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

Things

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.