Year: 2012

Mathematics theorems for philosophers?

I mentioned that Mark Colyvan, by way of epilogue to his Introduction to the Philosophy of Mathematics has a list of choice “mathematical results that have some philosophical interest, or in some cases are just very cool pieces of mathematics”. Some might be interested in knowing about his list. It is divided into three main parts (leaving aside the epilogue to the epilogue on “Some interesting numbers”). So we have, with his dates,

1. Philosophers’s Favourites:  The Tarski-Banach Theorem (1924), Löwenheim-Skolem Theorem (1922), Godel’s Incompleteness Theorems (1931) , Cantor’s Theorem (1891),  Independence of Continuum Hypothesis (1963) , Four-Colour Theorem (1976),  Fermat’s Last Theorem (1995) Bayes’ Theorem (1763), The Irrationality of Square root of 2 (ca. 500 BC), The Infinitude of the Primes (ca. 300 BC).

2. The Under-Appreciated Classics: The Borsuk-Ulam Theorem (1933), Riemann Rearrangement Theorem (1854), Gauss’s Theorema Egregium (1828), Residue Theorem (1831), Poincare Conjecture (2002), Prime Number Theorem (1849). The Fundamental Theorems of Calculus (ca. 1675), Lindemann’s Theorem (1882),  Fundamental Theorem of Algebra (1816),  Fundamental Theorem of Arithmetic (ca. 300 BC).

3. Some Famous Open Problems: The Riemann Hypothesis, The Twin Prime Conjecture, Goldbach’s Conjecture, Infinitude of the Mersenne Primes.

Well, that strikes me as a fairly random list. And Colyvan’s comments are a pretty mixed bag too. For example, I’d have thought that one of the things of interest about the Prime Number Theorem is that, though it originally looked ‘deep’, something that required some serious apparatus to prove, it has latterly been shown to have an elementary proof (indeed can be proved in IDelta_0 + exp). Now questions about ‘depth’ of proof, and what can be revealed by proofs of different kinds, are surely of some philosophical interest, but Colyvan misses the chance to hint at them. Again, what’s the philosophical interest of the proof of the Poincaré conjecture? Colyvan gives us some human interest gossip about Grigori Perelman turning down the Fields medal and so on: but what’s the philosophical point? Perhaps Colyvan has the Poincaré conjecture on the list because it is “cool”: but then why is it especially cool (apart from the fact that it resisted proof for a long time?).

However, I’m certainly not at all averse to Colyvan’s project of giving a list of conceptually interesting mathematical problems and proofs: it could be a fun and illuminating project. What would you put on the list and why? [For more initial thoughts, see the comments on the previous blog post.]

Book note: Mark Colyvan’s Intro to Phil. Maths

Ever since my Gödel book was published unexpectedly as one of the  “Cambridge Introductions to Philosophy”, I’ve kept an interested eye on what else has appeared in the series. The latest addition is Mark Colyvan’s An Introduction to the Philosophy of Mathematics. I have to say that this is rather disappointing.

The blurb says “The book is suitable for an undergraduate course in phil. maths”. But not for many, I would hazard. Such courses are nearly always upper-level courses (well, that is certainly the case in the UK), but the discussions in this book are very much at an elementary level, and very little is pursued in any depth at all. Chapter 2, for example, is on “The limits of mathematics”. There are just five pages on the L-S theorem and Skolem’s Paradox, of which three and a bit are purely expository. Then, rather bizarrely, there are just three pages on Gödel’s Theorems, and that includes telling the reader what they are.

The sense of rush continues. The whole book (minus the epilogue which lists a number of interesting mathematical theorems which Colyvan thinks that any philosopher of mathematics should know about) comes in at 150 pages, and rather spaciously set pages at that. So this just hasn’t the space for the kind of coverage and argumentative sophistication that you’d want in a book that is going to provide the backbone for an upper level undergraduate course.

