ABSTRACT
John Bigelow begins his review of Penelope Maddy, Realism in mathematics, Oxford 1990, with the following remark:
Mathematics has three striking features. It discovers truths which are necessary, not contingent. It proceeds by a priori methods, and does not justify its claims by appeal to experience. It has a distinctive subject matter, not material objects, nor plants, nor animals, nor thoughts and feelings, nor societies-but numbers and other things of that sort. (Bigelow, 1992, p. 235)
Now the third point, the question whether mathematics possesses objects of its own, is itself an object of extremely controversial debates. Nominalism has always denied that there are mathematical objects at all. And since 1870 when it was formulated the following definition of mathematics has met with a lot of approval: ‘Mathematics is the science which draws necessary conclusions. This definition of mathematics is wider than that which is ordinarily given, and by which its range is limited to quantitative research. The ordinary definition, like those of other sciences, is objective; whereas this is subjective’ (Peirce, 1965, Vol. 4, par. 229).
