Thephilosophy of mathematics studies the nature of mathematical truth, mathematicalproof, mathematical evidence, mathematical practice, and mathematicalexplanation.

Three philosophicalviews of mathematics are widely regarded as the ‘classic’ ones.Logicism holds that mathematics is reducibleto principles of pure logic.Intuitionism holds that mathematics is concerned with mental constructionsand defends a revision of classical mathematics and logic. Finally,formalism is the view that much or allof mathematics is devoid of content and a purely formal study of strings ofmathematical language.  

In recentdecades, some new views have entered the fray. An important newer arrival isstructuralism, which holds thatmathematics is the study of abstract structures. Anon-eliminative version of structuralism holds that there exist such things asabstract structures, whereas aneliminativeversion tries to make do with concrete objects variously structured.Nominalism denies that there are anyabstract mathematical objects and tries to reconstruct classical mathematics accordingly.Fictionalismis based on the idea that, although most mathematical theorems are literallyfalse, there is a non-literal (or fictional) sense in which assertions of themnevertheless count as correct.Mathematicalnaturalismurges that mathematics be taken as asui generis discipline ingood scientific standing.

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