What strikes me as I wade through Part II tripos papers (well, the thing that strikes me that it wouldn't be out of place to comment on, here and now) is that students are tending to write less than they used to. And I suspect that part of the explanation is this: few students actually ever write at length by hand any more, from one examination season to the next. Weekly essays are invariably word-processed; note-taking in lectures is a dying art (since people often give copious handouts, and students can then just make short scribbles on those); more and more students take their laptops to the libraries to make notes there. So the business of sitting down for three hours pushing pen across paper, fast and furious, must be an unaccustomed physical challenge, for a start. And it’s no doubt even more of a challenge to compose something straight onto the page in a quite different way to what students are now used to. I certainly wouldn't like to have to do it.

But of course the shorter an answer, the more difficult it is to shine (especially if an answer starts with a bit of routine exposition). Heaven knows what we do about this: but we are surely -- sooner rather than later -- going to have to come up with a system that doesn't quite so favour those who have happened to acquire the philosophically irrelevant antique skill of being able to write fast.

## Wednesday, May 30, 2007

### Hand engraved examination scripts

Posted by Peter Smith at 9:23 PM 6 comments

## Tuesday, May 29, 2007

### Church’s Thesis 7: Physical computability

As light relief from tripos marking, back to commenting on two more papers in the Olszewski collection: “Church’s Thesis and physical computation” by Hartmut Fitz, and “Did Church and Turing have a thesis about machines?” by Andrew Hodges. But I’ll be very brief (and not very helpful).

Both papers are about what Fitz calls the Physical Church-Turing Thesis (a function is effectively computable by a physical system iff it is Turing machine computable), and its relative the Machine Church-Turing Thesis (a function is effectively computable by a machine iff it is Turing machine computable). Hodges argues that the founding fathers, as a matter of historical fact, endorsed MCT. I’ve already noted, though, that Copeland in his piece has vigorously and rather convincingly criticized Hodges’s line, and I've nothing more to add.

Fitz, however, isn’t so much interested in the historical question as in the stand-alone plausibility of MCT and PCT. He covers a lot of ground very fast in his discussion: some of what he says I found obscure, some points look good ones but need more development, and I'm not sure I'm getting a clear overall picture. But in any case, I can’t myself get very worked about MCT and PCT once we’ve granted that neither is implied by the core Church-Turing Thesis, so I'm probably not paying Fitz quite enough attention. Anyway, I'm cheerfully going to skip on to the next papers.

Posted by Peter Smith at 9:00 PM 1 comments

Labels: Logic

## Monday, May 28, 2007

### Tesco’s shows unexpected taste and discrimination

Wow. Tesco Entertainment is selling my Gödel book. Look for piles by the checkouts when publication arrives! I’m booking a world cruise on the expected sales right now.

And on the same page they are advertising another ‘hot product’ (their words), Carol Vorderman’s Maths Made Easy: Ages 3-5, Preschool Shapes and Patterns. Well, I have tried to make my book reasonably accessible: but I'm not quite so sure the markets will overlap.

I think tripos marking is making me lightheaded.

Posted by Peter Smith at 10:30 PM 0 comments

## Friday, May 25, 2007

### At last, again ...

At long last, the final final version of my Gödel book went to the publishers this afternoon. I’ve been taking advantage of a quite unexpected chance to make changes until the last moment -- and I think that some that I’ve made in the last month have been quite significant improvements. But now I really do have to stop. And indeed a pile of tripos papers arrived to mark which will prevent me neurotically worrying about stuff that it is too late to change anyway!

It’s an odd feeling, finally letting go. On the whole,as I've said before, I’m pretty pleased with the result. To be sure, if I were starting afresh I’d handle a few things a bit differently; and it will be interesting to see if the few sections that still bug me as being really too skimpy technically or philosophically are the ones which reviewers pick up on. But I'm not sure that I could really have sorted them out without making a long book (already at the limits of CUP's patience) quite a bit longer. So the book will have to take its chances as is!

