Logic Matters

The blog is dead .... long live the blog!

2009-11-08T10:12:00.004Z

After almost 500 posts, this will be the last post here, meaning at this URL ....

.... but I'll be continuing the Logic Matters blog at logicmatters.net (and all the posts here at Blogger have been imported to that address, though the aesthetics are at the moment a bit primitive).

Geeky explanation: At very long last, I'm joining the cool kids and am using the Wordpress platform on a hosted site. That's not in fact to make blogging easier -- I rather like the undistracting minimalism of Blogger -- but because Wordpress works as a nice content management system to build/maintain the rest of the Logic Matters website which I've rather neglected of late (thanks to The Daughter for a very helpful advice about why it would -- after the transition -- make updating much easier).

Gödel Without Tears -- 5

2009-11-06T15:16:00.003Z

Here now is the fifth episode on the idea of a primitive recursive function. The preamble explains why this matters and where this is going. [As always, I'll be very glad to hear about typos/thinkos.]

The previous episodes are available:

Episode 1, Incompleteness -- the very idea (version of Oct. 16)
Episode 2. Incompleteness and undecidability (version of Oct. 26)
Episode 3. Two weak arithmetics (version of Nov. 1)
Episode 4. First-order Peano Arithmetic (version of Nov. 1)

Ruse gets a beta minus.

2009-11-04T23:00:00.003Z

Philosophers don't get asked often enough to write for the newspapers and weeklies: so it is really annoying when an opportunity is wasted on second-rate maunderings. Michael Ruse writes in today's Guardian on whether there is an "atheist schism". And he immediately kicks off on the wrong foot.

As a professional philosopher my first question naturally is: "What or who is an atheist?" If you mean someone who absolutely and utterly does not believe there is any God or meaning then I doubt there are many in this group.

Eh? Where on earth has that "or meaning" come from? In what coherent sense of "meaning" does an atheist have to deny meaning?

It gets worse. Eventually a lot worse.

If, as the new atheists think, Darwinian evolutionary biology is incompatible with Christianity, then will they give me a good argument as to why the science should be taught in schools if it implies the falsity of religion? The first amendment to the constitution of the United States of America separates church and state. Why are their beliefs exempt?

That is so mind-bogglingly inept it is difficult to believe that Ruse means it seriously. Does Ruse really, really, think that the separation of church and state means that no scientific fact can be taught if it happens to be inconsistent with some holy book or religious dogma?

Ruse is upset by the stridency of Dawkins and others, and there is indeed a point to be argued here. But it is ironic that philosophers often complain that Dawkins misrepresents too many practising Christians (or Muslims, or whatever). For related misrepresentations -- if that's what they are -- are to be found in more or less any philosophy of religion book. I blogged here a while back about the Murray/Rea introduction, and remarked then about the unlikely farrago of metaphysical views it foisted upon the church-goer, views which have precious little to do with why you actually go to evensong or say prayers for dying, and which indeed deserve to be well Dawkinsed.

The Autonomy of Mathematical Knowledge -- Chap. 2, §§3-5

2009-11-04T11:18:00.003Z

To return for a moment the question we left hanging: what is the shape of Hilbert's "naturalism" according to Franks? Well, Franks in §2.3 thinks that Hilbert's position can be contrasted with a "Wittgensteinian" naturalism that forecloses global questions of the justification of a framework by rejecting them as meaningless. "According to Hilbert … mathematics is justified in application" (p. 44), and for him "the skeptic's path leads to the death of all science". Really? But, to repeat, if that is someone's basic stance, then you'd expect him to very much want to know which mathematics is actually needed in applications, and to be challenged by Weyl's work towards showing that a "sceptical" line on impredicative constructions in fact doesn't lead to the death of applicable maths. Yet Hilbert seems not to show much interest in that.

At other points, however, Franks makes Hilbert's basic philosophical thought sound less than a claim about security-through-successful-applicability and more like the Moorean point that the philosophical arguments for e.g. a skepticism about excluded middle or about impredicative constructions will always be much less secure than our tried-and-tested methods inside mathematics. But in that case, we might wonder, if the working mathematician can dismiss such skepticism, why engage in "Hilbert's program" and look for consistency proofs?

Franks' headline answer is "The consistency proof … is a methodological tool designed to get everyone, unambiguously, to see [that mathematical methods are in good order]." (p. 36). The idea is this. Regimenting an area of mathematics by formalisation keeps us honest (moves have to be justified by reference to explicit axioms and rules of inference, not by more intuitive but risky moves apparently warranted by intended meanings). And then we can aim to use other parts of mathematics that aren't under suspicion -- meaning, open to mathematical doubts about their probity -- to check the consistency of our formalized systems. Given that formalized proofs are finite objects, and that finitistic reasoning about finite objects is agreed on all sides to be beyond suspicion, the hope would be to give, in particular, finitistic consistency proofs of mathematical theories. And thus, working inside mathematics, we mathematically convince ourselves that our theories are in good order -- and hence we won't be seduced into thinking that our theories need bolstering from outside by being given supposedly firmer "foundations".

In sum, we might put it this way: a consistency proof -- rather than being part of a foundationalist project -- is supposed to be a tool to convince mathematicians by mathematical means that they don't need to engage in such a project. Franks gives a very nice quotation from Bernays in 1922: "The great advantage of Hilbert's procedure rests precisely on the fact that the problems and difficulties that present themselves in the grounding of mathematics are transformed from the epistemological-philosophical domain into the domain of what is properly mathematical."

Well, is Franks construing Hilbert right here? You might momentarily think there must be a deep disagreement between Franks with his anti-foundationalist reading and (say) Richard Zach, who talks of "Hilbert's … project for the foundation of mathematics". But that would be superficial. Compare: those who call Wittgenstein an anti-philosopher are not disagreeing with those who rate him as a great philosopher! -- they are rather saying something about the kind of philosopher he is. Likewise, Franks is emphasizing the kind of reflective project on the business of mathematics that Hilbert thought the appropriate response to the "crisis in foundations". And the outline story he tells is, I think, plausible as far as it goes.

It isn't the whole story, of course. But fair enough, we're only in Ch.2 of Franks' book! -- and in any case I doubt that there is a whole story to be told that gives Hilbert a stably worked out position. It would, however, have been good to hear something about how the nineteenth century concerns about the nature and use of ideal elements in mathematics played through into Hilbert's thinking. And I do want to hear more about the relation between consistency and conservativeness which has as yet hardly been mentioned. But still, I did find Franks' emphases in giving his preliminary orientation on Hilbert's mindset helpful. To be continued

The Autonomy of Mathematical Knowledge -- Chap. 2, §§1 & 2

2009-11-02T15:57:00.012Z

Hilbert in the 1920s seems pretty confident that classical analysis is in good order. "Mathematicians have pursued to the uttermost the modes of inference that rest on the concept of sets of numbers, and not even the shadow of an inconsistency has appeared .... [D]espite the application of the boldest and most manifold combinations of the subtlest techiniques, a complete security of inference and a clear unanimity of results reigns in analysis." (p. 41 -- as before, references are to passages or quotations in Franks' book.) These don't sound like the words of a man who thinks that the paradoxes cause trouble for 'ordinary' mathematics itself -- compare Weyl's talk of the "inner instability of the foundations on which the empire is constructed" (p. 38). And they don't sound like the words of someone who thinks that analysis either has to be revised (as an intuitionist or a predicativist supposes) or else stands in need of buttressing "from outside" (as the authors of Principia might suppose).

Franks urges that we take Hilbert at his word here: in fact, "the question inspiring [Hilbert] to foundational research is not whether mathematics is consistent, but rather whether or not mathematics can stand on its own -- no more in need of philosophically loaded defense than endangered by philosophically loaded skepticism" (p. 31). So, on Franks' reading, Hilbert in some sense wants to be an anti-foundationalist, not another player in the foundations game standing alongside Russell, Brouwer and Weyl, with a rival foundationalist programme of his own. “[Hilbert’s] considered philosophical position is that the validity of mathematical methods is immune to all philosophical skepticism, and therefore not even up for debate on such grounds” (p. 36). Our mathematical practice doesn’t need grounding on a priori principles external to mathematics (p. 38). Thus, according to Franks, Hilbert has a “naturalistic epistemology. The security of a way of knowing is born out, not in its responsibility to first principles, but in its successful functioning” (p. 40).

Functioning in what sense, however? About this, Franks is (at least here in his Ch. 2) hazy, to say the least. “The successful functioning of a science … is determined by a variety of factors -- freedom from contradiction is but one of them -- including ease of use, range of application, elegance, and amount of information (or systematization of the world) thereby attainable. For Hilbert mathematics is the most completely secure of our sciences because of its unmatched success.” Well, ease of use and elegance are nice if you can get them, but hardly in themselves signs of success for theories in general (there are just too many successful but ugly theories, and too many elegant failures). So that seemingly leaves (successful) application as the key to the “success”. But this is very puzzling. Hilbert, after all, wants us never to be driven out of Cantor’s paradise where -- as Franks himself stresses in Ch. 1 -- “mathematics is entirely free in its development", meaning free because longer tethered to practical application. Odd then now to stress application as what essentially legitimises the free play of the mathematical imagination! (Could the idea be that some analysis proves its worth in application, and hence the worth of the mathematical methods by which we pursue it, and then other bits of mathematics pursued using the same methods get reflected glory? But someone who takes that line could hardly e.g. be as quickly dismissive of the predicative programme as Hilbert was or Franks seems to be at this point -- for Weyl, recall, is arguing that actually applicable analysis can in fact all be done predicatively, and so no reflected glory will accrue to classical mathematics pursued with impredicative methods since those methods are not validated by essentially featuring in applicable maths.)

So what does Hilbert’s alleged “naturalism” amount to? To be continued.

Gödel Without Tears -- 4

2009-11-02T13:20:00.005Z

Here now is the fourth episode [slightly corrected] which tells you -- for those who don't know -- what first-order Peano Arithmetic is (and also what Sigma_1/Pi_1 wffs are). A thrill a minute, really. Done in a bit of a rush to get it out to students in time, so apologies if the proof-reading is bad!

Here are the previous episodes:

Episode 1, Incompleteness -- the very idea (version of Oct. 16)
Episode 2. Incompleteness and undecidability (version of Oct. 26)
Episode 3. Two weak arithmetics (version of Nov. 1)

Gödel Without Tears -- 3

2009-10-26T11:14:00.005Z

Here's the third episode (slightly updated to take account of some initial comments). Not anywhere near so exciting as the first two -- but after all that arm-waving generality, we do need to get our hands dirty looking at some actual formal theories of arithmetic, mildly tedious though that is! And you really ought to know, e.g., what Robinson Arithmetic is.

The Autonomy of Mathematical Knowledge -- Chap. 1

2009-10-20T15:10:00.005Z

As I said, I'm planning to blog, chapter by chapter, about Curtis Franks’s new book on Hilbert, The Autonomy of Mathematical Knowledge (all page references are to this book). Any comments on my comments will of course be welcome!

Let's take ourselves back to the "foundational crisis" at beginning of the last century. Mathematicians have, over the preceding decades, freed themselves from the insistence that mathematics is tied to the description of nature: as Morris Kline puts it, "after about 1850, the view that mathematics can introduce and deal with arbitrary concepts and theories that do not have any immediate physical interpretation ... gained acceptance" (p. 11). And Cantor could write "Mathematics is entirely free in its development and its concepts are restricted only by the necessity of being non-contradictory and coordinated to concepts ... introduced by previous definition" (p. 9). Very bad news, then, if all this play with freely created concepts in fact gets us embroiled in contradiction!

As Franks notes, there are two kinds of responses that we can have to the paradoxes that threaten Cantor's paradise.

We can seek to "re-tether" mathematics. Could we confine ourselves again to applicable mathematics which has, as we'd anachronistically put it, a model in the natural world so must be consistent? The trouble is we're none too clear what this would involve (remember, we are back at the beginning of the twentieth century, as relativity and quantum mechanics are getting off the ground, and any Newtonian confidence that we had about structure of the natural world is being shaken). So put that option aside. But perhaps (i) we could try to go back to find incontrovertible logical principles and definitions of mathematical notions in logical terms, and try to reconstruct mathematics on a firm logical footing. Or (ii) we could try to ensure that our mathematical constructions are grounded in mental constructions that we can perform and have a secure epistemic access to. Or (iii) we could try to diagnose a theme common to the problem paradoxical cases -- e.g. impredicativity -- and secure mathematics by banning such constructions. Of course, the trouble is that the logicist response (i) is problematic, not least because (remember where we are in time!) logic itself isn't in as good a shape as most of the mathematics we are supposedly going to use it to ground, and what might count as logic is obscure. Indeed, as Peirce saw, "a mature science like mathematics, with a history of successful elucidation and problem solving, was needed in order to develop logic" (p. 20); and indeed "all formal logic is merely mathematics applied to logic" (p. 21). The intuitionistic line (ii) depends on an even more obscure notion of mental construction, and in any case -- in its most worked out form -- cripples mathematics. The predicativist option (iii) is perhaps better, but still implies that swathes of seemingly harmless classical mathematics will have to be abandoned. So what to do? What foundational programme will rescue us?
Well, perhaps we shouldn't seek to give mathematics a philosophical "foundation" at all. After all, the paradoxes arise within mathematics, and to avoid them we just ... need to do mathematics better. As Peirce -- for example -- held, mathematics risks being radically distorted if we seek to make it answerable to some outside considerations (from philosophy or logic) rather than being developed "by the continuous confrontation with and the creative solution of ordinary mathematical problems" (p. 21). And we don't need to look outside for a prior justification that will guarantee consistency. Rather we need to improve our mathematical practice, in particular improve the explicitness of our regimentations of mathematical arguments, to reveal where the fatal mis-steps must be occurring, and expose the problematic assumptions.

Now enter Franks's Hilbert. We are perhaps wont to read Hilbert as belonging to Camp (1), advancing a fourth philosophical foundationalist position, to sit alongside (i) to (iii). We see his "finitism" as aiming to impose more constraints on "real" mathematics from outside mathematics. And, taking such a perspective, most mathematicians and many philosophers would agree with Tarski's dismissal of Hilbert's supposed philosophy as "theology", and insist on a disconnect between the dubious philosophy and the proof-theory it inspired.

But Franks is having none of this. His Hilbert is a sort-of-naturalist like Peirce (in sort-of-Maddy's sense of "naturalist:), and he is firmly situated in Camp (2). "His philosophical strength was not in his ability to carve out a position among others about the nature of mathematics, but in his realization that the mathematical techniques already in place suffice to answer questions about those techniques -- questions that rival thinkers had assumed were the exclusive province of pure philosophy. ... One must see him deliberately offering mathematical explanations where philosophical ones were wanted. He did this, not to provide philosophical foundations, but to liberate mathematics from any apparent need for them." (p. 7).

So there, in outline -- and we don't get much more than outline in Chap. 1 -- is the shape of Franks's Hilbert. So, now let's read on to Chap. 2 to see how well Franks makes the case for his reading. To be continued.

Curtis Franks: The Autonomy of Mathematical Knowledge

2009-10-19T11:40:00.007Z

On Saturday, from the new books stand the CUP bookshop, I picked up a copy of Curtis Franks's The Autonomy of Mathematical Knowledge: Hilbert's Program Revisited.

Two quick grumbles. First, the book is short: just a hundred and ninety very generously spaced pages, maybe 60,000 words in all? Well, I'm all for short books, and I'm trying myself to write one now. But £45/$75? Much as though I love CUP, that really is more than a tad extortionate (and I probably wouldn't have coughed up but for a big discount as a press author). Secondly, I can't say that I particularly like Franks' prose style, which tends to the unnecessarily flowery and slightly contorted, which makes you occasionally too aware of the medium rather than the message.

But having got those grumbles off my chest, let me say that the book looks very interesting indeed -- a must read for anyone interested in matters round and about Hilbert's Programme, which means pretty much any philosopher of mathematics. So order for your library today. And I plan to blog about this book, chapter by chapter, starting here tomorrow ... (promises, promises!).

Gödel Without Tears -- 2

2009-10-17T15:21:00.003Z

As promised, Episode 2 of Gödel Without Tears (in which we prove sufficiently strong theories are undecidable and incomplete -- just like that!)

As explained, I'm writing these notes as just-after-the-event handouts for weekly lectures. And each week I'll be checking through the previous handout (and no doubt finding small corrections to make) before I give the next lecture. So here's the latest version of Episode 1, dated 16 October.

Modal logic, with a lot more tears than necessary

2009-10-14T20:34:00.003Z

The logic crew were minded to do some more modal logic. And, casting around for a modern book that might link up with recent stuff on e.g. second order modal logic, I suggested that in our reading group we tried Nino Cocchiarella and Max Freund's Modal Logic (OUP, 2008). Mea culpa. I confess I didn't look at it closely enough in advance. Today was the first meeting, and it fell to me to introduce the first couple of chapters.

This really is a poorly written book, and it is pretty difficult to imagine for whom it is written. Although it is subtitled "An introduction to its syntax and semantics", no one who hasn't already done some modal logic is going to get anything much out of the opening chapters. For this is written in that style of hyper-formalization and over-abstraction that philosophers writing logic books still too often affect. Why? Who is it supposed to impress? (It is as if the authors are trying to prove that they aren't really weedy soft-minded philosophers, but can play tough with the big boys. The irony is that the big boys, the good mathematicians, don't play the game this way.)

Here's a trivial example. If you or I were introducing a suitable language for doing propositional modal logic, we might say: OK, we need an unlimited supply of propositional atoms, and here they are, P, P', P'', P''', etc.; we want a couple of propositional connectives, say → and ¬; and the Box as a necessity operator. Then we'd remark, parenthetically, that of course the precise choice of symbolism is neither here nor there. Job done. For of course, sufficient unto the day is the rigour thereof.

