Explanation, Independence and Realism in Mathematics

This elegant little identity connects a fundamental constant of analysis, e, with equally fundamental constants of arithmetic, algebra and geometry. It would be hard for a philosopher to pass it by. Steiner distinguishes mere existence (in Quine's sense of being the value of a variable) from two types of reality—ontic and epistemic reality. We will concern ourselves here with the epistemic reality of mathematical objects, for Steiner not only appeals to Euler's equation in connection with their epistemic reality, he also devotes most of his discussion to that topic. Steiner [1983] proposes that an object is real in the epistemic sense in case it has independent descriptions. In mathematics this means that there are two different descriptions of the object and a proof that they are coreferential, but no explanatory proof'that they are. On Steiner's view, pi is real in the epistemic sense because Euler's identity shows that pi, first described geometrically as the ratio of the circumference of a circle to its diameter, answers to the analytic description of arg{— 1), where