Phew! A glass or three of Caol Ila seems in order ...

Posted by Peter Smith at 8:50 PM 0 comments

## Wednesday, May 23, 2007

### The Symposium

A couple of days ago in an Oxfam shop I picked up a second-hand copy of a beautiful parallel text of the Symposium, with a racy and highly readable translation by Tom Griffith and wonderfully evocative wood engravings by Peter Forster. It’s a fun read (though I found the experience tinged with regret at the more or less total loss of my Greek).

But is it still philosophically important? Philosophical interest is not a timeless feature of a text, it seems. No doubt the Symposium is a great source for those looking for clues about the mores of ancient Athens (and that is a fascinating subject: put James Davidson's Courtesans and Fishcakes on your reading list if you don't know that terrific exploration). But does the Symposium really tell you anything serious about its ostensible subject, love? Indeed, how do you ‘philosophize’ about that? Read the English poets instead!

Posted by Peter Smith at 6:23 AM 3 comments

## Tuesday, May 22, 2007

### Church’s Thesis 6: Concepts, extensions, and proofs

Continuing my blogview of the papers in Church’s Thesis After 70 Years, I'll skip the contribution by Hartmut Fitz to return to later, and next look at Janet Folina’s ‘Church’s Thesis and the variety of mathematical justifications’ because I'm interested in her main topic, the variety in the idea of proof (I'm just looking one last time at the concluding few sections of my Gödel book, where I talk about this too).

Folina’s paper, though, gets off on the wrong foot. She writes: “Rather than a claim about mathematical objects such as numbers or sets, CT (insofar as it is an assertion) is a claim about a concept. Assessing it requires conceptual analysis.” Which makes it sound as if arguing about CT is like arguing whether the concept of knowledge can be analysed as the concept of justified true belief. But whatever the history of this, nowadays CT is, precisely, a claim about mathematical objects (in the broad, non-Fregean, sense of ‘object’): it’s the claim that a function is effectively computable if and only if it is recursive -- or equivalently, if and only if it is Turing computable. And the truth of that extensional claim doesn't depend on any claim about whether mere conceptual analysis can reveal e.g. that the effectively computable functions are Turing computable.

(Aside: In fact, it is as clear as anything is in this area that, pace Turing, mere conceptual analysis couldn't reveal such an equivalence, since there is nothing in the notion of an effective computation that demands that the ‘shape’ of our workspace stays fixed during a computation -- quite the contrary, we often throw away scratch working, reassemble pages of a long computation etc. -- whereas a Turing machine operates on a fixed workspace. It requires a theorem, not conceptual analysis, to tell us that what is computable in a more dynamically changeable workspace is also Turing computable.)

But as I say, Folina’s main remarks are about the notion of ‘proof’ involved in the majority claim that CT is unprovable, and in the minority claim that it is, or might be, provable. She insists that there can be mathematical reasons for a proposition that aren’t proofs, properly so called. Fair enough. But she seems to have in mind the sort of grounds we might have for believing Goldbach’s conjecture. And it isn't clear to me what she’d say about e.g. the diagonal argument that there are effectively computable functions which aren’t primitive recursive. This isn’t just a quasi-inductive argument, and we’d indeed normally announce it as a proof even though it doesn't fall under what Folina calls the “Euclidean” concept of mathematical proof, i.e. “a deductively valid argument in some axiomatic (or suitably well defined) system”, given that it involves the intuitive idea of an effectively computable function.

But say what you like here. For even if we allow Folina to reserve (hijack?) the term ‘proof’ for the “Euclidean” cases, the question remains in place whether there can be an argument for CT which is as rationally compelling as the argument that shows that there are effectively computable functions which aren’t p.r., or whether ultimately the grounds are more like the quasi-inductive reasons that support a belief in Goldbach’s conjecture. And that’s the real issue at stake about the status of CT (not whether we call such an argument, if there is one, a proof).

