tag:blogger.com,1999:blog-234786892018-06-19T09:49:24.629+00:00Logic MattersNB : You have reached the blog's old addressPeter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.comBlogger490125tag:blogger.com,1999:blog-23478689.post-70562710175553969852009-11-08T10:12:00.004+00:002009-11-08T12:30:45.926+00:00The blog is dead .... long live the blog!After almost 500 posts, this will be the last post here, meaning at this URL ....<br /><br />.... but I'll be continuing the Logic Matters blog at <a href="http://www.logicmatters.net/">logicmatters.net</a> (and all the posts here at Blogger have been imported to that address, though the aesthetics are at the moment a bit primitive).<br /><br />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 <span style="font-style: italic;">rest</span> 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).Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com4tag:blogger.com,1999:blog-23478689.post-32970079675847281792009-11-06T15:16:00.003+00:002009-11-06T15:20:40.484+00:00Gödel Without Tears -- 5Here now is <a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT05.pdf">the fifth episode</a> 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.]<br /><br />The previous episodes are available:<br /><ol><li><a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT01.pdf">Episode 1</a>, Incompleteness -- the very idea (version of Oct. 16)</li><li><a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT02.pdf">Episode 2</a>. Incompleteness and undecidability (version of Oct. 26)</li><li><a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT03.pdf">Episode 3</a>. Two weak arithmetics (version of Nov. 1)</li><li><a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT04.pdf">Episode 4</a>. First-order Peano Arithmetic (version of Nov. 1)</li></ol>Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com0tag:blogger.com,1999:blog-23478689.post-7273372712242831792009-11-04T23:00:00.003+00:002009-11-04T23:44:23.266+00:00Ruse 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. <a href="http://www.guardian.co.uk/commentisfree/belief/2009/nov/02/atheism-dawkins-ruse">Michael Ruse writes in today's Guardian </a>on whether there is an "atheist schism". And he immediately kicks off on the wrong foot.<br /><blockquote>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.</blockquote>Eh? Where on earth has that "or meaning" come from? In what coherent sense of "meaning" does an atheist have to deny meaning?<br /><br />It gets worse. Eventually a lot worse.<br /><blockquote>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?</blockquote>That is so mind-bogglingly inept it is difficult to believe that Ruse means it seriously. Does Ruse really, <span style="font-style: italic;">really</span>, 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?<br /><br />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 <span style="font-style: italic;">philosophers</span> 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.Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com6tag:blogger.com,1999:blog-23478689.post-58887145135891885012009-11-04T11:18:00.003+00:002009-11-04T12:11:18.940+00:00The Autonomy of Mathematical Knowledge -- Chap. 2, §§3-5To 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 <i>is</i> someone's basic stance, then you'd expect him to very much want to know <i>which</i> 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 <i>doesn't</i> lead to the death of applicable maths. Yet Hilbert seems not to show much interest in that.<br /><br />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?<br /><br />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 <i>mathematical</i> 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 <i>mathematically</i> convince ourselves that our theories are in good order -- and hence we won't be seduced into thinking that our theories <i>need</i> bolstering from outside by being given supposedly firmer "foundations".<br /><br />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."<br /><br />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 <i>kind</i> of philosopher he is. Likewise, Franks is emphasizing the <i>kind</i> 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.<br /><br />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. <i>To be continued</i>Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com1tag:blogger.com,1999:blog-23478689.post-59115065558890185722009-11-02T15:57:00.012+00:002009-11-03T23:13:48.741+00:00The Autonomy of Mathematical Knowledge -- Chap. 2, §§1 & 2Hilbert 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 <span style="font-style: italic;">Principia </span>might suppose).<br /><br />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 <span style="font-style: italic;">anti</span>-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).