Monday, June 30, 2008

An Introduction to Gödel's Theorems revamped!

I've just, at last, got hold of a printed copy of the revised version of An Introduction to Gödel's Theorems which came into stock with CUP about six weeks ago. Of course, as soon as it went to press, three or four people told me of errors that still need correcting! -- but this version has significantly fewer typos, and also a small number of passages have been improved. The corrections page at the book's website will tell you what's been changed, and what still needs changing.

(So, tell your libraries that they just must have the shiny new improved version! And to answer Richard Zach's question -- the quick way of telling the versions apart is to glance at the imprints page, i.e. the verso of the title page. The later version notes, halfway down the page, "Reprinted with corrections 2008''.)

Saturday, June 28, 2008

Galois connections again

For some reason, I found myself minded to spend a bit of time tidying up the pages on posets and Galois connections that I was working on some weeks ago. They have the form of the first two chapters of something longer, something that may or may not get written -- but these pages should have some stand-alone interest. Here's the new version. As always, comments gratefully received. (Thanks so far to Tim Button and Luca Incurvati -- and in particular to Nathan Bowler who gave a talk in which he explained the basic idea of syntax/semantics as a Galois connection, which inspired me to write these pages.)

[Later: Monday 30th June] And that link is now to a better version, with some paragraphs about quantifiers as adjunctions, and some superfluous material removed for later use.

Wednesday, June 25, 2008

The Review of Symbolic Logic

The first issue of The Review of Symbolic Logic is out and available online (if your library has the right subscription). And two Cambridge friends have papers in it, both on set theory. Luca Incurvati has a piece on Kripke semantics in set theory, and Thomas Forster discusses the iterative conception. Excellent stuff.

Conference: Computation and Cognitive Science

There's an upcoming conference in Cambridge on Computation and Cognitive Science with an impressive line-up. The plan is for the papers to be made available in advance, for there to be no formal presentations, but for the sessions to be devoted to discussion. The papers are beginning to appear online here.

Saturday, June 21, 2008

Back in Cambridge. Sigh.

So this is the borgo we've left behind for a while (the little white patch in the centre distance in the centre of the photo is the distant dome of the duomo in Siena, or rather its covers during restoration works). Ah well. Back in September, we hope. It has taken us a few days to re-adjust to Cambridge (which was particularly grey and damp when we returned). But the sun is out today, and the place is almost at its best, so that is cheering. Logic postings will resume here in the next day or two, now I'm back in the mood again!

Tripos results came out, both for philosophy and for maths, just as I got back, posted on the Senate House boards. It was good to see that all the names I looked for of students I thought ought to get a first were there in the right class. Justice, of course, is always done ...

Monday, June 16, 2008

Postcard from Siena - 8

The meteo predicts that really good weather will start on Thursday. Since we are leaving on Wednesday, this is just a bit galling. This morning it was so cold we put the heating on again. And jazz last night in the little village piazza under our window was good, but not the balmy June night under the stars we might have expected, and the well-wrapped-up audience was understandably a bit thin.

Siena itself is like Cambridge at least in this respect: the tourists tend to stick to a small part of the city. So it can be very busy round the Campo and the Duomo. But other sights, even those the guide books warmly praise, can be more or less deserted. We did make one nice discovery a couple of days ago when it was dry in the afternoon. We found ourselves at the botanical gardens which we'd never visited before (and, predictably, they were more or less empty of people). They are very fine, cool under the trees, tumble down a steep slope, and so the views out of the city are beautiful. Recommended.

Thursday, June 12, 2008

Awodey's Category Theory: Ch. 2

The second chapter of Awodey's book is called 'Abstract Structures'. It gives the usual abstract category-theoretic definitions of epis and monos, of sections and retractions, of initial and terminal objects, of products, and so on. This would certainly be tough going if it was the first time you'd ever encountered these notions. Even as revision/consolidation it's a bit of a bumpy ride. But for all that, I did get a fair bit out the chapter (Awodey's clusters of illustrative examples can be very illuminating).