But Cocchiarella and Freund are having none of this. In fact they don't tell us what any actual modal language looks like. Rather they introduce some metalinguistic names for the atoms, whatever they are; and then there are three other symbols named c, n and l, whatever they might be, to serve as a conditional, negation and necessity operator. And the rest of the discussion proceeds at one remove, without us ever actually meeting an object language modal sentence. (Well, actually there's another problem: for on their account it would be jolly hard to meet one, as for them a modal sentence is a set of sets of sets of numbers and symbols. Despite their extreme pernicketiness about formal matters, they are cheerfully casual about identifying set-theoretic proxies with the real thing -- but let that pass.)

OK, what does their formalistic fussing get us? Nothing that I can see. The surface appearance of extra generality is spurious. And in fact, Cocchiarella and Freund soon stop any pretence at generality. For example, when the wraps are off, they require any logistic system based on the conditional and negation to have a bracket-free Polish grammar, where logical operators are prefix. And they require any derivation in such a system to be in linear Hilbert style, without rules of proof or suppositional inferences. Those requirements combined make most modal logical systems you've ever seen not count as such according to them.

Consider your old friend, von Wright's M. As we all learnt it in the cradle from Hughes and Cresswell, and ignoring the fact that they go for particular modal axioms and a rule of substitution rather than using axiom schemata, their system has two rules of inference, modus ponens and a rule of necessitation that allows us to infer BoxA if we've proved A from no assumptions. But such a rule of course isn't allowed if derivations all have to be Hilbert style, with conclusions always being derived by the application of rules to previous wffs, not to previous (sub)proofs. This means that Hughes and Cresswell's M is not a modal system according to Cocchiarella and Freund. And when they talk about M, since they only have modus ponens as an inference rule, they have to complicate the axioms, by allowing us to take any of Hughes and Cresswell's axioms and precede it by as many necessity operators as you want. They then prove what they call the rule of necessitation, which tells us that if there is a proof of A from no assumptions in their system M, then there is also a proof of BoxA in their system. But note, the C&F "rule of necessitation" is quite different from H&C's rule. In fact the C&F rule stands to H&C's rule pretty much as the Deduction Theorem stands to Conditional Proof.

Now, I don't particularly object to Cocchiarella and Freund doing things this way. But I do object to their doing it this way without bothering to tell us what they are doing, how it relates to the more familiar way, and why they've chosen to do things their way. Why is the reader left trying to figure out which deviations from the familiar might be significant, and which not?

Anyway, we certainly weren't impressed. The grad students -- a very bright and interested bunch -- uniformly found the style rebarbative and entirely off-putting. There was no general will to continue. And democracy rules in the reading group!

Gödel Without Tears -- 1

2009-10-12T10:40:00.004Z

Here, as promised, is the first of a series of lecture handouts (roughly weekly, and about twelve in all) encouragingly titled Gödel Without Tears -- 1. As is the way with lecture handouts, this was dashed off at great speed, and I don't promise that this is free of either typos or thinkos. So do please let me know of any needed corrections, or indeed of any passage which is too unclear/could do with just a little amplification. Enjoy!

Later: I've already replaced the first version with a slightly better one ...

Gowers's conversation about complexity lower bounds

2009-10-11T07:10:00.002Z

I should have mentioned before that Tim Gowers's blog is running installments of a "conversation" on complexity lower bounds. It's structured as a dialogue between three characters, a cheerful mathematical optimist who likes to suggest approaches to problems, a more sceptical mathematician who knows a bit of theoretical computer science (and is tagged with a "cool" smiley), and an occasionally puzzled onlooker who chips in asking for more details and gives a few comments from the sidelines. We're just on instalment IV, and there are oodles of comments on the previous instalments.

This is fascinating stuff for philosophers of maths, in both form and content -- though I don't begin to pretend to be following all the ins and outs. In form, because it's always intriguing to see mathematical work-in-progress, exploring ideas, guesses, dead-ends (live mathematics as an activity, if you like, as opposed to the polished product presented according to the norms for "proper" publication). And in content, because you begin to get a sense of why something that initially seems as though it ought to be easy to settle (P = NP?) is really hard.

Mullova/Dantone play Bach

2009-10-10T21:13:00.004Z

The Bach recital that Viktoria Mullova gave at the Wigmore Hall last week was simply terrific. Up there with my all-time great concerts, including some Brendel, Holzmair singing Die Schöne Müllerin, and the Lindsays (often). Mullova finished up playing the great Chaconne from the second Partita. And she didn't attack it as some do. As a reviewer said, "... many violinists try to match its immensity with a heroic sound. But Mullova often went the other way, becoming light and dancing where most violinists would be losing bow-hairs in an effort to wring a bigger sound from the instrument ... totally convincing." Certainly, she stunned the audience who sat in silence for some moments after she finished.

But the revelation for me was the two sonatas she played with Octavio Dantone. I didn't know their recording of the sonatas on Onyx (I like Rachel Podger's recording quite a bit, and hadn't sought out another). But their performances last week bowled me over too, and so I sent off for the discs. And yes, hugely recommended!

Oh, the delights of term again ....

2009-10-10T17:20:00.003Z

Well, that's the beginning of term survived, and I hope to pick up the philosophical threads here next week.

It's been back to first year logic lectures, for what I guess -- with retirement looming -- will be the penultimate time. The opening two lectures went tolerably well. Drat. Just getting the hang of doing this and I'm having to stop! Lecture pacing is an odd thing, though: there are fewer lectures in the course this year, and I need to push things on. So I've put the admin stuff in a hand-out, cut out some other slides from the Beamer presentations, and felt I was cracking on faster. Yet I'm exactly where I got to last year after two lectures. Ah well: maybe it is good not to put the foot on the accelerator too hard too soon. But we must push on next week.

The other course I'm starting this term, which I'm planning to repeat when I get to NZ, is a dozen lectures on Gödel's (Incompleteness) Theorems for third year undergrads and postgrads. This is much more difficult to get right. Last year, I just did talk and chalk, introducing chunks of my book. But that didn't really work: there was too much gap between what I had time to do in relaxed chat, and what's in the book. So maybe use Beamer presentations for this course too? After one class I think this isn't going to work either -- or at least, the effort put into writing the presentation would be much better used writing a couple of pages of lecture handouts as a more careful/comprehensive intro that can be followed up in the book, better filling the gap between lecture chat and the book. OK, down to it then, and I'll write some weekly handouts, Gödel Without Tears. Watch this space ...

The logical highpoint of the week, though, was the first Logic Seminar, where Fraser MacBride was talking about neo-logicism. He gave an terrific impromptu intro for the surprising number of third-years who turned up, quite innocent of the debates, and then he had a persuasive bash at the latest Hale/Wright effort, ‘The Meta-Ontology of Abstraction’. Fraser set the bar pretty high for the rest of term. Excellent stuff.

Research Excellence Bullshit

2009-09-24T20:57:00.014Z

So, there's another consultation document on the Research Excellence Framework -- "the new arrangements for the assessment and funding of research in UK higher education institutions that will replace the Research Assessment Exercise (RAE)". A wonderful document indeed, literate and elegantly written, revealing much thought and reflection on the nature of the university in the best traditions of Arnold and Leavis. Of course. Still, perhaps it isn't quite what we might hope for.

Ok, ok, I jest. It isn't at all what we might hope for, though it is the sort of egregious crap we've come to expect. How about this, for example: "As an indication of our current thinking we propose the following weightings" (between different components of assessment); "Outputs: 60 per cent. Impact: 25 per cent. Environment: 15 per cent." Hold on! Impact? Impact? What's that?

Well, the document gives "a common menu of impact indicators" under various headings to help us out. Here are the headings ...

Delivering highly skilled people [as evidenced e.g. by "Staff movement between academia and industry, Employment of post-doctoral researchers in industry or spin-out companies".]
Creating new businesses, improving the performance of existing businesses, or commercialising new products or processes
Attracting R&D investment from global business
Better informed public policy-making or improved public services
Improved patient care or health outcomes
Progress towards sustainable development, including environmental sustainability
Cultural enrichment, including improved public engagement with science and research
Improved social welfare, social cohesion or national security
Other quality of life benefits

Right. Let me see if I understand. If you are a medieval historian, an editor of Euripides, a Shakespeare scholar, or indeed just a logician trying to understand the philosophical significance of Gentzen's work on the consistency of arithmetic, then 25% of your score in son-of-RAE is going to be for "impacts" utterly irrelevant to your projects and concerns?

I'm being unfair, you say: arts subjects at least get into the frame under the heading "Cultural enrichment". You might think so: but in fact we are told that possible indicators of that are -- I kid you not -- "Increased levels of public engagement with science and research (for example, as measured through surveys). Changes to public attitudes to science (for example, as measured through surveys). Enriched appreciation of heritage or culture (for example, as measured through surveys). Audience/participation levels at public dissemination or engagement activities (exhibitions, broadcasts and so on). Positive reviews or participant feedback on public dissemination or engagement activities." Yep, and we are also told that impact does not include "we do not intend to include impact through intellectual influence on scientific knowledge and academia".

Ah, there's a chink of light perhaps: not everyone is to be ranked for impact, if I've got it right? -- a department's return will rather involve a series of "case-studies" of impactful individuals. Well, yes, you can just see the guys and gals in M&E sitting around trying to find a smidgin of impact somewhere between them.

Brilliant. Well, I know will happen; you know what will happen; HEFCE no doubt know what will happen when traditional humanities departments come to fill in the impact case studies on which 25% of their overall rating is going to depend.

They'll have to bullshit.

Added later. My jest about the M&E contingent having a bit of difficulty cooking up an impact statement was truer than I realized. Eric Schliesser, currently at Leiden, writes in a comment on the Leiter blog that "in places where 'impact' is already playing a prominent role (say, in Netherlands and Flanders), certain subjects (e.g., analytic metaphysics,) have very little chance to receive coveted research grants (now almost the sole source for PhD funding). Yesterday, Michael della Rocca gave a terrific talk on the three-dimensionalism vs four-dimensionalism debate. It generated great discussion. But the people in attendance were hard-pressed to name a sole Dutch philosopher who is working on the topic ... Of course, other subjects (e.g., philosophy of technology, applied ethics, decision theory, semantics, logic, normative ethics, etc) have an easier time in articulating the impact factor and are generously funded."

Schubert's Piano Trios

2009-09-23T19:29:00.008Z

I buy few new CDs these days, as I already have ridiculously many (and multiple recordings of most pieces that I really care about it). But I was driving home from my aged mother's the other day, and the BBC were playing the Schiff/Shiokawa/Perényi recording of the E flat trio D 897, and I was bowled over. The double CD with the other trio, the D897 Notturno and the Arpeggione Sonata came out in 1997, Schubert's bicentennial year, but -- though I've always admired Schiff's Schubert playing -- I'd missed this record. But, still in stock at Amazon, it arrived a couple of days ago.

And it indeed is terrific. The performances could hardly be bettered it seems to me -- there's a flow to the playing and a rapport between the three that gives new life to the pieces after years of listening to the Beaux Arts' recordings. The Gramophone review agrees. (I can't imagine though why, after a decade, this hasn't been reissued in a cheaper version: it deserves to be on everyone's shelves.)

Praise for Just and Weese!

2009-09-18T19:22:00.004Z

From time to time I do get more than a bit critical here about books of one sort or another: so it's good to give praise for once!

Over the last couple of days I've been reading the first volume of Winfried Just and Martin Weese's Discovering Modern Set Theory (AMS, 1996), with an eye to moving on to the second volume. Well, I just loved the style, and think it works very well. I don't mean the occasional (sightly laboured?) jokes: I mean the in-the-classroom feel of the way that proofs are explored and motivated, and also the way that teach-yourself exercises are integrated into the text. For instance there are exercises that encourage you to produce proofs that are in fact non-fully justified, and then the discussion explores what goes wrong and how to plug the gaps. My grip on set theoretic niceties is patchy enough to be find this kind of reinforcement of understanding pretty helpful from time to time, even at the elementary level of the first volume. So I'll be rather warmly recommending the book to students.

Student evaluations

2009-09-12T21:48:00.010Z

I remember, quite a few years ago, giving the same introductory logic course two years running, as far as I could tell doing as a good a job each time. But my student evaluations plummeted between one year and the next. Why? I could only put it down to the fact that the first year I gave the course in relaxed casual dress; the next year (because a committee was scheduled the same afternoons) I wore a rather serious suit. So I supposedly came across as remote, unhelpful, and harder to understand.

I was reminded of that experience -- which made me permanently a tad sceptical about the worth of student evaluations -- when I read these two scepticism-reinforcing pieces^*, by the philosophers Michael Huemer and Clark Glymour. I was particularly amused (in a world-weary sort of way) by this excerpt from the former:

[There was a] study, in which students were asked to rate instructors on a number of personality traits (e.g., "confident," "dominant," "optimistic," etc.), on the basis of 30-second video clips, without audio, of the instructors lecturing. These ratings were found to be very good predictors of end-of-semester evaluations given by the instructors' actual students. A composite of the personality trait ratings correlated .76 with end-of-term course evaluations; ratings of instructors' "optimism" showed an impressive .84 correlation with end-of-term course evaluations. Thus, in order to predict with fair accuracy the ratings an instructor would get, it was not necessary to know anything of what the instructor said in class, the material the course covered, the readings, the assignments, the tests, etc.

Williams and Ceci conducted a related experiment. Professor Ceci, a veteran teacher of the Developmental Psychology course at Cornell, gave the course consecutively in both fall and spring semesters one year. In between the two semesters, he visited a media consultant for lessons on improving presentation style. Specifically, Professor Ceci was trained to modulate his tone of voice more and to use more hand gestures while speaking. He then proceeded, in the spring semester, to give almost the identical course (verified by checking recordings of his lectures from the fall), with the sole significant difference being the addition of hand gestures and variations in tone of voice (grading policy, textbook, office hours, tests, and even the basic demographic profile of the class remained the same). The result: student ratings for the spring semester were far higher, usually by more than one standard deviation, on all aspects of the course and the instructor. Even the textbook was rated higher by almost a full point on a scale from 1 to 5. Students in the spring semester believed they had learned far more (this rating increased from 2.93 to 4.05), even though, according to Ceci, they had not in fact learned any more, as measured by their test scores. Again, the conclusion seems to be that student ratings are heavily influenced by cosmetic factors that have no effect on student learning.

So now you know: bounce in optimistically, wave your hands around confidently, and you can sell the kids anything ...

And I should say that these days I always wear a suit to lecture (so I've a cast-iron excuse for any poor evaluations, of course).

Added For a bit of judicious balance, do read Richard Zach's second contribution (Comment 12 below), and the linked paper.

^*Links from twitter, thanks to John Basl and Allen Stairs

Math logic reading list (updated)

2009-09-08T12:35:00.009Z

I've spent the last couple of days reorganizing and rewriting the reading list for the Part II Math Logic paper (that's a third year undergraduate paper for philosophers). It was a rather minimalist affair, and I've taken a step or two towards its becoming an annotated study guide.

The paper is something of a Cambridge institution, pretty much unchanged in its basic syllabus since when I took it a long time ago. It rather distinctively mixes an introduction to the "greatest hits" as far as formal results are concerned, with a look at some of the philosophical issues arising.

Anyway, having had some initial comments here and from local grad students, you can now download my third shot at an updated list. All comments and suggestions for further improvement (within the current, fixed, syllabus) will still be very welcome.

Congratulations to Thomas Forster

2009-09-08T10:54:00.002Z

Another logic-seminar regular hits the big time! It is good to see that Thomas's “The Iterative Conception of Set”, published last year in the new Review of Symbolic Logic was judged one of the ten best papers of 2008 by the Philosopher's Annual. Here's a link.

Grumpy old man, #42

2009-09-08T10:18:00.007Z

I think I'm turning into a grumpy old man ...

[Cue suppressed laughter off stage, murmurings of "Turning? Turning? Happened years ago", etc. But I shall ignore these scurrilous interruptions.]

... and the latest cross-making irritation (especially galling for a long-time Analysis editor) is the effort by some OUP copy-editor to improve a forthcoming Analysis paper by Luca Incurvati and myself, by inter alia, removing all the contractions, replacing "don't"s by "do not"s etc.

Now, it is one thing to replace American spelling by English spelling (or vice versa), or to replace "...ize" by "...ise", for example. But to replace "don't" (one long syllable) by "do not" (two short staccato syllables) is to change the rhythm of a sentence. The use of "don't" can smooth the reading of a sentence, slightly modulating the emphasis. Has the OUP editor being paying attention to such matters? Somehow I think not. My bet is that the changes have been made without thinking, slavishly following some semi-literate "style book".

And to make such changes wholesale is to arbitrarily change the authorial tone of voice: which is just impolite (to put it mildly -- especially when some of us put quite a bit of effort into getting the tone we want).

Harrumppph.

School maths, from the distant past

2009-09-02T12:39:00.014Z

I found myself yesterday in a small-town bookshop, kicking my heels for half an hour. Prompted by recent press discussion of the standard of A-levels (the UK 18+ end-of-high-school examination), I browsed through some books intended for A-level further maths students. I must say that they did seem really rather noddy to me, though of course it is only too easy to be seduced into the thought that things are going to the dogs!

Still, that prompted me, just for fun, to look out the papers I sat aged seventeen and a bit, to get into Cambridge, back when the world was young. So here's a small selection of some of the shorter questions:¹ click to enlarge. (There were four three-hour papers with ten questions apiece: as I recall you aimed to get out at least half-a-dozen a time).

