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An Introduction to the Philosophy of Mathematics

Mark Colyvan
Cambridge University Press
Publication Date: 
Number of Pages: 
Cambridge Introductions ot Philosophy
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The Basic Library List Committee suggests that undergraduate mathematics libraries consider this book for acquisition.

[Reviewed by
Mark Hunacek
, on

Many years ago, as a college freshman, I wound up registered in an Introduction to Philosophy course. This was not a voluntary decision; the college I attended had a long list of required courses, and I thought philosophy sounded more interesting than alternatives like sociology. Nevertheless, I approached the course with considerable misgivings. This was 1968, the era of hippies and sit-ins, and I was sure I was in for a semester of listening to my fellow students (and perhaps the professor) uttering vacuous profundities about life. In fact, I couldn’t have been more wrong: the teacher ran a tightly focused and fairly rigorous course that compared and contrasted specific philosophical views of famous people like Kant and Descartes, and I wound up so pleasantly surprised that I even found myself taking other courses offered by the philosophy department, mostly about logic.

One course that I never did get to take — either because it wasn’t offered or because of a schedule conflict, I forget which — was the philosophy of mathematics, but I did get exposed to some of the issues in this area in a wonderful course I took a year or so later on the foundations of geometry. As I advanced still further in my studies I wound up hearing about people like Brouwer and the various controversies about the foundations of mathematics. This all struck me as interesting but I never seemed to have the time to sit down and really study this in any kind of systematic way.

How nice, therefore, for me to now have the opportunity to read Colyvan’s lively and entertaining book on the philosophy of mathematics. This is a chatty, interesting book with an agenda that sets it somewhat apart from many other books on the subject, and which the author addresses early in the book when he announces that he will be, in the words of a section title, “[s]kipping through the big isms”.

To be more specific: the “big isms” are formalism, logicism, and intuitionism, all three of which were important movements in the foundations of mathematics. The author briefly defines these ideas but does not dwell on them, instead pointing out that discussions about these tended to dry up after the first half of the 20th century and were replaced by more contemporary areas of discussion, largely influenced by the work of Paul Benacerraf. It is these more contemporary issues, rather than the three “isms”, that are addressed in the remainder of the book.

This seems like a good idea, for several reasons. For one thing, not being au courant in the current philosophical research literature, I am strongly inclined to accept at face value the word of an expert on what is, and what is not, of contemporary interest. But even if Colyvan is incorrect and plenty of people are still interested in the foundational movements identified above, the fact still remains that there are other books which discuss them at some length — discussion of the three “isms”, for example, take up much of Bostock’s Philosophy of Mathematics. As the author points out, “This material is already covered in a number of readily accessible places. I don’t have anything to add to what has already been said (many times) elsewhere.” A book which blazes a new trail can be a very interesting and useful find.

Another aspect of this book that makes it interesting is that the author does not shy away from actual mathematics. As he puts it in chapter 1:

[M]any treatments of the philosophy of mathematics focus on simple, pedestrian arithmetic statements such as 7 + 5 = 12, with perhaps a bit of set theory thrown in. That’s a bit like trying to keep your trip to Europe simple by visiting only Glasgow. As interesting as Glasgow is, it is not representative of all of Europe. Traveling can be inconvenient, tiring and sometimes downright hard work, but the rewards are great. If you want to see and learn about Europe, you need to venture beyond Glasgow. To hell with the pedestrian, live adventurously!

Consistent with this quote (which also demonstrates, by the way, that Colyvan has a way with words), we see in this book some actual mathematical proofs (including Cantor’s proof that a set has less cardinality than its power set, and Euclid’s proof of the infinitude of primes) as well as mathematical objects like differential equations, the Klein bottle and the Dirac delta function (wonderfully described as “too weird to live, and too useful to die”).

So, just what are the contemporary topics that are discussed in the rest of the book? Chapter 2 addresses the limits of mathematical reasoning, discussing in some detail both the mathematics and philosophical significance of the Lowenheim-Skolem theorem and Gödel’s incompleteness theorem. I thought the discussion here was significantly clearer and more accessible than that in, say, Crossley’s What is Mathematical Logic?, notwithstanding the fact that the latter book purported (incorrectly, I think) to be addressed to people with no mathematical training. (Of course, Crossley attempted to actually prove the results.)

