Thursday, August 30, 2007

Philosophical atheists

I must not buy more books. I must not buy more books.

Well, that's what I try to tell myself in bookshops, but to little avail. Today, having sat in Heffers for half-an-hour or more reading the first couple of essays in Philosophers Without Gods: Meditations on Atheism and the Secular Life, I just couldn't resist, and bought a copy. It is so far a wonderful read -- twenty essays by philosophers including Simon Blackburn, David Lewis and Stewart Shapiro, writing about their atheism, in some cases with a moving personal slant. Highly recommended!

But I really must not buy so many books. I suppose I ought to institute a friend's rule "one book in, one book out" which is his way of coping with a similarly tiny Cambridge house. But I really don't think I can bring myself to do that. There seems to be nothing else for it other than to get yet another wall fitted with bookshelves ...

Monday, August 27, 2007

Three books

"God has all possible perfections." Ah yes, I would say sagely (in first-year supervisions of yesteryear), so He has perfect conductivity, perfect insulation, 20/20 eyesight and a first-class honours in social anthropology ...

The joke, of course, is Michael Frayn's. And it is very good to see a new selection of his old pieces for the Guardian and Observer just published, called simply Collected Columns. There's lots more lovely philosophy-by-jokes scattered around (didn't Wittgenstein say to Norman Malcolm that you could write a philosophy book containing just jokes?). Perhaps my all-time favourite remains 'The monolithic view of mirrors' about the debate in the Carthaginian Monolithic Church on the vexed question of the use of rear-view mirrors. After all

looking backwards while travelling forwards is categorically and explicitly forbidden by God, since it was for doing this that He visited instant fossilisation on Lot's wife. In this context 'looking back' has always been interpreted as frustrating the natural forward gaze of the traveller, whether by turning the head (visus interruptus) or by the imposition of a mechanical device such as a mirror. ...

Such pieces -- some of them forty years old now -- still explode theological bollocks wonderfully effectively as well as being exceedingly funny.

So I've been rereading Frayn over the last few days. I also finished Melvin Fitting's book, which I think is terrific, though I'll want to reread it more carefully before writing more of a review here. (For those wondering about getting it/reading it, let me just say that although it takes little for granted, it is quite compressed and is perhaps best for readers with a reasonable amount of mathematical sophistication -- perhaps one step up, then, from e.g. Smullyan's Gödel's Incompleteness Theorems and two steps up from my book).

And talking of my book, Keith Frankish's posted comment helped wonderfully in recovering my sense of proportion about that silly mistake about ACA0! Thanks!

Wednesday, August 22, 2007

Corrections page started

Following on from the last post, I have started a corrections page on the Gödel book's website.

Staring in disbelief ...

Sigh. Glancing through Chapter 22 of my Gödel book late last night, my eye was caught by a sentence on p. 197. I stared in disbelief. It says that ACA0 is the second-order theory you get when you restrict the φ(x) we can substitute into the Comprehension Scheme to those which lack second-order quantifiers. That's fine. But there in four black and white words it also says -- as if it is the same thing -- that the φ(x) must belong to LA (the language of first-order arithmetic). Which is of course plain wrong. The φ(x) might contain second-order free variables/parameters.

Aaargghh! How on earth did that stray false clause get in? Checking an earlier version of the book, it wasn't there: so it must have been a later 'helpful' addition!

The psychology of this kind of "thinko" (I can hardly plead that it is a typo!) is intriguing. How is it possible to write, and then no doubt let pass on another reading or two, something you know perfectly well to be false? Sigh.

Sunday, August 19, 2007

Forthcoming book of interviews on philosophy of mathematics

I stumbled across a link to this, announcing a forthcoming book Philosophy of Mathematics, 5 Questions, in which a pretty impressive line-up of people (from Jeremy Avigad, Steve Awodey, John L. Bell alphabetically through to Philip Welch, Crispin Wright, and Edward N. Zalta) respond to five questions about the philosophy of mathematics. Some quite extended excerpts from answers are available on the website, and they indicate that the questions posed were these: "1. Why were you initially drawn to the foundations of mathematics and/or the philosophy of mathematics? 2. What examples from your work (or the work of others) illustrate the use of mathematics for philosophy? 3. What is the proper role of philosophy of mathematics in relation to logic, foundations of mathematics, the traditional core areas of mathematics, and science? 4. What do you consider the most neglected topics and/or contributions in late 20th century philosophy of mathematics? 5. What are the most important open problems in the philosophy of mathematics and what are the prospects for progress?" Some authors seem to have answered point by point, others written reflecting more generally. It looks as if the result will be very readable and will provide an interesting snap-shot of the state of the philosophy of mathematics at the moment.

