Year: 2010

TTP, 1. Introduction: Platonism vs ‘naturalized epistemology’

Let me begin by setting the scene, embroidering only a little on Weir’s opening pages.

Consider then the following claims, ordinarily regarded as mathematical truths:

  1. 3 is prime.
  2. The Klein four-group is the smallest non-cyclic group.
  3. There is an uncountably infinite set of nested subsets of Q, the set of rationals.

On the surface, (1) looks structurally very similar to ‘Alan is clever’. The latter is surely about some entity, namely Alan, and is true because that entity has the property attributed. Likewise, we might initially be pretty tempted to say, (1) also is about something, namely the number three, and is true because  that thing has the property attributed. Similarly (2) is about something else, the Klein four-group. And (3) is about the set of rationals, and claims there is a further thing, an uncountably infinite family of sets of rationals nested inside each other.

So what  kind of things are three, the Klein group, the set of rationals? Not things we can see or kick, but non-concrete things, surely — i.e. abstract objects. And given the standard view that (1), (2) and (3) don’t just happen to be true but are necessarily true, it would seem that these abstract objects must be necessary existents.

But how can we possibly know about such things? Once upon a time (Weir might have reminded us), the thought seemed attractive that we are made in God’s image, and — albeit to a limited extent — partake in his rational nature (for Spinoza, indeed, ‘the human mind is part of the infinite intellect of God’). And God, the story went, can just rationally see all the truths of mathematics: sharing something of his nature, in a small way we can come to do that too. Thus Salviato, speaking for Galileo in his Dialogue, says that in grasping some parts of arithmetic and geometry, the human intellect ‘equals the divine in objective certainty, for here it succeeds in understanding necessity’. And Leibniz writes `minds are … the closest likenesses of the first Being, for they distinctly perceive necessary truths’. (For more on this theme, see Ch. 1 of Edward Craig’s wonderful The Mind of God and the Works of Man (OUP, 1987), from which the quotations are taken.)

However, this conception of ourselves as approximating to the God-like has lost its grip on us. So, getting back to Weir, given the sort of beings we now think we are, with the sorts of limited and ramshackle cognitive powers with which evolution has provided us to enable us to survive in our small corner of the universe, the question becomes pressing: how come that we can possibly get to know anything about supposedly necessarily existent abstract entities (entities in Plato’s heaven, as they say)? How are we supposed to cognitively ‘lock on’ to such things? Indeed, we might wonder, how do we ever manage even to frame concepts of things apparently so remote from quotidian experience?

Now, note that to find this sort of question pressing it isn’t that we already have to bought into the idea that epistemology should be ‘naturalized’ in some strong sense, or that a Quinean ‘naturalized epistemology’ exhausts the legitimate parts of what used to be epistemology. And though Weir does talk about mathematical platonism being “put to the test by naturalized epistemology”, he officially means no more than that our conception of ourselves as natural agents without God-like powers “imposes a non-trivial test of internal stability” (p. 3) when combined with views like platonism. The problem-setting issue, then, is an entirely familiar one: as Benacerraf frames it in his classic paper, ‘a satisfactory account of mathematical truth … must fit into an over-all account of knowledge in a way that makes it intelligible how we have the mathematical knowledge that we have’.

So far, then, so good. We have a familiar but still pressing question, and in the next post, I’ll say something about various lines we might take in response and indicate how, in rest of his Introduction, Weir situates on the map the position he wants to defend.

