Towards a Philosophy of Real MathematicsIn this ambitious study, David Corfield attacks the widely held view that it is the nature of mathematical knowledge which has shaped the way in which mathematics is treated philosophically and claims that contingent factors have brought us to the present thematically limited discipline. Illustrating his discussion with a wealth of examples, he sets out a variety of approaches to new thinking about the philosophy of mathematics, ranging from an exploration of whether computers producing mathematical proofs or conjectures are doing real mathematics, to the use of analogy, the prospects for a Bayesian confirmation theory, the notion of a mathematical research programme and the ways in which new concepts are justified. His inspiring book challenges both philosophers and mathematicians to develop the broadest and richest philosophical resources for work in their disciplines and points clearly to the ways in which this can be done. |
Contents
| 1 | |
PART I Human and artificial mathematicians | 35 |
PART II Plausibility uncertainty and probability | 101 |
PART III The growth of mathematics | 149 |
PART IV The interpretation of mathematics | 233 |
Appendix | 271 |
| 274 | |
| 286 | |
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Common terms and phrases
algebraic number algebraic topology algorithm allow analogy analysis axiomatisation axioms Bayesian calculation category theory century chapter claim complex concept conjecture construction Dedekind degrees of belief diagrams dimension domain ematics entities equations equivalent evidence example factorisation finite formula function functors generalisation groupoids hard core heuristic higher-dimensional algebra Hilbert homotopy groups homotopy theory Hopf algebras ideal ideas imagine important inductive integers isomorphic knot Lakatos Lakatos’s language lemmas logic Mac Lane manifolds mappings math mathematical research mathematicians models monoidal n-category natural numbers notation notion number fields number theory objects P´olya path philosophers philosophy of mathematics philosophy of science physicists physics plausible Poincaré Pólya polynomial prime problem PROGOL Proofs and Refutations prove quantum field theory quantum gravity real mathematics reasoning representation research programmes result Riemann scientific sense spin foams structure theorem provers topological spaces University Press vector