The philosophy of mathematics studies the nature of mathematical truth, mathematical proof, mathematical evidence, mathematical practice, and mathematical explanation.
Three philosophical views of mathematics are widely regarded as the ‘classic’ ones. Logicism holds that mathematics is reducible to principles of pure logic. Intuitionism holds that mathematics is concerned with mental constructions and defends a revision of classical mathematics and logic. Finally, formalism is the view that much or all of mathematics is devoid of content and a purely formal study of strings of mathematical language.
In recent decades, some new views have entered the fray. An important newer arrival is structuralism, which holds that mathematics is the study of abstract structures. A non-eliminative version of structuralism holds that there exist such things as abstract structures, whereas an eliminative version tries to make do with concrete objects variously structured. Nominalism denies that there are any abstract mathematical objects and tries to reconstruct classical mathematics accordingly. Fictionalism is based on the idea that, although most mathematical theorems are literally false, there is a non-literal (or fictional) sense in which assertions of them nevertheless count as correct. Mathematical naturalism urges that mathematics be taken as a sui generis discipline in good scientific standing.
|On the more traditional views, it is hard to beat the selection of readings in Benacerraf & Putnam 1983. Non-eliminative structuralism is defended in Resnik 1997, Shapiro 1997, and Parsons 2007. A modal version of eliminative structuralism derives from Putnam 1967 and is developed in Hellman 1989. Two classic defenses of logicism are Frege 1884/1950 and Russell 1919. A neo-Fregean programme was initiated in Wright 1983; see the essays collected in Hale & Wright 2001 and, for critical discussion, Dummett 1973. Nominalism is often driven by the epistemic challenge due to Benacerraf 1973 and Field 1989, ch. 1 and 7. Field’s classic attempt to vindicate nominalism is Field 1980. For a comprehensive overview of the subject, see Burgess & Rosen 1997. On fictionalism, see Yablo 2010. The indispensability argument derives from Quine but crystallized in Putnam 1971; for a recent defense, see Colyvan 2001. On mathematical naturalism, see Maddy 1997 and Maddy 2007.