The questions do seem tougher than anything I saw in the contemporary text for further maths. But it would be interesting to know from anyone with their finger more on the pulse how many reasonably bright school kids are in a position to tackle this sort of thing these days. Or indeed -- though the answer could be depressing -- how many of their teachers.

¹ I don't guarantee my proof reading in copying the questions!

Simpson's SOSOA

2009-08-11T21:47:00.002Z

The rather long awaited new edition of Simpson's wonderful Subsystems of Second Order Arithmetic is out with CUP.

Added: I've now looked at a copy in the CUP bookshop, and this is a corrected reprinting of the first-edition, without new material. So if you (or your library) already have a copy of the first edition, then just print out a copy of the corrections page and you won't be missing anything. But the original edition had become very difficult to get hold of, so it is good to have the book back in print.

Disappearing logic again

2009-08-04T14:59:00.002Z

A footnote to my post, Logic disappearing over the horizon. I've just been reading Stephen Simpson's "Unprovable Theorems and Fast-Growing Functions" (an introductory piece in the 1987 AMS Contemporary Mathematics Logic and Combinatorics volume that contains some important papers on provably computable functions -- it is a pity that Simpson's very helpful and accessible survey isn't more readily available, e.g. on his website). I was struck by this remark:

Like most good research in mathematical logic, the results which I am going to discuss had their origin in philosophical problems concerning the foundations of mathematics.

And that's right: the most interesting work in mathematical logic is bound up with problems and projects of a more philosophical kind concerning the foundations of mathematics. All the more worrying, then, the seeming trend I was remarking on for logic courses to be less and less available even to graduate philosophy students. If the wonderfully fruitful long dialogue since Frege between philosophers and mathematicians (or often, between the philosophical and mathematical sides of the same individual) is to continue, then some philosophers at any rate do need to be logically well-educated!

More logic books available online ...

2009-08-04T14:31:00.005Z

Richard Zach, over at LogBlog, has posted this:

Exciting developments! The Association of Symbolic Logic has made the now-out of print volumes in the Lecture Notes in Logic (vols. 1-12) and Perspectives in Mathematical Logic (vols. 1-12) open-access through Project Euclid. This includes classics like
Shoenfield's Recursion Theory,
Lindström's Aspects of Incompleteness in the LNL,
Sacks' Higher Recursion Theory,
Hájek and Pudlák's Metamathematics of First-order Arithmetic,
Shelah's Proper and Improper Forcing,
Barwise's Admissible Sets and Structures, and
Barwise and Feferman's Model-theoretic Logics in the PiML.
I'm especially excited about the Hájek/Pudlák and Barwise/Feferman volumes, which are chock-full of useful material!

This is indeed an excellent development (I'm not sure why Project Euclid puts the books up in chapter-length chunks and then complains if you download too many chunks at once: but let's not sound ungrateful, because I'm certainly not!).

Looking around online, you can in fact find a large number of logic books available, though most of them are there contrary to copyright. Frankly, I don't feel guilty about having a bootleg e-book on my laptop if the hard copy acquired with hard cash is sitting on my shelves. But it would be wonderful if this is the beginning of a trend for out-of-print classics to be made freely available in high-quality PDFs.

Conceptual mathematics again

2009-08-03T21:48:00.002Z

Newly in to the CUP bookshop today, a second edition of Lawvere and Schanuel's Conceptual Mathematics. This has a little new material over and above what was in the first edition: that looks a good move, as I found when new to category theory that the first version ended too soon, without enough pointers forward to where we were we being taken.

Logic disappearing over the horizon ....

2009-07-29T11:26:00.003Z

I've just had an invitation to give a talk at the University of X, a distinguished place, with a philosophy graduate community of about fifty (according to their website). So I checked out how much logic/phil maths is going on, what I could reasonably take as given. Zilch. Apart from a first year course perhaps approaching the level of my intro logic book, nothing at all, as far as I can tell. Which leaves me a bit bereft of anything to go to talk about. But more to the point, it means that for students at X a central swathe of the work of lasting value from the last hundred years has disappeared over the horizon. Which is, shall we say, a pity.

My sense is that this is happening more and more in UK universities. I'd be delighted to learn that I'm wrong.

World-class again

2009-07-25T16:01:00.009Z

I got the very good news eight or nine days ago of an award under the AHRC Research Leave scheme, to complete a book on Gentzen's proof(s) of the consistency of arithmetic (how the best versions work -- not obvious -- and what their philosophical significance is -- not at all obvious). As quite a few people have said to me, there's a very real need for such a book, and I hope I can make a decent job of it. I'm aiming to write something that is as accessible as my Gödel book as far as the technicalities are concerned (why is it that books on proof theory can be such tough going?); in other words, I want at least beginning grad students in philosophy who've done an intro math logic course to be able to follow it. And as for the philosophical commentary and critical discussion as we go along, well again I hope that will be accessible to the same audience too. (As my Explaining Chaos and Gödel books should show, I'm all for maximum accessibility: there's no point in trying to write a book for a readership of eleven, if only because no publisher these days would touch it.)

Now, I posted here a week ago, saying that I'd got the grant (and praising the AHRC for a conspicuous lack of ageism). But I added a remark -- in what was supposed to be tone of world-weary amusement -- about the fact that my research proposal was ranked "an outstanding proposal meeting world-class standards of scholarship, originality, quality and significance", suggesting that "world-class" was going it a bit.

Where two or three philosopher are gathered together these days, we often bemoan the exaggerations that have become routine in writing references, commenting on grant proposals, etc. etc. A student, to get into a US grad school, has to be the best you've taught in a dozen years; a planned piece of work has to be of ground-breaking originality, with the world waiting breathlessly. (In fact, I wrote here about reference inflation just a few weeks ago: we all know the phenomenon only too well).

Well, I don't know about you, but to me "world-class" means really, really, outstanding. How many world-class philosophers are there active in the UK? How many would you put into your world first eleven? Ok, let's be generous, your world first twenty-five all-stars? The fingers of one hand would be enough to count them, surely.

And one thing is for certain, by my lights most of us who get AHRC grants are not "world-class". We are trying to usefully move things on just a bit; we hope our stuff might get onto reading lists and get talked about a bit in its area. In other words, we try to be decently interesting and make some good new points. But in my idiolect, as in that of most philosophers, that hardly makes the work discipline-changing world-class stuff.

So, I said I was amused by the seeming gap between "world-class" and the useful, pushing-things-on-a-bit book that I'm writing. And I lamented the way that exaggerations of that kind have become rife in political and management discourse (ok, I used that philosopher's term of art "bullshit").

Well, what was supposed to be a weary old lag's comment on a linguistic decline has apparently caused serious offence. In particular, it has been suggested that describing my planned book as doing for Gentzen what I tried to do for Gödel, i.e. "explain clearly and make a few philosophical comments along the way" was inconsistent with its being proper research, with the implication that I shouldn't be getting the grant and had been deceiving the AHRC. But "making a few philosophical comments along the way" was of course exaggerating in that understated Cambridge way, to counterbalance the "world-class" exaggeration in the opposite direction. So I do apologize if someone got the wrong end of the stick. Of course what I'm doing is serious business, pushing things on as best I can. That should go without saying: but it seems that I need to say it.

Ok, back to thinking about provably terminating computations ...

Congratulations to Wilfrid Hodges!

2009-07-22T13:37:00.003Z

Logicians will be delighted to see that Wilfrid Hodges has been elected a Fellow of the British Academy.

One to cross off your list

2009-07-22T08:36:00.006Z

Suppose an undergraduate wrote this:

The first [incompleteness theorem] says that there are truths of arithmetic that are not provable in a consistent first-order logic that can express arithmetic.

You'd patiently explain that there are four things wrong with this. First, it confuses "logic" with "theory". Second, as you'd remind the student, the first theorem was originally proved for a higher-order theory, and applies to any theory which is properly axiomatized, whether first-order or not. Third, this statement confuses the conditions for semantic and syntactic versions of the first theorem. If you only assume the theory can express enough arithmetic, then you need to assume soundness to derive incompleteness; if you only assume consistency, as in the standard syntactic version, then you have to assume that the theory can represent enough arithmetic (where this is a matter of proving, rather than merely expressing, enough). And fourth, the statement is fatally ambiguous between (a) there are truths not provable in any consistent theory which can represent enough arithmetic, and (b) for any consistent theory etc. there are truths that that theory can't prove. Given that folk misinterpretations of Gödel trade on that ambiguity, you drill into your students the importance of clearly avoiding it.

Your student goes on to write:

The standard view is that we cannot prove CON(PA), period. (I use CON(PA) as an abbreviation for the sentence that expresses the consistency of Peano Arithmetic.) ... However, all that follows from the Gödel theorems is that we cannot prove CON(PA) with mathematical certainty.

Again, you'd start by patiently reminding the student that there is no such thing as the sentence that expresses the consistency of Peano Arithmetic -- and that matters because, as Gödel himself later observed, there are sentences that arguably in some sense express the consistency of PA which are provable in PA. In headline terms, it matters for the Second Theorem which consistency sentence you construct, in a way that it doesn't matter for the First Theorem which Gödel sentence you construct. But second, and much more importantly, it certainly is not the standard view that we cannot prove CON(PA), period. Any good treatment emphasizes that unprovability in PA is not unprovability period. Third, you'd add that it doesn't follow from Gödel theorems either that "we cannot prove CON(PA) with mathematical certainty". What's wrong, for example, in proving CON(PA) from PA plus the Pi_1 reflection schema for PA? If you are mathematically certain about PA and its implications, why wouldn't you be equally certain about the result of adding instances of the reflection schema? Arguably you should be: but in any case, nothing follows from Gödel theorems about that issue. And fourth, you might quiz your student about what he makes of the Gentzen proof.

Ah, he says,

We need transfinite inductions along a well-ordered path of length epsilon_0 to prove CON(PA) [in Gentzen's way]. The issue, then, is this: if human minds know the truth of CON(PA) with mathematical certaintly, is the only method by which we do it the use of infinitely long derivations?

But there seems to be a bad misunderstanding here: you'd remind your student that a proof by transfinite induction is not an transfinitely long derivation: it is just a proof assuming that a certain ordering is well-founded.

Unfortunately, those quotations -- and there's more of the same -- are not from a student essay but from a book, one published by MIT Press no less, Jeff Buechner's Gödel, Putnam, and Functionalism (see p. 8 fn. 8; p. 33; p. 39). The book turned up as I dug through the archeological layers on my desk in my Big Book Clear Out: I was sent it some time ago to review. But with garbles like that in a book one of whose main topics is the implications of Gödel for functionalist mechanism about the mind, I'm not encouraged to read any further. Life being short, I probably won't.

Added later: The Reviews Editor is twisting my arm in a flattering kind of way. Maybe I will review this after all.

Time-travelling with Google maps

2009-07-17T21:29:00.004Z

Late last night, I walked down the street where I lived from the age of four until I was fourteen. A virtual walk, using Street View in Google maps. A rather sad experience though. For what a visual mess so much of the English urban landscape has become!

When I lived there -- we are talking about the outer London suburbs in Surrey -- the streets were quietly respectable, each house (built in the 30s, I think) separated from the road by a small front garden, neatly enclosed in privet hedges, with the obligatory flowerbeds and small patch of lawn. The houses along the road were identical apart from the colours of the front door. No one could call them particularly elegant. But there was a quiet uniformity; and the trees along the grass verges to the road, together with the hedges and front gardens, softened the rather dour drabness of the brick houses, so the overall effect was pleasing enough. As home-owning (as opposed to renting) became more common among the middle middle classes after the war, it was just the kind of street people aspired to live in.

Now, of course, the hedges and front lawns and hollyhocks have nearly all gone: where there were gardens, there is concrete and asphalt and paving stones, so cars can be parked two abreast with their noses up against the front windows. The long green verges to the road have been paved over too, so people can drive their cars across, with the few remaining trees isolated on little patches. The houses themselves have suffered from scattered cheap replacement windows, a new porch here, a differently tiled roof there. It all looks more than a bit scruffy: there is nothing along the road to soothe the eye, no riots of flowers to cheer the heart. I can't imagine anyone positively aspiring to live there.

Yet the houses now change hands for a third of a million pounds.

Twitter is sometimes not entirely frivolous

2009-07-14T09:44:00.006Z

I was amused by the kids twittering away at the Joint Session (like third-formers passing notes in the back row) -- some very good pics too. Mildly fun to see what I was missing. If not exactly useful.

However, a recent tweet from Robbie Williams did very interestingly point out that the first vol. of Saul Kripke's papers has been announced. I'll believe it, though, when I see it.

Gödel sorted ... until the next time

2009-07-06T19:14:00.002Z

Well, I've just this morning sent off the PDF for the next reprinting of my Gödel book: it should be available mid-August. I've eliminated quite a few more typos (thanks to everyone who let me know of errors in the earlier printings), made a scattered selection of small clarifications/stylistic improvements, and corrected the major errors that were still lurking even in the first corrected reprint. The biggest change is the one I mentioned a couple of posts ago, concerning Rosser's Theorem.

Being a reprint though, I couldn't change the "extent" of the book, the number of sheets that need printing. (Fortunately, the previous version finished on a recto page, so I had the verso to play with, and needed the extra side to correct the treatment of Rosser's Theorems). But given a new edition, I'd like another dozen or fifteen pages. First, I'd give a nicer proof that Robinson Arithmetic can capture all p.r. functions. Second, I'd fill in all the details of the proof that Prf(m, n) is p.r. (the final stages of which are just sketched). Third, I'd fill out the proof of the derivability conditions for theories with a smidgin of induction (again, I give an arm-waving sketch). As to the second and third, I originally didn't worry about giving sketches, as filling out the details is pretty tedious and unexciting (what you surely want to get across are the proof-ideas: and I referred masochists and completists to places where they can get the full gory story). But a number of readers have thought I should have gone the extra mile and risen to the challenge of giving full, though still maximally accessible, proofs. OK: if there's a second edition with a slightly longer page budget I'll have a bash ... But that will have to wait a bit.

Apple Preview: fail

2009-07-06T17:16:00.010Z

This might just save a few Mac-based LaTeX users some grief. Suppose you use TeXShop to typeset

\documentclass[11pt,a4paper]{article}
\begin{document}
$\{0\} \to \{y = 0\}$
\end{document}

The result looks just fine in the TeXShop preview window onscreen (which calls on the Mac pdf Preview engine). But print it out and the spacing after the arrow is wrong. You get

{0} →{ y = 0}

Use Adobe Reader to print the pdf and all is well:

{0} → {y = 0}

The problem seems to be that the use of '}' as a printing character before a binary operator can mess up the printing though not the viewing of a pdf by Preview. Very, very odd. But apparently there are other known issues with Preview printing pdfs and missing/misplacing symbols. So the moral is: when it matters, print using Adobe Reader.

Stumbling over the bug caused me some hours of annoyed and mystified head-scratching. The invaluable comp.text.tex newsgroup helped me find a minimal example and pin the blame on Preview rather than on my (occasionally ropey) LaTeX coding.

Rum and reason

2009-06-30T19:23:00.002Z

Long time readers will remember I used to link to The Daughter's cooking blog, BakeMeHappy (that's still on line, full of good things, and beautifully written -- but apart from a little late flurry, it really came to a halt a year ago). She's now left Italy, and is cooking up a storm in the Bahamas: and to go with that move, there's a new blog, Rum & Reason, promising more great food and another good read. Enjoy!

Rosser's Theorem

2009-06-26T16:01:00.006Z

Wolfgang Rautenberg's very nice A Concise Introduction to Mathematical Logic states Gödel's first theorem in the usual sort of way. Roughly: given a suitably axiomatized omega-consistent theory T containing enough arithmetic, there's a Pi_1 undecidable sentence. He proves this by appeal to the diagonalization lemma applied to the predicate not-Prov (where Prov suitably expresses provability in T). A fixed point G for this is neither provable nor disprovable. And being equivalent to not-Prov('G') that's undecidable too. But not-Prov by construction is Pi_1, so that gives us a Pi_1 undecidable sentence.

So far, so very familiar! He then goes on to say (p. 195) that the assumption of omega-consistency in Gödel's theorem can be weakened to that of consistency. And he gives the usual Rosser construction. Define a Rosser proof-predicate RProv (defined from the usual Sigma_1 proof-relation Prf, so it is satisfied by the Gödel number for a wff if the wff has a proof, and there is no "smaller" proof of its negation). Then, just on the assumption of consistency, a fixed point R for not-RProv is neither provable nor disprovable. Great.

But oops, that isn't quite enough to prove the weakening of Gödel's theorem as stated. For Rautenberg hasn't checked that there's a Pi_1 undecidable sentence that you can get in this way. And note that not-RProv (as he defines it from the Sigma_1 Prf) isn't explicitly Pi_1, so the argument from before doesn't carry over. So there's a gap in the proof here: some more work needs to be done.

There's the same gap it seems in Per Lindström's Aspects of Incompleteness (p. 26). He just cheerfully starts off, in effect, "Let R be a Pi_1 fixed point of not-RProv." Why should there be one? He doesn't explain.

At least one book, though, does worse. That is to say, it notes the need for an argument that there is a Pi_1 undecidable sentence that you can get from Rosser's kind of construction. But then it proceeds to give a bad argument, claiming not-RProv can be manipulated into an equivalent Pi_1 formula, but giving an evidently hopeless "proof" for that claim. Oh dear.

And the culprit was me. So I'm very grateful to Adil Sanaulla for alerting me to the fact that something is badly amiss in An Introduction to Gödel's Theorems at the foot of p. 178. So I've needed to go back to the drawing board.