<br /><br />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 <i>success</i> for theories in general (there are just too many successful but ugly theories, and too many elegant failures). So that seemingly leaves (successful) <i>application</i> 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 <i>that</i> 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 <i>applicable</i> analysis can in fact all be done predicatively, and so <span style="font-style: italic;">no</span> reflected glory will accrue to classical mathematics pursued with impredicative methods since those methods are not validated by essentially featuring in applicable maths.)<br /><br />So what <i>does</i> Hilbert’s alleged “naturalism” amount to? <span style="font-style: italic;">To be continued.</span> <span style="font-style: italic;"> </span>Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com0tag:blogger.com,1999:blog-23478689.post-52542393538961349122009-11-02T13:20:00.005+00:002009-11-02T16:41:27.297+00:00Gödel Without Tears -- 4Here now is <a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT04.pdf">the fourth episode</a> [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!<br /><br />Here are the previous episodes:<br /><ol><li><a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT01.pdf">Episode 1</a>, Incompleteness -- the very idea (version of Oct. 16)</li><li><a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT02.pdf">Episode 2</a>. Incompleteness and undecidability (version of Oct. 26)</li><li><a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT03.pdf">Episode 3</a>. Two weak arithmetics (version of Nov. 1)</li></ol>Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com2tag:blogger.com,1999:blog-23478689.post-76983743110523241092009-10-26T11:14:00.005+00:002009-11-02T13:25:10.785+00:00Gödel Without Tears -- 3Here's <a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT03.pdf">the third episode</a> (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 <span style="font-style: italic;">do</span> 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.Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com7tag:blogger.com,1999:blog-23478689.post-4133227942910800292009-10-20T15:10:00.005+00:002009-11-03T22:55:47.707+00:00The Autonomy of Mathematical Knowledge -- Chap. 1As I said, I'm planning to blog, chapter by chapter, about Curtis Franks’s new book on Hilbert, <span style="font-style: italic;">The Autonomy of Mathematical Knowledge</span> (all page references are to this book). Any comments on my comments will of course be welcome!<br /><br />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!<br /><br />As Franks notes, there are two kinds of responses that we can have to the paradoxes that threaten Cantor's paradise.<br /><ol><li>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?<br /></li><li>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 <span style="font-style: italic;">mathematics</span> 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. </li></ol>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.<br /><br />But Franks is having none of this. <span style="font-style: italic;">His</span> 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 <span style="font-style: italic;">about</span> 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).<br /><br />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. <span style="font-style: italic;">To be continued.</span>Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com2tag:blogger.com,1999:blog-23478689.post-37074878505188485762009-10-19T11:40:00.007+00:002009-10-26T13:05:47.854+00:00Curtis Franks: The Autonomy of Mathematical KnowledgeOn Saturday, from the new books stand the CUP bookshop, I picked up a copy of Curtis Franks's <a href="http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521514378"><span style="font-style: italic;">The Autonomy of Mathematical Knowledge: Hilbert's Program Revisited</span></a>.<br /><br />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<span style="font-style: italic;"> £45/$75</span>? Much as though I love CUP, that really <span style="font-style: italic;">is</span> 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.<br /><br />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!).Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com2tag:blogger.com,1999:blog-23478689.post-25397991695191357862009-10-17T15:21:00.003+00:002009-10-17T20:49:02.457+00:00Gödel Without Tears -- 2As promised, <a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT02.pdf">Episode 2 of Gödel Without Tears</a> (in which we prove sufficiently strong theories are undecidable and incomplete -- just like that!)<br /><br />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 <a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT01.pdf">here's the latest version of Episode 1, dated 16 October</a>.Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com5tag:blogger.com,1999:blog-23478689.post-4903799239204893722009-10-14T20:34:00.003+00:002009-10-14T22:32:44.970+00:00Modal logic, with a lot more tears than necessary<span style="font-style: italic;"></span>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 <span style="font-style: italic;">Modal Logic</span> (OUP, 2008). <span style="font-style: italic;">Mea culpa</span>. 