New Directions in the Philosophy of Mathematics

accepted algebra algorithms analogy argue argument arithmetic axiom of choice axiomatic set theory axiomatic system axioms binary string calculations color complexity computer proofs concept conjecture consider constructions continuum hypothesis culture deductive definition discovery equations error essay Euclidean Euclidean geometry Euler example existence fact falsifiers finite formal axiomatic system formal proof formal system formalist foundations of mathematics four-color Frege function geometry Gödel Gödel's theorem Haken Hilbert history of mathematics idea Imre Lakatos induction infinite informal proofs intuition intuitionism intuitionist Lakatos language mathe mathematical change mathematical knowledge mathematical logic mathematical practice mathematical proof mathematical truth mathematicians matics methods natural numbers natural science notion number theory objects paradigm paradox philosophy of mathematics physical Platonism possible principle priori problem prove Putnam quasi-empirical quasi-empirical methods question real numbers reasoning recursive reference rigorous scientific change sense set theory structure symbols tion triangle true University Press verify