But first just a word or two more about Weir’s enthusiasm for naturalism. He goes on to write “I accept, as a general methodological maxim, the prescription that one should push a naturalistic approach as far as it will go.” (p. 5) And what does that involve? “The methodological naturalist … prescribes that one ought to follow scientific method, at a level of sophistication appropriate to the problem at hand, whenever attempting to find out the truth about anything.” Really? If using the methods of science is construed more narrowly as involving the careful rational weighing of evidence to test specific, antecedently formulated, empirical conjectures, etc., then of course this isn’t the only way to discover truths. Evolution has thankfully provided us with other quick-and-dirty ways of fast-tracking to the truth reliably enough to avoid fleet-footed predators often enough! While if the methods of science are understood in a more relaxed and embracing way, as whatever goes into the mix as we develop our best overall theory of nature, then (a familiar old-Quinean point) these methods would seem to subsume the methods of (much) mathematics which seem so entangled, and ‘methodological naturalism’ in itself has no special bite again platonism (over and above Benacerraf’s problem, which doesn’t depend on the naturalism).

But we really don’t want to start off on that debate again, about the appropriate formulation of a possibly-defensible ‘naturalism’. So let’s re-emphasize the key point that I think Weir would make, which his rhetoric hereabouts could possibly obscure: we don’t need to endorse any strong form of naturalism, or have any commitment to naturalized epistemology as the one legitimate residuary legatee of the epistemological tradition, to find troubling the combination of platonism with our conception of ourselves as limited creatures epistemically geared to the sublunary world. That‘s enough of a problem to get Weir’s project going.

Truth Through Proof, 0. Preamble

I am eventually going to be writing a (short) review for  Mind of Alan Weir’s new book  Truth Through Proof: A Formalist Foundation for Mathematics (OUP, 2010). The blurb on the publisher’s website gives  you an idea what of what the book is about. The clue is in the subtitle — but note, this is a philosophy book, not a technical book in foundational studies.

To help fix my ideas,  I’ll be posting a (much longer) series of discussion notes here, as I sporadically work through the book. Given other commitments, however, I’ll have to take things pretty slowly over the coming weeks.

As with similar series of postings on other books, I expect that these notes will weave around and about Weir’s book (henceforth  TTP) in a more free-ranging way than would be appropriate in a review: but I’ll try to make it clear when I’m summarizing TTP, when I’m directly commenting on Weir’s views, and when I am striking out more on my own account. And more generally, I’ll try to keep things accessible to students, even if that sometimes means including more background explanations than some other readers here will feel they need.  All comments as we go along will be very gratefully received!

After the Introduction, Weir’s chapters are divided into sections numbered off with roman numerals: so here ‘3.III’ means Chapter 3, §III. Double quotation marks are reserved to signal quotations from TTP (and otherwise unattributed page numbers of course refer to the book too).

OK. With that by way of brisk preamble, let’s dive in … tomorrow!

The Bertrand Russell Chair

That well-known website LeiterLeaks tells the world

Huw Price, Challis Professor of Philosophy at the University of Sydney and a two-time winner of the lucrative Australian Federation Fellowships, has been offered the Bertrand Russell Professorship in Philosophy at Cambridge University (presently held by Simon Blackburn, who will be retiring this year).  Price has written widely in philosophy of science and physics, metaphysics, and philosophy of language.

Hmmm. I thought this was all hush-hush, with things depending on the University’s further discussions (in the Cambridge system, the appointing committee makes an election, but then it is up to quite other bodies in the University to come to terms with the hoped-for appointee). So given it’s confidential still, I couldn’t possibly comment.

Oh well, all right …

Just a bit …

Let’s just say it seems to me that if Huw Price does indeed take up the offer, that will be a very good outcome for the Faculty.

Florence, for body and soul

Piazza del Duomo

Florence in winter is a delight. You can get even into the Uffizi without queuing. Stand quietly in front of your favourite paintings or frescos for as long as you like without crowds around you all the time.  And when you feel like feeding body rather than soul, get into decent restaurants which are half-empty. It can certainly be cold: but with luck, you’ll be able to walk around under blue skies. We hadn’t expected snow, though! Nor had the Florentines, judging by the way everything ground to a complete halt for a day. But it did all look wonderful.

