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.

ClarkPeter,

If you will graciously overlook my being off topic, there’s a recent unusually philosophic paper from Don Martin you might enjoy if you have not already done so.

http://logic.harvard.edu/EFI_Martin_CompletenessOrIncompleteness.pdf

No shortage of typos in this draft.

Aldo AntonelliI have not seen Mark’s book, but I am intrigued by the list of theorems philosophers should know about, as this is a topic to which I have occasionally given some thought. There are some obvious ones (Gödel’s incompleteness, Löwenheim-Skolem, etc) all remarkable for their (alleged) philosophical consequences. And there is a potential ambiguity as to what sense of “know” is intended here, where one could equally well mean “know (and understand) the statement of” the theorem or “have complete and detailed command of the proof of” the theorem.

In any case, I’d like to put in my favorite: Gödel’s

completenesstheorem for the first-order predicate calculus. I’ll even go as far as saying that every philosophy graduate student should at some point work his or her way through the proof. Not only because it is a deep result about an essential item in the philosopher’s toolbox (though it is), but because it is perhaps the only example of a deep mathematical proof that many such students will ever see. It is characteristic in establishing the extensional equivalence of two very different notions (one an existential quantification over finite configurations of sign, the other a universal quantification over the totality of the possible interpretations of the language), and thus surprising, and interesting. Philosophy graduate students, no matter their field, should have at least some conception of what a real mathematical proof looks like, and this one is (with work) within their grasp. (By the same token they should also all have taken a crack at the firstCritique, etc.).Finally, let me bring up another pet peeve: there is a conception that some of these results, on their own, have wide-ranging philosophical consequences. That seems wrong to me. Gödel’s incompleteness theorem establishes a fact about what is and is not provable within a formal theory of arithmetic. It is, at heart, a theorem about the natural numbers. No philosophy is going to come out of that, unless the theorem itself is supplemented by philosophical premises (that’s just Craig Interpolation, as my advisor used to point out, Interpolation itself being a candidate for the list).

Anyways, I’d be interested in hearing what’s on people’s list of theorems philosophy graduate students should know about.

Rowsity MoidSince so many things can be encoded as natural numbers, I don’t think it’s so clear that nothing of philosophical significance will come out of something like the incompleteness theorem, though you may be right that some philosophical premises would also be needed.

Anyway, some theorems not mentioned by Colyvan that it might be worth knowing about …

The central limit theorem.

The compactness theorem.

Tarski”s theorem on the undefinability of truth.

That the second-order Peano axioms are categorical.

Perhaps Löb’s theorem.

Perhaps Lindström’s theorem.

David AuerbachAgreement and amplification: When teaching the completeness theorem, particularly in its compactness guise, I point out to the students that although in general ∀∃ doesn’t imply ∃∀ many important proofs establish something particular of that form.

Decades ago, the first thing I ever wrote made Aldo’s last point wrt G2.

Charles YoungRegarding “philosophical consequences” (especially of the Goedel results): Boolos used to say, “When I hear the words ‘philosophical significance,’ I reach for the safety on my revolver.”

IagoOn the relation between mathematics and structuralism I suggest take a look at the book

The Artist and the Mathematicianby the mathematician Amir D. Aczel. Mainly, it is about the history of the group Bourbaki and its relations with Levi-Strauss and other structuralist researchers like Jean Piaget or Jacques Lacan.JasonGiven the negative review of Colyvan’s book, what would you recommend to the neophyte as a good “Intro” to the philosophy of mathematics? Thanks much

Peter SmithThe best place to start is, I think, Stewart Shapiro’s admirable

Thinking about Mathematics(OUP, 2000), which is very clear, well-structured, fair and sensible. And then there is Marcus Giaquinto’s splendidThe Search for Certainty(OUP, 2002).