The book is announced for October: no price seems to be given, though previous books in the series have been very inexpensive.

Thursday, August 16, 2007

Gödel forums

The comment on the last post prompts me to get round to doing what I've been meaning to do for a while. I've set up a forum at godel.freeforums.org for discussion of my Gödel book. This is very much an experiment; and I really have no clue how things will go. So we'll just have to see how things develop!

Eat your heart out again

As a bit of a break from absolute generality, I've just started reading Melvin Fitting's short book Incompleteness in the Land of Sets. I'll write some comments here when I've finished it (which shouldn't take long, as there are just 134 pages before the endmatter): but the book so far promises to be absolutely excellent.

Meanwhile, for anyone passing by Siena this summer (and to make Tim Crane even more envious), a couple more recommendations. An old favourite is Bottega di Lornano (though be warned, locals out for a special meal seem expected to have huge appetites): the pici with a pork ragu with fennel seeds and pine nuts is amazing. While twenty yards from our door, La Porta del Chianti is less hardcore, seems to have more of a tourist clientele in the summer, but is still pretty good and they care a lot about their wines.

It will have to be marmite on toast for a fortnight to recover ...

Monday, August 13, 2007

Dievole

Normal philosophical service will be resumed soon. Too soon ... But meanwhile, if you get a chance to sample Dievole's "Novecento" Chianto Classico riserva 2003, then you really should splurge out on a bottle! (Recommended after an excellent afternoon's wine-tasting at their stunningly situated estate: their other wines are excellent too.)

Tuesday, August 07, 2007

Absolute Generality 7: Hellman on talking donkeys

The final section of Hellman's paper is called 'Making do with "less"', and concerns strategies that the sceptic about the coherence of absolutely general quantification can use to makes sense of (true!) assertions like 'there are no talking donkeys' or 'there are no gods' which seem on the face of it to make absolutely general claims.

I found his discussion murky. (At least as far as philosophy is concerned, I'm with Isabella Dale in The Small House at Allington [ch. xliv]: 'I hate books I can't understand,' said Bell, 'I like a book to be as clear as running water, so that the whole meaning may be seen at once.') But the main point Hellman makes, if I'm understanding him aright, is what strikes me as the obviously right point. 'There are no talking donkeys' stands or falls with 'No animal is a talking donkey', and there are no problems about the restricted quantification involved there. Issues about indefinite extensibility are beside the point because 'animal' (unlike 'ordinal' or 'set') is not indefinitely extensible; and issues about relativity to alternative conceptual schemes are beside the point since, in talking about donkeys at all, we are already talking within a certain scheme that recognizes animals.

'There are no gods' can't be handled quite so straightforwardly (pace Hellman): but not because of issues about absolute generality so much as because of issues about the lack of clear content of 'gods'. (Who knows what a boojum is? Especially if boojums are described as having all sorts of daft and seemingly incompatible properties. If I then impatiently say there are no boojums, I'm not making a bold speculation about the contents of the universe but rather rejecting -- though perhaps not in the most transparent way -- the presupposition that there is any clear content to claim that there are boojums. It is pretty similar with gods.)

Sunday, August 05, 2007

Negative Type Theory

Stephen Simpson repeatedly talks about certain subsystems of second-order arithmetic as 'natural' or as 'arising naturally' (SOAS, e.g. pp. 33, 43, etc.). In a similar context, John Burgess contrasts 'artificial examples' with 'theories that it is natural to consider' (Fixing Frege, p. 54). But what idea of naturalness is at work here? There's more than one sort of naturalness that can be in play when we talk about mathematical theories. Which is a trite point, but which bears some discussion. In fact, there's a number of things to be said. But here's a nice case which is perhaps relatively unfamiliar but which very vividly illustrates one basic, preliminary, distinction we need to draw.