One query. In the sections on products, Awodey starts carefully, talking of a product as an object together with a pair of arrows, and rightly referring to the object A x B as part of a product. And mostly what he says about products reflects this understanding of what products are. But on p. 42 he says that any object A is the unary product of A with itself one time. Is that right? The unary product is surely not just the object but the object with its self-identity arrow.

And one suggestion. The first stage of the two-stage proof at the top of p.27 is surely unnecessarily. Just start in f(-n) = f(-n) * u = f(-n) * g(0) = f(-n) * g(n + -n) etc. [Actually the first stage too illustrates one of Awodey's quirks, a tendency to occasionally slightly abuse notation without explanation.]

Postcard from Siena - 7

Here, everyone has to park outside the walls of the old part of the borgo. But that's no hardship. There's stone and gravel put down between the olive trees just under the house, and you park the car among them, leaving it to quietly admire the views for miles over the hills.

The trees have been brutalized since last year, obviously scaring the living daylight out of them, and as a result they are beginning to fruit like mad. I can report that the local olive oils vary, but from merely very good indeed to the amazing. (And judging from the ages on the gravestones in the village cemetery, they must have magically life-extending properties.)

Wednesday, June 11, 2008

Parsons's Mathematical Thought: Sec. 7

Back in Sec. 1, Parsons says "Roughly speaking, an object is abstract if it is not located in space and time and does not stand in causal relations." In the last section of the first chapter, he returns to question of characterizing abstract objects, and suggests a distinction among them between pure abstract objects (e.g. pure sets) and those which "have an intrinsic relation to the concrete" -- Parsons calls the latter quasi-concrete.

As a paradigm example of the quasi-concrete, Parsons takes the example of sentence types: "what a sentence [type] is is a matter of what physical inscriptions are or would be its tokens". (Actually, just as an aside, I suppose we might wonder whether sentence types might be a counter-example to the claim that abstract objects lack temporal location. We might ask: did the sentence type "the cat is on the mat" really exist in 2000 BC before anyone spoke English?)

But how should we generalize from this case? Parsons writes "What makes an object quasi-concrete is that it is of a kind which goes with an intrinsic, concrete 'representation'". The scare quotes are there in Parsons -- and you can see why. Should we really say, for example, that a sentence token is a representation of its type? Your first response might be: the token isn't about the type, so isn't a representation of it. But, reading on, it becomes clear that Parsons doesn't mean representation but representative. And then, yes, we might say that the token is a representative of the type. Parsons also writes "Although sets in general are not quasi-concrete, it does seem that sets of concrete objects should count as such; here the relation of representation would be just membership." (no scare quotes!). Again, we might say the spoon in my coffee cup is a representative of the set of cutlery (though not a representation).

How clear is the idea of "having a concrete representative"? You might have supposed that the Earth's equator is a candidate for belonging with sentence types as tangled with the concrete. But does the equator have a concrete representative? Could it? What about that old Fregean example, the direction of a line. Of course there can be physical lines with that direction; but it doesn't seem quite natural to me to say a particular line is a representative of the direction. (We might say the equator or a direction could have a representation, painted on the ground!)

Parsons's discussion here thus seems to me to be rather undercooked. To be sure, it is plausible to say that some abstract objects are more purely abstract than others, but I don't think he has given a sharp characterization of the phenomenon.

But let's go, for the moment, with his notion of the quasi-concrete. Then he raises the question, are numbers quasi-concrete? We might be tempted to say yes, suggesting that the number five, for example, has the concrete representatives like: ||||| . Parsons makes two Fregean points against this. First, to take that block as representative, we have already to take it as a set or sequence of strokes (rather than as a single grid, for example). So the representative here is not strictly concrete but itself quasi-concrete. Perhaps then we can say that numbers are quasi-quasi-concrete (meaning they have quasi-concrete representatives). But second, that can't be the whole story, as numbers can number anything, including the purely abstract. (Parsons says he is going to return to talk about this in Chapter 6, so I'll say no more for the moment.)