In fact, sorting out things out took just a bit of thought. To recap, the sort of argument that I use in the book (and that e.g. Rautenberg uses) to get an undecidable sentence just assuming consistency relies on proving that a fixed point for not-RProv is undecidable. But that doesn't immediately give us the result that there's a Pi_1 undecidable sentence. On the other hand, e.g. Raymond Smullyan in Chap. VI of his wonderful Gödel's Incompleteness Theorems uses a slightly different construction that does get us a Pi_1 undecidable sentence: it is perhaps closer to Rosser's original, but it doesn't use the now standard Rosserized proof predicate. The two arguments are evidently very closely related, however. But exactly how?

Well, this is hardly a deep expository problem! But I confess it did take me a while, after a false start, to make the obvious connection. The result is here: an updated section for the book (which will appear in the next revised printing). It still gives much the same original version which shows that a fixed point for not-RProv is undecidable, and then I hope smoothly segues into showing how to give what is in essence the original Rosser/Smullyan version that yields a Pi_1 undecidable sentence. Obvious when you see how, and I kick myself that this wasn't in the previous printings of the book.

Do please let me know if you still spot some errors!!!

Functions and gunctions

2009-06-22T09:54:00.002Z

Tim Gowers has a very nice piece on his blog about functions, multivalued functions, relations and the like, called "Why aren't all functions well-defined?".

The book problem

2009-06-21T07:51:00.005Z

So you start buying books -- I mean academic, work-related, books of one kind or another -- in your late teens. As retirement age looms you've been doing it for more than forty-five years. Suppose you average a couple of books a month. Not that very difficult to do! You buy a few current books on topics that you are working on; books for reference; books you feel you should read anyway, given the ripples they are producing; books for seminars or reading groups you belong to; books it is useful to have to hand for teaching (the textbooks the kids are reading, or just useful collections of articles, before the days when everything was online). It very soon mounts up. Add in a few review copies, freebies, books given by friends, serendipitous finds rescued from the back of obscure second-hand bookshops (I got a set of Principia Mathematica that way). Then without any effort at all your modest library is steadily growing at thirty books a year or more. But go figure: that's already around 1400 books as you get to the end of your career. I've been a bit more incontinent than some, but actually not a lot (especially as my interests have rather jumped about). Say I've acquired 1750 over the years. I've got rid of a few books from time to time, of course, though I've been absurdly reluctant to let them go: but overall, I've still probably got not far short of 1500. Which, I agree, is a stupid number to end up with -- but (as we've seen!) it's easy enough to end up there without a ridiculously self-indulgent rate of book-buying as you go along.

Soon enough, I'm going to lose an office; and we're trying to declutter at home anyway. So over the coming weeks and months I need to cut that number down. A lot. Near halving is the order of the day. What to do?

Most of the Great Dead Philosophers and the commentaries can go -- I can't see myself ever being overwhelmed by a desire to re-read Locke's Essay, for example (and anyway I can always get the text online). But that doesn't make much of a dent, as I was never much into the history of philosophy anyway. I can get rid of some of the books-for-teaching, and old collections of articles whose contents are now instantly available on Jstor. But that doesn't help particularly either. So now it gets difficult.

It could just be neurotic attachment of course! But I like to think that there is a bit more to it than that. I'm sure I'm never going to seriously work on chaos again, so -- though it was great fun at the time -- I guess I will let the chaotic dynamics books go fairly easily. I'm also pretty sure that I'm never going to seriously work on the philosophy of mind again, and I've never done anything in epistemology: but just axing the phil. mind and theory of knowledge books seems to go clean against how I think of philosophy, as the business of trying to understand “how things in the broadest possible sense of the term hang together in the broadest sense of the term” as Sellars puts it. And anyway, some of the issues I'd like to understand better in the philosophy of mathematics seem to hang together with broader issues about representation and about knowledge. So perhaps I need to hang on to all the mind and knowledge books after all ....?

No, no, that way madness lies (or at any rate, swamping by unnecessary books). After all, Cambridge is not exactly short of libraries, even if I do dump something I later find myself wanting to read again! So, I'm just going to have to be brutal. A few old friends apart, if I haven't opened it in twenty years, it can certainly go. If it is just too remote from broadly logicky/phil mathsy stuff, it really better go too. Sigh.

PS Before more readers write asking for books, I should say that I have charitable plans!

Gowers on Razborov's theorem about monotone circuit complexity

2009-06-20T21:40:00.002Z

I mentioned before that I'd been to Tim Gowers's lectures on computational complexity. As I noted before, videos of his lectures are available here. But, as promised, he has also written up the proof which was the topic of the first part of the course, namely Razborov's demonstration that the monotone circuit complexity of the clique function is superpolynomial. You can read it here. Tim Gowers puts a lot of effort into making the ideas seem reasonably "natural". Enjoy! -- if you have a taste for this sort of thing.

It's tough (reprise)

2009-06-19T12:41:00.003Z

I was on leave last Easter term, and so not examining. But the previous year I commented here, under the heading "It's tough being a philosophy student",

It strikes me again while marking that it's quite tough being a philosophy student these days: the disputes you are supposed to get your head around have become so sophisticated, the to and fro of the dialectic often so intricate. An example. When I first started teaching, Donnellan's paper on 'Reference and Definite Descriptions' had quite recently been published -- it was state of the art. An undergraduate [indeed, a final year undergraduate] who could talk some sense about his distinction between referential and attributive uses of descriptions was thought to be doing really well. Just think what we'd expect a first class student to know about the Theory of Descriptions nowadays (for a start, Kripke's response to Donnellan [on our first year reading list!], problems with Kripke's Gricean manoeuvres, etc.). True there are textbooks, Stanford Encyclopedia articles, and the like to help the student through: but still, the level of sophistication we now expect from our best undergraduates is daunting.

The same basic point struck me just as forcibly this year. Except perhaps now I'd say that textbooks and the Stanford Encyclopedia in some ways make things even tougher for students. Here's a very good, well-briefed student: they've got their head round X's excellent text book presentation. They write four and a half crisp sides on topic, organizing the necessary points covered by X to answer the question set. Before the textbook appeared, we'd have been delighted with the answer. Now we read the same script and think, yeah, fine, a competent rehearsal of X's treatment -- so nothing outstanding. So how is the poor student to really impress? It gets harder.

Thank heavens that's over ...

2009-06-19T07:53:00.006Z

I've been chair of the examining boards for Parts IB and II of the Philosophy Tripos (so that's the second and third [final] year exams here). The process is now over, reasonably painlessly for me.

But it's not so painless for a good handful of disappointed students. For we still have to do the increasingly pointless task of dividing performances into "first class", "upper second", etc. This was, of course, always an artificial business. But at least once upon a time the division at the top approximately corresponded to the distinction between the really outstanding and the rest (and very few expected/hoped for a first). Now, with grade inflation, a first is more in reach, with that first/upper second divide coming further down the rank order. But it typically seems to fall bang in the middle of a bunch of really pretty good if not quite outstanding students, some of whom were just that bit luckier with the way the exams went for them than the others. It makes no defensible sense.

Another very warm recommendation

2009-06-03T21:50:00.007Z

Hard on the heels of Viktoria Mullova's re-recording of the Bach Partitas, and Angela Hewitt's re-recording of the Well-Tempered Clavier, Imogen Cooper is re-recording late Schubert. I love her earlier recordings on Ottavo; but this first disk in the projected new series is even better.

David Cairns in the Sunday Times gets it right: "The intervening years have seen a deepening understanding of this wonderful repertoire. The range of colour, the subtle details, the singing line, the freedom of tempo within the driving momentum, the haunting and haunted beauty, are greater than ever." Revelatory in fact. Getting off the marking treadmill and listening to this has made a wonderful pause.

Oh, and another musical delight. I just can't think what prompted me idly to visit the Wigmore Hall website again this afternoon (five minutes, in fact, after getting the cheering e-mail from CUP to say they were putting things right), but I was amazed to find that a little block of the very best seats for Viktoria Mullova's Bach recital in September had become available. So I was able to snap up a couple after all! I am thrilled to bits.

All's well that ends well

2009-06-03T21:36:00.003Z

It was by the sheerest fluke that I discovered that there had been a third printing of my Gödel book. I was in Waterstone's in Bloomsbury with time to kill, looking through the maths books, and there were a couple of copies of my book. And (as you do!) I was flicking through it, and found that (a) there had been another reprint without checking with me to see if I wanted to correct anything, and (b) in fact the reprint reproduced the original version, not the corrected second printing. On the scale of world disasters, this perhaps doesn't register (I tell myself). But I was damned cross all the same.

I'm pleased to report, then, that CUP have been more than apologetic. They have hung, drawn and quartered the culprit; are changing procedures so it can't happen to anyone else's book; are going to withdraw all the copies and pulp them; and I'm going to get a fourth reprint with whatever further corrections I want to make. Which is, all-in-all, an excellent outcome. Though, as I said, all the result of a fluke discovery.

The Wisdom of Wikipedia

2009-05-29T16:32:00.007Z

Ye gods. The Wikipedia entry for Definite Descriptions until a moment ago read:

Bertrand Russell ... proposed according to his 'theory of descriptions' that when we say "the present King of France is bald", we are making three separate assertions:
1. there is an x such that x is the present King of France.
2. for every x that is the present King of France and every y that is the present King of France, x equals y (i.e. there is at most one present King of France).
3. for every x that is the present King of France, x is bald.

So Russell solved a problem about a sentence with a non-denoting description by analysing it into a conjunction of three sentences with the very same non-denoting description. Terrific.

I thought about leaving it as a bear trap for unwary students cribbing essays. But I couldn't, and it's been minimally corrected. It will interesting to see how long that lasts before some idiot changes it back.

Praise where praise is due

2009-05-24T09:31:00.002Z

I'm finishing marking a stack of dissertations and "assessed essays", submitted for examination in the Philosophy Tripos. I have some philosophical grumbles (of course!), but it is a beautiful late spring day, and I'm feeling cheerful, so let me give praise where praise is due. For I'm struck by the fact that so many are so well written. Students having to read a great deal of sharply written analytical philosophy; being forced to write supervision essays week in, week out; having essays ruthlessly criticized hour after hour for lack of clarity and cogency -- all that seems to produce after a couple of years some excellent writers of spare, readable, transparently lucid English prose. It's good to see we are doing something right!

It's that time of year again ...

2009-05-21T14:27:00.006Z

Things will continue to be quiet here for a while. Tripos starts tomorrow, with piles of marking to come. But in one way or another I've already been marking all week (reading dissertations and submitted essays, interspersed with looking at the work submitted by shortlisted candidates for the Analysis studentship). Let's just say it's been a pretty mixed experience.

To keep myself going, I've just got Angela Hewitt's new version of Bach's Well-Tempered Clavier to listen to between scripts. It seems to be the season for re-recordings by artists who have already made classic disks. As I noted here, Viktoria Mullova has released another version of the Partitas (stunning); Imogen Cooper is starting to release another Schubert cycle (very warmly reviewed, and I've just sent off for the first disks). And here Hewitt is giving us another version of the 48. Her previous version was about my favourite: this one might take some getting used to, as it is more ‘more expressive’, ‘more elastic’, than the earlier one. But on a first listen to the first couple of disks, I think I could warm to it.

Quantum computation again

2009-05-13T20:22:00.002Z

The audience is thinning ... but we've got through Peter Shor's cunning quantum algorithm for factorizing numbers exponentially faster than the best classical algorithm. Fascinating stuff (even if the biggest number a real-world implementation has so far has managed to factorize is 15, which isn't yet too alarming for those worrying about the quantum algorithm being used for busting public key cryptography!).

Tim Gowers warmly recommended Michael Nielsen and Issac Chuang's book Quantum Computation and Quantum Information which does indeed seem very nicely put together, is pretty readable, and is distracting me from all the things I should be doing (like marking philosophy dissertations).

Maddy on logic

2009-05-13T19:32:00.004Z

My blog postings on Maddy's Second Philosophy came to an early halt, largely due to the pressure of other commitments. But our reading group is continuing to work through the book. I liked the partly historical, scene-setting, first part of the book a good deal. But I think we all found the second part of the book, on truth, unsatisfactory (not least because it was unclear what was distinctively second-philosophical about this part of the enterprise). We are now discussing her treatment of logic in the third part of the book.

She aims to give what she regards as a naturalized version of a Kantian account of the status of logic:

As a first approximation, then, the Second Philosopher hopes to develop an account of logical truth with two components: (1) logic is true of the world because of its underlying structural features, and (2) human beings believe logical truths because their most primitive cognitive mechanisms allow them to detect and represent the aforementioned features of the world. As soon as these two ideas are laid down, it's natural to hope that they can be further reinforced by a connection between them: (3) human begins are so conﬁgured cognitively because they live in a world that is so structured physically.

But this way of putting things is surely going to ring alarm bells with well-brought-up logicians! Logic, we have learnt to say, is not about a special class of worldly truths, but is about what follows from what. Of course, regiment the rules about what follows from given assumptions in standard kinds of ways, and we'll find ourselves saying that certain propositions follow from no assumptions at all: call those the logical truths. But note that this account of the logical truths emerges as, so to speak, a spin-off from something else, namely a prior account of logical inference. And it's the account of logical inference which has to come first. So Maddy's Second Philosopher is starting in the wrong place. Or so it will seem to many.

Maybe Maddy could retort that her way of putting things is recommends itself just for simplicity and to make connections with an earlier tradition. She could -- couldn't she? -- have equally well started:

As a slightly better approximation, the Second Philosopher hopes to develop an account of logic with two components: (1') logical inference is reliably truth-preserving because of features of the way the world is structured, and (2') human beings accept certain inference rules because their most primitive cognitive mechanisms allow them to detect and represent the aforementioned features of the world.

But why is that supposed to be the obvious route for the hyper-naturalist Second Philosopher to take? After all there is a familiar enough alternative, much explored, which at a similarly broad-brush level runs:

However the worldly facts go, whatever structure the world has, our thoughts don't always track those facts. We get things wrong. And thoughtful agents need a way of explicitly acknowledging they've got things wrong -- we need a negation operator in our language.

Likewise, sometimes we can only narrow down the facts to some options. And thoughtful agents need a way of expressing the options without committing to any -- so we need a disjunction operator as well. We need negation and disjunction, then, not because the world is full of negative and disjunctive facts which we have to track (whatever such facts might be), but because of our cognitive limitations. And so it goes, mutatis mutandis, for other logical operators too.

Now, what makes something a negation operator, a disjunction operator, etc.? Meaning is use! It's what we do with the operator (in the jargon, the practice codified in the introduction and elimination rules) that fixes which operator is which, and shows it is apt for rejecting something as wrong, for narrowing down options, or whatever. Of course there are constraints -- we can't pair up introduction and elimination rules willy-nilly: that's the lesson of tonk. But the constraints aren't so to speak external, world-imposed, ones: in a naturalistically anodyne sense, they are a priori constraints of harmony imposed by sensible conservativeness requirements etc., needed to keep the enquiry game from falling apart.

The harmonious inferential rules, then, are meaning-fixing -- we read off the content of complex sentences involving the operators from the rules in just such a way as to ensure that the rules are truth-preserving. So we don't need to look at the way the world is structured to determine that they are truth-preserving. But there's nothing naturalistically suspect about all this: we've indicated why limited cognitive agents have need of the likes of negation and disjunction working in the way they do.

Of course, I'm not saying that such an inferentialist story is utterly unproblematic. Far from it. But it is the obvious foil to Maddy's sort of story. So good Second Philosophy methodology might suggest she should at least be taking the stories in parallel and devising some nice "crucial experiments" to decide between them.

But that isn't how Maddy proceeds. In fact, she just doesn't mention the well-trodden inferentialist path at all. Maybe she associates it with e.g. Dummett, who she would have marked down as a modern First Philosopher par excellence. But I don't see that there is in fact anything especially first philosophical about inferentialism (in Tennant's hands, his inferentialist treatment is bound up with what look to be rather Second Philosophical concerns -- evolutionary considerations, thoughts about the inferential practices necessary for good science).

So I'm left rather puzzled about Maddy's confidence that treating logical laws like particularly general laws of nature is evidently the way to go for the naturalist. It surely isn't.

Gowers online!

2009-05-11T17:49:00.004Z

Tim Gowers's lectures are being recorded: the first five lectures are now fully online in various formats (including iPod friendly video) here, with the rest presumably to follow.

Quantum Computing Since Democritus

2009-05-06T21:54:00.004Z

Tim Gowers this morning recommended Scott Aaronson's notes for a lecture course 'Quantum computing since Democritus.' I'm half way through, and the notes are a great read and highly illuminating. So I add my recommendation, if you want to get to know more about computational complexity and about quantum mysteries too. (Aaronson also has a fun blog.)

The State of the Nation

2009-05-04T19:27:00.008Z

I've just finished helping do the shortlisting for the Analysis Studentship. Without inappropriately giving things away, what have I learnt about the state of the nation, philosophically speaking? (A reminder: the studentship is intended for those who are finishing or who have recently finished a PhD in the UK, to give them another year in which to have a second stab at applying for JRFs, post-docs, or other posts that will keep them in philosophy. So the applications give a partial snapshot of what finishing/recently completed UK-based grad students are up to.)