Posted by Peter Smith at 9:56 AM 2 comments

Labels: Logic

## Monday, May 21, 2007

### England in May

Often we think about going to live in Italy in a couple of years or so, but it would be a wrench not to see the wooded English countryside in May (we forgot to take the camera when walking last weekend in a quite unregarded area of Bedfordshire -- but it was stunning in its quiet way, as even a phone snap hints).

One of the less welcome delights of May, however, is tripos-marking, which starts for me at the end of this week. And before then, I have the tough task of ranking thirty six applications for the Analysis studentship, many of them impressive. So this blog will probably now slow down a bit after the recent flurry of postings.

Posted by Peter Smith at 12:27 PM 0 comments

## Sunday, May 20, 2007

### Church’s Thesis 5: Effective computability, machine computability

The sixth paper in the Olszewski collection is Jack Copeland's “Turing Thesis”. Readers who know Copeland's previous writings in this area won't be surprised by the general line: but truth trumps novelty, and this is all done with great good sense. To take up a theme in my last posting, Copeland insists that 'effective' is a term of art, and that

The Entscheidungsproblem for the predicate calculus [i.e. the problem of finding an effective decision procedure] is the problem of finding a humanly executable procedure of a certain sort, and the fact that there is none is consistent with the claim that some machine may nevertheless be able to decide arbitrary formulae of the calculus; all that follows is that such a machine, if it exists, cannot be mimicked by a human computer.

And he goes on to identify Turing's Thesis as a claim about what can be done 'effectively', meaning by finite step-by-step procedure where each step is available to a cognitive agent of limited (human-like) abilities, etc. Which seems dead right to me (right historically, but also right philosophically in drawing a key conceptual distinction correctly, as I've said in previous posts).

Copeland goes on to reiterate chapter and verse from Turing's writings to verify his reading, and he critically mangles Andrew Hodges's claims (later in this volume and elsewhere) that Turing originally had a wider thesis about mechanism and also that Turing changed his views after the war about minds and mechanisms. I'm not one for historical minutiae, but Copeland seems clearly to get the best of this exchange.

Posted by Peter Smith at 6:34 AM 0 comments

Labels: Logic

## Thursday, May 17, 2007

### Church’s Thesis 4: Computability by any means

The next paper is “The Church-Turing Thesis. A last vestige of a failed mathematical program” by Carol E. Cleland. Oh dear. This really is eminently skipable. The first five sections are a lightning (but not at all enlightening) tour through the entirely familiar story of the development of analysis up to Weierstrass, Dedekind and Cantor, the emergence of a set theory as a foundational framework, the ‘crisis’ engendered by the discovery of the paradoxes, Hilbert’s formalizing response, the Entscheidungsproblem as a prompt to the development of a theory of effective computation. No one likely to be reading the Olszewski collection needs the story rehearsing again at this naive level.

And when Cleland comes to the Church-Turing Thesis she without comment runs together two importantly different ideas. On p. 133 the claim is [A] one about the ‘effectively computable’ numerical functions -- which indeed is the version of the Thesis relevant to the Entscheidungsproblem. But by p. 140 the Thesis is being read as a claim [B] about the functions which are ‘computable (by any means)’. And these are of course distinct claims, requiring distinct arguments. For example, suppose you think that the kind of hypercomputation that exploits Malament-Hogarth spacetimes is in principle possible: then, on that view, there indeed can be computations which are not effective in the standard sense as explicated e.g. by Hartley Rogers, i.e. involving algorithmic procedures which terminate after some finite number of steps. And the questions we can raise about the Hogarth argument are highly relevant to [B] but not to [A].

Cleland’s last section offers some weak remarks about whether computation ‘by any means’ goes beyond Turing computability; but (I'm afraid) nothing here seriously advances discussion of that topic.

Posted by Peter Smith at 7:37 PM 0 comments

Labels: Logic

## Wednesday, May 16, 2007

### Paris 1967, Paris 2007

Well, time to turn to serious matters after such the logical diversions ... No doubt you've all supported the campaign to absolve Paris Hilton from her prison sentence: after all, in the words of the petition, “She provides hope for young people all over the U.S. and the world. She provides beauty and excitement to (most of) our otherwise mundane lives.” I couldn't have put it better.

