Philosophy of Mathematics

Do numbers sets and so forth exist? What do mathematical statements mean? Are they literally true or false or do they lack truth values altogether? Addressing questions that have attracted lively debate in recent years Stewart Shapiro contends that standard realist and antirealist accounts of mathematics are both problematic. As Benacerraf first noted we are confronted with the following powerful dilemma. The desired continuity between mathematical and say scientific language suggests realism but realism in this context suggests seemingly intractable epistemic problems. As a way out of this dilemma Shapiro articulates a structuralist approach. On this view the subject matter of arithmetic for example is not a fixed domain of numbers independent of each other but rather is the natural number structure the pattern common to any system of objects that has an initial object and successor relation satisfying the induction principle. Using this framework realism in mathematics can be preserved without troublesome epistemic consequences. Shapiro concludes by showing how a structuralist approach can be applied to wider philosophical questions such as the nature of an object and the Quinean nature of ontological commitment. Clear compelling and tautly argued Shapiro''s work noteworthy both in its attempt to develop a full-length structuralist approach to mathematics and to trace its emergence in the history of mathematics will be of deep interest to both philosophers and mathematicians.