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.<br /><br />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.)<br /><br />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.<br /><br />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 <span style="font-weight: bold;">c</span>, <span style="font-weight: bold;">n</span> and <span style="font-weight: bold;">l</span>, whatever <span style="font-style: italic;">they</span> 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 <span style="font-style: italic;">is</span> 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.) <br /><br />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.<br /><br />Consider your old friend, von Wright's <span style="font-style: italic;">M</span>. 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 Box<span style="font-style: italic;">A </span>if we've proved <span style="font-style: italic;">A</span> 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 <span style="font-style: italic;">wffs</span>, not to previous (sub)proofs. This means that Hughes and Cresswell's <span style="font-style: italic;">M</span> is not a modal system according to Cocchiarella and Freund. And when they talk about <span style="font-style: italic;">M</span>, 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 <span style="font-style: italic;">they</span> call the rule of necessitation, which tells us that if there is a proof of <span style="font-style: italic;">A </span><span style="font-style: italic;"></span>from no assumptions in their system <span style="font-style: italic;">M</span>, then there is also a proof of Box<span style="font-style: italic;">A </span>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.<br /><br />Now, I don't particularly object to Cocchiarella and Freund doing things this way. But I <span style="font-style: italic;">do</span> 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?<br /><br />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! <span style="font-style: italic;"></span>Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com0tag:blogger.com,1999:blog-23478689.post-5203297651324759972009-10-12T10:40:00.004+00:002009-10-13T15:03:52.413+00:00Gödel Without Tears -- 1Here, as promised, is the first of a series of lecture handouts (roughly weekly, and about twelve in all) encouragingly titled <a href="http://www.phil.cam.ac.uk/teaching_staff/Smith/blogstuff/GWT01.pdf">Gödel Without Tears -- 1.</a> 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!<br /><br /><span style="font-style: italic;">Later</span>: I've already replaced the first version with a slightly better one ...Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com3tag:blogger.com,1999:blog-23478689.post-12929284048217728362009-10-11T07:10:00.002+00:002009-10-11T07:35:09.041+00:00Gowers's conversation about complexity lower boundsI should have mentioned before that <a href="http://gowers.wordpress.com/">Tim Gowers's blog </a>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.<br /><br />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 <span style="font-style: italic;">why</span> something that initially seems as though it <span style="font-style: italic;">ought</span> to be easy to settle (P = NP?) is <span style="font-style: italic;">really hard</span>.Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com0tag:blogger.com,1999:blog-23478689.post-88868975711337100142009-10-10T21:13:00.004+00:002009-10-13T15:06:33.008+00:00Mullova/Dantone play Bach<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_wgASjh7v8gE/StD5Za5V9eI/AAAAAAAAAGk/_BK_RvvEj18/s1600-h/Mullova-Dantone.jpg"><img style="margin: 0pt 0pt 10px 10px; float: right; cursor: pointer; width: 265px; height: 265px;" src="http://1.bp.blogspot.com/_wgASjh7v8gE/StD5Za5V9eI/AAAAAAAAAGk/_BK_RvvEj18/s320/Mullova-Dantone.jpg" alt="" id="BLOGGER_PHOTO_ID_5391082969074365922" border="0" /></a>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 <span style="font-style: italic;">Die Schöne Müllerin</span>, and the Lindsays (often). Mullova finished up playing the great <span style="font-style: italic;">Chaconne</span> 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.<br /><br />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!Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com0tag:blogger.com,1999:blog-23478689.post-56871053945980027262009-10-10T17:20:00.003+00:002009-10-11T07:42:36.935+00:00Oh, 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.<br /><br />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 <span style="font-style: italic;">exactly</span> 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 <span style="font-style: italic;">must</span> push on next week.<br /><br />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 <span style="font-style: italic;">much</span> 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, <span style="font-style: italic;">Gödel Without Tears. </span>Watch this space ...