Colyvan starts the book by saying what he isn’t going to be talking about — the familiar menu of “isms”, logicism (old and new), formalism, intuitionism. Instead we get Platonism, structuralism, nominalism, fictionalism. That reflects the concerns of a lot of post-Benaceraff philosophy of mathematics, but it is certainly not all gain. If you ask which philosophical debates actually engage with the concerns of reflective mathematicians (or at least, with questions you can get them interested in over coffee), then the list will cross-cut the old and new isms. Eyes will glaze over if you try to amuse your local mathematicians with (neo)logicism or with fictionalism. But questions about what can be rescued from Hilbert’s program (cp. the reverse mathematics program), related issues about how little it takes to do how much, questions about the idea of structure (and what category theory brings to the party when thinking about structure), issues about which set theories are worth taking seriously (NF anyone?), etc., can still produce animation. Now, I’m not suggesting that all those latter topics should be touched on in a first phil. maths course. But I do think that there is something to be said for shaping the introductory menu with an eye to laying the groundwork for moving on to ‘real’ debates (as opposed to the rather regrettable in-house obsessions of recent philosophers).

The book moves on to discuss four topics beyond the isms  — so you can see how fast Colyvan must be going! It discusses the idea of mathematical explanation (both explanation within mathematics, and apparent cases of the mathematical explanation of the extra-mathematical), the “unreasonable effectiveness of mathematics” in applications, there’s a chapter entitled “Who is afraid of inconsistent mathematics?”, and finally there is a chapter that says it is about ‘notation’ but is actually about something a bit deeper concerning representations (e.g. the use of cartesian vs. homogeneous coordinates for the plane).

Two comments. First on (intra)mathematical explanation: yes, this is an intriguing topic. The trouble is the Colyvan tries to discuss this with too few actual examples of mathematical proofs (note that his epilogue is a catalogue of results, not of proofs, so is going to be no help here). One of the few proofs he gives is (a rather heavy-handed version of) Euclid’s proof of the infinitude of primes. Now, this is the first of the Proofs from The Book in Aigner and Ziegler, who go on to give five other reasonably elementary proofs of the same result: it would have been fun to look at one or two of these too and do a compare-and-contrast for “explanatoriness”. As it is, the restricted diet of examples leaves the discussion floundering.

Second, on inconsistent theories. Leaving aside the special case of the rise, fall, and rise of infinitesimal analysis, talk of mathematicians working with inconsistent theories can be much overdone. Colyvan early on in the book writes of Russell “proving that the foundational mathematical theory, set theory, was inconsistent”. But we are (of course!) not told in just what sense “set theory” was “foundational”, or indeed just which set theory is in question. Here’s a useful exercise. Take a look at William and Grace Young’s wonderfully lucid The Theory of Sets of Points, published in 1906 (and still in print). They are doing foundational work in one good sense. Ask yourself how and why they can be unfazed by Russell’s paradox. And of course it isn’t because they are proto paraconsistent logicians of the kind Colyvan talks about here!

Given the lack of depth because of covering so much in such a short space, I can’t really see this book being much used as a course book (it won’t trump options like e.g. Marcus Giaquinto’s exemplary The Search for Certainty for part of a course).  It is, however, very attractively and mostly pretty clearly written even if it skips past too fast: so I suppose Colyvan would be a good recommendation for not-too-challenging pre-course vacation reading.

Teach yourself logic, #3: Beginning model theory

(For later, better, thoughts, see the most recent version of the TYL Guide.)

I have been working away on the second edition of my Gödel book. The current task: giving a more lucid proof showing Robinson arithmetic can represent all primitive recursive functions. In the first edition I cheated by taking a clever trick from Burgess, Boolos and Jeffrey. I do now regret that. But I can certainly sympathise with my earlier self for taking the easy way out!

By way of diversion, then, and as an exercise in constructive procrastination, here is the draft third instalment of my slowly developing ‘teach yourself logic’ guide. So far we’ve covered (1) standard first-order logic, at an introductory level, and (2) some basic modal logic. Today’s new list (3) looks at the path forward from what’s covered in a standard first-order logic course on towards full-blown model theory. [The ordering of the instalments here is going to be henceforth a bit arbitrary; but I hope a tolerably sensible structure will emerge in the final composite Guide!]