If you happen to be in Tuscany before January 23, do go to the Bronzino exhibition at Palazzo Strozzi. It is a unique opportunity to see some 80% of Bronzino’s surviving paintings together. And the exhibition is quite beautifully put together and wonderfully presented (the contrast with the Uffizi’s dreadfully drear hanging could hardly be greater). And if you have the chance, get yourself on one of the weekly tours of the Contini Bonacossi collection acquired by the Uffizi a dozen years ago and still not on general display. It includes the wonderful Veronese portrait of Iseppo da Porto and his young son which we liked so much at the Louvre last year.

We ate well (surprise, surprise). These are the three places we’ll definitely go back to (and they will still be there after January 23rd, so make a note!):

  • Cibrèino (Trattoria Cibrèo) Via de’ Macci 122/r. This shares a kitchen of the very expensive Cibrèo restaurant next door: but you will eat as well for less than half the price. (You can’t book, and might have to share a table if there are only two of you.)
  • Olio & Convivium Via Santo Spirito 4. A modern take on Tuscan food — we liked the atmosphere, the food, and the stunning wine list.
  • Cantinetta dei Verrazzano Via dei Tavolini, 18/r. Fantastic place if you want a light (not cheap!) lunch.

A room with a view

Well, that isn’t supposed to happen. Heavy snow in Florence before Christmas. The city cut off. The airports closed.

But there are far worse places to be forced to stay a couple more days than planned. The galleries and the churches were still open more or less when they were supposed to be; the restaurants still a delight; and our welcoming and very comfortable hotel had no problem letting us stay on a couple of days (and we indeed had the very room that was used in the film A Room With a View: here’s the Ponte Vecchio from our terrace, the day after the snow.)

There were of course other tourists still around; but we were a sprinkling among real Florence out and about doing its Christmas shopping. Even the Uffizi was quiet. Still, after six nights, you begin to suffer from visual overload, and even I find that the thought of yet another meal out begins to pall. So it was good to get back last night to an equally snowy Cambridge.

Jane Austen and moral philosophy

I’m one of the panelists on the Ask Philosophers website. And I’m amused to notice that three particular favourites of mine happen to be the topics of my last three responses, namely the philosophy of maths, wine, and Jane Austen.

And how, you ask, does Austen come in for a mention? Well, someone who’d evidently enjoyed reading Camus’ The Plague was asking for suggestions of other novels with “philosophical underpinnings”. So — perhaps not at all the sort of answer s/he was expecting — I first offered Jane Austen (perhaps setting aside Northanger Abbey, any of her novels will do, though my favourite is Emma). Now,

I am not going to try to make out that Jane Austen was a philosopher or even a philosopher manquée. But … she was interested from the south side in some quite general or theoretical problems about human nature and conduct in which philosophers proper were and are interested from the north side.

The very titles of some her novels indicate their moral concerns. Thus

Sense and Sensibility really is about the relations between Sense and Sensibility or, as we might put it, between Head and Heart, Thought and Feeling, Judgement and Emotion, or Sensibleness and Sensitiveness.

And there are correspondingly thematic framings in other novels, even those without abstract nouns in their titles. Thus

If cacophony had not forbidden, Emma could and I think would have been entitled Influence and Interference. Or it might have been called more generically Solicitude. Jane Austen’s question here was: What makes it sometimes legitimate or even obligatory for one person deliberately to try to modify the course of another person’s life, while sometimes such attempts are wrong? Where is the line between Meddling and Helping? Or, more generally, between proper and improper solicitude and unsolicitude about the destinies and welfares of others? Why was Emma wrong to try to arrange Harriet’s life, when Mr Knightley was right to try to improve Emma’s mind and character? Jane Austen’s answer is the right answer. Emma was treating Harriet as a puppet to be worked by hidden strings. Mr Knightley advised and scolded Emma to her face. Emma knew what Mr Knightley required of her and hoped for her. Harriet was not to know what Emma was scheming on her behalf. Mr Knightley dealt with Emma as a potentially responsible and rational being. Emma dealt with Harriet as a doll. Proper solicitude is open and not secret. Furthermore, proper solicitude is actuated by genuine good will. Improper solicitude is actuated by love of power, jealousy, conceit, sentimentality and so on.