The good news is that there are some really rather impressive-looking young philosophers starting out, already publishing in good places.
The bad news is that there are a lot of good but not quite so impressive-looking philosophers starting out. Surely far too many to ever get permanent jobs in this country. To be sure, some of the applicants are of overseas origin and might eventually want to return home. But I can't help feeling that there are going to be an increasing number of people who have given (say) seven or eight years of their lives to postgraduate work in philosophy, doing a MA and a PhD followed by some temporary employment, and who are then faced at 30 with an unenviable and depressing choice between hanging on in a sequence of very temporary jobs or starting over in some other career. (That would still be the situation even if everything were rosy in the economy, given the numbers now coming out of UK grad schools: but things are only going to be made worse by the financial plight of universities here and in the USA.)
I would have predicted that one effect of the drive for early publications would be a kind of scholasticism. There were some signs of this -- philosophers who seemed to know a lot about rather little (Professor X's views about Y) and coming at it from a narrow angle too. But in the event, this happily wasn't too much in evidence.
Some topics were over-represented. Predictably, issues to do with consciousness and knowledge of one's own mental states loomed large (the hottest topics of a decade ago evidently became the routine topics of starting graduate students four or five years back). Other topics were rather unpredictably under-represented. For example, mainstream philosophical logic or indeed straight philosophy of language surprisingly featured hardly at all.

Overall, though, the future of UK philosophy looks cheeringly bright.

Tim Gowers's lectures

2009-05-01T22:47:00.003Z

I'm going to Tim Gowers's graduate course on Computational Complexity, and we're four sessions in. It must be the first time I've followed a full-blown maths lecture course since I was a Part III student (once upon a time, when the world was young), and I'm enjoying it hugely. Partly because the topic is fascinating, and I'm being prompted to read around a bit. And partly because the performance is terrific.

At five past the hour, Tim Gowers picks up a piece of chalk, and -- without notes, it seems -- proceeds to explain ideas and proof-strategies and outline proof-steps. Not using overheads or a data projector means that things go at a pace you can take in, and there's a sense of the proofs being re-created in real time which is very engaging. And his running commentary of incidental comments can be extremely illuminating. For Tim Gowers characteristically wants to show that proof-ideas aren't just rabbits to be pulled out of the hat in a mysterious way, but are in fact rather natural ideas to try. And that -- I warmly agree -- is how mathematical proofs should be presented.

Lies, damned lies, and references

2009-04-30T11:56:00.002Z

I'm in the midst of reading through a pile of applications for the Analysis Studentship. There are some impressive looking candidates. But I'm frankly not too impressed with some of my colleagues in various universities who are writing references. Indeed I'm pretty damned irritated. For two reasons.

First, the advert for the Studentship plainly says the "the successful candidate will have a CV which would make him or her a strong contender for a Junior Research Fellowship". How come then that too many colleagues are agreeing to write references for people they must know perfectly well wouldn't haven't a snowball in hell's chance in a JRF competition. They should just have the honesty to say straight out "Sorry, you are batting out of your league here; I don't think you should waste your time or the time of the Studentship Committee in applying; so I can't support you on this."

Second, it can't be that every PhD student is one of the top 5% of students the referee has taught, etc. , etc., etc. The inflationary guff that you get in too many references is now just ridiculous. And prompts in this reader the sceptical response "Oh yeah?", so is in fact counterproductive. Irritating your reader is not a good way to promote your students.

Tim Gowers on "correct" proofs

2009-04-26T07:23:00.002Z

Catching up with Tim Gowers's blog, I notice he makes some interesting remarks in passing about the idea of a proof. He'd initiated/co-ordinated a "polymath" project -- a collaborative all-comers group effort trying to find a combinatorial proof of a result in Ramsey theory previously known from a proof in ergodic theory. After hundreds on contributions, but of course before any attempt to put the bits and pieces together into a conventionally written-up proof, Gowers says "I am basically sure that the problem is solved (though not in the way originally envisaged)."

Why do I feel so confident that what we have now is right, especially given that another attempt that seemed quite convincing ended up collapsing? Partly because it’s got what you want from a correct proof: not just some calculations that magically manage not to go wrong, but higher-level explanations backed up by fairly easy calculations, a new understanding of other situations where closely analogous arguments definitely work, and so on. And it seems that all the participants share the feeling that the argument is “robust” in the right way.

So, interestingly, getting a "correct" proof in his sense involves more than getting a proof in the austere logician's sense of a correct proof as just ("magically") getting out the desired result without logical gaps -- it goes with explanations and understanding.

Let's not fuss about terminology. It undoubtedly is the case that a notion of a correct proof in something like Gowers's sense functions centrally in mathematicians' thinking. But how good are the attempts by philosophers of mathematics to elucidate this notion?

Three lectures on incompleteness

2009-04-23T14:44:00.002Z

Here, in a very slightly revised form -- and I'm not going to have time to revise them properly for a while -- are the handouts for three "Back to Basics" lectures on incompleteness that I gave at the Cameleon workshop a few weeks ago. There's nothing excitingly original here, with the possible exception of mistakes! But maybe if you are looking for one story about the shape of the wood which doesn't get too distracted by all the trees, then the handouts could be useful.

Viktoria Mullova's new Bach

2009-04-22T20:31:00.005Z

"To hear Mullova play Bach is, simply, one of the greatest things you can experience," wrote Tim Ashley in the Guardian of a concert.

She's now recorded the solo sonatas for the first time and re-recorded the partitas (this time using gut strings and a baroque bow on a stunning-sounding 1750 Guadagnini). I liked the previous recording of the Partitas a lot, and that was already one of my favourite versions. But this time the result I think is simply amazing, one of those recordings that immediately imposes its vision -- compelling you to think this is how the music ought to be played. It is indeed a great thing.

Awodey on Sets, Types and Categories

2009-04-16T07:38:00.003Z

Steve Awodey has a quite short but fascinating paper now available online called 'From Sets to Types to Categories to Sets', inter alia saying something tolerably accessible about the significance of his recent technical work. Here's the first paragraph:

Three different styles of foundations of mathematics are now commonplace: set theory, type theory, and category theory. How do they relate, and how do they differ? What advantages and disadvantages does each one have over the others? We pursue these questions by considering interpretations of each system into the others and examining the preservation and loss of mathematical content thereby.

And here's the last paragraph. Awodey has been noting the set theory gives constructions features that are not mathematically salient -- 'no topologist or algebraist is concerned with the logical type or ordinal rank of a manifold or module' -- though 'they can serve a useful purpose in foundational work by providing the concrete data for specifications and calculations, facilitating constructions and proofs.'

By contrast, the purely structural approach of category theory sometimes offers comparatively little such "extra" structure to hold on to. Practically speaking, it can be harder to give an invariant proof. That is why it's good to know that such logical structure can always be introduced into a category when needed; the devices of introducing an internal logic or a set theoretic structure into a category, as sketched in the foregoing sections, were originally developed in order to benefit from their advantages, much like introducing local coordinates on a manifold for the sake of calculation. The analogy is quite a good one: no one today regards a manifold as involving specific coordinate charts, and one generally works with coordinate free methods so that the results obtained will apply directly|this is the modern, structural approach. But at times it can still be useful to introduce coordinates for some purpose, and this is unobjectionable, as long as the results are invariant. So it is with categorical versus logical foundations: category theory implements the structural approach directly. It admits interpretations of the conventional logical systems, without being tied to them. Category theory presents the invariant content of logical foundations.

Anyone interested in foundational issues will want to read the paragraphs in between the first and last! (Thanks to Richard Zach, over at LogBlog, for the link.)

An Introduction to Formal Logic, Reprinted!

2009-04-14T11:47:00.005Z

At long last, the much corrected reprint to my An Introduction to Formal Logic (CUP, originally 2003) has arrived on my desk and is in stock at the publishers. It's been a bit of a saga, but worth the effort. All the needed corrections which I originally listed on the web pages for the book have been made, and there are dozens of other small improvements scattered through the book.

Students: it will be much less irritating getting the corrected version even if it costs a bit more than a second-hand copy of the first printing. The paperback is relatively cheap anyway, and the new version is significantly better.

Colleagues: if you are using the book to teach from, don't worry. Even though the new version is an improvement, and some things are better explained, you won't have to change your lectures and classes much if at all!

Phew! Well, thank heavens that's over. I just hope it is more than three days before I notice the first misprint in the new version ...

Recommended: Alimentum

2009-04-13T08:09:00.004Z

This is only going to be of interest to locals (or those passing through Cambridge). But I warmly recommend Alimentum. The best meal out we've had in Cambridge, ever. By miles. In fact the best meal out since we were last in Italy. Modern euro style -- fantastic quality venison, beef. Share a plate of the cheeses too before moving on to dessert. The wine was quite excellent as well, though I'm not going to say what we drank, in case that means people finish up their stock! -- but let's just say that Gambero Rosso were right about it.

Note to students: this has to be the place to get your ever-loving visiting parents to take you to, if they are used to London prices and won't curl up in shock (though The Daughter, who has dined out a lot in London, thought Alimentum compared well with many supposedly classy London places).

How not to present Gödel's Theorems

2009-04-07T15:14:00.007Z

I was wondering whether to spruce up my very rapidly written hand-outs on Gödel's Theorems which I produced for the Cameleon weekend and publish them here. I was beginning to think I wouldn't bother, as there are other things I want to be getting on with. But I just now read the chapter on incompleteness in Shawn Hedman's A First Course in Logic, and that reminded me how crummy some textbook treatments can be, and hence of the need for crisp and clear presentations. Not that Hedman is technically wrong, of course (well, I haven't read him that carefully, but the details look ok). But I defy any beginning student to take away from his chapter a really clear sense of what the key big ideas are, or of how to distinguish the general results from the hack-work needed to show that they apply to this or that particular theory. So back to those hand-outs!

Gödel book corrections

2009-04-06T20:07:00.003Z

I've just uploaded a new version of the corrections sheet for my Gödel book, at www.logicmatters.net. I'm still cogitating how best to handle some further suggestions/complaints (and how best to respond to nagging worries I have about my presentation of the Second Theorem) -- but I thought I should at least upload an interim report while I was in the mood to get at least some of the job done.

I hope I'll eventually have the chance of another corrected reprint. Part of me would like a whole new edition; there's stuff I'd do differently now in ways that would require more than local changes. But new editions aren't always better (for the temptation is to pack in more material, and the book is already too long).

Back to Monica Vitti

2009-04-02T15:29:00.004Z

It's all been getting a bit monochrome and serious here lately (memo to self: lighten up a bit). So as a distraction from the rigours of logic here's another in that occasional series of photos of Monica Vitti. This one -- click to enlarge -- was taken in Rome by the photographer Willy Rizzo in 1960, the year of L'avventura, when she was 28.

A lot has been written about that elusive film: this, by the critic Gregory Solman, perhaps gets something right about it.

I don't get to the cinema much these days: but I wonder what comparable films are being made that can and will be watched, repaying the same serious attention, almost fifty years on. Any suggestions?

Starting from the beginning

2009-04-01T12:23:00.004Z

Am I alone in this? Suppose I want to look at someone's discussion of topic X in (as it might be) Chap. 7 of some techie book on logic. I find I just can't stop myself reading Chaps 1 to 6 first. Even if most of that is relatively elementary stuff that in a sense I know backwards. Ok, it slows me down: but on the other hand -- if it is a good enough author for me to want to read Chap. 7 in the first place -- I almost never regret going over the stuff: I almost always learn something interesting, get a new angle on this, a cute approach for doing that, see connections I hadn't fully appreciated before.

Today's topic X is ordinals, and the aim is to check out Thomas Forster's discussion in Chap. 7 of his idiosyncratic and insightful Logic, Induction and Sets. I flicked through this when it came out. But I've started reading again more carefully from the beginning of the book, and good fun it is too. Recommended.

^{Though if you follow the recommendation, check out the corrections page.}

Rejection and valuations

2009-03-30T18:31:00.003Z

After the enjoyable Second Cambridge Graduate Conference on the Philosophy of Logic and Mathematics back in January, I wrote up some remarks I gave responding to a talk by Julien Murzi and Ole Hjortland, based on their 'Inferentialism and the Categoricity Problem: Reply to Raatikainen' (which is coming out in Analysis). Luca Incurvati independently shared my worries about what they say about Smiley and rejection, and had more points to make about their remarks on intuitionism. Putting our thoughts together, Luca and I have come up with a joint paper "Rejection and valuations", now also forthcoming in Analysis: it has, we hope, some stand-alone interest, if you want a steer on what is going on in Smiley's original paper on Rejection. Enjoy!

Cameleon report

2009-03-29T16:14:00.004Z

Well, the Cameleon weekend is over. Something of a success, I think (though the main CMS building at the weekend is a rather bleak empty space, not very conducive to socializing between papers, and the catering-arrangements could have been better; these things matter -- don't they? -- to getting a really good feel to an occasion).

I talked about the incompleteness theorems. The exercise of trying to pack some headline news into three sessions was very useful (to me, at any rate) -- though I'm afraid that I became rather conscious of some inadequacies in my book in the process. Enough of the audience seemed gratifyingly surprised by simple observations that e.g. generic Gödel sentences (fixed points for ¬Prov) can be false, and that there are provable "consistency" sentences, to make it worthwhile going over some basics again. I wrote a 43 page handout, though I think I'd like to tidy it up just a bit before publishing here. So watch this space.

Thomas Forster talked about countable ordinals, and there's a rough-and-ready version of some notes here. I've heard him talk about these things before, but I like his way of thinking about ordinals, and I want to get clearer still about these things before getting back to writing about Gentzen.

John Truss presented some work on countably categorical structures, Fraissé theory, and the "classification of countable homogeneous multipartite graphs", all with enviable lucidity. I don't know enough -- or have the right interests -- to really understand why this might be interesting. I had a sense that denizens of a mathematical zoo were being pointed out and classified (giving, as it were, the natural history of part of the abstract realm). I guess my tastes run to more abstract theory.

For me, though, the high points of the weekend were Wifrid Hodges's talks on the history of logic. His question was: why did modern logic take so long to arrive? And his tour through various episodes from the history of logic, and his diagnosis of some causes for stagnancy from Aristotle to Leibniz, was absolutely fascinating. Do visit his website and look at some of the other historical pieces there too.

An Introduction to Formal Logic

2009-03-23T15:12:00.002Z

The long saga of the reprint of IFL continues, and the happy ending is in sight. At last, I went in this morning to the CUP offices to check the print-out of the revised version. Famous last words, but it looked OK. I'd forgotten just how much I've changed (though mostly in small ways), but I think the improvements are well worth the effort put in last summer. Their cumulative effect is to make me look with a more kindly eye on the book ... until I spot some additional horror I've now introduced!

When the reprint comes into stock, I'll tell the world (loudly) and then you can all adopt it for your courses/get multiple new versions for the library/buy it as birthday presents ...

Philosophy RAE yet again

2009-03-21T08:10:00.003Z

I'm sure most people are getting a little bored with this -- but the detailed RAE results for philosophy are now out. I can't conceive why the results weren't given in this form in the first place (three months ago). For as you'll see, the initially published "Overall Quality Profile" scores can be pretty significantly misleading as to relative gradings for the actual Research Outputs we care about. (E.g. a four point difference with 4* ratings for the latter can turn into a fifteen point difference in 4* ratings for the former.)

Maddy, Carnap, Quine, etc.

2009-03-18T19:46:00.002Z

I meant by now to have written here about Maddy on Carnap and Quine. It's not that I've lost interest. To the contrary. I'm finding myself wanting to follow up some of the other literature on Carnap and Quine that Maddy refers to (and more besides). This could take a while. And first, I must write some lectures on Gödel -- which I fear are, at least in level, inevitably going to be way out of step with the other workshop lectures for the upcoming Cameleon meeting. Gulp.

It's only a theory

2009-03-17T22:26:00.002Z

It's Only a Theory is a new group-blog on the philosophy of science which looks as though it could develop very nicely. There's certainly an impressive line of contributors signed up (and me as well).

TeX update and SyncTeX

2009-03-13T23:28:00.005Z

\begin{geeky procrastination when I should be writing some Gödel lectures}

I keep the TeXShop editor updated, but the background LaTeX installation just works so smoothly that I haven't bothered to touch it for a while. But I did today get round to uploading the MacTeX 2008 installation. The process is simplicity itself, and then the Tex Live Utility will update the update. (By the way, none of this overwrites the previous installation: you can swap back if need be, and delete the old version later when you are happy to do so.)

Who knows what other under-the-bonnet improvements there are! But the distribution now includes SyncTeX. Tell TeXShop to use this, and this seems to notably improve the synchonization between the editing window and the preview window. Worth the (small) effort of upgrading by itself.

In a TeXie mood, I also played just a bit with the rebarbatively named XeLaTeX (basically, this just adds wonderfully easy font handling to LaTeX). I might report back on this in due course.

\end{geeky procrastination when I should be writing some Gödel lectures}

Maddy's Second Philosophy: I.4 Kant's transcendentalism

2009-03-10T20:24:00.009Z

Of course, there are philosophers who have delved into the Critique of Pure Reason and returned to tell a tale in plain prose with crisp arguments. In particular, I remember reading much of Jonathan Bennett's delightfully combative Kant's Analytic with admiration and enjoyment. But the fact that different travellers come back with such very different tales is not a great encouragement to follow them by going back to Kant yourself. After all, if the quite incompatible positions A and B can each be read into Kant by some smart and well-informed commentator, then evidently his writings can't be the best unambiguous developments of A and B. If we are interested in philosophy rather than history, and want to know which of those positions about space or causation or perception or whatever is true, then we might very reasonably think that it is better to seek out the best available modern version instead. So I for one have long since crossed Kant off my reading list.

But it's worse: at least when it comes to the Big Kantian Claims (as opposed to local discussions of particular topics) I don't really grasp the import of supposed cleaned up versions either. Some years ago, I was taking my usual cheerfully realist line on something or other, and a then colleague remarked -- with a patronizing air of scorn -- that obviously I hadn't understood how Kant had shown my sort of realism to be naive and untenable. When I asked how so, I just didn't follow the answer. And that's the trouble: Kantians intimate that their hero has revealed Deep and Difficult (nay, even Transcendental) truths which philistine naturalists fail to grasp. But they seem to have a lot of difficulty explaining these supposed insights in straightforward terms the rest of us can understand. So I simply don't "get" them.