But Guy Debord did, forty years ago (albeit in a French style which isn't quite mine!):

Behind the glitter of spectacular distractions, a tendency toward making everything banal dominates modern society the world over, even where the more advanced forms of commodity consumption have seemingly multiplied the variety of roles and objects to choose from. ... The celebrity, the spectacular representation of a living human being, embodies this banality by embodying the image of a possible role. As specialists of apparent life, stars serve as superficial objects that people can identify with in order to compensate for the fragmented working lives that they actually live. Celebrities exist to act out in an unfettered way various styles of living ... They embody the inaccessible result of social labour by dramatizing its by-products of power and leisure (magically projecting them). The celebrity who stars in the spectacle is the opposite of the individual, the enemy of the individual in herself as well as in others. Passing into the spectacle as a model for identification, the celebrity renounces all autonomous qualities ...

And there is much more in the same vein, in his La société du spectacle, which I've been looking at again (so long after my mispent youth in various leftist groups interminably debating such ideas). Paris 1967 anticipates Paris 2007.

Posted by Peter Smith at 7:58 PM 0 comments

### Church’s Thesis 3: Constructivism, informal proofs

The third, short, paper in the Olszewski collection is by Douglas S. Bridges -- the author, with Fred Richman, of the terrific short book Varieties of Constructive Analysis. The book tells us a bit about what happens if you in effect add an axiom motivated by Church's Thesis to Bishop-style constructive analysis. This little paper says more on the same lines, but I don't know enough about this stuff to know how novel/interesting this is.

The fourth paper is “On the Provability, Veracity, and AI-Relevance of the Church-Turing Thesis” by Selmer Bringsjord and Konstantine Arkoudas. At least these authors get it right about what the core Thesis is: a numerical function is effectively computable if and only if it is Turing-computable. But that's about as good as it gets. The first main section is a bash against Mendelson's argument against “the standard conception of the thesis as mathematically unprovable”. Now, although I am sympathetic to Mendelson's conclusion, I'd want to argue for it in a rather different way (and do so in the Gödel book). But Bringsjord and Arkoudas's objections just seem badly point-missing about the possibility of a Mendelsonian line. Their argument (p. 69) depends on a bald disjunction between proofs in formal systems and what they call “empirical evidence” for CTT. But of course, tertium datur. Take, for example, the familiar theorem that there are effectively computable functions which aren't primitive recursive. I'm not being tendentious in calling that a theorem -- that's how the textbooks label the result. And the textbooks, of course, give a proof using a diagonalization argument. And it is a perfectly good proof even though it involves the informal notion of an effectively computable function (the argument isn't a fully formalizable proof, in the sense that we can't get rid of the informal notions it deploys, but it doesn't just give “empirical” support either). Now, the Mendelsonian line is -- I take it -- precisely to resist that dichotomy formal/formalizable proof vs mere quasi-empirical support and to remind us that there are perfectly good informal proofs involving informal concepts (and Mendelson invites us to be sceptical about whether the informal/formal division is in fact as sharp as we sometimes pretend). And the key question is whether it is possible to give an informal mathematical proof of CTT as compelling as the proof that not all effectively computable functions are primitive recursive. Reiterating the rejected dichotomy is of course no argument against that possibility.

That's not a great start to the paper (though the mistake is a familiar one). But things then go further downhill. For much of the rest of the paper is devoted to a discussion of Bringsjord's claim that membership of “the set of all interesting stories" (!!) is effectively decidable but not recursively decidable (Gödel number the stories and we'd have a counterexample to CTT). And what, according to Bringsjord, is the rationale behind the claim that members of that set is effectively decidable? “The rationale is simply the brute fact that a normal, well-adjusted human computist can effectively decide [membership]. Try it yourself!” Well, you might be able to decide in many cases (always? just how determinate is the idea of an interesting story?): but who says that it is by means of implementing a step-by-step algorithm? Effective decidability is a term of art! It doesn't just mean there is some method or other for deciding (as in: ask Wikipedia or use a cleverly tuned neural net). It means that there is an algorithmic procedure of a certain sort; and in trying to judge whether a story is interesting it most certainly isn't available to inspection whether I'm implementing a suitable algorithm. This whole discussion just seems badly misguided.