In fact, in reworking the first two instalments of the Guide — you can now download an expanded version of them here — I have rethought the division between what is to go in instalment (1) and this new one. So take the treatment of first-order logic in (1) now to get just as far the completeness proof but really no further (so, that’s pretty much the content of e.g. Chiswell and Hodges’s terrific Mathematical Logic).

So where next, if you want to move on from those first intimations of classical model theory in the completeness theorem to something of a grasp of the modern theory? There is a very short old book, the very first volume in the Oxford Logic Guides series, Jane Bridge Beginning Model Theory: The Completeness Theorem and Some Consequences (Clarendon Press, 1977) which takes on the story just a few steps pretty lucidly. But very sadly, the book was printed in that short period when publishers thought it a bright idea to save money by photographically printing  work produced on electric typewriters. So, used as we now are to mathematical texts beautifully LaTeXed, the look of the book is decidedly off-putting. So let’s set that aside (as the first recommendation covers much of the same ground anyway).

Here, then, are two natural and rather complementary places to start:

  1. Dirk van Dalen Logic and Structure (Springer 4th edition 2004). In instalment (1) I warmly recommended reading this modern classic text up to and including Section 3.1, for coverage of basic first-order logic. Now read the whole of Chapter 3, for a bit of revision and then for the Löwenheim-Skolem theorems and some basic model theory.
  2. Wilfrid Hodges’s `Elementary Predicate Logic’, in Handbook of Philosophical Logic, Vol. 1, ed. by D. Gabbay and F. Guenthner, (Reidel 2nd edition 2001). This is an expanded version of the essay in the first edition of the Handbook, written with Hodges’s usual enviable lucidity. Over a hundred pages long, this serves both as an insightful and fresh overview course on basic first-order logic (more revision!), and as an illuminating introduction to some ideas from model theory.

For a more expansive treatment (though not really increasing the level of difficulty, nor indeed covering everything touched on in Hodges’s essay) here is a still reasonably elementary textbook:

  1. Maria Manzano, Model Theory (OUP, 1999). I seem to recall, from a reading group where we looked at this book, that the translation can leave something to be desired. However, the coverage as far as it goes is good, and the treatment accessible. I like the way it starts off by talking about relationships among structures in general before talking about structures-as-models-of-theories.

This might already be about as far as most philosophers will want to go. But if you do press on, the choice at the next level up is surely self-selecting:

  1. Wilfrid Hodges A Shorter Model Theory (CUP, 1997). Deservedly a modern classic — under half the length of the encyclopedic original, but still full of good things, going a good way beyond Manzano. It gets tough as the book progresses, but the earlier chapters should be manageable.
  2. Rather different in focus is another older book, which is particularly elegant (though perhaps you will need more mathematical background to really appreciate it) is J. L, Bell and A. B. Slomson’s Models and Ultraproducts (North-Holland 1969; Dover reprint 2006). As the title suggests, this focuses particularly on the ultra-product construction.

Finally, though probably this is looking over the horizon for most readers of this list, at a further notch up in difficulty and mathematical sophistication, there is another book which has also quickly become something of a standard text:

  1. David Marker, Model Theory: An Introduction (Springer 2002). Rightly very highly regarded. (But it isn’t published in the series ‘Graduate Texts in Mathematics’ for nothing!)

So that is my main list. What have I missed out? Well, you could still get a lot out of C. Chang and H. J. Keisler’s classic Model Theory (North Holland, 2nd edition 1977). This is leisurely, very lucid and nicely constructed with different chapters on different methods of model-building. You could well look at quite a bit of this before or alongside reading Hodges’s book. There’s a short little book by Kees Doets Basic Model Theory (CSLI 1996), which concentrates on Ehrenfeucht games which could appeal to enthusiasts. And then, of course, many Big Books on Mathematical Logic have chapters on model theory: a good treatment of some central results seems to be that in Shawn Hedman, A First Course in Logic (OUP 2004), Chs 4–6 which could be perhaps read after (or instead of) Manzano.

Comments and suggestions?

