polkingorne to you). He taught me quantum mechanics a long time ago, and he was a terrific lecturer and expositor. Since then, he's become perhaps a more famous theologian than scientist, and keeps writing books trying to square his distinctly conservative theology with science. They are philosophically pretty awful. For a fun read, try my colleague Simon Blackburn lambasting a couple of Polkinghorne's books.

Thursday, September 11, 2008

Parsons's Mathematical Thought: Secs 31, 32, Numbers as objects

Chapter 6 of Parsons's book is titled 'Numbers as objects'. So: what are the natural numbers, how are they "given" to us, are they objects available to intuition in the kinds of ways suggested in the previous chapter?

Sec. 31 tells us that a partial answer to its title question 'What are the natural numbers?' is that they are a progression (a Dedekind simply infinite system). But "might we distinguish one progression as being the natural numbers, or at least uncover constraints such that some progressions are eligible and others are not?". The non-eliminative structuralism of Sec. 18 is Parsons's preferred answer to that question, he tells us. Which would be fine except that I'm still not clear what that comes to -- and since it is evidently important, I've backtracked and tried reading that section another time. Thus, Parsons earlier talks on p. 105 of "the conclusion that natural numbers are in the end roles rather than objects with a definite identity", while on p. 107 he is "most concerned to reject the idea that we don't have genuine reference to objects if the 'objects' are impoverished in the way in which elements of mathematical structures appear to be". So the natural numbers are, in the space of three pages, things to which we can make genuine reference (hence are genuine objects, given that "speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification to make serious statements"), but also are only impoverished 'objects', and are roles. I'm puzzled. This does seem to be metaphysics done with too broad a brush.

Anyway, Parsons feels the pressure to say more: "our discussion of the natural numbers will be incomplete so long as we have not gone into the concepts of cardinal and ordinal". So, cardinals first ...

Sec. 32 'Cardinality and the genesis of numbers as objects'. This section outlines a project which is close to my heart -- roughly, the project of describing a sequence of increasingly sophisticated arithmetical language games, and considering just what we are committed to at each stage. (As Parsons remarks, "The project of describing the genesis of discourse about numbers as a sequence of stages was quite foreign to [Frege]", and, he might have added, oddly continues to remain foreign to many.)

We start, let's suppose, with a grasp of counting and a handle on 'there are n Fs'. And it would seem over-interpreting to suppose that, at the outset, grasp of the latter kind of proposition involves grasping the second-order thought 'there is a 1-1 correspondence between the Fs and the numerals from 1 to n'. Parsons -- reasonably enough -- takes 'there are n Fs' to carry no more ontological baggage than a first-order numerical quantification '∃nxFx' defined in the familiar way. Does that mean, though, that we are to suppose that counting-numerals enter discourse as indices to numerical quantifiers? Even if ontologically lightweight, that still seems conceptually too sophisticated a story. And in fact Parsons has a rather attractive little story that treats numerals as demonstratives (in counting the spoons, I point to them, saying 'one', 'two', 'three' and so on), and then takes the competent counter as implicitly grasping principles which imply that, if the demonstratives up to n are correctly applied to all the Fs in turn, then it will be true that ∃nxFx.

So far so good. But thus far, numerals refer (when they do refer, in a counting context) to the objects being counted, and then recur as indices to quantifiers. Neither use refers to numbers. So how do we advance to uses which are (at least prima facie) apt to be construed as so referring?

Well, here Parsons's story gets far too sketchy for comfort. He talks first about "the introduction of variables and quantifiers 'ranging over numbers'" -- with the variables replacing quantifier indices -- which we can initially construe substitutionally. But how are we to develop this idea? He mentions Dale Gottlieb's book Ontological Economy, but also refers to the approach to substitutional quantification of Kripke's well-known paper (and as far as I recall, those aren't consistent with each other). And then there's the key issue -- as Parsons himself notes -- of moving from a story where number-talk is construed substitutionally to a story where numbers appear as objects that themselves are available to be counted. So, as he asks, "in what would this further conceptual leap consist?". A good question, but one that Parsons singularly fails to answer (see the middle para on p. 197).

At the end of the section, Parsons returns to the Fregean construal of 'there are n Fs' as saying that there is a one-one correlation between the Fs and the Gs (with 'G' a canonical predicate such that there are n Gs). He wants the equivalence between the two kinds of claim to be a consequence of a good story about the numbers, rather than the fundamental explanation. I'm sympathetic to that: and if I recall, Neil Tennant has pushed the point.