Posted by Peter Smith at 3:26 PM 1 comments

Labels: Logic

## Tuesday, May 15, 2007

### Church’s Thesis 2: What’s an algorithm?

Andreas Blass and Yuri Gurevich’s paper “Algorithms: A Quest for Absolute Definitions” really covers too much too fast to be very satisfactory. The first part is a quick review of the separate histories of Church’s Thesis and Turing’s Thesis, followed by a quick overview of the path from Turing’s original analysis of what we might call a classical algorithmic procedure to its generalization in the work of Kolmogorov and Uspenskii, and Schönhage. But they also say

In fact the notion of algorithm is richer these days than it was in Turing’s days. And there are algorithms ... not covered directly by Turing’s analysis, for example, algorithms that interact with their environments, algorithms whose inputs are abstract structures, and geometric or, more generally, non-discrete algorithms.

And they go on to describe very briefly -- or at least, too briefly for this reader -- some of work on more abstract general notions of computation.

But, by my lights, once we go beyond Kolmogorov and Uspenskii we lose touch with discussions that are directly relevant to the Church-Turing Thesis, construed as a claim about effectively computable functions (where the notion of effective computability is elucidated in the traditional way, in terms of what can be done by a sequential, step-by-deterministic-step procedure, where each small step is available to a computing agent of limited cognitive abilities, etc.). And indeed, Blass and Gurevich themselves don't challenge the Thesis: their concern is more with extensions of the core concept of an algorithm -- or at least, that’s how I prefer to describe what they are up to.

Posted by Peter Smith at 9:29 AM 0 comments

Labels: Logic

## Monday, May 14, 2007

### Church’s Thesis 1: CTT, minds and supertasks

I mentioned a few posts ago the collection Church’s Thesis After 70 Years edited by Adam Olszewski et al. Since the editors helpfully refrain from suggesting a sensible reading order, I'm just going to dive in and read the contributed papers in the order they are printed (they are arranged alphabetically by the authors' names). And to keep my nose to the grindstone, I've promised myself to post comments and thoughts here -- so it will be embarrassing to stop doing my homework! Here goes, then, starting with Darren Abramson, “Church’s Thesis and Philosophy of Mind”.

Abramson identifies “the Church-Turing Thesis” with the claim “[...] no human computer, or machine that mimics a human computer, can out-compute a universal Turing machine”. That doesn't strike me as a terribly helpful move, for it runs together two claims, namely (i) no human computer can (in a reasonable sense) compute something that is not effectively computable (by a finite, step-by-step, algorithmic process), and (ii) whatever is effectively computable is Turing-computable/recursive. In fact, in my Gödel book, I call just (ii) “the Church-Turing Thesis”. But irrespective of the historical justification for using the label my way (as many do), this thesis (ii) is surely to be sharply separated from (i). For a start, the two claims have quite different sorts of grounds. For example, (i) depends on the impossibility of the human performance of certain kinds of supertask. And the question whether supertasks are possible is quite independent of the considerations that are relevant to (ii).

I didn't know, till Abramson told me, that Bringsjord and Arkoudas have argued for (i) by purporting to describe cases where people do in fact hypercompute. Apparently, according to them, in coming to understand the familiar pictorial argument for the claim that lim n → ∞ of 1/2^n is 0 we complete an infinite number of steps in a finite amount of time. Gosh. Really?

Abramson makes short shrift of the arguments from Bringsjord and Arkoudas that he reports. Though I'm not minded to check now whether Abramson has dealt fairly with them. Nor indeed am I minded to bother to think through his discussion of Copeland on Searle's Chinese Room Argument: frankly, I've never felt that that sort of stuff has ever illuminated serious issues in the philosophy of mind that I might care about. So I pass on to the next paper ....