<br /><br />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.Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com0tag:blogger.com,1999:blog-23478689.post-39712624515430822222009-09-24T20:57:00.014+00:002009-09-25T21:37:30.698+00:00Research Excellence BullshitSo, there's another consultation document on the <a href="http://www.hefce.ac.uk/pubs/hefce/2009/09_38/09_38.pdf">Research Excellence Framework</a> -- "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<span style="font-style: italic;"> quite</span> what we might hope for.<br /><br />Ok, ok, I jest. It isn't <span style="font-style: italic;">at all</span> 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? <span style="font-style: italic;">Impact? </span>What's that?<br /><br />Well, the document gives "a common menu of impact indicators" under various headings to help us out. Here are the headings ...<br /><ul><li>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".]<br /></li><li>Creating new businesses, improving the performance of existing businesses, or commercialising new products or processes</li><li>Attracting R&D investment from global business</li><li>Better informed public policy-making or improved public services</li><li>Improved patient care or health outcomes</li><li>Progress towards sustainable development, including environmental sustainability</li><li>Cultural enrichment, including improved public engagement with science and research</li><li>Improved social welfare, social cohesion or national security</li><li>Other quality of life benefits</li></ul>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?<br /><br />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 <span style="font-style: italic;">that</span> 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 <span style="font-style: italic;">not</span> include "we do not intend to include impact through intellectual influence on scientific knowledge and academia".<br /><br />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.<br /><br />Brilliant. Well, I know will happen; <span style="font-style: italic;">you</span> 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.<br /><br />They'll have to bullshit.<br /><br /><span style="font-style: italic;">Added later</span>. 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, <a href="http://leiterreports.typepad.com/blog/2009/09/the-raes-successor-the-ref.html#comments">writes in a comment on the Leiter blog</a> 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."Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com9tag:blogger.com,1999:blog-23478689.post-24704185414796783652009-09-23T19:29:00.008+00:002009-10-10T21:16:49.700+00:00Schubert's Piano Trios<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_wgASjh7v8gE/Srp5s1BNMyI/AAAAAAAAAGc/DIU9T_yqWJI/s1600-h/SchubertTrios.jpg"><img style="margin: 0pt 0pt 10px 10px; float: right; cursor: pointer; width: 265px; height: 265px;" src="http://1.bp.blogspot.com/_wgASjh7v8gE/Srp5s1BNMyI/AAAAAAAAAGc/DIU9T_yqWJI/s320/SchubertTrios.jpg" alt="" id="BLOGGER_PHOTO_ID_5384750115528192802" border="0" /></a>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.<br /><br />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 <span style="font-style: italic;">Gramophone</span> review <a href="http://www.gramophone.net/Issue/Page/December%201997/90/746157/">agrees</a>. (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.)Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com0tag:blogger.com,1999:blog-23478689.post-91275426258198237002009-09-18T19:22:00.004+00:002009-09-18T23:07:20.029+00:00Praise 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!<br /><br />Over the last couple of days I've been reading the first volume of Winfried Just and Martin Weese's <span style="font-style: italic;">Discovering Modern Set Theory</span> (AMS, 1996), with an eye to moving on to the second volume. Well, I just loved the <span style="font-style: italic;">style</span>, 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.Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com3tag:blogger.com,1999:blog-23478689.post-31964701067829346742009-09-12T21:48:00.010+00:002009-09-16T17:50:36.948+00:00Student evaluationsI 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.<br /><br />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<sup>*</sup>, by the philosophers <a href="http://home.sprynet.com/%7Eowl1/sef.htm">Michael Huemer</a> and <a href="http://www.hss.cmu.edu/philosophy/glymour/glymour-universityFCE2003.pdf">Clark Glymour</a>. I was particularly amused (in a world-weary sort of way) by this excerpt from the former:<br /><blockquote>[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.<br /><br />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.</blockquote>So now you know: bounce in optimistically, wave your hands around confidently, and you can sell the kids anything ...<br /><br />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).<br /><br /><i>Added</i> For a bit of judicious balance, do read Richard Zach's second contribution (Comment 12 below), and the linked paper.