Wednesday, September 10, 2008

LHC: so far, so good ...

The BBC reports that start-up runs of the Large Hadron Collider at Cern are going well. That's great. I do feel pangs though, reading about all this. In a very close possible world, I'd have taken up the offer of a research studentship with the high energy particle physics group in DAMTP and worked with the likes of Jeffrey Goldstone and John Polkinghorne and been around in the days of the birth of the Standard Model which is being tested to the limit at Cern (tested to destruction?? -- we'll see).

Of course the choice I made to jump to philosophy was mine, and it's been quite fun. But these days, especially here in Cambridge, students get a lot more mentoring than I ever had -- and given better advice, I'd have probably not made the decision I did.

Burgess reviews Parsons

Luca Incurvati has just pointed out to me that John Burgess has a review of Parsons forthcoming in Philosophia Mathematica, and an electronic pre-print is available here (if your library has a subscription). Burgess is very polite, but reading between the lines, maybe he had some of the problems I'm having. For example, "[Parsons's] own version of structuralism is only rather sketchily indicated", and Burgess is himself pretty sketchy about Parsons on intuition.

I hope to return to Parsons here tomorrow; but in fact the pressure is off for me. It turns out that Bob Hanna and Michael Potter here are going to be running a reading group on the book this coming term, so I've arranged for the delivery date of my critical notice to be delayed until the end of term, after I've had the benefit of hearing what others think about some of what I'm finding obscure.

Thursday, September 04, 2008

The Princeton Companion to Mathematics

I'm really looking forward to getting a copy of the Companion (Tim Gowers has a nice podcast about the project). Looking at the table of contents and the author list, you should certainly order this for your local philosophy library too.

Wednesday, September 03, 2008

Quine's Mathematical Logic revisited

I recently unpacked a box of old books that I'd stored away in the garage, which included my missing copy of Quine's Mathematical Logic. I've just found myself (re)reading the first part -- that's the initial hundred pages on propositional and quantificational logic. And it's mostly still a great read -- though I do wonder how on earth anyone got to think that Principia style dots were a great idea for bracketing?! The brief end-of-section historical notes are sometimes particularly interesting. So actually, I'd recommend any beginning graduate student brought up on natural deduction and/or trees to spend a morning zipping through these chapters, both for the historical perspective they bring, and also to prompt some thoughts about what's gained and what's lost by doing things the modern ways.

I rather wish I'd found my copy before I sent off the revised version of IFL to the press. I might well have stolen a sentence here and an example there! Oh well, next time ...

Parsons's Mathematical Thought: A footnote on intuition

Qn: "You do seem increasingly out of sympathy with Parsons's book. So why are you spending all this effort blogging about it?"

Ans: "Well, as I think I said at the outset, I have promised to write a review (indeed, a critical notice) of the book, so this is just my way of forcing myself to read the book pretty carefully. And I'm not so much unsympathetic as puzzled and disappointed: I'm finding the book a much harder read than I was expecting. The fault could well in large part be mine. However, I do think that the prose is too often obscure, and the organization of thoughts unclear, so a bit of impatience may by now be creeping in (and talking to one or two others, I don't think my reaction to Parsons's writing is in fact that unique). But these ideas are certainly worth wrestling with: so I'm battling on!"

One thing I didn't comment on before was Parsons's motivation for pushing the notion of intuition and intuitive knowledge. "Intuition that," he says, "becomes a persuasive idea when one reflects on the obviousness of elementary truths of arithmetic. Two alternative views have had influential advocates in this century: conventionalism ... and a form of empiricism according to which mathematics is continuous with science, and the axioms of mathematics have a status similar to high-level theoretical hypotheses." Carnapian(?) conventionalism is, Parsons seems to think, a non-starter: and Quinean empiricism "seems subject to the objection that it leave unaccounted for precisely the obviousness of elementary mathematics." An appeal to some kind of intuition offers the needed account.

But I'm not sure that the Quinean should be abashed by that quick jab. For the respect in which the axioms of mathematics are claimed to have a status similar to high-level theoretical hypotheses is in their remoteness from the observational periphery, in their central organizational roles in a regimentation of our web of belief by logical/confirmational connections. That kind of shared status is surely quite compatible with the second-nature "obviousness" that accrues to simple arithmetic -- for some of us! -- due to intense childhood drilling and daily use. Logical position in the web, a Quinean would surely say, and degree of entrenched obviousness something else.

