{"id":"philosophy-of-mathematics","domain":"history-philosophy","type":"Concept","title":"Philosophy of Mathematics","slug":"philosophy-of-mathematics","url":"/mathematics/history-philosophy/philosophy-of-mathematics/","summary":"Studies foundational questions: do mathematical objects exist independently of minds? Three major schools: Platonism, Formalism, Intuitionism.","key_positions":["Platonism — mathematical objects exist objectively (Gödel)","Formalism — mathematics is symbol manipulation per rules (Hilbert)","Intuitionism — mathematics is mental construction (Brouwer)","Logicism — mathematics reducible to logic (Frege, Russell)","Structuralism — studies structures not object identity (Benacerraf, Shapiro)"],"added_to_codex":"2026-05-27","last_verified":"2026-05-27","sources":[{"tier":1,"citation":"Shapiro, S. (2000). Thinking About Mathematics. Oxford University Press."},{"tier":1,"citation":"Benacerraf, P. and Putnam, H. (eds.) (1983). Philosophy of Mathematics: Selected Readings. 2nd ed. Cambridge University Press."},{"tier":2,"citation":"Lakatos, I. (1976). Proofs and Refutations. Cambridge University Press."},{"tier":3,"citation":"Wigner, E. (1960). The unreasonable effectiveness of mathematics. Communications on Pure and Applied Mathematics, 13(1), 1-14."}],"relationships":[{"type":"related_to","target":"godels-incompleteness-theorems","label":"Gödel's Incompleteness Theorems — challenged Formalism"},{"type":"related_to","target":"set","label":"Set — Russell's Paradox challenged Logicism"}],"entry_status":"live","reading_levels_available":["curious","exploring","deep_dive"],"completeness":1.0,"codex_version":"1.6"}