The next chapter of Colyvan shows that there are still plenty of “isms” to go around, and discusses Platonism (also known as realism), which is the school of thought that mathematical statements about objects (such as prime numbers, the example used in the text) are true by virtue of the existence of these objects.

Of course this raises a host of sticky questions about the meaning of existence, and these are addressed in this chapter, as is the so-called “indispensability argument” in favor of Platonism, which essentially states that (I am oversimplifying here) since mathematics is indispensable to scientific theory and since we should have “ontological commitment” to objects that are indispensable to scientific theory, we should have such ontological commitment to mathematics.

As a syllogism, this is impeccable, but some mathematical philosophers have disputed the premises. This leads the author to discuss, in the next chapter, some claims of the “fictionalists”, including the influential work of Hartry Field and his “nominalization” program, in which he attempts to demonstrate that, for example, areas of science like Newtonian gravitational theory can be done without mathematics. The material in these chapters was fairly technical but quite interesting; one of the more elementary, but amusing, comments made by the author was that a fictionalist would claim that the equation 7 + 5 = 12 was false, simply by virtue of the fact that 7 and 5 don’t exist as entities; on the other hand, the statement “there is no largest prime” would be accepted as true: there being no primes as actual entities, there certainly is no largest one.

The four remaining chapters address independent but related issues in the philosophy of mathematics. In chapter 5, the author looks at the issue of mathematical explanation, and discusses questions like: Is a proof by contradiction an “explanatory” proof? Does mathematics “explain” the physical world, and, if so, does that provide support for the indispensability argument for realism? Chapter 6 addresses philosophical issues raised by the applicability of mathematics to other areas of science: “why is mathematics, which is developed primarily with broadly aesthetic considerations in mind, so crucial in both the discovery and the statement of our best physical theories?” The next chapter addresses some philosophical questions posed by inconsistent mathematical theories: naïve set theory, after all, is known to be inconsistent, yet many professional mathematicians use it routinely. Finally, chapter 8 discusses some issues involving mathematical notation, which the author, by means of carefully chosen examples involving some non-trivial mathematics, shows can actually advance mathematics as well as explain it.

Perhaps I shouldn’t have said “finally” in the last sentence. Although chapter 8 marks the last chapter really addressing philosophical issues, it is not the last chapter of the book: there is an “epilogue” (also labeled chapter 9, and entitled “Desert Island Theorems”) in which the author discusses some of his favorite results in mathematics, divided into three categories: philosophers’ favorites, under-appreciated classics, and open questions (so, the use of the word “theorems” in the title is not quite correct). I was a bit puzzled by the inclusion of this chapter, particularly since some of the results mentioned (for example, the Poincaré conjecture or Gauss’s Theorema Egregium in differential geometry) seemed to have no particular philosophical significance that I could discern (the author’s brief discussion of each theorem doesn’t provide any either). I enjoy reading about interesting mathematics as much as the next person, but this epilogue almost seemed like a space-filler; it didn’t really advance the agenda of the text. Even when some of the other results mentioned, like the Four-Color Theorem, have some obvious philosophical import (in this case, to the question of whether a computer-based argument constitutes a proof), the issue was just mentioned and not really discussed in any depth.

And this brings me to my one real concern about the possible use of this book as a text. This is quite a slim book, just about 150 pages of text (not counting the appendix), and while I found it entertaining and informative, I also wonder whether it contains, by itself, sufficient material and adequate depth for a full semester-long course on the subject. Having never taught a course in philosophy, I can’t really offer a definitive opinion, but just from my experiences as a teacher of mathematics I get the feeling I would be hard-pressed to spend an entire semester on the material covered here. But of course this is not necessarily a fatal flaw; one can always supplement a text. (Each chapter ends with a list of further reading. In addition, each chapter contains discussion questions that could certainly be used to pursue matters further.)

The book certainly has considerable value apart from its use as a text; a mathematician might enjoy reading this (as I did) as a way of learning, in a painless and entertaining way, about interesting ideas that he or she never get around to studying before. What is covered here, is covered quite nicely. I enjoyed this book, and recommend it.

Mark Hunacek ( teaches mathematics at Iowa State University.

1. Mathematics and its philosophy
2. The limits of mathematics
3. Plato's heaven
4. Fiction, metaphor, and partial truths
5. Mathematical explanation
6. The applicability of mathematics
7. Who's afraid of inconsistent mathematics?
8. A rose by any other name
9. Epilogue: desert island theorems.