Monday, September 01, 2008

Parsons's Mathematical Thought: Secs 27-30, Intuition, continued

I've been trying to make good sense of the rest of Parsons's chapter on intuition, and have to confess failure. We might reasonably have hoped that we'd get here a really clear definitive version of the position on intuition that he has been developing for the better part of 30 years; but I'm afraid not. Looking for some help, I've just been rereading James Page's 1993 Mind discussion 'Parsons on Mathematical Intuition', which Parsons touches on, and David Galloway's 1999 Philosophical and Phenomenological Research paper 'Seeing Sequences', which he doesn't mention. Those papers show that it is possible to write crisply and clearly (though critically) about these matters: but Parsons doesn't pull it off. Or at least, his chapter didn't work for me. Although this is supposed to be a pivotal chapter of the book, I'm left rather bereft of useful things to say.

Sec. 27, 'Toward a viable concept of intuition: perception and the abstract' is intended to soften us up for the idea that we can have intuitions of abstracta (remember: intuitive knowledge that, whatever exactly that is, is supposed to be somehow founded in intuitions of, where these are somehow quasi-perceptual). There's an initial, puzzling, and inconclusive discussion of supposed intuitions of colours qua abstract objects: but Parsons himself sets this case aside as raising too many complications, so I will too. Which leaves the supposed case of perceptions/intuitions of abstract types (letters, say): the claim is that "the talk of perception of types is something normal and everyday". But even here I balk. True, we might well say that I see a particular squiggle as, for example, a Greek phi. We might equivalently say, in such a case, that I see the letter phi written there (but still meaning that we see something as an instance of the letter phi). But I just don't find it at all normal or everyday to say that I see the letter phi (meaning the type itself). So I'm not softened up!

Sec. 28, 'Hilbertian intuition' rehashes Parsons's familiar arguments about seeing strings of strokes. I won't rehash the arguments of his critics. But I'm repeatedly puzzled. Take, just for one example, this claim:

What is distinctive of intuitions of types [now, types of stroke-strings] is that the perceptions and imaginings that found them play a paradigmatic role. It is through this that intuition of a type can give rise to propositional knowledge about the type, an instance of intuition that. I will in these cases use the term 'intuitive knowledge'. A simple case is singular propositions about types, such as that ||| is the successor of ||. We see this to be true on the basis of a single intuition, but of course in its implications for tokens it is a general proposition.
A single intuition? Really? If I'm following at all, I'd have thought that we see that proposition to be true on the basis of an intuition of ||| and a separate intuition of || and something else, some kind of intuitive (??) recognition of the relation between them. What is the 'single' intuition here?

Or for another example, consider Parsons's wrestling with vagueness. You might initially have worried that intuitions which are "founded" in perceptions and imaginings will inherit the vagueness of those perceptions or imaginings (and how would that square with the claim that "mathematical intuition is of sharply delineated objects"?). But Parsons moves to block the worry, using the example of seeing letters again. The thought seems to be that we have some discrete conceptual pigeon-holes, and in seeing squiggles as a phi or a psi (say), we are pigeon-holing them. The fact that some squiggles might be borderline candidates for putting in this or that pigeon-hole doesn't (so to speak) make the pigeon-holes less sharply delineated. Well, fair enough. I'm rather happy with a version of that sort of story. For I'm tempted by accounts of analog non-conceptual contents which are conceptually processed, "digitalizing" the information. But such accounts stress the differences between perceptions of squiggles and the conceptual apparatus which is brought to bear in coming to see the squiggles as e.g. instances of the letter phi. Certainly, on such a view, trying to understand our conceptual grip here in terms of a prior primitive notion of "perception of" the type phi is hopeless: but granted that, it is remains entirely unclear to me what a constructed notion of "perception of" types can do for us.