Maddy evidently is a lot more patient about wading into the Kantian mire than I am. But in fact she ends up in much the same position: the Second Philosopher too "finds no compelling motivation for his extra-empirical [transcendental] inquiry", and also regards its supposed methods as obscure and confused.

In more detail: Maddy first distinguishes two kinds of interpretation of Kant's transcendentalism, 'harsh' and 'benign'. The harsh reading takes Kant to be making a distinction between appearances or empirical objects and things in themselves, where these are quite distinct, and the latter are "unknowable but they affect our sensibility to produce appearances, which we can and do know". The harsh reading, in Strawson's words, makes "Kant ... closer to Berkeley than he acknowledges". On the benign reading,

the appearance and the thing in itself aren't two separate items -- one mental, one extra-mental -- but a single object regarded in two different ways. ... The object as appearance is subject to our human forms and categories; it is non-mental, spatiotemporal, subject to causal laws. Thus Kant is an empirical realist, as opposed to Berkeley’s empirical idealist ... On the other hand, the object as it is in itself is not subject to our forms and categories; these are impositions of our minds, not features of the transcendental object. Thus Kant is also a transcendental idealist.

So on this reading, empirical enquiry and transcendental enquiry are two kinds of investigation of the same thing.

Maddy says little about the Kant of the harsh reading (which does seem to give him an entirely unattractive and arguably incoherent position), and concentrates her discussion on the benign Kant. So what is empirical enquiry and what is transcendental enquiry? The first, according to Maddy, is just science as we know and love it:

Unlike Descartes, who thinks ordinary scientific theorizing [at least, in the then prevailing style] needs justification and revision, Kant takes scientific methods to be entirely in order, for the purposes of empirical inquiry. Kant’s message to the Second Philosopher is not that she needs to reform her empirical investigations, but that she should add to them a level of transcendental enquiry. Where Descartes is only selling a new method, Kant is promoting a new purpose.

But what is this new purpose? It's here that the Second Philosopher struggles. By her lights there is just one kind of enquiry, empirically constrained, and she needs to be persuaded that there is another non-empirical level available to be pursued. What is its topic? What are its methods?

Not that she dismisses the very idea of two levels just because it is "unscientific" (for she is, as we've noted before, unhappy about attempts to police boundaries in the sort of way that suggests -- she is always open to new ideas and new methods). So she is happy enough to "respectfully ask Kant to explain what it is that his transcendental psychology studies, and how this study is to be conducted". But she doesn't find the answers persuasive.

Transcendental psychology, we are told, is the study of the nature of the discursive intellect. Sure, says the Second Philosopher, but how we do know that human cognition is discursive in the defined sense without empirical work? Ah, no, we are here engaged in an apriori enquiry: 'This transcendental reﬂection is a duty from which no one can escape if he would judge anything about things a priori’ (A263/B319). To which the Second Philosopher might well riposte, borrowing Reichenbach's words, 'The evolution of science in the last century may be regarded as a continuous process of disintegration of the Kantian synthetic a priori.' She is very unimpressed by the track-record of so-called a priori enquiries (outside, perhaps, mathematics). It seems that Kant's method for pursuing them just isn't reliable, and she hasn't been told how else transcendental enquiry is supposed to be carried out.

But surely, the Kantian will respond, there are questions about the necessary conditions that must be satisfied if the world is to be experienceable, and philosophers can surely reflect on those. The Second Philosopher needn't disagree, however: of course there are very general facts (as we've discovered them to be) about how our minds work, that have implications about what sorts of worlds can be experienced by minds like those. But she will regard the exploration of such general facts to be just part of psychology done at at a level of abstraction from implementation details: part of science, broadly construed, not part of a different enterprise done at a different level with a different methodology.

The Second Philosopher knows enough of the history of philosophy to be alert to the seductive dangers of neat dualisms, and is rightly suspicious of sales talk about two distinct levels of enquiry here, empirical and transcendental. So pending further enlightment, she rests content with her one-level, empirically constrained, mode of enquiry. And I'm with her on that!

Papers for papers

2009-03-06T23:14:00.004Z

Like everyone else, I download quite a few journal articles from current issues or from the Jstor archive. Question: just how do you keep the heaps of PDFs organized? What do you use to search across them?

Well, I've belated just discovered Papers (Mac OSX only, I'm afraid), a sort of iTunes for your PDFs. It has been around for over two years -- there's an old explanatory poster here -- and it knocks spots off the various previous solutions I've tried. Here are some high points:

You fire up Papers and there in its designated folder is your library of PDFs, neatly listed and sorted. Papers uses the Spotlight engine to do very fast searches. You can then click on items to read them from within Papers (and you can write notes too). And you can open different papers in different tabs, rather than have a clutter of windows.
That's very pretty, but Papers really comes into its own when e.g. you want to search and download papers e.g. from Jstor. You can search Jstor from within Papers (early releases of the program only knew about science databases: being able to work with a wider set of sources is the big new feature added in later versions). You can just store the found paper details for later: but click on the found title you want to fetch, and you get to the paper's Jstor download page. Download the paper, and it arrives in your library, with the author/title/year/journal etc. metadata all neatly listed -- and Papers systematically changes the title of the PDF file itself, from Jstor's to your preferred naming system. (I use author, date, first five words of title). So looking in the library folder itself from the Finder, you see a neatly and usefully named set of files.
How to you move some previously obtained paper into the library? Or a paper newly downloaded from a current journal issue? No problem, drop the paper onto the Papers icon in the dock, and it will appear in the Library. If it is recent with a DOI identifier, again Papers extracts the metadata and renames the file according to your system. Otherwise you give Papers e.g. the author name and a word or two from the title, and Papers asks Google scholar to find a match: click on the match, and -- whoosh! -- the paper is neatly filed and renamed again.
With a library full of papers you can then sort them in various ways, and make various "collections" (smart ones too, if you want). As I said, you can of course search your library from within Papers. And because the PDF library remains just that (i.e. the files aren't messed about with) you can inspect it from e.g. DevonThink if you want to do some much fancier "intelligent" searches.
The thing is a joy to use, as I'm discovering. You can of course export BibTex and other citation data if you want. And the interface is really neat. It was won an award for being quintessentially Mac -- which it is, to the point that they don't bother to provide a manual apart from a short getting-started video. Just remember to control-click on any likely-looking button or sidebar item, and you'll get a drop down menu of options.
And oh, if you really want, you can sync a collection of your papers to Papers for the iPhone or iPod Touch, to read some PDFs on the move.

All in all, I'm a newly converted enthusiast. Terrific.

Maddy's Second Philosophy: I.3 Hume's naturalism

2009-03-03T12:14:00.012Z

Has the Second Philosopher been too sanguine in resolutely dismissing scepticism? Here's a thought that might give her pause.

Hume starts off the Treatise as a cheerful naturalist. OK, he's not quite a naturalist of the Second Philosopher's stripe. She "won't follow Hume in placing the science of man (what she thinks of as psychology, sociology, linguistics, etc.) at the foundation of Natural Philosophy (what she thinks of as physics, chemistry, etc.) -- she tends to regard these various natural sciences as complementary parts of one big puzzle": but still she recognizes this Hume as a forerunner. Yet by the end of Book I, Hume's naturalistic starting point has seemingly driven him to a despairing skepticism about the senses. Is there something in his naturalistic stance that contains the seeds of its own undoing? Does this apply to naturalism more generally?

Some of course have argued so. Latterly, for example, Plantinga has argued that the contemporary naturalist will have to regard our cognitive mechanisms as quick-and-dirty adaptive devices which have evolved to get us successfully feeding, fighting, fleeing, and reproducing. And being success-conducive is one thing, truth-conductive is something else. The naturalist, by her own lights, has no good reason to suppose his deep-rooted cognitive habits are better than merely useful, and so should be sceptical about her own scientific endeavours, for scientific reasons.

I'll not discuss this (rather dismal) argument here, and Maddy doesn't either. But she does worry about how things went amiss, as she supposes, for Great Uncle David.

In the most general terms, his line of thought is familiar. We must distinguish between those perceptions and objects "which preserve a continued existence and identity" whether perceived or not. For arguments from the variability of our perceptions are supposed to show that "our sensible perceptions are not possessed of any [external] existence" (for, presenting inconsistent appearances, they can't all have "external existence", and since like effects have like causes, none of them do). The trouble then is, as Hume puts it in the Abstract, "the belief [in external objects], which attends experience" can't be defended by reason, so we are left with "nothing but a peculiar sentiment, or lively conception produced by habit. ... Philosophy wou'd render us entirely Pyrrhonian, were not nature too strong for it."

But what is Hume's conclusion here? That not all our beliefs are based on reason (we are not God-like enquirers, with rational insight into the nature of the world); many of our basic beliefs about the world around us are just produced in us by natural mechanisms operating at the same level as with other animals. So if 'knowledge' is conceived of as requiring reasoned beliefs, then we don't have much 'knowledge'. The naturalist can cheerfully agree with that, however. She doesn't share that concept of knowledge, and so she won't regard Hume's conclusion as sceptical in any worrying sense (for she's long since taken aboard the lesson that reasonable beliefs need not be reasoned beliefs).

Can we get something more alarming out of Hume, however? The Second Philosopher thinks not, or at least easily. Certainly, if some rather plausible-looking naturalist assumptions do lead to scepticism in the worrying sense, the sensible naturalist's first reaction will be to look around to see which dodgy planks on Neurath's boat are revealed on closer inspection to be letting in water, and do some running repairs if she can. For example, Hume writes

’Tis this principle [in effect, our ordinary cognitive norms], which makes us reason from causes and effects; and ’tis the same principle, which convinces us of the continued existence of external objects, when absent from the senses. But though these two operations be equally natural and necessary in the human mind, yet in some circumstances they are directly contrary, nor is it possible for us to reason justly and regularly from causes and effects, and at the same time believe the continued existence of matter.

But now why suppose those conflicting principles really are both 'necessary', and then embrace a worrying scepticism (if that's what they lead to in Hume, contrary to the suggestion above)? As Maddy puts it:

[The Second Philosopher's] spirit of open inquiry suggests a different course [from Hume's]: if our cognitive norms lead to a contradiction, shouldn’t we re-examine our cognitive norms? Why should we assume it is impossible to amend or correct them?

Which seems evidently right. Though Maddy adds

... Regarded second-philosophically, the despairing Hume hasn’t fully lived up to his guiding naturalistic impulse.

And that's arguably too flat-footed about Hume. As hinted above I'd say that what the real Hume was giving up on was our supposed capacity for a God-like 'knowledge' as imagined by some of his predecessors (see Edward Craig's The Mind of God and the Works of Man for a lot more on this). And there's nothing despairing about rejecting that: it is exactly what a persistent naturalist needs to do.

Be that as it may, however, as a claim about Hume. And anyway, there are more twists and turns in Maddy's discussion as she confronts more than one Hume found by different interpreters. But she ends up in the same place each time: at least in Hume's writings, "[C]ommon sense and its scientific refinements have not been convicted of undercutting the reasonableness of their own methods."

That looks right. Though, advocating a sublunary naturalistic line on the reasonable, Hume could probably have agreed.

Maddy's Second Philosophy: I.2 Neo-Cartesian skepticism

2009-03-02T16:58:00.015Z

The Descartes we met in the first section is a methodologist of science with a bad idea about how to improve our scientific success rate. He isn't so very far removed, then, from the Second Philosopher in terms of his general intellectual concerns. She can warmly approve of this Descartes's overall aim of doing science better: she just rejects his failed attempt at securing that aim.

But there's another Descartes (a fictional figure perhaps, but occupying a familiar location on the philosophical landscape). To quote Stroud,

By the end of his First Meditation, [this] Descartes finds that he has no good reason to believe anything about the world around him and therefore that he can know nothing of the external world.

For this Descartes -- or better, for the 'neo-Cartesian sceptic' -- doubt has become more-than-merely methodological. He is not just setting aside received but fallible beliefs about the world, reasonable and unreasonable alike, to locate an indubitable residue: he thinks he has an argument that even the most 'reasonable' received beliefs in fact just aren't reasonable. How does the naturalistically inclined philosopher respond?

Now, speaking for myself, I'm all for a cheerfully resolute response. We have every good reason to suppose that in ordinary life, when our usual criteria indicate that we are wide awake and not dreaming, we are wide awake and not dreaming. And if the sceptic asks me how I rule out the possibility that, for all that, my whole ordinary life is in fact just one big coherent dream (unlike any ordinary dream, of course, a put-up job engineered by an evil demon), I just riposte that I have been given not the slightest reason that to suppose that that is a live possibility. It is one thing to use the evil demon fantasy as Maddy's Descartes does, as an imaginative tool in trying to locate some core of infallible beliefs: it is something else entirely to imagine that the honest enquirer needs to rule out the evil demon scenario before he can regard science as a reasonable enterprise. The scientist just doesn't have to waste her time ruling out myriads of daft hypotheses she hasn't the slightest reason to suppose are true. And for me, pretty much end of story.

Speaking for her Second Philosopher, Maddy is rather more patient in tangling with the neo-Cartesian sceptic (though whether the sceptic will appreciate her efforts is a moot question). I won't go into all the ins and outs. She agrees, of course, that "she can’t justify anything without appeal to her familiar beliefs and methods", but so what? Ah, says Stroud,

... in philosophy we want to understand how any knowledge of an independent world is gained on any of the occasions on which knowledge of the world is gained through sense-perception. So, unlike ... everyday cases, when we understand the particular case in the way we must understand it for philosophical purposes, we cannot appeal to some piece of knowledge we think we have already got about an independent world. [My emphasis]

So now, as Maddy puts it, "it's freely admitted that the skeptic is engaged in a peculiarly philosophical project, distinct from mundane concerns". But what notion of 'philosophy' is in play here? I for one just haven't much grip on what sort of coherent project might be meant here, this special 'philosophical' enquiry that isn't part of science broadly conceived. Maddy offers the following from Stroud:

All of my knowledge of the external world is supposed to have been brought into question in one fell swoop ... I am to focus on my relation to the whole body of beliefs which I take to be knowledge of the external world and to ask, from 'outside' as it were ... whether and how I know it ...

But that gives the game away. If the 'philosophical' project is supposed to involve jumping outside my beliefs and methods, and trying to squint sideways at them from 'outside' as it were, to see how they match up to an external world, then that is just incoherent. (If the sceptic is complaining that the project can't be pulled off, then he's quite right the project is impossible, but quite wrong to complain.)

Maddy puts it this way,

Our Second Philosopher freely acknowledges one poignant aspect of the human condition: we can't step outside our system of beliefs and methods and justify them from an external perspective; the only perspective we can occupy is our own.

But why is that particularly 'poignant'? -- (dictionary: 'evoking a keen sense of sadness or regret'). You can't have a settled feeling of regret that you can't be both 'inside' and 'outside' your own thoughts (thinking them, but squinting at them sideways too) any more than you can sensibly regret that two and two isn't five, or regret that you can't back and kill your own grandfather before he sired your parent. Stroud himself talks of

the desire to get outside that knowledge and that condition, as it were, while somehow retaining all the resources needed to see them as they are.

But again, that is just an incoherent aspiration. And if it is a persistent aspiration, then that just means that it's a bit of intellectual bindweed, and the Second Philosopher shouldn't feel any sadness or regret about resolutely taking the flame-thrower to it.

Maddy's Second Philosophy: I.1 Descartes's first philosophy

2009-03-01T08:55:00.008Z

Here's a thumbnail sketch of one Descartes.

This Descartes's aim, by the end of Meditations, is -- as Maddy puts it -- to replace the reigning Scholastic Aristotelianism with his own Mechanistic Corpuscularism. For he believes that the then dominant systematic story of the world is in deep error. He has a diagnosis, too, of the source of error -- he sees Aristotelianism as springing from some deeply embedded childhood habits of thought. Radical measures are required to prise us out of such deep-rooted error. The 'Method of Doubt' provides the once-in-a-lifetime jolt needed to shift us out of certain childish thought-habits and to get us adopt better intellectual methods and open the way to improved science. Faced with even the most reasonable-seeming presumptions,

I must withhold my assent from these former beliefs just as carefully as I would from obvious falsehoods, if I want to discover any certainty in the sciences.

For this Descartes, it is not that our former beliefs are all unreasonable: but we are to play along with fantasies about dreams and evil demons as an instrumental step to help us sort out the safe beliefs from the contaminating dross. And then, as Maddy puts it,

The hope is that once we set aside all our ordinary beliefs, reasonable or not, some absolutely indubitable foundational beliefs will then emerge, on the basis of which science and common sense can then be given a ﬁrm foundation. The Method of Doubt is the one-time expedient that enables us to carry out this difﬁcult task.

So this Descartes's hope and belief is that, after his strategic retreat, he will still be left with enough indubitable foundational beliefs to provide a secure bridgehead from which he can fight back and recover those common-sense beliefs that don't carry the baggage of disputable theory, and then go on to ground a secure corpuscularian science.

Is this thumbnail sketch true to the historical René Descartes? For all I know, yes. And Maddy appeals to the work of the Descartes scholar Janet Broughton in support of her reading. But I don't suppose that -- as a Second Philosopher -- it matters particularly to Maddy whether the reading does get the real René right. She is after the truth about what there is and how we know it: and getting at those truths about the world is only indirectly aided, if at all, by pursuing the second-order issue of the precise truth about what some relatively remote struggling enquirer happened, with uneven degrees of clarity, to think was the truth about the world.