Tuesday, June 10, 2008

Postcard from Siena - 6

It's sunnier and warmer (for a while). This year the excellent restaurant just a few steps across the piazza has put a few tables outside, and will bring you a coffee and cornetto from when they open up in the morning, or an aperitivo in the afternoon. A great idea, but so far the weather has been such that we've only made use of it a few times. But this morning, sitting in the sun at half-past nine, it was already pretty hot. At last.

Parsons's Mathematical Thought: Sec. 6, 'Being and existence'

At the outset of this section, Parsons writes that one point at which "reservations about standard first-order logic as the universal measure of ontology can affect the notion of mathematical object is the ancient question whether reference to objects is necessarily reference to objects that exist."

A comment before proceeding. Note that Parsons had earlier (Secs 1 to 4) proposed that (1) "speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification" to make serious, and indeed true, statements. And defending that view about, so to speak, the measure of what objects we are committed to falls short of saying that (2) standard first-order logic is the universal measure of ontology in general. Resisting the more sweeping claim is quite consistent with accepting Parsons's initial Fregean claim about objects. Not that I'm suggesting that Parsons thinks otherwise. I'm just emphasizing that if (e.g. as a Fregean) you are not persuaded by Parsons Sec. 5 suggestions, and hold that we are committed to entities that are not objects, then you can accept formulation (1) without accepting (2).

Anyway, what of reference to objects that is not reference to objects that exist? Parsons discusses Meinongian views in some detail (this is one of the longest sections in the book). Here's part of his final summary of the discussion.

We are left with the question whether the "true" meaning of the existential quantifier is [i] the permissive Meinongian one [allowing quantification over objects that do not exist], [ii] existence that allows freely for abstract objects but that rules out impossibilia, or [iii] something like actuality. The logic based concept of object does not decide between these alternatives, although, once it has been set forth, the case for [iii] is weakened. But in order to understand the notions of object and existence in mathematics we have to put more flesh on the bare form given by formal logic. We need to fill out the logic-based conception by looking at cases. ... [C]onsiderations proper to mathematics will not lead us to favour [i] over [ii]. General as the notion of object in mathematics is, there is still a constraint of possibility, coherence, or consistency that objects postulated in Meinongian theories are allowed to violate.

The talk here of having to "fill out the logic-based conception" might initially seems surprising given what has gone before. But, though he is not entirely clear, I assume that what Parsons means is simply this: the Fregean thesis is that objects are just whatever are we have to construe terms that behave in the right sorts of way in true sentences as referring to. So, to fill out that general template view about objects, we have to say what kinds of sentences we do in fact accept as being true. If we e.g. take statements like "Sherlock Holmes is more famous than any living detective" and "There's a fictional detective who is more famous than any living detective" at face value as true claims then (the suggestion goes) we have to accept (i) the Meinongian line that there are objects that do not exist. If we paraphrase away apparent talk of fictional objects and the like, but accept that there are true mathematical statements talking of numbers, sets, etc., then (ii) we are not committed to non-existent objects, but have to accept that there are abstract objects which aren't "actual". If we insist on also paraphrasing away apparent straight talk of numbers (e.g. construing it as governed by an operator "in the arithmetical fiction ..."), then perhaps (iii) we may only be committed to actual objects.

Parsons is sceptical about whether we have any need "to admit into the range of our quantifiers such objects as the golden mountain, the round square, Pegasus and Sherlock Holmes", though it is not his concern to argue for this here. But he does argue that "considerations proper to mathematics" don't give any impetus for preferring the Meinongian views (i) over (ii). Mathematics doesn't countenance impossibilia like the round square, or present itself as fictional discourse. As to (iii), I assume Parsons thought is that a critic of our common-or-garden standards of mathematical truth on the basis of a metaphysical repudiation of abstract objects is (in danger of) getting things upside down, at least by the lights of the truth-first, "logic-based conception" of objects, according to which we don't have a handle on the notion of an object except via a prior grip on the notion of truth for the relevant object-referring statements.

If this reading of Parsons is right, then I agree with him.