Posted by Peter Smith at 2:50 PM 1 comments

Labels: Logic

### Interrupted categories

Well, the reading group which was slowly working through Goldblatt's Topoi got about half-way through but has collectively decided that enough is enough! In retrospect I probably suggested the wrong thing to read -- perhaps, after all, the much shorter Lawvere and Rosebrugh's Sets For Mathematics would have been the better bet, as at least we'd have had the satisfaction of getting through it. But Goldblatt's book promised much more meat for logicians to chew on. But most of the group lost its faith that ploughing on was going to deliver more insights. One thing is clear, the book that is going to persuade the generality of logicians or those interested in the foundations of mathematics that it really is important and rewarding to get on top of a substantial amount of category theory has yet to be written. Still, even if I was in the minority, I was left wanting to know more -- so I'm sure I'll return here to the categorial theme in due course.

Posted by Peter Smith at 1:50 PM 0 comments

## Friday, May 11, 2007

### Kleeneness is next to Gödelness

This is a pretty shameless trailer for my forthcoming book (which I confess I'm still fiddling with, since the final final version doesn't have to with the Press for a week more).

It's fun and illuminating to show that the First Incompleteness Theorem can be proved without any appeal to sentences that "say" that they are unprovable, or indeed without any appeal to the apparatus of Gödel numbering, Diagonal Lemmas and the like. This can be done in various ways, of course. But there is a simple argument from Kleene's Normal Form theorem to Incompleteness, which doesn't seem to be well known. Here's a version extracted from the book -- a version that relies on Church's Thesis, but only to save labour and make for cuteness. Enjoy! Pre-order the book on Amazon for lots more where that came from!

Posted by Peter Smith at 10:34 PM 1 comments

Labels: Logic

### Wandering stars

Well, if James can link to a YouTube video, I guess I can! Hardly Shaun the Sheep, and perhaps not adding to the gaiety of nations in quite the same way .... but here's a rare recent sighting of a certain group, for late-night listening, from an unannounced performance in Bristol in February. Even so stripped down, the magic still works.

If Beth Gibbons' words here have always seemed hauntingly evocative, maybe that is because she is drawing from -- or should I say sampling? -- the King James Version rendering of Jude 13.1: "Raging waves of the sea, foaming out their own shame; wandering stars, to whom is reserved the blackness of darkness for ever."

There still surface rumours of another CD, 10 years after the last studio album.

Posted by Peter Smith at 2:15 PM 1 comments

## Thursday, May 10, 2007

### Kenodoxia

I've belatedly discovered another Cambridge blog, James Warren's Kenodoxia (I've added a link alongside). It's terrific, ranging from Xenophanes to Shaun the Sheep, via a sideswipe at Jane Austen. Not that I quite go along with his rating the latter as "pants" ...

Posted by Peter Smith at 11:09 PM 0 comments

## Tuesday, May 08, 2007

### Philosophical archeology

I'm having to try to sort out my philosophy library -- I can't start shelving yet another wall at home -- and that's a painful business. It's not just that I've always had a rather self-indulgent book-buying habit and so there is a ridiculous number to sort through. It's also a matter of encountering long-past philosophical selves, and not quite wanting to wave them goodbye. In the seventies, I was mostly interested in the philosophy of language (though there is a lot of ancient philosophy books too dating from then, and a lot of Wittgenstein-related stuff); from the eighties there is a great number of books on the philosophy of mind; from the nineties a lot of philosophy of science and metaphysics. Digging through these archeological layers I'm reminded of past enthusiasms -- not just of mine, but quite widely shared enthusiasms which seemed philosophically rewarding at the time, but some of which now seem rather remote and even in some cases quite odd misdirections of energy. What creatures of fashion we are!