Sec. 29, 'Intuitive knowledge: a step toward infinity' Can we in any sense see or intuit that any stroke string can be extended? Parsons has discussed this before, and his discussions have been the subject of criticism. If anything -- though I haven't gone back to check my impression against a re-reading of his earlier papers -- I think his claims may now be more cautious. Anyway, he now says (1) "If we imagine any [particular] string of strokes, it is immediately apparent that a new stroke can be added." (2) "Although intuition yields one essential element of the idea that there are, at least potentially, infinitely many strings ... more is involved in the idea, in particular that the operation of adding an additional stroke can be indefinitely iterated. The sense, if any, in which iteration tells us that is not obvious." But (3) "Although it will follow from considerations advanced in Chapter 7 that it is intuitively known that every string can be extended by one of a different type, ideas connected with induction are needed to see it." We could, I think, argue about (1). Also note the slide from "imagine" to "intuition" between (1) and (2): you might wonder about that too (Parsons is remarkably quiet about imagination). But obviously, the big issue is going to come later in trying to argue that ideas "connected with induction" can still be involved in what is "intuitively known". We'll see ...

Finally, I took little away from Sec. 30, 'The objections revisited', so I won't comment now.

Sunday, August 31, 2008

Contributing off the cuff?

Relatively recently, we've started having an occasional in-house one-day faculty colloquium, where staff and grad students give papers on their current work. I've just been asked if I'd like to talk to the next one, so I offered to chat about induction in second-order arithmetics (introducing some of the themes from this paper). But the organizer wasn't sold on the suitability of the idea: a non-expert member of the audience might get to see what the issues are, but "could not him/herself hope to contribute". So I'm off the hook.

But that response got me thinking. Once upon a time -- in my philosophical lifetime, indeed -- you could "keep up" over quite a wide front, and so dive in and intelligently discuss issues across quite a range with colleagues and visiting speakers. But really, how possible is that nowadays? Editing a journal made me vividly aware that, with almost any narrow topic, there's a now serious, sophisticated, well-developed, very clever literature out there, where the moves, counter-moves, counter-counter-moves are analysed and explored many levels deep. So I wonder if there is any way in which the non-expert can seriously hope to "contribute" off the cuff in response to a talk (unless that just means asking intelligent questions for further elucidation). The problem is obvious with the technical philosophy of maths, for example: but isn't it now actually the same pretty much right across the board? Is the conception of a wide-ranging colloquium with discussions to which the audience generally might hope to "contribute" past it sell-by date? I rather suspect so. Or do younger and more energetic philosophers feel differently?

Saturday, August 30, 2008

Honest Toil

I'm just catching up with the (fairly) new and excellent blog Honest Toil from Chris Pinnock.

Naturalism in the Philosophy of Mathematics

Alex Paseau has a new entry in the Stanford Encyclopedia on 'Naturalism in the Philosophy of Mathematics'. If I was being picky, I'd perhaps say that student readers will find that it dives in the deep end a bit quickly. But I found Alex's useful distinction-making and rather sceptical reflections very helpful.

Monday, August 25, 2008

Mediocrity and bullshit

"I have never pretended to political correctness, so I can happily abhor the sanctimonious politically correct bullshit that made the British contribution to the Olympic closing ceremony so appalling. The most dull dancing imaginable, completely unsuited in scale to the ceremony, and mismatching the Royal Ballet with (wait for it) a South London Hip-Hop ensemble and a dance group featuring able and disabled dancers. The quality produced was risible - it would not have graced a county fair, let alone the Olympics.

The PR bullshit said we were "honouring diversity". No, we were honouring mediocrity, and then apparently honouring Hello magazine by introducing Leona Lewis and David Beckham. I think I should run in the 100 metres in 2012, thus honouring diversity by vastly increasing our representation of overweight and unhealthy middle-aged men."

My thoughts exactly. Or rather, Craig Murray's thoughts, but I concur wholeheartedly.

Back to Sammartini

An old friend has just given me, burnt on to a CD, some symphonies of Sammartini, taken from an old Saga LP from the 1960s. Instantly takes me back to my student rooms in Trinity. And quite wonderful to hear again.

Saga was one of the cheap record labels of the time (another was the terrific Supraphon, from which I got to know a lot of East European and Russian music). It wasn't that cheap though, by modern standards: buying a record was still quite an event. If I'm remembering right, Saga records were about 12/6 -- that's twelve shillings and sixpence to you! -- when full-price records were about 32/6. To put things into perspective, that was then about the cost of three Penguin books, five pints of beer, or twenty five Mars bars. Compare now when Naxos CDs are cheaper than Penguins, cost not much more than two pints, or a dozen Mars bars. And of course there's about twice as much music on a classical CD compared with an LP, and they aren't instantly damaged by dodgy student turntables!

