Contents -- Preface -- List of contributors -- Contents -- Introduction -- 1 Benacerraf's worry -- 2 Mathematical knowers and mathematical knowledge -- What is the problem of mathematical knowledge? -- 1 What is distinctive about the mathematical case? -- 2 Implicationism all the way down? -- 3 Mother theories -- 4 Understanding and truth -- 5 Why externalism? -- 6 The route to knowledge -- 7 Benacerraf's problem -- 8 Benacerraf's problem generalized -- 9 The real problem -- Mathematics, Memory, and Mental Arithmetic -- 1 Introduction -- 2 The problem of induction in mathematics -- 3 The seemingly inappropriate use of modal language by mathematicians -- 4 Explicating informal mathematical terminology -- 5 Memory -- 6 Direct memory, competence, and generated memory -- 7 Two proofs and how one remembers them -- 8 Mental arithmetic and the concept of width -- 9 How does the notion of width apply to proofs? -- 10 Concluding remarks -- Is there a problem of induction for mathematics? -- 1 Introduction -- 2 Enumerative induction and discovery -- 3 The descriptive question: Two case studies -- 4 Hume's problem of induction -- 5 The normative question: Is enumerative induction in mathematics rationally justified? -- 6 Re-examining the descriptive question -- 7 Conclusions -- The cognitive basis of mathematical knowledge -- 1 Numerical cognition is innate and distinct from other cognitive skills -- 2 Numbers and language -- 3 Different uses of the same numeral -- 4 Limits of cognitive science -- 5 Concluding remarks -- What's there to know? -- 1 Trading ontology for modality -- 2 Fictionalism and nominalism -- 3 A new Benacerraf problem? -- 4 A new indispensability argument? -- 5 Defending the consistency of ZFC and PA -- 6 Conclusion -- Mathematical recreation versus mathematical knowledge -- 1 Empiricism in the philosophy of mathematics.

2 An empiricist account of mathematical knowledge -- 3 Unapplied mathematics as mathematical recreation -- 4 Is all mathematics recreation? -- 5 Empiricism revisited -- Scientific Platonism -- 1 Introduction -- 2 Preliminaries -- 3 The pragmatic and indifference objections -- 4 Weak and strong scientific platonism -- 5 For the indifference objection -- 6 The publication test -- 7 General principles of scientific method -- 8 Scientific grounds and the actual content of mathematics -- 9 Conclusion -- On quantifying into predicate position -- 1 Basic idea and project outline -- 2 Fixing the meanings of the quantifiers -- 3 Extreme neutralism -- 4 A neutralist heuristic -- 5 Comprehension -- 6 Incompleteness -- 7 Impredicativity -- 8 Appendix: Abstractionist Mathematical Theories -- Bibliography -- Index -- A -- B -- C -- D -- E -- F -- G -- H -- I -- J -- K -- L -- M -- N -- O -- P -- Q -- R -- S -- T -- U -- V -- W -- Y -- Z.