Making room for mathematicians in the philosophy of mathematics
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- Tymoczko, T. The Mathematical Intelligencer (1986) 8: 44. doi:10.1007/BF03025789
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We began this essay with the concept of mathematician and the realization that a concept of mathematician could be idealized in various ways. We drew an elementary distinction between private and public conceptions. Private theories of mathematicians applied, in principle, to the case of a single mathematician practicing in isolation. Public theories made essential use of the fact that mathematicians occur in communities. We argued that tradition encourages us to prefer private theories and that competition among traditional schools in the philosophy of mathematics is competition among private theories of mathematicians. The rest of the essay tried to establish public theories as an alternative in serious philosophy of mathematics. We considered the practice of teaching and argued that it was an important feature of mathematical experience, intimately related to other central aspects of mathematics and finally, that teaching could only be accounted for by public models. Teaching is not traditionally regarded as a central topic in the philosophy of mathematics, but the nature of mathematical proof is. So we turned to the concept of proof and tried to suggest how it might appear from the perspective of public epistemology. We found that the proofs of a mathematical community could be characterized very differently from the formal proofs of private, isolated mathematicians. These arguments are very suggestive. They raise for the philosophy of mathematics some issues that are basic to contemporary epistemology and the philosophy of science. More important, they offer to philosophy a new way of looking at and characterizing mathematics, one that better coheres with the experience of mathematicians. In making room for the concept of mathematicians in the philosophy of mathematics, perhaps we are also making more room for mathematicians themselves.