Sunday, June 08, 2008

Postcard from Siena - 5

We normally never watch breakfast TV, but here we have the excuse of trying to pick up more Italian: and actually it isn't at all bad. The weekend show we watch has a nice slot visiting different places around Italy and talking at length about their local produce, and demonstrating a characteristic recipe. That -- followed by walking through the woods onto the estate of Villa Arceno and alongside their vineyards -- worked up appetites for Sunday lunch at a favourite restaurant, La Bottega di Lornano. But by then the weather was getting too threatening again to eat outside (even under their big awning). Still, a terrific meal as always, in Tuscan quantities, and we drank a favourite wine, Dievole's Broccato. Prices in Italy are going up, and the pound is going down against the euro, so this is not quite the stunning bargain it would have seemed three years ago. But we still ate much better for less than the cost of a second-rate chain restaurant meal in England. Which is why we very rarely bother to eat out at home.

Awodey's Category Theory: Ch. 1

I bought Steve Awodey's book Category Theory (Oxford Logic Guides, Clarendon Press, 2006) when it first came out. Awodey says that his book is aimed, inter alia, at "researchers and students" in philosophy; I'd been impressed and intrigued by a couple of his lucid contributions to Philosophia Mathematica, and had hoped for an equally approachable book. But, whatever Category Theory's virtues, easy approachability isn't one of them, and after reading a fair bit of it, I had to put the book aside for when I had enough time to work through it again more slowly. At last, I've got back to it, and I'll give some reactions here.

I have to say immediately (as in fact I said here before) that I can't imagine that there are many philosophers who would be equipped to dive straight in and cope with this book. Meeting Cayley's Theorem (about representing groups as permutations on sets) at p. 11 or free monoids at p. 16 is going to be quite a challenge to those without a background in mathematics. It isn't that those ideas are intrinsically very difficult; but you surely won't grasp their point or feel comfortable with the ideas just from their brisk presentations here. Likewise, I bet no one will understand Remark 1.7 (p. 12) on concrete categories who hasn't already met the idea of "test objects" from elsewhere. By the time the reader gets to the first example of a "universal mapping property" at pp. 17-18, most philosophers surely will be floundering: Awodey's explanations of what is going on are too terse to help the not-so-mathematical. And things seem only to get worse as the book progresses. I'm pretty sure, then, that this book wouldn't work as a first introduction to category theory e.g. for philosophy graduate students interested in logic and the philosophy of maths (unless they have an unusually strong background in pure maths already). Although Awodey says in the preface that, if Mac Lane's book is for mathematicians, his is for 'everyone else', in fact Category Theory is actually orientated to students who are, as they say, 'mathematically mature'.

So, from now on, I'll be taking the book as in fact operating at (so to speak) a level up from the one Awodey says that it is designed for, i.e. as a follow-up text for mathematically ept readers, to read after mastering e.g Lawvere and Rosebrugh's Sets for Mathematicians -- a follow-up which starts again from scratch to consolidate some basic ideas and then pushes things on deeper and further.

How does the introductory first chapter work on this level? Well, to be frank, still not entirely brilliantly. For example, the whys and wherefores of the first example of a universal mapping property are not really explained that well (nor why we should be particularly interested in free categories). However, on the other side, I like the way that the idea of a functor between categories is introduced early; and some of the illustrative examples of categories and functors between categories in the chapter are illuminating. And the idea of "forgetful functors" comes across nicely.

Saturday, June 07, 2008

Parsons's Mathematical Thought: Sec. 5

Parsons has been proposing the view that "speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification". And the focus so far has been on first-order quantification. But what about generalizations about properties, the sort of generalization involved in familiar mathematical statements like the induction principle for arithmetic, or the separation axiom in set theory? Should we construe those as involving generalization over something like Frege's "unsaturated" concepts, entities which aren't objects? Or is the commitment here just to more objects? I'll try to outline some of Parsons's discussion (though I did not find it always easy to construe).