Let me, by way of aside, say just a bit more about this (speaking here for myself). Why should the philosopher be any more especially interested in the history of her subject than the physicist is in the history of hers? If you take a broadly naturalist line, then I think the answer, to a first approximation, is: there is no good reason. The physicist and philosopher alike should start from the hard-won available theoretical options in their best-developed forms. Of course, philosophy is difficult, there's a danger of foreclosing options too soon, and it is a good to remind ourselves that there may be more theoretical options than the currently most explored ones: the Great Dead Philosophers might provide a useful source we can mine for alternative ideas. So, less approximately, the naturalistic philosopher -- being grateful for all the help she can get in her pursuit of truth -- might occasionally delve into the history of philosophy for inspiration (and she supposes that she's more likely to get inspiration from something like the lines of thought actually pursued by her best predecessors than from straw positions created by incompetent exegesis). Still, by my lights, the naturalistic philosopher's interest in the history of her subject should remain relatively minor and completely instrumental. It perhaps feeds into her thinking about causation or knowledge, or whatever: but it is causation and knowledge that she cares about, and she is interested in Descartes or Hume or Kant only insofar as they offer useful pointers. And as soon as she finds herself at the edge of interpretative swamps -- which is in practice rather soon -- the naturalistic philosopher will typically lose interest: let the historians amuse themselves, and come back and tell her if and when they manage to dredge up any new nuggets of wisdom that will actually help her with her present philosophical problems. She's certainly not holding her breath.

Back though to Maddy's Descartes. How should the Second Philosopher regard his project? Well,

[She] will agree that many of her childhood beliefs were false, and that the judgments of common sense often need tempering or adjustment in light of further investigation, but she will hardly see these as reasons to suspend her use of the very methods that allowed her to uncover those errors and make the required corrections!

And she isn't much impressed with Descartes's assumption that (prior to engaging in the Method of Doubt) we are inevitably stuck in childish ways of thinking. The very existence of modern science seemingly gives the lie to that. So she isn't at all persuaded that pursuing the project of the Meditations is an essential prolegomenon to getting a successful science. Still, the Cartesian meditator does promise something she agrees would be worth having, i.e. a secure method for science. So, even though the Second Philosopher is now very suspicious about whether his methods are going to deliver, in her open-minded way

she might well think it proper procedure to read past the ﬁrst Meditation, to see what comes next. The unconvincing arguments that follow will quickly conﬁrm her expectation that there is no gain to be found in this direction.

The Second Philosopher, then, sees no reason to follow Descartes steps into the mess he gets himself into. Rather, "she will continue her investigation of the world in her familiar ways, despite her encounter with Descartes and his meditator." As Maddy puts it,

She will ask traditionally philosophical questions about what there is and how we know it, just as they do, but she will take perception as a mostly reliable guide to the existence of medium-sized physical objects, she will consult her astronomical observations and theories to weigh the existence of black holes, and she will treat questions of knowledge as involving the relations between the world -- as she understands it in her physics, chemistry, optics, geology, and so on -- and human beings -- as she understands them in her physiology, cognitive science, neuroscience, linguistics, and so on.

Which seems to me, I must say, just the right way to go!

But others might complain that the Second Philosopher has rather thumpingly missed the true lesson we should draw from her encounter with Descartes. Whether this is so is the topic of the next section.

Maddy's Second Philosophy: Introduction

2009-02-28T21:47:00.015Z

As announced, I'm (belatedly!) planning to comment here section-by-section on Penelope Maddy's Second Philosophy: A Naturalistic Method (OUP, 2007). Let me say straight away, having read on a hundred pages or so, that I have nothing but praise for Maddy's writing style, which I can only envy. This is a remarkably readable book. And perhaps I should add that I am also highly sympathetic to the kind of naturalistic project that she is pursuing: so it might turn out that my remarks here won't be that interesting (vigorous and acerbic criticism always being so much more fun to read!). But let's see ...

Who is the 'Second Philosopher'? She espouses a kind of 'naturalism' -- but that descriptive label has been used so variously, "to mark little more than a vague science-friendliness", as to become unhelpful. So over the opening chapters of the book, Maddy aims to characterize the Second Philosopher, not by giving a sharp definition of her position, but by means of a "character sketch", illustrating her philosophical disposition. But this much is clear from the outset: in her pursuit of truth, the Second Philosopher "rejects authority and tradition as evidence, she works to minimize prejudices and subjective factors that might skew her investigations." Rather, "she uses what we typically describe with our rough and ready term 'scientiﬁc methods'" though she avoids trying to give a definitive account of what that entails. She is open minded about how best to increase her knowledge; "she simply begins from commonsense perception and proceeds from there to systematic observation, active experimentation, theory formation and testing, working all the while to assess, correct, and improve her methods as she goes".

That perhaps makes it sound as if the Second Philosopher is just a scientist. But she has, as we will see in due course, a taste for some very general questions about the world and our epistemic relation to it and the concepts we bring to describe it, as well as for more local questions. And it is this (a matter of the range of her interests rather than of her intellectual methods) that distinguishes her from run-of-the-mill scientists. Unlike the working chemist, for example, she addresses "a wide range of questions we would ... typically regard as 'philosophical'" -- but she doesn't sense a crashing of the gears as she turns from narrower scientific questions to the more sweeping 'philosophical' ones. She has no criterion for sharply demarcating science from philosophy, and she aims to bring to bear the same methods of theory building and testing in her pursuit of truth, whether on the local or more global scale.

We recognize the tone of voice here! Here's Reichenbach (later quoted by Maddy):

Modern science ... has refused to recognize the authority of the philosopher who claims to know the truth from intuition, from insight into a world of ideas or into the nature of reason or the principles of being, or from whatever super-empirical source. There is no separate entrance to truth for philosophers.

The Second Philosopher, enlisting with Reichenbach, finds no place for a First Philosophy, prior to science, which can supply a priori principles of justification for science.

But in keeping with her anti-dogmatic stance, her open-minded willingness to test out any ideas that have some hope of advancing knowledge, it isn't that the Second Philosopher is refusing the very possibility of First Philosophy on the basis of some priori argument (after all, she isn't that keen on a priori arguments!). Rather, "[h]er reaction to extra-scientiﬁc philosophy is puzzlement; she asks methodically after its standards and goals, and assesses these by her own lights." She looks case-by-case at what is on offer from those who do have ambitions towards a First Philosophy and she finds herself quite unpersuaded by their offerings.

This is, in particular, her view of Descartes's project, as he engages in his Meditations on First Philosophy. He is promising, after all, a new method to put all future science on a firmer footing. So of course the Second Philosopher will sit up and take notice: she's all for anything that will improve science. As Maddy puts it,

When Descartes proposes that she adopt his Method of Doubt, she doesn’t reject it as 'unscientific'; impressed by the promised pay-off -- a firmer foundation for her beliefs -- she’s quite willing to give his proposal a try; she eventually discards it only as it proves ineffective.

The story is spelt out in more detail in Maddy's first chapter, so let's turn to that.

One hundred, not out.

2009-02-27T00:04:00.003Z

I started contributing to the Ask Philosophers website just over a year ago. I've just posted my hundredth response -- not one of my most exciting efforts, but of course obviously true like all the rest! You can read my collected works here. Leaving aside my own contributions, though, there are some really excellent contributions to the site. Certainly well worth pointing philosophical beginners towards.

Back to Gödel

2009-02-25T15:08:00.003Z

I've foolishly agreed to give a mini-mini-course on matters to do with incompleteness at a Cameleon weekend that Thomas Forster is organizing in four weeks time. I ought to find some new Deep and Interesting Things to say that aren't in my book -- but that's a trifle difficult because if it was interesting and I knew it, I'd surely have put it in. I think I'm going to bluff my way through talking about iterated consistency extensions, but I need to do a lot more homework before I'll feel comfortable, and at least the next two weeks are going to be very busy before I can really get round to that. Going to be a close-run thing!

Meanwhile, Arnon Avron has just been in touch with a rather embarrassing question about the book. Why didn't I just point out that Theorem 30.10 (The truths of L, the language of arithmetic, aren't r.e.) follows from Theorem 21.5 (given a sensible system of Gödel-numbering, no predicate of L can express the numerical property of numbering-a-truth-of-L).

Suppose for reductio there is a recursive function f that enumerates (the Gödel numbers for) the truths of L. Then we know that any recursive function can be expressed in L (put together Theorem 30.1 and the last remark in Sec 4.7). So in particular there is a formal wff F(x,y) which expresses the enumerating function f, and then by definition the formal wff Ex(Fx,y) will be satisfied by a number if and only if it numbers a truth of L. But then this is impossible by Theorem 21.5. Which proves Theorem 30.10.

Why the heck didn't I say that? I guess I was so keen to make a connection with the informal proof in Ch. 5, and so motivate the discussion of Turing machines, that I forgot to mention the simple proof. Oops!

Constructive ZF

2009-02-24T08:46:00.004Z

Just to note that there is a new entry in the Stanford Encyclopedia of Philosophy, on Set Theory: Constructive and Intuitionistic ZF, by Laura Crosilla. I'm out of my comfort zone here, but I found this a very interesting and helpful piece. The SEP really is going from strength to strength, and the logic/phil maths entries are most certainly of a fine standard.

The Leiter report is out

2009-02-23T23:50:00.002Z

The new Leiter report is out. No real surprises it seems, at least as far as the UK rankings are concerned. In particular, no surprise that we (= Cambridge) come out markedly better than on the RAE, for reasons it would be perhaps be inappropriate to go into here. Still, quite cheering.

Forthcoming attractions: Second Philosophy

2009-02-18T23:37:00.003Z

The last few weeks, since finishing Parsons, our Wednesday reading group has been talking about Jeffrey King's The Nature and Structure of Content. But I confess that, after three chapters, I've stirred up a rebellion (for I find King's project mystifying and am at a complete loss as to what the rules of the game he thinks he is playing are). So we are going to ditch King, and move on to read Penelope Maddy's Second Philosophy: A Naturalistic Method.

And I plan to comment here, section by section as I did on Parsons (at least that gets me to read the book carefully). Though I hope I finish the job rather more speedily this time. In fact, I predict I will, if I can project from my experience reading the first four chapters: for Maddy writes beautifully, with enviable clarity and style -- and I guess that I'm also intellectually much more at home with her naturalism. I'm looking forward to it.

So watch this space ...

Fools, damned fools, and the designers of online forms

2009-02-16T21:00:00.005Z

I've a sabbatical coming up next Lent term, and I really ought to apply for matching "double your research leave" funding from the AHRC to have another term off. Not that I hold out much hope of getting it rather too near retirement, but it Looks Good with the powers that be to make the effort. But I'm struggling with the idiocies of the on-line application form, which seems purpose designed to raise your blood pressure to the point where you decide that it would be better for your health to give up and bang your head against a wall instead.

I'm planning to write a shortish book on matters to do with the consistency of arithmetic, which is just underway now, and should be very well-advanced by the end of next Lent term, given I've promised to talk about the stuff in various places in the meantime. But "if you are seeking funding for the completion of a monograph, you need to identify the chapter headings and contents in the Case for Support." Eh? Am I just supposed to lie? How best to organize and chunk up your material might only become clear pretty late in the game, especially if -- like me -- you favour rather short snappy chapters. I haven't a clue what the chapter headings will turn out to be. Have the people designing this form ever written a book? The hell they haven't.

Oh well, set that aside for the moment: let's try to tackle the easy stuff. "You must state the reason why relief is required from the teaching, administration, examining and/or managerial duties detailed." Ah, that's straightforward: because I can't do two things at once. But then they know that, and my reason is exactly the same as everyone else's. So why the hell are they asking? Are they idiots?

Ah, now I have to list my current teaching. Fair enough. Make a list of current lecture courses. "Hours per week?" And what on earth does that mean? Official contact hours? Contact hours plus preparation time? Contact hours plus preparation time plus time going to grad seminars that aren't exactly duties but it would be a Very Bad Thing if no lecturers turned up to? Contact hours plus preparation time plus grad seminar time and background reading to be able to talk informally to various grads? Where do "teaching" hours stop? Who is to say? I press "Help". Which doesn't give any clue at all. More idiocy.

Well, I'll just have to see if our Faculty admin officer -- who is of a very calm and reasonable disposition, and knows about such things -- can hold my hand through the process tomorrow without me blowing a gasket.

Fortunately, I guess the dolts at the AHRC responsible for these forms don't read academic blogs.

Another week goes by ....

2009-02-14T08:55:00.009Z

A busy week. A short but interesting talk by Bob Hale at the Moral Sciences Club on Tuesday on the very idea that there might be possible worlds where the laws of logic are different -- what idea of possibility does that involve? Then on Wednesday afternoon, we've moved on from reading Charles Parsons to looking at Jeffrey King's The Nature and Structure of Content. I can't say I'm much attracted by his project, or by his theory of propositions -- but I don't think I'm exercised enough about the issues to put the energy into blogging about it here.

On Wednesday evening, a fun talk from Rob Trueman, one of our M.Phil. students, at the Serious Metaphysics Group. Suppose you reject the composition-is-identity thesis (as Rob thinks you should), so the train can't just be identified with the carriages. Then why does moving the train take no more force than moving the carriages? Rob was arguing that the natural answers we might try have surprising consequences. I wasn't convinced, but it was a fun evening (as 'Serious' certainly doesn't mean 'Solemn').

I've been trying to keep the Thursday Logic Seminar accessible to the handful of undergraduate students actually doing our Math Logic paper. So this week and next we are looking at four chapters from Marcus Giaquinto's The Search for Certainty, discussing Hilbert's Programme and the impact of Gödel's theorems on the Programme. I slightly disagree with Marcus's reconstruction of Hilbert (see my discussion of two routes for getting from the consistency an ideal theory to its real-soundness on p. 256 of my Gödel book): but that's very minor. Marcus's book is terrific.

On Friday, a prolonged Staff-Student Committee meeting. Every so often, for one reason or another, the disconnect between what many students think they want ("more continental philosophy") and what most professional philosophers in serious departments think they should get becomes dramatically evident. That's happened from time to time everywhere I've been. Most departments engineer some muddled compromises, but the strains can tell.

Top 50 philosophy blogs?

2009-02-09T16:54:00.002Z

Here's a new list of philosophy blogs -- a bit random, but such lists can turn up serendipitous finds, so I pass on the link.

Pohler's Proof Theory

2009-02-07T19:04:00.005Z

I've started struggling through Wolfram Pohler's recent Proof Theory: The First Step Into Impredicativity. And it is, I'm afraid, a struggle -- for a textbook exposition "pitched at undergraduate/graduate level", it is really quite unnecessarily hard going. For example, I can't imagine that anyone who hasn't already encountered the idea would have much hope of cottoning on to what is going on with the Veblen hierarchy from the discussion in Sec. 3.4. And even if you are very familiar with completeness proofs for first-order logic, it's made ridiculously hard work to see what's going in the completeness proof in Sec. 4.4.

I'll certainly keep ploughing on through, given the book's coverage. But not with relish. Why on earth write like this, without the introductory informal motivating comments and explanations of concept definitions and proof ideas that you'd give in lectures? I'm quite baffled that anyone can think that this is the right way to write a book intended for a student readership.

Anonymous comments: I think this is the usual way of writing a math book. Most, if not all, math books are written this way.

I don't think that's entirely true. I'm sitting here surrounded my shelves of relatively recent maths books, both pure and applied maths (dating from when I was writing my chaos book ten years ago, or teaching the philosophy of space-time theories). They do illustrate a whole spectrum of modes of presentation from rather relaxed to take-no-prisoners relentless formality. To be sure, there are too many of the latter kind -- but it is possible to do better!

Pro-vice-chancellorian bollocks

2009-02-04T22:13:00.005Z

Once upon a very long time ago, I spent a year or more of my life doing nothing else much but mathematics problems, preparing for the then entrance scholarship to get into Trinity to do maths. The standard was fairly stratospheric, and it was quite difficult to find enough practice questions on e.g. projective geometry. But one source was selected questions from various university examinations. Yes, degree-level questions from elsewhere being used as Cambridge entrance paper practice ... which rather undermined any belief I might have had in the myth of close equality of level of enterprise across different universities.

Well, things have no doubt changed in all sorts of ways since those distant days. But one thing remains the same. As almost everyone knows perfectly well, there are wide differences between what is intellectually required to get (say) an upper second-class degree in a particular subject at different UK universities.

But there are always those concerned to deny the obvious. Thus, in the correspondence column of the Guardian today, the Pro-vice-chancellor of London South Bank University writes

All universities work to a common understanding of degree classification (supported by an external examiner system which is common to all).

And another correspondent writes

There is, in fact, close equivalence of degree-level standards across institutions. It is effectively maintained through the Quality Assurance Agency, which asserts that the standard "should be at a similar level across the UK". The much-admired external examiner system is a major feature of this.

Which is all, of course, complete bollocks.

For a start, external examiners are very largely swapped between universities at similar levels in the pecking order. I bet, for example, that London South Bank University rarely uses examiners from Cambridge, or vice versa. And moreover, even when you do act as an external examiner for another university, your main job is to certify that the examining board there is behaving properly in following its own rules, and is behaving in a transparent and principled way. You might hope, where appropriate, for some rough comparabilities -- but only pretty rough. And rough comparability is patently not transitive. Standards at A might be roughly comparable with standards at B, and standards at B might be roughly comparable with standards at C, and so on down the chain. But standards at A may of course be in a different ballpark to standards at Z.