Book note: Nick Smith’s new Logic: The Laws of Truth

A long time ago, when the world was young and UK philosophy departments nearly all taught an amount of formal logic to their first year students, Peter Millican (I think it was) wrote round — yes, this was before email — to ask what text books we used in our courses. The majority answer was “Lemmon but …”, as in “I use Lemmon, but the students find it tough/I have to supplement it with handouts/I don’t really like it because …” There wasn’t a huge amount of choice back then, however, and Lemmon’s 1965 Beginning Logic was indeed notably better than most of the alternatives.

Nowadays, by contrast, there’s a ridiculous number of introductory logic books to choose from. If you ask around, no book stands out (partly no doubt because there’s no longer any kind of consensus about what should go into an entry-level logic course). However, the form of the answer still tends to be the same — “I use XYZ, but …”. Teachers of first year logic do seem to be a very picky bunch, rarely satisfied by someone else’s text. The unwise and perhaps over-confident among us of course think we could do better. The really unwise actually try do so. We spend an inordinate amount of time writing our introductory masterpiece, not believing the friends who kindly warn us that it will all take three times longer than we’d ever planned. And then an ungrateful world quite mysteriously fails to rush to welcome the result of all our efforts as the One True Logic Text.

So it goes. I speak from hard-won experience. It is with warm fellow-feeling for the author, then, that I note the arrival on the scene of Yet Another Introductory Logic Book (I rather wanted to use that as the title for mine, but thought better of it). My namesake Nicholas J. J. Smith’s Logic: The Laws of Truth is now out with Princeton UP: the publisher’s web page for the book is here. If you are thinking of adopting a new logic text, you can get an e-inspection copy. Full disclosure: Nick has very kindly indeed sent me a hardback copy.

I’ve browsed through quite a bit, dipping in and out. In general terms, I do much like the tone and the level and the coverage and the writing. And there’s the right amount of more ‘philosophical’ discussion alongside the formal work. Though I’m now going to be picky. Of course. Read “below the fold” for some comments. Still, the headline news is that if you are a teacher in the market for new logic text, or a student looking for very helpful reading, this could indeed be the book for you.

Back to Schubert …

I was going to post a rave recommendation for this recording a few weeks ago: but it was about the time of BBC Radio 3’s ten day, wall-to-wall, 24hr, Schubert season, so I thought I should leave it a while, till you could bear to hear yet more of Franz.

I’ve become a great fan of Viktoria Mullova’s recent CDs (her Bach, the Beethoven sonatas with the fortepianist Kristian Bezuidenhout, her revelatory Vivaldi). So, a bit speculatively, I sent off for her 2005 recording of the Schubert Octet, not quite knowing what to expect. Well, this is fantastic. Don’t be put off by the quite misleading insert photo: this isn’t an unbalanced star vehicle, Mullova plus anonymous band. All the players are just terrific and the ensemble playing is all you could ask for. But it’s more than that. In the past, I can’t say I’ve ever really been bowled over by the Octet — if you’d asked me, I’d have said “happy enough tootling, but …”. However, this performance strikes me as being in a different league to those I’ve heard before, and brings the music much closer to the world of the A minor and D minor quartets of the same year. For example, there’s a seriousness in the Adagio that other recordings I know don’t really bring out.

But don’t take my word for it. I’ve just thought to look up the Penguin Guide’s recommendations for the Octet, and they write “This newest version of the Octet … is not only the best since the celebrated 1957 Vienna Octet version … but it actually surpasses it. These artists find a greater depth than their predecessors in a work we all think we know well. There is … grace, pathos and tenderness here …” Exactly.


Teach yourself logic, #2: Modal logic

An expanded, improved, version of the full Guide to teaching yourself various areas of logic is now here.

So let’s move on to looking at books to read on modal logic.

The ordering of some of the instalments here is necessarily going to be a little bit arbitrary. But I’m putting this one next for two reasons. First, the basics of modal logic don’t involve anything mathematically more sophisticated than the elementary first-order logic covered in the first instalment. Second, and more much importantly, philosophers working in many areas surely ought to know a little modal logic, even if they can stop their logical education and manage without knowing too much about some of the fancier areas of logic we are going to be looking at later.

