## Sunday, November 08, 2009

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

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).

## Friday, November 06, 2009

### Gödel Without Tears -- 5

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:

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

## Wednesday, November 04, 2009

### Ruse gets a beta minus.

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

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

## Monday, November 02, 2009

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

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

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:

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

## Monday, October 26, 2009

### Gödel Without Tears -- 3

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.

## Tuesday, October 20, 2009

### The Autonomy of Mathematical Knowledge -- Chap. 1

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.

1. 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?
2. 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.

## Monday, October 19, 2009

### Curtis Franks: The Autonomy of Mathematical Knowledge

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!). ## Saturday, October 17, 2009 ### Gödel Without Tears -- 2 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. ## Wednesday, October 14, 2009 ### Modal logic, with a lot more tears than necessary 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! ## Monday, October 12, 2009 ### Gödel Without Tears -- 1 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 ... ## Sunday, October 11, 2009 ### Gowers's conversation about complexity lower bounds 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. ## Saturday, October 10, 2009 ### Mullova/Dantone play Bach 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 .... 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. ## Thursday, September 24, 2009 ### Research Excellence Bullshit 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." ## Wednesday, September 23, 2009 ### Schubert's Piano Trios 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.) ## Friday, September 18, 2009 ### Praise for Just and Weese! 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. ## Saturday, September 12, 2009 ### Student evaluations 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 ## Tuesday, September 08, 2009 ### Math logic reading list (updated) 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 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 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. ## Wednesday, September 02, 2009 ### School maths, from the distant past 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! ## Tuesday, August 11, 2009 ### Simpson's SOSOA 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. ## Tuesday, August 04, 2009 ### Disappearing logic again 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 ... 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 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. ## Monday, August 03, 2009 ### Conceptual mathematics again 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. ## Wednesday, July 29, 2009 ### Logic disappearing over the horizon .... 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. ## Saturday, July 25, 2009 ### World-class again 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 ... ## Wednesday, July 22, 2009 ### Congratulations to Wilfrid Hodges! Logicians will be delighted to see that Wilfrid Hodges has been elected a Fellow of the British Academy. ### One to cross off your list 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. ## Friday, July 17, 2009 ### Time-travelling with Google maps 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. ## Tuesday, July 14, 2009 ### Twitter is sometimes not entirely frivolous 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. ## Monday, July 06, 2009 ### Gödel sorted ... until the next time 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 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}
{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.

## Tuesday, June 30, 2009

### Rum and reason

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!

## Friday, June 26, 2009

### Rosser's Theorem

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!!!

## Monday, June 22, 2009

### Functions and gunctions

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?".

## Sunday, June 21, 2009

### The book problem

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!

## Saturday, June 20, 2009

### Gowers on Razborov's theorem about monotone circuit complexity

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.

## Friday, June 19, 2009

### It's tough (reprise)

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 ...

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.

## Wednesday, June 03, 2009

### Another very warm recommendation

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

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.

## Friday, May 29, 2009

### The Wisdom of Wikipedia

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.

## Sunday, May 24, 2009

### Praise where praise is due

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!

## Thursday, May 21, 2009

### It's that time of year again ...

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.

## Wednesday, May 13, 2009

### Quantum computation again

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).

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.

## Monday, May 11, 2009

### Gowers online!

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.

## Wednesday, May 06, 2009

### Quantum Computing Since Democritus

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.)

## Monday, May 04, 2009

### The State of the Nation

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.