As background, recall the structure of a simple theory of types. The entities in its universe are divided into levels. At level 0, there are individuals. At level 1, there are sets of individuals. Then at level 2, sets of level 1 sets; at level 3, sets of level 2 sets; and so on up an unending but non-cumulative hierarchy. The two key principles structuring this hierarchy are an extensionality principle for sets,and a comprehension principle to the effect that given any condition satisfied by zero or more level n − 1 entities, there exists a level n set containing just those entities.

Now, to get an interesting system in which we can actually do some mathematics, we'll also need to add a third independent principle governing the hierarchical universe, an axiom of infinity which tells us that there is an infinite set. And why do we need a substantive axiom of infinity? Because, to labour the obvious point, there are only a finite number of predecessor levels below any given level in the hierarchy. So if there are only finitely many individuals at ground level, then – although the population at each succeeding level grows exponentially – when we reach the given level the population need still only be finite.

So far, all so very familiar. But the alert mathematician might pick up on the point I've just laboured, and wonder whether we can't after all get a type theory which guarantees infinitely populated levels by changing our structural assumptions about the levels, and imagining the hierarchy continuing infinitely downwards as well as upwards.

So now think of the denizens of level 0, and the lower levels, as more sets. It is easy to see that comprehension alone will guarantee an infinite population at every level.For example, comprehension guarantees that there is an empty set ∅n at level n. And so, at level zero, again by comprehension, there are all the following sets: ∅0 , {∅−1 }, {{∅−2 }}, {{{∅−3 }}}, . . . , which by extensionality are all distinct. So we've conjured infinite populations just out of the structure of the hierarchy, without making any additional substantive assumption. We've shown how to get more out of less.

Now, that is indeed very cute! And, after all, if we are considering a typed universe, with the levels indexed by N, what – in one good sense – is more natural than to explore by way of contrast, on the one hand, a restricted universe with only a finite number of levels and, on the other hand, a richer universe with the levels indexed by Z? These formal variants on a standard simple theory of types immediately suggest themselves for mathematical investigation. But while the idea of a 'negative type theory' is a formally natural one – and there is indeed some fun to be had exploring the resulting theory, which can readily be proved consistent – it is of course conceptually highly artificial. For the original idea of a type hierarchy, after all, involved the conception of building from the ground up by repeatedly applying an operation of set-formation. Remove the distinguished ground level and it seems that we can make no conceptual sense of what is supposed to be going on in the resulting hierarchy. And we can make even less sense of the theory when we find e.g. that, by a little theorem of Thomas Forster (1989), all its models are nonstandard in the sense that the population of level n + 1 is not reliably of the expected standard size in a type theory, i.e. not the size of the power set of level n.

This lack of a clear conceptual motivation is presumably why Hao Wang, who first noted the possibility of a negative type theory in his (1952), calls it 'a kind of curiosity'. So let's take this as a cautionary tale, to illustrate the contrast we surely do need to draw between what we might call formal naturalness and conceptual naturalness. Or, as we might be tempted to put it in some cases, between purely mathematical and philosophical naturalness. Though note, talking the second way might suggest that we are presupposing that there's a place for a 'first philosophy', critically assessing mathematics from outside. But that, I'd argue, is the wrong picture: conceptual naturalness is itself already one kind of mathematical virtue, well recognized in the actual practice of mathematicians.

It seems rather easy to be seduced by a Bourbarkiste fantasy, and to fall into thinking that mathematics – real mathematics – is, or ought to be, an entirely formal game, with theories evolving purely by linear deduction from definitions and axioms. But as Imre Lakatos, for one, reminds us, that isn't how mathematics in fact proceeds. There is, rather, more of a to-and-fro between intuitive ideas, analogies, pattern-recognitions, informal proofs, and the development of sharpened concepts and the more rigorous proofs which deploy them (though sufficient unto the day is the rigour thereof ). The conceptually unnatural theories are the ones where, in the process of formalizing, we lose touch with, or distort too far, the informal mathematical ideas that were originally supposed to be guiding the theory-building. More on all this anon.

Wednesday, August 01, 2007

Bella Toscana

The view from the kitchen table from which I'm blogging for the next week or two ...