Again, the plan is to offer a list of books of increasing range and difficulty, choosing those which look most promising for do-it-yourself study. The place to start is clear, I think:

  1. Rod Girle, Modal Logics and Philosophy (Acumen 2000, 2009). Girle’s logic courses in Auckland, his enthusiasm and abilities as a teacher, are justly famous. Part I of this book provides a particularly lucid introduction, which in 136 pages explains the basics, covering both trees and natural deduction for some propositional modal logics, and extending to the beginnings of quantified modal logic.

Also pretty introductory, though perhaps rather brisker than Girle at the outset, is

  1. Graham Priest, An Introduction to Non-Classical Logic (CUP, much expanded 2nd edition 2008): read Chs 2–4, 14–18. This book — which is a terrific achievement and enviably clear and well-organized — systematically explores logics of a wide variety of kinds, always using trees in a way that can be very illuminating.

If you start with Priest’s book, then  at some point you will need to supplement it by looking at a treatment of natural deduction proof systems for modal logics. A possible way in would be via the opening chapters of

  1. James Garson, Modal Logic for Philosophers (CUP, 2006). This again is intended as a gentle introductory book: it deals with both ND and semantic tableaux (trees). But — on an admittedly rather more superficial acquaintance —  this doesn’t strike me as being as approachable or as successful.

We now go a step up in sophistication:

  1. Melvin Fitting and Richard L. Mendelsohn, First-Order Modal Logic (Kluwer 1998): also starts from scratch. But — while it should be accessible to anyone who can manage e.g. the Hodges overview article on first-order logics that I mentioned before — this goes quite a bit more snappily, with mathematical elegance. But it still also includes a good amount of philosophically interesting material. Recommended.

Getting as far as Fitting and Mendelsohn will give most philosophers a good enough grounding. Where, if anywhere, you go next in modal logic, broadly construed, would depend on your own further concerns (e.g. you might want to investigate provability logics, or temporal logics).  But if  you want to learn more about mainstream modal logic, here are some suggestions (skipping past older texts like the estimable Hughes and Cresswell, which now rather too much show their age). Though note, further technical developments do tend to take you rather quickly away from what is likely to be philosophically interesting territory.

  1. Sally Popkorn, First Steps in Modal Logic (CUP, 1994). The author is, at least in this possible world, identical with the mathematician Harold Simmons. This book, entirely on propositional modal logics, is written for computer scientists. The Introduction rather boldly says ‘There are few books on this subject and even fewer books worth looking at. None of these give an acceptable mathematically correct account of the subject. This book is a first attempt to fill that gap.’ This perhaps oversells the case: but the result is still  illuminating and readable — though its concerns are not especially those of philosophers.
  2. Going further is Patrick Blackburn, Maarten de Ricke and Yde Venema, Modal Logic (CUP, 2001). One of the Cambridge Tracts in Theoretical Computer Science. But don’t let that put you off. This text on propositional modal logics is (relatively) accessibly and agreeably written, with a lot of signposting to the reader of possible routes through the book, and interesting historical notes. I think it works pretty well. However, again this isn’t directed to philosophers.
  3. Alexander Chagrov and Michael Zakharyaschev’s Modal Logic (OUP, 1997) is a volume in the Oxford Logic Guides series and also concentrates on propositional modal logics. This one is probably for real enthusiasts: it tackles things in an unusual order, starting with a discussion of intuitionistic logic, and is pretty demanding of the reader.  Still, a philosopher who already knows just a little about intuitionism might well find the opening chapters illuminating.
  4. Nino B. Cocchiarella and Max A. Freund, Modal Logic: An Introduction to its Syntax and Semantics (OUP, 2008). The blurb announces that ‘a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight’. That sounds hopeful, and the authors are right about the unusually wide range. As noted, the previous three books only deal with propositional logics, while many of the more challenging philosophical issues about modality tangle with quantified modal logic. So the promised coverage makes the book potentially of particular interest to philosophers. However, when I looked at this book with an eye to using it for a graduate seminar, I didn’t find it appealing: I suspect that many readers will indeed find the treatments in this book uncomfortably terse and rather relentlessly hard going.
  5. Finally, in the pretty unlikely event that you want to follow up even more, there’s the giant Handbook of Modal Logic, ed van Bentham et al., (Elsevier, 2005). You can get an idea of what’s in the volume by looking at the opening pages of entries available online here.

