I hesitated to write a review of this book as I am neither a philosopher nor a mathematician. However, as the broad outlines of the arguments are accessible to someone who, like myself, has no more than high-school math, I felt justified in sharing the excitement and pleasure I experienced in reading Why is there a Philosophy of Mathematics at all? The book explores two themes:
1. The nature of mathematical proof
2. The applicability of mathematics (obviously to physics and economics, but also, as Hacking points out, across fields within mathematics itself).
The answer to the question Why is there a Philosophy of Mathematics is, I take it, that proof and applicability are issues that both mathematicians and philosophers feel compelled to return to again and again (notwithstanding the many practitioners who do not feel so compelled). In one sense, the answer is psychological: the astonishment generated by these two aspects of mathematics continually draws mathematicians and philosophers into a discussion that is, not itself, mathematical. It is, for lack of a better home, philosophical.

A historical perspective informs all the arguments. To summarize the thesis of the book: “A central aspect of the Ancient answer to the why question is none other than Proof. A central aspect of the Enlightenment answer is one notion of applying mathematics, namely Kant's (p 83).” Indeed, the entire book can be read as an essay in historical epistemology. And it is not history in a crude historicist sense: there is a full recognition that the actual course of historical development is the product of complex contingencies. One interesting example he cites is the successful development of mathematics in ancient China without the Greek (Euclidean) notion of proof. Thus raising the question of whether the latter is fundamental to foundations, or simply a cultural artifact.

Early on in the book the author suggests two different conceptions of proof that he terms the Cartesian and the Leibnizian. The former is associated with the sense of an “aha” moment and the concept of apodeictic certainty in the Critique of Pure Reason. The latter is associated with an algorithmic approach in which the steps are too numerous to be comprehended in a single “aha” moment. I have over-simplified. Nonetheless, one of the many fascinating parts of this work is the way Ian Hacking illustrates the manner in which the “gold standard” of proof is itself subject to the contingencies of history (though for every practitioner it must stand outside of history).

Ian Hacking teases out three different ways in which we can explore the concept of applicability. First, the applicability of one branch of mathematics to another; secondly, the application of mathematics to physics (with references to Wigner's famous essay: The Unreasonable Effectiveness of Mathematics in the Natural Sciences) and, lastly, mathematics and engineering. The latter is used as a further illustration of the complexity of historical contingencies (as in: compare and contrast Göttingen and the École Polytechnique).

In a work on the philosophy of mathematics, the question of the the putative existence of mathematical objects (platonism vs nominalism ) is unavoidable. Ian Hacking treats the question seriously and describes the major fault lines among modern mathematicians, taking care to reveal the complexities and subtleties of the question (among the archetypal representatives: Tim Gowers and Alain Connes). He uses a quotation twice (from Robert Langlands) to illustrate the ambiguities implicit in the way mathematicians approach the question. Speaking of the notion that “mathematics, and not only its basic concepts, exists independently of us. This is a notion that is hard to credit, but hard for a professional mathematician to do without.” (quoted on pages 41 and 256). Although the initial provisional answer to the question why is, Proof and Applicability, the perennial arguments over ontology suggest that they also form part of the answer. The author is well aware that the ontological question is probably irrelevant to the practice of mathematics...but, nonetheless, it keeps on reoccurring (moths to a light!)

There is something about the rapid, dense style of this book that is infectious: it effectively conveys the author's enthusiasms, interests and perplexities. If one test of a good book is that it inspires readers to explore further, then Why is there Philosophy of Mathematics at all? deserves the highest possible rating. And the book contains an excellent bibliography. ( )