One way of perhaps resisting the Fregean line arises from noting that we can easily parlay quantification into predicate position into just more quantification into subject position (or so it seems). Suppose, using Parsons's notation, we use '(Ox)Fx' to denote some object corresponding to the Fregean concept expressed by 'F...'. And suppose we use '$' for an appropriate copula ('has' if the object is a property/quality, 'is a member of' if the object is a set, etc.) Then we have Ft if and only if t $ (Ox)Fx. And so, given a context when we are minded to quantify into the position held by 'F' we could instead first nominalize and then quantify into the position held by the singular term '(Ox)Fx' instead. It seems then that we can treat quantification over properties (as we might initially put it) as just more quantification over a kind of object. This after all seems common mathematical practice, as when we familiarly regiment second-order arithmetic as a theory of numbers and sets of numbers.

Still, at least two objections to the nominalizing strategy as an across-the-board way of eliminating 'direct' quantification into predicate position readily suggest themselves (as Parsons notes). First, the claim that Ft if and only if t $ (Ox)Fx is, itself, intended as a generalization, to express which we need to generalize into predicate position in a way that can't be nominalized away. And second, that generalization in any case has to be restricted or else or we could instantiate with the predicate '¬x $ x', and paradox ensues.

However, that's not yet game set and match to the Fregean. Can't the force of the first objection be turned by adding the device of semantic ascent to our armoury? We can, for example, generalize about the possibility of nominalization by saying that for any predicate 'F' (and term 't'), 'Ft' is true if and only if 't $ (Ox)Fx' is true.

Ah, it will be protested, the device of semantic ascent still doesn't really allow us fully to capture what we want to say by means of quantifications over properties. Compare for example the familiar thought that the content of the full informal arithmetic induction axiom is not captured by semantically ascending and saying that all instances of the first-order schema are true. Reply: that familiar thought is true, if we confine the instances to a fixed language. But suppose we treat the schema in an open-ended way, available to be instantiated however we extend our language (as Parsons puts it, "In practice, in any language in which we talk about natural numbers, we are prepared to affirm induction for any predicate of that language"). Then, by treating the schema as open-ended we arguably recapture the intended sweep of the informal axiom still without taking on ontological commitments to Fregean concepts.

And as to second objection against the nominalizing strategy, the threat of paradox only arises if we take the reference of '(Ox)Fx' as an object that is, so to speak, already in the original domain of objects (i.e. of subjects of predication). But we could take the moral here to be that objects segregate into different types, the references of nominalized predicates being of a different type to the references of common-or-garden singular terms.

So where does this take us? Parsons summarizes: "the present discussion does show that considerations about predication do not lead inevitably to our taking second-order logic as our canonical framework and admitting, as values of our second-order variables, entities that are not objects."

Three comments about all this. First, about semantic ascent and the open-ended nature of our commitment e.g. to the induction schema. Just why do we stand prepared to take on all-comers and instantiate the schema with any novel predicate we care to extend our language with? Kreisel suggested long since that we accept the instances of the induction schema because we already accept the full second-order induction axiom. I think there are issues about that claim (which I can't pursue here and now). But the claim is a familiar one that many have found persuasive. And a fuller defence of the idea that we can avoid taking second-order quantifications at face value would require Parsons to say more about this.

Second, about avoiding paradox on the nominalizing strategy. The Fregean might well riposte that saying that the way to go is to segregate objects into different types just sounds like theft of Frege's key insight rather than an alternative story. After all, speaking with the vulgar, the Fregean will say that what he is arguing for is precisely a distinction among "entities" between saturated and unsaturated types, between objects and concepts. So he has a principled type story to tell. And, he will add, once the distinction is made in the right way, the temptation to pursue the nominalizing strategy, putting all the work of unifying propositions into a copula, should evaporate. And what is the alternative principled story supposed to be?

Third, I'm left unclear exactly how Parsons thinks about the relationship between the two ways of avoiding second-order quantification that he discusses (i.e. the routes via nominalization and ascent). He does say that "The laws of logic have a certain dialectical character, in that the method of nominalization and the method of semantic ascent can both be used to state them, and neither can completely displace the other." I've wrestled with this a bit, and I don't have a clear grasp of the point. (And helpful comments on that here would be welcome!)