The minor characters in Emma too are ‘systematically described in terms of their different kinds or degrees of concernment or unconcernment with the lives of others.’ And Austen’s treatment of her themes is guided by what we might call an Aristotelian conception of the gradations of the many virtues as opposed to a black and white, saint vs sinner, Calvanist morality.

[T]he Aristotelian pattern of ethical ideas represents people as differing from one another in degree and not in kind, and differing from one another not in respect just of a single generic Sunday attribute, Goodness, say, or else Wickedness, but in respect of a whole spectrum of specific week-day attributes. A is a bit more irritable and ambitious than B, but less indolent and less sentimental. C is meaner and quicker-witted than D, and D is greedier and more athletic than C. And so on. A person is not black or white, but iridescent with all the colours of the rainbow; and he is not a flat plane, but a highly irregular solid. He is not blankly Good or Bad, blankly angelic or fiendish; he is better than most in one respect, about level with the average in another respect, and a bit, perhaps a big bit, deficient in a third respect. In fact he is like the people we really know, in a way in which we do not know and could not know any people who are just Bad or else just Good.

Jane Austen’s moral ideas are, with certain exceptions, ideas of the Aristotelian and not of the Calvinist pattern. Much though she had learned from Johnson, this she had not learned from him. When Johnson is being ethically solemn, he draws people in black and white. So they never come to life, any more than the North Pole and the South Pole display any scenic features. Jane Austen’s people are, nearly always, alive all over, all through and all round, displaying admirably or amusingly or deplorably proportioned mixtures of all the colours that there are, save pure White and pure Black. If a Calvinist critic were to ask us whether Mr Collins was Hell-bound or Heaven-bent, we could not answer. The question does not apply. Mr Collins belongs to neither pole; he belongs to a very particular parish in the English Midlands. He is a stupid, complacent and inflated ass, but a Sinner? No. A Saint? No. He is just a ridiculous figure, that is, a figure for which the Calvinist ethical psychology does not cater. The questions Was Emma Good? Was she Bad? are equally unanswerable and equally uninteresting. Obviously she should have been smacked more often when young; obviously, too, eternal Hell-fire is not required for her.

You can tell from persons being “he” and the cheery talking of smacking that that the essay from which all those sane and insightful quotations are taken wasn’t written quite yesterday. But the piece is by a good philosopher who is perhaps less read now than he should be. For a treat, for those who don’t know his wonderful essay ‘Jane Austen and the Moralists‘, which argues in particular for the influence of Shaftesbury, I’ve linked to a PDF. (And if after reading further, you still don’t recognize the author’s voice, then a quick Google search will reveal him.)

Exercise: now write in Austenian style brief reflections on the morality of thus further expediting the circulation of an old piece that is already only a couple of clicks away from any knowledgeable searcher.

Students are right to be pissed off …

… about the proposed “reforms” to higher education funding. I’ve started a couple of times to write a blog-post adding my two-pennyworth of comment. But firstly, I get too depressed musing more generally about the awfulness of various education “reforms” over the last forty or so years (at my most charitable, let’s say they are quite spectacular object lessons in the Law of Unintended Consequences). And second, much of what I might say has in fact already been said, and said very well, by others — e.g. by Stefan Collini on ‘Browne’s Gamble’, John Sutherland, writing under the cheery title English degrees for £27k – who’s buying?, and particularly by Iain Pears on ‘How the Humanities’.

Iain Pears’s point about re-centering the business of humanities departments on teaching strikes a real chord with me, as I approach the finishing-line with my job. It is difficult to credit now, but when I started as a lecturer we really had only half a day’s “induction” course — and one element of that was a talk on Arnold, Newman and Leavis on the idea of a university (can you imagine?). Yet that didn’t seem out of place. We did mostly thought of ourselves as university teachers with a commitment to “pass it on” (as Alan Bennett puts it in The History Boys). So we took it for granted that we would spent a lot of time talking with our students. ‘Research’ (as opposed to ‘scholarship’, i.e. keeping up our reading and thinking to inform our teaching) was something to be done in our — admittedly generous — spare time. Certainly, the idea that research in the humanities — yet another article in some minor passing debate, yet another unnecessary book? — should be at the very centre of everything, and teaching something to be avoided as much as possible (by getting research grants) was a long way in the future.