Sunday, August 24, 2008

MacBook Air, one month on

Anyone out there who is wavering about getting a MacBook Air might be interested in some comments from a new owner. Everyone else can, of course, just cheerfully ignore this posting! (The headline summary is: get one! -- though perhaps not quite today as there is rumour of a chip upgrade soon.)

  1. The portability is fantastic. No question. Just to compare: I've had 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 just too heavy/bulky to make that particularly convenient. I very rarely bothered to take them elsewhere, e.g. to a coffee shop. (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 in the kitchen for a change of scene, or answer emails with a computer on my knees in the living room in the evening. And partly I wanted 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.
  2. Some reviews complained about the MBA's footprint, saying that it isn't a genuine ultraportable. Well, true, the footprint isn't in fact that much smaller that the 15" machine, and I can imagine e.g. that very frequent fliers would find it a pain to use in the cramped conditions of an airline seat. But that sort of issue doesn't arise for me. The thinness means that you can carry it so comfortably in a hand, and otherwise the footprint goes with the stunningly good, uncramped, screen and the generous keyboard.
  3. I don't use the MBA to watch movies, or do anything else very processor-intensive. So I've never had a temperature-induced core shut-down. 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 -- unlike the small brick for the 17" -- the MBA's charger is very small and portable (though I've bought a second one for the office, 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 an irritating issue. So this is plenty fast enough.
  5. And the reason, when I previously traded up, I chose the 17" MBP model was to have enough "real estate" to have a TeXShop editing window and the PDF output 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 this is manageable, and the screen quality is really terrific. Of course it is nicer e.g. for extended on-screen reading to plug in an external monitor as well. But that's a luxury, not any sort of necessity.
  6. What about the paucity of ports, mentioned critically by all the reviewers, or the absence of an onboard CD drive? With one caveat, I've found those features no problem at all. Just not been an issue for me since day one. (So the one caveat indeed concerns day one. 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 the 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, this is a bit noisier (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 (and external monitor) as a main, quasi-desktop, set-up. In fact I find myself increasingly heavily favouring the MBA. I've hardly used 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. Or one modestly decent bottle of Chianti Classico a fortnight if you prefer. Put like that, how can you resist?
  10. And then, of course, there is the "Wow!"-factor ...

Saturday, August 23, 2008

Parsons's Mathematical Thought: Secs 24-26, Intuition

Chapter 5 of Parsons's book is called "Intuition". And I guess I should declare an interest (or rather, lack of interest!) here. I've never really understood talk about intuition: and I'm certainly not helped when Parsons writes "I shall be concerned to develop a conception of mathematical intuition that is in a general way Kantian", since Kant is pretty much a closed book to me. So perhaps I'm not the best reader for this chapter! But still, let's proceed ...

Sec. 24, "Intuition: Basic distinctions". Parsons distinguishes supposed intuition of objects from intuition that such-and-such is the case. And he stresses that in his usage, intuition that isn't factive. So is an intuition that such-and-such just a non-inferential belief? Well note, for example, that "knowledge without observation" of our own bodily movements is non-inferential, but is not normally counted as intuitive. So what differentiates intuition properly so-called? Parsons promises an answer by a "development of the concept ... in the Kantian tradition".

Sec. 25, "Intuition and perception". Now, the headline suggestion here is that "It is hard to see what could make a cognitive relation to objects [intuition of] count as intuition if not some analogy with perception" (cf. e.g. Gödel). Further, intuition that is intimately connected with intuition of, rather as perception that is grounded in perception of. Well, fair enough: but that, of course, already does make claims about intuitions of mathematical objects very puzzling. Which leads to ...

Sec. 26, "Objections to the very idea of mathematical intuition". Start with the following point. Ordinary perception is (so to speak) evident to the subject -- when I see an object, my computer screen say, "there is a phenomenological datum here". But "it is hard to maintain that the case is the same for mathematical objects ... [Are] there any experiences we can appeal to in the mathematical cases that are anywhere near as indisputed as my present experience of seeing the computer screen?" This seems to undermine any alleged analogy between "intuition of mathematical entities" and ordinary perception. So how are we to defend the analogy, given the different phenomenologies? Unfortunately, Parsons next remarks here are Kantian obscurities I can do nothing with. So I'm left stumped.

(Parsons also raises a question about the relation between structuralist thoughts and claims about intuition. The worry seems to be one about how a particular intuition can latch on to a particular object, if mathematical objects are indentified by their places in structures. The point, however, is rather rushed. But since I think Parsons is going to return to these matters, I won't say more at the moment.)