Friday, June 06, 2008

Postcard from Siena - 4

I need to reread Parsons's next section before posting on that, as I'm not sure I have the measure of it (his book is evidently the distillation of a lot of thought over a long time, so it isn't going to make for a quick read).

Meanwhile, we went yesterday to the Archivio di Stato in Siena (which does guided visits three times a morning). The interest there -- apart from the great ranks of volumes of documents -- is an exhibition of the Tavolette di Biccherna. These are painted wooden panels that were produced as covers for bundles of civic account books, starting in 1258 with the practice continuing to the eighteenth century. The earlier ones, in particular, are fascinating (particularly interesting to see secular art of the time). Very definitely worth a visit: we enjoyed it great deal. There were exactly two other people there when we went.

Thursday, June 05, 2008

Postcard from Siena - 3

Even in rainy first light, Tuscany is beautiful, and the views from our windows remain wonderful. But yesterday started bright and dry, and we set off over the hills via Asciano by backroads to Montalcino to meet up with a friend. The sign-posting of Italian roads is characteristically awful even on major roads. And they here don't seem to have an equivalent either of the wonderful Ordnance Survey maps. So using backroads has in the past been a recipe for getting lost, getting cross, and (shall we say) disharmony in the car. But we've the use of a sat-nav system for a while. It quite unproblematically got us took us via a very circuitous route which we'd never have attempted before, well off the beaten track, over the hills south-east of Siena. I'm an instant convert: a must buy for Italy.

As always, the last mile or so climbing up to Montalcino itself was stunning, worth the journey in itself. (And if, when you get there, you want somewhere to eat more than a snack but less than a big lunch, try the Enoteca Osteria Osticcio in Via Matteoti. The tables inside at the back have the most wonderful panoramic views over miles and miles of countryside, the people are friendly, and food very good.)

Wednesday, June 04, 2008

Parsons's Mathematical Thought: Secs. 1-4

Right, as promised, time to make a start commenting on Charles Parsons's long awaited Mathematical Thought and Its Objects (CUP, 2008).

For those who haven't had a copy in their hands, this is a pretty substantial volume (pp. xx + 378). Its chapters extensively "draw on", "incorporate material from", "overlap considerably with", or "are expanded versions of" papers published over the last twenty-five or so years; but glancing ahead the material indeed seems to be reworked into a continuous book. The nine chapters are divided into 55 sections numbered continuously through the book, and those divisions will be very handy here: I'll aim to comment on small groups of sections (from one to three or four) at a time. From what I've seen so far, the book needs and repays slow reading.

Chapter 1 is entitled Objects and Logic. And the claim to be defended is that "Speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification to make serious statements". Thus construed, the idea of objects in general is loosened from ties with the idea of actuality (Kant's Wirklichkeit) -- where this has something to do with "act[ing] on our senses or at least producing effects which may cause sense-perceptions as near or remote consequences" (to quote Frege). Talk of objects is also loosened from ties with ideas of intuitability (whatever that Kantian idea comes to: things are left pretty murky at this stage, but then Parsons is going to talk a lot about intuition later in the book). Consequently, endorsing the logical conception of an object will "defuse too-high expectations of what the existence of objects of some mathematical type such as numbers would entail." The suggestion is that those who are inclined to deny abstract objects, or find them puzzling, are illegitimately(?) imposing requirements on being an object that go beyond those captured in the logical conception.

Now, I'm entirely sympathetic to the Fregean line Parsons is following here. He says that "its most important advocates in more recent times are Carnap and Quine". But I would have added Dummett's name to the list, starting with his early paper on nominalism: and Dummett initiated the most sophisticated development of the Fregean line in the hands of Crispin Wright in his Frege's Conception of Numbers as Objects, and then particularly Bob Hale's Abstract Objects (neither of which Parsons mentions here).

I'm not sure, though, in quite what spirit Parsons is proposing "the view that the most general notion of object has its home in formal logic".