But at least in those more academically relaxed days I could follow my then interests wherever they led or didn't lead (I never got bored). Young colleagues now don't have the luxury: to get even their first permanent job they have to specialize, concentrate their resources, carve out a niche, build a research profile: and it takes more of the same to get promoted. The structures that we philosophers have allowed to be imposed on "the profession" (as we are now supposed to think of it) have thus come to be in real tension with the free-ranging cast of mind that gets many people into philosophy in the first place. What Marxists used to call a contradiction ...

Posted by Peter Smith at 11:05 PM 0 comments

## Sunday, May 06, 2007

### What are sets for?

Yiannis Moschovakis on p. 1 of his very useful Notes on Set Theory writes that one "basic property of sets" is that

Every set A has elements or members.And then, on p. 2, he writes

Somewhat peculiar is the empty set ∅ which has no members.

But of course he can't have it both ways. Either every set has elements (and there is no empty set) or there is an empty set (and so not every set has elements).

I offer this as another example to my esteemed colleagues Alex Oliver and Timothy Smiley who have a lot of fun with this sort of thing at the beginning of their recent paper "What are sets and what are they for?" (in John Hawthorne, ed., Metaphysics), and who give lots of other examples of set theorists' arm-waving introductory chat about sets being similarly hopeless. But what are we do about that? Alex and Timothy take the stern line that we should take such set theorists at their introductory word, and if that word is confused, then so much the worse for them and for the very idea of the empty set (and for the idea of singletons too, if sets are defined to be things with members, plural). Pending a secure argument for the overwhelming utility of postulating them, we should do without empty sets and singletons and indeed without the whole universe of pure sets.

But it doesn't seem good strategy to me to take set theorists at their first word -- any more than it would be a good strategy to take quantum theorists at their first word (if that introductory word involves a metaphysical tangle about particles-cum-waves). However, I'm with Alex and Timothy that the question "What are sets for?" surely is just the right question to ask. Moschovakis indicates one sort of answer (indeed, they mention it): the universe of sets provides a unified general framework in which we can give "faithful representations" of systems of mathematical objects by structured sets. Now this is, of course, the sort of thing that category theorists aim to talk about: in their terms, there will be structure-preserving functors from other (small) categories to the category of (pure) sets -- roughly because the category of sets has such a plenitude of objects and morphisms to play with. What we need to do here is think through what this kind of use for sets -- a use that can be illuminated in category-theoretic terms -- really comes to. My hunch is that we'll get to a rather different place than Alex and Timothy.

I've been a bit slow to get thinking about set theoretic matters since being back in Cambridge (there's the decidedly daunting prospect of having Michael Potter and Thomas Forster locally breathing down my neck ...). But Alex and Timothy's provocations are hard to resist. So something else to add to the list of things to think about.

Posted by Peter Smith at 1:20 PM 1 comments

Labels: Logic

## Thursday, May 03, 2007

### Two books

Church’s Thesis After 70 Years edited by Adam Olszewski, Jan Wolenski and Robert Janusz looks as if it might be a very useful collection. The bad news is that the hard copy is a steep 129 Euros. But the good news is that you can download a PDF version for just 22.50 Euros. Even better, although the publishers' site says "Windows only", that actually isn't true as I quickly discovered. Buy it and then send in your Adobe reader digital ID and you'll be sent a version that runs on your Mac. Great. Surely an unmissable bargain!

A tiny grumble though. Why do editors of collections such as this one too often take the easy way out and just print pieces in the alphabetical order of the authors' names? They know their way around the contributions, and know what a sensible reading order would look like ....

My Introduction to Formal Logic is perhaps not quite so terrific. But it has its moments, and those who have used it quite like it. But the sales figures I got this morning aren't very hot. There would seem to be two options. Either sigh, wish I'd listened to the wise advice not to bother to write Yet Another Logic Book for an overcrowded market, and let it go. Or try to persuade CUP to have a relaunch with a revised improved edition two or three years hence. Of course, the former is the wise move! But the proud parent would inevitably like to see the tottering toddler do better ...

Posted by Peter Smith at 1:37 PM 0 comments