It wouldn’t be such a bad thing — and will be the least we owe to the kids who have to mortgage more of their futures to study with us — if in this one respect at any rate we went forward to the past.

Cambridge Principia Symposium

A while back, a number of us had the thought that we ought to do something in Cambridge to mark the centenary of Principia. But organizing blockbuster conferences is always a real pain; and to hold them here you need arrange everything years in advance; and in any case, to be frank, big conferences are very often just not worth the effort. So we thought, well, let’s perhaps organize something small and relatively informal nearer the time. Then time passed and the centenary date got nearer. And more time passed. And in fact the occasion would in the end have gone unmarked, except that Nik Sultana — a grad student in Comp. Sci.– took up the idea late in the day and made it all happen. All credit to him. Here’s the programme, some abstracts, and links to some short papers.

Given Nik’s roots in computer science, there was a somewhat different spin to this symposium than it would have had if organized just by a logic-minded philosopher or a philosophy-minded logician. Though, truth to tell, I’m not sure I learnt a great deal from the comp. sci. orientated talks. Formal proof checkers are developing apace, and this work on formal proofs is beginning to impact on ‘mainstream’ mathematics. But I basically knew that, though I got to hear about a few interesting details.

Three random thoughts. (1) When people talk about “type theories” they tend to really mean theories-with-type-disciplines or theories-about-theories-with-type-disciplines: I wonder why “typed theories” didn’t catch on? Anyway I do plan though to go off and have a look at the history of type theories co-written by Fairouz Kamareddine, from which Fairouz extracted a lighting tour for the symposium. (2) Like others who have learnt from Great Uncle Frege, I did find the wobbling between talk of functions and talk of expressions-for-functions by some people rather uncomfortable. (3) [Arising from some responses to Byeong-uk Yi’s talk], people are typically really unhappy with the idea that plural terms might refer, plurally, to several things rather than being disguised singular reference to a set: singularist prejudice goes deep.

The paper I enjoyed most, though, was my colleague Michael Potter’s talk about two ideas of ambiguity in Principia — the idea of “real” (i.e. unbound) variables as being ambiguous or variable names, and the idea of (most) “theorems” of PM in fact being type-ambiguous. The first evidences a step back from Frege’s clarity about the role of “variables” as anaphoric pronouns. The second has the authors of Principia struggling towards the idea of using schemas in the modern sense. Michael had some interesting quotes from correspondence between Whitehead and Russell, in which Whitehead is pressing towards the idea of a generalization using a metalanguistic schema, in using which you stand outside the given formalized language, and Russell (more wedded to a logical monism?) resisting. This was interesting stuff, with some nice historical detail, and I hope Michael works it up further.

Cuts, consistency and axiomatized theories

In the Wednesday Logic Reading Group, where we are working through Sara Negri and Jan von Plato’s Structural Proof Theory, I today introduced Chapter 6, ‘Structural Proof Analysis of Axiomatic Theories’. In their commendable efforts to be brief, the authors are sometimes a bit brisk about motivation. So I thought it was worth trying to stand back a bit from the details of this action-packed chapter as far as I understood it in the few hours I had to prepare, and to try to give an overall sense of the project. These are the notes I wrote for myself. As often with such middle-of-term efforts dashed off in a couple of hours, I both would have liked to have been clearer and do more justice to what we are reading, but I also just don’t have time to do more now than make a few corrections to the first version. The logic enthusiasts at the seminar seemed to find the remarks useful, though, so for what they are worth (and you don’t have to have read the book to get the gist) here there are. The usual warning applies: caveat lector.

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