Actually, as an aside, I'd remark that that surely isn't the happiest way of summing up the view. After all, suppose we translate back from first-order logical notation into a disciplined core fragment of English -- the sort of regimented English whose sentences are equivalent to the content of the logical wffs (and indeed the sort of English which we use in giving determinate content to the artificial language in the first place). Then here too we will find the core devices of singular terms, predication, identity and quantification. And the Quinean will presumably say that our commitments to objects are revealed equally well by rendering our theory of the world into the idioms of this disciplined core of ordinary language. Or if that's not exactly right, because we can never quite discipline English enough (e.g. we can't quite ensure that "It is not the case that ..." always expresses propositional negation), then this is not, so to speak, a deep failing of the vernacular. Formal languages don't magically do what ordinary language can't do: they just do ordinary things like use singular terms and quantify in tidier ways. So turning to "formal logic" doesn't really give us a different take on the general notion of object. Surely Parsons spoke better when he expressed the position he is proposing as the view that "speaking of objects just is using the linguistic devices of singular terms, predication, identity and quantification" to make serious, and indeed true, statements.

But to continue, as I said, I'm not sure in quite what spirit this view is being advanced. The fully Fregean line would be to insist that objects are what are referred to by singular terms in true sentences, and a singular term is whatever walks, quacks, and swims like a singular term in a disciplined way. We can't first pick out a class of genuine objects and then locate the genuine singular terms as those that refer to them: it goes the other way about (e.g. from identifying true sentences by the appropriate mathematical criteria, via identifying the singular terms in those sentences by their compositional behaviour, to insisting that those singular terms functioning in truths refer to mathematical objects).

But suppose you rejected that line. You might still think, in a Quinean spirit, that such is the mess and conversational plasticity in our various ordinary ways of talking that to determine when we are committed to objects of one kind or another, the best thing to do is to see how things look when we regiment our claims into a well-understood disciplined core discourse of singular terms, predication, identity and quantification -- the apparatus formalized in first-order logic.

A couple of Parsons's remarks suggest the stronger and more contentious Fregean line. But then it is perhaps odd that he doesn't more explicitly argue for it, and engage with the Dummett/Wright/Hale defence.

Monday, June 02, 2008

Postcard from Siena - 2

In the little piazza beneath our window, children have been celebrating their first communion. Being Italy, the occasion is marked before and after by a lot of noise, clanging bells and a brass band, and the inevitable gathering for food and wine. There are proud parents and grandparents, and the youth of the village dressed more for a party than for a solemn occasion. No doubt, it all means different things to different people: but these occasions are just part of village life, and I suspect that many of the participants are just comfortable through long familiarity with participating in religious services (with more or less regularity, more or less enthusiasm), and don't worry too much about what it all means. It is what you do, and it ceremoniously links the occasions of life with the eternal verities.

It's a salutary reminder for philosophers who are wont to over-intellectualize religion. That was part of my beef about the Murray/Rea book on the philosophy of religion which I blogged about here. They seem to take Christian belief, at any rate, as essentially replete with detailed metaphysical commitments (commitments articulated by early Councils of the Church imbued with late Greek philosophy), and so feel that defending the coherence of religious practice involves having to dodge and weave their way through some pretty murky metaphysics. Somehow I don't think that the local signore going along to say their rosary think of it quite like that.

Anyway, back to logic. I've pretty much finished correcting IFL: I'll just make a new pdf file of the whole book from the FrameMaker files and check through that again and then I'll send it off and forget about it. So there's time for the serious stuff again -- after all, I am supposed to be on sabbatical research leave! I've brought a couple of books with me for when I'm in a logical mood, Steve Awodey's Category Theory (because I want to give it a second chance and get another perspective of the role of the concept of an adjoint functor in category theory), and Charles Parsons's Mathematical Thought and Its Objects (because I have agreed to write a critical notice of it). So the plan of action is to comment a bit here on the Awodey book -- but just as a consumer, so to speak, representing one segment of his target audience (i.e. someone who knows a little logic and wants to get to know a bit more about category theory). And I'll start blog-reviewing the Parsons book too, which looks as if it should be a pretty rewarding read. So watch this space.