<br /> <br /><sup>*</sup><small>Links from twitter, thanks to John Basl and Allen Stairs</small>Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com13tag:blogger.com,1999:blog-23478689.post-63858205694297746832009-09-08T12:35:00.009+00:002009-10-08T19:55:08.507+00:00Math 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.<br /><br />The paper is something of a Cambridge institution, pretty much unchanged in its basic syllabus since when I took it a <span style="font-style: italic;">long</span> 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.<br /><br />Anyway, having had some initial comments here and from local grad students, you can now <a href="http://www.phil.cam.ac.uk/u_grads/Tripos/Math_Logic/Reading_List/PHIRL_II07.pdf">download my third shot</a> at an updated list. All comments and suggestions for further improvement (within the current, fixed, syllabus) will still be very welcome.Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com6tag:blogger.com,1999:blog-23478689.post-15983738888139301952009-09-08T10:54:00.002+00:002009-09-08T11:05:55.359+00:00Congratulations to Thomas ForsterAnother 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 <span style="font-style: italic;">Review of Symbolic Logic</span> was judged one of the ten best papers of 2008 by the Philosopher's Annual. <a href="http://www.pgrim.org/philosophersannual/pa28articles/forsterset.pdf">Here's a link</a>.Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com0tag:blogger.com,1999:blog-23478689.post-2625552140017854572009-09-08T10:18:00.007+00:002009-09-11T13:12:25.399+00:00Grumpy old man, #42I think I'm turning into a grumpy old man ...<br /><br />[Cue suppressed laughter off stage, murmurings of "Turning? <span style="font-style: italic;">Turning</span>? Happened years ago", etc. But I shall ignore these scurrilous interruptions.]<br /><br />... and the latest cross-making irritation (especially galling for a long-time <span style="font-style: italic;">Analysis</span> editor) is the effort by some OUP copy-editor to improve a forthcoming <span style="font-style: italic;">Analysis</span> paper by Luca Incurvati and myself, by inter alia, removing all the contractions, replacing "don't"s by "do not"s etc.<br /><br />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".<br /><br />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).<br /><br />Harrumppph.Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com1tag:blogger.com,1999:blog-23478689.post-65864787069765319732009-09-02T12:39:00.014+00:002009-09-02T22:26:35.797+00:00School maths, from the distant past<a onblur="try {parent.deselectBloggerImageGracefully();} catch(e) {}" href="http://1.bp.blogspot.com/_wgASjh7v8gE/Sp6FNd57VrI/AAAAAAAAAGU/B6HfPjZW6fQ/s1600-h/Oldqns.png"><img style="margin: 0pt 0pt 10px 10px; float: right; cursor: pointer; width: 226px; height: 320px;" src="http://1.bp.blogspot.com/_wgASjh7v8gE/Sp6FNd57VrI/AAAAAAAAAGU/B6HfPjZW6fQ/s320/Oldqns.png" alt="" id="BLOGGER_PHOTO_ID_5376881471539336882" border="0" /></a>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 <span style="font-style: italic;">did</span> 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!<br /><br />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:<sup>1</sup> 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).<br /><br />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.<br /><br /><small><sup>1</sup> I don't guarantee my proof reading in copying the questions!</small>Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com18tag:blogger.com,1999:blog-23478689.post-9407178462343139372009-08-11T21:47:00.002+00:002009-08-13T12:38:54.729+00:00Simpson's SOSOAThe rather long awaited <a href="http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=9780521884396">new edition of Simpson's wonderful <span style="font-style: italic;">Subsystems of Second Order Arithmetic</span></a> is out with CUP.<br /><br /><span style="font-style: italic;">Added</span>: 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 <a href="http://www.math.psu.edu/simpson/sosoa/typos.pdf">corrections page</a> 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.Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com6tag:blogger.com,1999:blog-23478689.post-81074725432930972182009-08-04T14:59:00.002+00:002009-08-04T16:43:20.344+00:00Disappearing logic againA footnote to my post, <a href="http://logicmatters.blogspot.com/2009/07/logic-disappearing-over-horizon.html">Logic disappearing over the horizon</a>. I've just been reading Stephen Simpson's "Unprovable Theorems and Fast-Growing Functions" (an introductory piece in the 1987 AMS Contemporary Mathematics <span style="font-style: italic;">Logic and Combinatorics</span> 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:<br /><blockquote>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.</blockquote>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!Peter Smithhttp://www.blogger.com/profile/03957579588136008664noreply@blogger.com4