Hacking, Ian. Why Is There Philosophy of Mathematics at All

HACKING, Ian. Why Is There Philosophy of Mathematics At All. New York: Cambridge University Press, 2014. xv + 290 pp. Cloth, $30.99--At first glance it would seem that the answer to the question posed in the title can be given in a sentence or two. Plato long ago recognized that the theorems of geometry are about things that are not in the material world: ideal circles, angles, and lines. For him--and for all mathematicians and philosophers since--this poses the question of just what are these things and what is their reality. So the obvious answer to the author's question is, "Because we want to know the ontological status of mathematical objects." Clearly if they are, for example, ideas in God's mind, that tells us something important. Plato's answer, of course, is that geometrical theorems refer to ideal objects, and indeed they are, for him, one proof of his theory of forms. So what then is the purpose of the book? It is rather a discussion of the nature of mathematics, loosely centered around the question of what mathematics is. The book has chapters with titles such as "What makes mathematics mathematics?", "Why is there a philosophy of mathematics?", "Proofs," "Applications," and two dealing with Platonic views. These would appear to be very meaty topics, but the author does not really confront them in the deep manner required, despite his obvious familiarity with the views of many philosophers. The chapters only loosely address their subject, and they are very episodic, jumping from one topic and philosopher's opinion to another, without developing any in a thoroughgoing manner. It reads as a collection of short essays rather than a monograph. Additionally, it is not, possible in the twenty-first century to discuss what mathematics is without in-depth treatment of Godel's Theorem and the question of provability and its relation to truth, which is not found in the book. The very language of mathematics is in terms of existence, postulation, and exploration: "Let X be a Hilbert Space," or "There exists a number y such that ...," or "There do not exist any integers a, b, c and n satisfying [a.sup.n] + [b.sup.n] = [c.sup.n], if n > 2 (Fermat's Last Theorem)." This should be a jumping-off point for any discussion of the nature of mathematics, but it is not for this author. Rather, we read about this or that philosopher's opinion. There is the obligatory nod to nominalistic interpretations of mathematics; but very few mathematicians have much interest in that approach since the very language of mathematics is quite contrary to nominalism. The author also makes some comparisons of modern mathematics with that of other cultures. …