And so they certainly ought to be. For example, here in Cambridge we spend a great deal of effort in trying to recruit the brightest and best; we then give them maybe seventy hours or more one-to-one 'supervisions' (i.e. tutorials) in philosophy over three years, not to mention all sorts of other formal and informal small group teaching on top of lectures -- a quantity and quality of provision that most universities can only dream about. It would simply be a scandal if our students by the end weren't in general performing at a level a good few notches above what it is reasonable to expect in many other places. And what we demand of them in Tripos quite rightly reflects that. That "close equivalence of degree-level standards" remains a myth.

Logic matters website updated

2009-02-04T13:31:00.003Z

I've not been at all attentive to the Logic Matters website recently -- it's one of those things that sometimes I'm quite in the mood to do, and at other times it seems to drop right off the bottom of the "to do" list.

Anyway, there is now a new page "Other logic" linking to some recent papers and expository handouts. There's more to be added, which I'll want to tidy very slightly before irrecoverably letting them lose onto the net. Putting this page together was a rather cheering experience -- I seem to have done more recently than I remembered!

I hope to get round to finishing the port of the LaTeX for Logicians pages from the old form to the new one sooner rather than later -- at the moment some internal navigation is a bit flakey, though the requisite info is mostly in there somewhere.

Tim Crane moving to Cambridge

2009-02-02T12:07:00.002Z

Since it is now announced on the Leiter blog, I guess there is no reason now not to say here that we're delighted that Tim Crane is moving to Cambridge from UCL as the new Knightbridge Professor, from the autumn.

Parsons, the whole story, at last

2009-02-02T09:14:00.002Z

I have been blogging on and off for quite a while about Charles Parsons's Mathematical Thought and Its Objects, latterly as we worked through the book in a reading group here. I've now had a chance to put together a revised (sometimes considerably revised) version of all those posts into a single document -- over 50 pages, I'm afraid. You can download it here.

I've learnt quite a bit from the exercise. I'll be very interested in any comments or reactions.

Parsons's Mathematical Thought: Sec 55, Set theory

2009-01-26T10:10:00.004Z

The final(!) section of Parsons's book is one of the briefest, and its official topic is about the biggest -- the question of the justification of set-theoretic axioms. But, reasonably enough, Parsons just offers here some remarks on how the case of justifying set theory fits with his remarks in the preceding sections.

First, on "rational intuition" again. We can work ourselves into sufficient familiarity with ZFC for its axioms to come to seem intrinsically plausible -- but such rational intuitions (given the questions than have been raised, by mathematicians and philosophers) "fall short of intrinsic evidence". Which isn't very helpful.

And what about Parsons's modified holism? In the case of set theory, is there "a dialectical relation of axioms and their consequences such as our general discussion of Reason would suggest"? We might suppose not, given that (equivalents of) the standard axioms were already "essentially in place in Skolem's address of 1922". Nonetheless, Parsons suggests, we do find such a dialectical relation, historically in the reception of the axiom of choice, and perhaps now in continuing debates about large cardinal axioms, etc., where "the role of intrinsic plausibility" is much diminished, and having the right (or at least desirable) consequences are an essential part of their justiﬁcation. But, Parsons concludes -- the final sentence of his book -- "apart from the purely mathematical difﬁculties, many problems of methodology and interpretation remain in this area". Which is, to say the least, a rather disappointing note of anti-climax!

Afterword Later this week, I'll post (a link to) a single document wrapping up all the blog-posts here into a just slightly more polished whole, and then I must cut down 30K words to a short critical notice for Analysis Reviews. I feel I've learnt a lot from working through (occasionally, battling with) Parsons -- but in the end I suppose my verdict has to be a bit lukewarm. I'm unconvinced about his key claims on structuralism, on intuition, on the impredicativity of the notion of number, in each case in part because, after 340 pages, I'm still not really clear enough what the claims amount to.

MacBook Air, six months on

2009-01-26T00:00:00.007Z

Anyone out there who is still wavering about getting a MacBook Air might be interested in some comments from a delighted owner who has now had one for six months (this updates my "after one month" post from last August). Everyone else can, of course, just cheerfully ignore this again!

As background, I make heavy academic use of a computer (particularly using LaTeX, and reading a lot of papers, even books, onscreen, as well of course as the usual surfing, emailing etc.) but don't really use one as a media centre except for light iPhoto use, and occasionally ripping CDs into iTunes for transfer to an iPod.

The portability is fantastic. No question. Just to compare: I've had in the past a 15" Titanium PowerBook, a 15" G4 PowerBook, and a 17" MacBook Pro before; and they've of course been portable in the sense I could heave them from home to my office and back. But all of them were honestly just too heavy/bulky to make that particularly convenient. I very rarely bothered to take them elsewhere, e.g. to a coffee shop, or even a library. (You might well ask why on earth, in that case, I had portables at all! Answer: Partly because our Cambridge house is very small, my "study" is the size of a large cupboard, and I very much like to be able to work on the kitchen table for a change of scene, or answer emails with a computer on my knees in the living room in the evening. So I certainly want "local" portability. And partly I need to be able to drive data projectors when lecturing.) Anyway, by contrast with the earlier portables, I can and do cheerfully tote the MBA (in its snug protective sleeve) anywhere, without really thinking about it, whether or not I'm definitely planning to use it. It just is so light and convenient. Much lighter than the new aluminium 13" regular MacBook which I've tried out in the Apple Store too. Yet the MBA always feels remarkably sturdy. There's not a sign really of six months of constant use.
Some early reviews complained about the MBA's footprint, saying that it isn't a genuine ultraportable. People still complain about that. Well, true, I can imagine e.g. it wouldn't be that easy to use in the cramped conditions of an airline seat. But that sort of issue just doesn't arise for me. It's not the footprint but the lightness and thinness which means that you can carry it so very comfortably in one hand, and of course the larger-than-ultra footprint goes with the stunningly good, uncramped, screen and the generous keyboard. In my kind of usage now, I've never found the footprint an issue.
I rarely use the MBA to do anything very processor-intensive for a prolonged period of time, and it normally remains cool -- though the fans can sometimes kick in a bit enthusiastically e.g. when backing up. And the battery life seems just fine: well over three hours for writing, text-browsing, reading. Recharging though is pretty slow: but if you need to take it with you, then the MBA's charger is very small and portable (though I've bought a second one for my office in the faculty, and so don't find in practice I need to carry it around).
One main reason I traded up a couple of years ago from the 15" G4 machine was that LaTeX ran pretty slowly: nearly 30 seconds to typeset my Gödel book on the G4, about 4 seconds on the new intel MacBook Pro. The MBA, despite its slower chip, seems almost as fast running LaTeX , and indeed in most other ways: occasionally, e.g. when opening an application, the MBA is noticeably slower -- but it has never been a particularly irritating issue. So this is plenty fast enough.
And the reason, when I traded up the previous time, I chose the 17" MBP model was to have enough "real estate" to have a TeXShop editing window and the PDF output window side-by-side and comfortably readable. Obviously, I'm now looking at 1280 x 800 pixels, rather than 1680 x 1050 (so that's just 58% as much). But I've surprised myself by getting very used to working with overlapping windows again, and the screen quality is really terrific. The best I've ever had by far. Of course it is nicer e.g. for extended on-screen reading to plug in an external monitor as well. But not any sort of necessity -- and indeed I seem these days pretty often not to bother even if I'm sitting next to the external monitor.
What about the paucity of ports, mentioned critically by all the reviewers, or the absence of an onboard CD drive? Really not an issue. I've a couple of times wished there were two USB ports, I bought a little one-to-two-port splitter, for very occasional home use, but in fact even when I don't have it with me, I've never been seriously annoyed. Of course, if I had one of the new version MBA's I'd be tempted with one of the new displays that also acts as a USB hub: but that would be an indulgence. I've latterly bought an external CD/DVD drive built for the MBA, for when I occasionally need one. (The one caveat concerns day one, long before I got the external drive. Since there is no firewire port, you can't migrate files from your old computer to your new MBA using the usual firewire connection. And using a wireless connection to migrate is both painfully slow and seems flaky. Is that a problem? I didn't find really it so. I installed new versions of necessary additional software, like the LaTeX installation, over the web, and then copied my documents folder and other bits and pieces from a SuperDuper! clone of the old hard disk on an external drive. Quick to do, and resulting in a clean and tidy MBA.)
So that's all very, very positive. Are there any negatives? The flat keyboard is surprisingly nice to use (much better than I imagined it would be). But, unlike the almost silent similarly flat new iMac keyboards, I find the MBA version to does seem a bit noisier (and a bit more so than the MBP keyboard). But that's a very marginal disappointment.
I thought, when I bought the MBA a month ago, I'd be using it very much as a second machine, carrying on using the 17" MPB (with external monitor) as a main, quasi-desktop, set-up. In fact I find I now almost never use the MPB.
So, assuming a three year life cycle (and it seems very well built so should last longer with a battery refresh after a while), the MBA after education discount costs much less than half a pint of beer a day. Put like that, how can you resist?
And then, of course, there is the "Wow!"-factor ...

Parsons's Mathematical Objects: Sec. 54, Arithmetic

2009-01-25T20:19:00.005Z

How does arithmetic fit into the sort of picture of the role of reason and so-called "rational intuition" drawn in Secs. 52 and 53?

The bald claim that some basic principles of arithmetic are "self-evident" is, Parsons thinks, decidedly unhelpful. Rather, "in mathematical thought and practice, the axioms of arithmetic are embedded in a rather dense network ... [which] serves to buttress [their] evident character ... so that in that respect their evident character does not just come from their intrinsic plausibility." Moreover, there is a subtle interplay between general principles and elementary arithmetical claims -- a dialectic "between attitudes towards mathematical axioms and rules and methodological or philosophical attitudes having to do with constructivity, predicativity, feasibility, and the like". Which, as Parsons notes, is all beginning to sound rather Quinean. How is his position distinctive?

Not by making any more play with talk of "rational intuition", which made its temporary appearance in Sec. 53 just as a way of talking about what is intrinsically plausible: indeed, the idea that the axioms of arithmetic derive a special status from being grounded in rational intuition is said to be "in an important way misleading". Where Parsons does depart from Quine -- and it is no surprise to be told, at this stage in the book! -- is in holding that some elementary arithmetic principles can be intuitively known in the Hilbertian sense he discussed in earlier chapters. And the main point he seems to want to make in this chapter is that, as we move to more sophisticated areas of arithmetic which cannot directly be so grounded, so "the conceptual or rational element in arithmetical knowledge becomes much more prominent", the web of arithmetic isn't thereby totally severed from intuitive knowledge grounded in intuitions of stroke strings and the like. It is still the case that "an intuitive domain witnesses the possibility of the structure of numbers".

Of course, how impressed we are by that claim will depend on how well we think Parsons defended his conception of intuitive knowledge in earlier chapters (and I'm not going to go over that ground again now, and nor indeed does Parsons). And what grounds the parts of arithmetic that don't get rooted in Hilbertian intuition? To be sure, those more advanced parts can get tied to other bits of mathematics, notably set theory, so there is that much rational constraint. But that just shifts the question: what grounds those theories? (There are some remarks in the next chapter, but as we'll see they are not very unhelpful.)

So where have we got to? Parsons's picture of arithmetic retains a role for Hilbertian intuition. And unlike an "all-in" holism, he wants to emphasize the epistemic stratification of mathematics (though his remarks on that stratification really do little more than point to the phenomenon). But still, "our view does not differ toto caelo from holism". And I'm left really pretty unclear what, in the end, the status of the whole web of arithmetical belief is supposed to be.

Parsons's Mathematical Objects: Secs 52-53, Reason, "rational intuition" and perception

2009-01-24T22:59:00.004Z

Back to Parsons, to look at the final chapter of his book, called simply 'Reason'. And after the particularly bumpy ride in the previous chapter, this one starts in a very gentle low-key way.

In Sec. 52, 'Reason and "rational intuition"', Parsons rehearses some features of our practice of supporting our claims by giving reasons (occasionally, he talks of 'features of Reason' with a capital 'R': but this seems just to be Kantian verbal tic without particular significance). He mentions five. (a) Reasoning involves logical inference (and "because of their high degree of obviousness and apparently maximal generality, we do not seem to be able to give a justification of the most elementary logical principles that is not in some degree circular, in that inferences codified by logic will be used in the justification"). (b) In a given local argumentative context, "some statements ... play the role of principles which are regarded as plausible (and possibly even evident) without themselves being the conclusion of arguments (or at least, their plausibility or evidence does not rest on the availability of such arguments)." (c) There is there is a drive towards systematization in our reason-giving -- "manifested in a very particular way [in the case of mathematics], though the axiomatic method". (d) Further, within a systematization, there is a to-and-fro dialectical process of reaching a reflective equilibrium, as we play off seemingly plausible local principles against more over-arching generalizing claims. (e) Relatedly, "In the end we have to decide, on the basis of the whole of our knowledge and the mutual connections of its parts whether to credit a given instance of apparent self-evidence or a given case of what appears to be perception".

Now, that final Quinean anti-foundationalism is little more than baldly asserted. And how does Parsons want us to divide up principles of logical inference from other parts of a systematized body of knowledge? His remarks about treating the law of excluded middle "simply as an assumption of classical mathematics" suggest that he might want to restrict logic proper to some undisputed core -- though he doesn't tell us what that is. Still, quibbles apart, the drift of Parsons's remarks here will strike most readers nowadays as unexceptionable.

Sec. 52, 'Rational inuition and perception', says a bit more to compare and contrast intuitions in the sense of statements found in a given context of reasoning to be intrinsically plausible -- call these "rational intuitions" -- and intuitions in the more Kantian sense that has occupied Parsons in earlier chapters. As he says, "intrinsic plausibility is not strongly analogous to perception [of objects]", in the way that Kantian intuition is supposed to be. But perhaps analogies with perception remain. For one thing, there is the Gödelian view that intrinsic plausibility for some mathematical propositions involves something like perception of concepts. And there is perhaps is another analogy too, suggested by George Bealer: reason is subject to illusions that, like perceptual illusions, persist even after they have been exposed. But Parsons only briefly floats those ideas here, and the section concludes with a different thought, namely there is a kind of epistemic stratification to mathematics, with propositions at the lowest level seeming indisputably self-evident, and as we get more general and more abstract, self-evidence decreases. Which is anodyne indeed.

Wittgenstein's Notes on Logic

2009-01-21T21:29:00.004Z

Just to say that my colleague Michael Potter's intriguing new book Wittgenstein's Notes on Logic is published tomorrow.

Nerdy stuff

2009-01-20T13:19:00.002Z

Just for fellow Macaholics ...

I've just noticed that a new version of TeXShop has been released in the last couple of weeks, with a couple of useful little tweaks.
I had a pre-release trial copy of Things for a while, and now the first proper release is out. Very neat and very simple: so it is, for once, "task management" software -- ok, a fancy way of keeping To Do lists -- that I actually do use.
Oh, and I've just got an Iomega eGo Helium external drive. Very small and no power block, so easy to tote, and no fan so very quiet. It's a bit sad to get even mildly pleased by a hard drive. But still, it is rather pretty ...

Leiter Report

2009-01-20T00:07:00.002Z

I notice that the Leiter Report pre-publication headlines about UK departments has Cambridge ranked third, after Oxford, and the Stirling/St Andrews show. A happier outcome than in the RAE -- and the PGR rankings are rather a better indicator for prospective grad students, given that the RAE carves up the Cambridge philosophers into their separate institutional units, and the PGR rankings clump us together.

Not that we place too much store by such things. Oh, not at all. Perish the thought.

Can Smiley be Carnapped?

2009-01-19T12:26:00.006Z

The Second Cambridge Graduate Conference on the Philosophy of Logic and Mathematics took place over the weekend. You can see what you missed here. It would be nice to drum up just a bit more support next time -- for I'm sure there will be a third in the series. The most substantial talk was, perhaps unsurprisingly, by Tim Williamson, who was running through some of the arguments of his piece Barcan Formulas in Second-Order Modal Logic.

I was responding to a talk by Julien Murzi and Ole Hjortland, based on their 'Inferentialism and the Categoricity Problem: Reply to Raatikainen' (which is coming out in Analysis). One part of their talk was about Timothy Smiley's bilateralist treatment of the Carnap problem, and that's what my comments focussed on. Here's a slightly expanded version of my comments, defending the local hero, rewritten though to be more stand-alone.

Parsons Mathematical Thought: Sec. 51, Predicativity and inductive definitions

2009-01-13T21:08:00.002Z

The final section of Ch. 8 sits rather uneasily with what's gone before. The preceding sections are about arithmetic and ordinary arithmetic induction, while this one briskly touches on issues arising from Feferman's work on predicative analysis, and iterating reflection into the transfinite. It also considers whether there is a sense in which a rather different (and stronger) theory given by Paul Lorenzen some fifty years ago can also be called 'predicative'. There is a page here reminding us of something of the historical genesis of the notion of predicativity: but there is nothing, I think, in this section which helps us get any clearer about the situation with arithmetic, the main concern of the chapter. So I'll say no more about it.

Travel broadens the mind ...

2009-01-13T19:38:00.004Z

When I was editing Analysis, I went to quite a few conferences in the line of fairly pleasurable duty, to find out what the bright young things were up to, what the hot topics were. But since then I've become a stay-at-home, going to a few conferences here in Cambridge, but otherwise not venturing out much. Philosophical globe-trotting for the sake of it has never much appealed. So, it's going a bit against type to have just agreed to spend a couple of months in New Zealand next year as a Visiting Erskine Fellow at the University of Canterbury. But by all accounts, the place is wonderfully welcoming to visitors, a gentle-paced sojourn in one place attracts me much more than the kind of whistle-stop tours some people delight in, and New Zealand is spectacularly beautiful. I'm beginning to look forward to it a lot.