Philosophy of Mathematics

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.

Introductory book: Shapiro 2000. Anthologies: Benacerraf & Putnam 1983, Hart 1996, Bueno & Linnebo 2009, and Marcus & McEvoy 2016 (with lots of historical material). Handbook: Shapiro 2005.