Abstract

This very interesting and extremely useful study raises the question, by virtue of its title and what it does not do, of what is, or ought to be, meant by the philosophy of mathematics. The author begins his study of Euclid with a brief discussion of Hilbert's axiomatization of geometry. The two main points in this discussion are: "Hilbertian geometry and many other parts of modern mathematics are the study of structure", i.e., of the interpretations of axiom-systems; and intuition of geometrical form becomes irrelevant to the content of mathematics. Greek mathematics, and specifically Euclid, cannot be understood in these terms; hence, Euclid's use of the axiomatic method differs from ours. One may also say that logic, "the theory of structure-preserving inference", becomes central to modern mathematics, as it was not for Euclid. But, the author continues, it does not follow that one cannot exhibit the logical structure employed by Euclid. The bulk of Mueller's book is concerned to do just this, thanks to a formalization of Euclid which attempts to be faithful to his principal concepts, while at the same time employing contemporary logical techniques.