Book note: Smullyan’s Theory of Formal Systems

When I wrote my Gödel book I did a lot of it from memory (on the principle that if I had to reconstruct proofs without too much cheating, the discipline of doing so would help me to explain the proofs). But lately, as I slowly work away at the second edition, I’ve been (re)reading some of the old literature around and about the incompleteness theorems.

I’ve had a copy of Smullyan’s Theory of Formal Systems (1961) for thirty years, but to be honest I don’t think I’ve ever read it properly before. And it does date from that time when many maths books were photo-printed from typewritten originals, and looked exceedingly uninviting. Moreoever, in some ways, this volume has been superseded by Smullyan’s Gödel’s Incompleteness Theorems (1992).

But still, as I’ve found with great enjoyment, the earlier book remains quite terrific stuff. It is just so very elegant and insightful in developing Post’s ideas on formal systems and in excavating one basic line of ideas underlying incompleteness proofs (I plan to write up some notes in due course). Impressive indeed — even more so when you remember the book is essentially Smullyan’s doctoral thesis. Still very worth reading after all this time.

(And how many philosophy books, as opposed to logic books, still stand up so well after fifty years?)

Book note: Tent and Ziegler’s A Course in Model Theory

The CUP bookshop marked its 20th birthday with a day offering a big discount on top of the usual discounts, so I came away with a small pile of half-price logic books (I wasn’t enticed by anything on the philosophy shelves, but that’s another story).

One of them, the book I’d actually gone in to buy (not knowing about the birthday celebrations), was Katrin Tent and Martin Ziegler’s new contribution to the ASL ‘Lecture Notes in Logic’ series, A Course in Model Theory. To declare an interest, I’ve wondered more than once about writing a book on model theory useful for philosophers. Well, according to the blurb, ‘This concise introduction to model theory begins with standard notions and takes the reader through to more advanced topics such as stability, simplicity and Hrushovski constructions.’ And according to the introduction, ‘This book aims to be an introduction to model theory which can be used without any background in logic. We start from scratch, introducing first-order logic, structures, languages etc. but move on fairly quickly to the fundamental results in model theory and stability theory.’

Well, for ‘fairly quickly’ read ‘very quickly’. In fact, I would have thought that a graduate student mathmo who really did have no background in logic would find this seriously tough going with stuff going by far too fast (what on earth would even a sophisticated such reader get out of the half a page on ultraproducts on p. 13?). And this quite certainly isn’t a book for graduate philosophers who want to learn some model theory.

It’s probably best, in fact, to think of it as something to read after having really mastered e.g. Wilfrid Hodges’s classic Shorter Model Theory, by way of consolidation and then extension. Taken in that spirit, it does indeed look useful for serious mathematicians (though I’m not going to have time to work through it further at the moment).  But this is probably not a book for beginners in model theory (and surely not one for graduate philosophers, even those of a logical bent).

Postcard from the Bahamas

20120428-125024.jpgA visit to family, and a much needed break. As you can see, the beach has been so very crowded, the skies so very gray …

For once, I have been away for the better part of two weeks without a laptop, and haven’t read a word of philosophy or logic, or indeed thought about it much. So it has been novels for me (some ‘serious’, some not so — I’ve become a much entertained fan of the Falco books). I’ve also been a very latecomer to Edmund de Waal’s The Hare with Amber Eyes, which — like so very many others — I thought quite wonderful (perhaps at some level, de Waal’s story of generations of his family living through fractured times speaks to our own deep anxieties about what is to come in our troubling times). Read it.

And that in turn put me in the mood to re-read an old favorite, Nabokov’s Speak Memory (by my lights, his best book). Then, having been talking about Tony Judt’s The Memory Chalet, I found myself re-visiting that as well. Ah, the joys of traveling with an iPad with books already aboard, and a huge library available at the touch of a button …

It’s Kurt Gödel’s birthday today: back to thinking about him tomorrow.

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