The essence of philosophy of mathematics is the study of the epistemology and ontology of mathematics: what are mathematical entities and how do know that mathematical statements are true?
Certain Austrian economists and libertarians influenced by Mises have an embarrassingly outdated and unconvincing Kantian view of mathematics and geometry, one which appears ignorant of the whole history of philosophy and epistemology as applied to mathematics over the past 130 years.
To see how rich and sophisticated modern philosophy of mathematics is we need only see a list of the various competing theories.
The first major division of theories in philosophy of mathematics is epistemological: how do we and can we know that mathematical statements are true, and how do we justify their truth?
The following is a first attempt at classifying epistemological theories in philosophy of mathematics:
I. Anti-RealismAnother classification of theories in philosophy of mathematics is ontological.(1) Logicism (Deductivism).II. Realism(1.1) Logicism (Frege, Russell);(2) Intuitionism (L.E.J. Brouwer, Arend Heyting, Michael Dummett);
(1.2) Conditionalism (if-thenism) (early Putnam, Hellman, Musgrave);
(3) Formalism (Haskell Curry);
(4) Predicativism (Solomon Feferman).(a) Platonic Realism(1) Platonism(b) Naturalist Realism/Empiricism (Anti-Platonist Realism)
(2) Gödel’s Platonism(1) John Stuart Mill (mathematical physicalism), Philip Kitcher.
(3) Quine, Putnam, Penelope Maddy’s Realism in Mathematics
(4) Social constructivism or social realism;
(5) quasi-empiricism (Imre Lakatos, Karl Popper)
That is, we have theories that explain what mathematical entities are and whether and in what sense they exist.
We could classify such ontological theories in philosophy of mathematics as follows:
I. RealismThe major division in both types of theories is that between (1) realist and (2) anti-realist theories of mathematics.(i) PlatonismII. Anti-Realism(1) Platonism;(ii) Realistic Anti-Platonism
(2) full-blooded Platonism;
(3) Aristotelian realism;
(4) Structuralism (Resnik, Shapiro)(1) Psychologism (Edmund Husserl, L.E.J. Brouwer, Arend Heyting);
(2) Physicalism (David Armstrong, J. S, Mill);
(i) Anti-Realist Nominalism(1) Paraphrase nominalism (or if-thenism, deductivism; Hilary Putnam, Geoffrey Hellman, Haskell Curry, Charles Chihara);
(3) Fictionalism (Hartry Field, Mark Balaguer, Gideon Rosen, Mary Leng, Stephen Yablo’s figuralism);
(4) neo-Meinongians (Richard Sylvan).
Some realists think that mathematical entities have a real existence independently of the human mind, and human beings discover, rather than invent, mathematics.
Anti-realists, by contrast, think we invent mathematics or construct it in some sense. Mathematical entities are therefore not independent of conscious rational minds capable of understanding and perceiving these concepts.
Of course there is overlap in both classifications, and I may have made some errors and omissions here, and anyone who has specialist knowledge of philosophy of mathematics is welcome to comment on and criticise both lists.
“Philosophy of Mathematics,” Stanford Encyclopedia of Philosophy, 2007 (rev. 2012)
“Philosophy of Mathematics,” Wikipedia
Brown, James Robert. 2012. Platonism, Naturalism, and Mathematical Knowledge. Routledge, New York and Abingdon.
Putnam, H. 1972. Philosophy of Logic. George Allen & Unwin, London.
Gray, Jeremy. 2008. Plato’s Ghost: The Modernist Transformation of Mathematics. Princeton University Press, Princeton, N.J. and Woodstock.