Mathematical KnowledgeMary Leng, Alexander Paseau, Michael D. Potter What is the nature of mathematical knowledge? Is it anything like scientific knowledge or is it sui generis? How do we acquire it? Should we believe what mathematicians themselves tell us about it? Are mathematical concepts innate or acquired? Eight new essays offer answers to these and many other questions. Written by some of the world's leading philosophers of mathematics, psychologists, and mathematicians, Mathematical Knowledge gives a lively sense of the current state of debate in this fascinating field. |
Contents
Introduction | 1 |
What is the problem of mathematical knowledge? | 16 |
Mathematics Memory and Mental Arithmetic | 33 |
Is there a problem of induction for mathematics? | 59 |
The cognitive basis of mathematical knowledge | 74 |
Whats there to know? | 84 |
Mathematical recreation versus mathematical knowledge | 109 |
Scientific Platonism | 123 |
On quantifying into predicate position | 150 |
| 175 | |
| 184 | |
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abstract objects accept applications argue argument arithmetic assumption axioms Benacerraf’s claim cognitive concepts Conjecture context derivation digits discussion distinction ematical empirical enumerative induction epistemic example existence explain fact fictionalist first-order higher-order logic hypothesis idea impredicative indifference objection indispensable instance interpretation intuitive justified language Leng logical knowledge logical notion logical possibility math mathematical beliefs mathematical knowledge mathematical objects mathematical proof mathematical theories mathematical truths mathematicians memorable modal model theory model-theoretic natural numbers nominalist one’s particular Peano axioms perfect numbers philosophical philosophy of mathematics physical platonist predicate prime numbers principle problem problem of induction proof proof theory proof-theoretic properties quantification question Quine Quine’s Quinean realism reason to believe role scientific grounds scientific platonism scientific platonist scientific standards endorse scientific theories scientists second-order logic semantic semantic externalism sense sentence set theory simply structure theorem thought tion true validity width