Making room for mathematicians in the philosophy of mathematics
- Thomas Tymoczko
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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.
- Curiously enough, the Platonist ideal might be impossible, at least if we regard the arithmetical version of “No one knows this statement” as a mathematical statement. Does the ideal mathematician know it or not? For an extended discussion of this point, see my “An Unsolved Puzzle About Knowledge”,The Philosophical Quarterly, October 1984.
- This criticism was forcibly pressed by Willard Quine in “Truth by Convention” in O. H. Lee (ed.),Philosophical Essays for A. N. White-head, (New York: Longmans, 1936).
- The idea that the role of the community in mathematics must be taken into account by the philosophy of mathematics has been argued recently by Philip Kitcher,The Nature of Mathematical Knowledge, (Oxford: Oxford University Press, 1983). It also figures prominently in Philip Davis and Reuben Hersh,The Mathematical Experience, (Boston: Birkhauser, 1982), and, of course, in the work of Raymond Wilder. One of the earliest expressions of this thesis was L. White, “The Locus of Mathematical Reality: An Anthropological Footnote”,Philosophy of Science, 17 (1947) 189-203.
- See Judith Grabiner, “Is Mathematical Truth Time-Dependent?”,American Mathematical Monthly, Vol. 81, No. 4, (1974) 354–365, reprinted in my anthologyNew Directions in the Philosophy of Mathematics, Boston: Birkhauser, forthcoming). Grabiner argues that it wasn’t always this way. Formerly, mathematicians were independently wealthy or patronized, like artists. CrossRef
- Reuben Hersh and Rene Thorn are two mathematicians who have stressed the philosophical relevance of teaching practices. See their respective essays, “Some Proposals for Reviving the Philosophy of Mathematics”,Advances in Mathematics 31, (1979) 31-50; and “Modern Mathematics: An Educational and Philosophic Error?”American Scientist, LIX, 6 (1971), both reprinted in Tymoczko,New Directions.
- For further discussion of this issue, see Anthony Ralston, “Computer Science, Mathematics and the Undergraduate Curricula in Both”,American Mathematical Monthly, Vol. 88, No. 7, (1981) 472–485. CrossRef
- Gottlob Frege,The Foundations of Arithmetic, (Oxford: Basil Black-well, 1968), preface.
- Jacque Hadamard,The Psychology of Invention in the Mathematical Field, (New York: Dover, 1954), 13.
- A graphic representation of this increase is given in Davis and Hersh,op. cit., especially 29-30.
- Willard Quine,Word and Object, (Cambridge: MIT Press, 1960), preface; Wittgenstein,Philosophical Investigations, (New York: MacMillan, 1953); Saul Kripke,Wittgenstein on Rules and Private Language, (Cambridge: Harvard, 1982).
- I have done so in my paper, Gödel, Wittgenstein and the Nature of Mathematical Knowledge” in P. Asquith, ed.,PSA 1984, forthcoming.
- Imre Lakatos,Proofs and Refutations, (Cambridge: Cambridge University Press, 1976). CrossRef
- Gödel’s work offers one reason to reject foundationalism, but foundationalism can be criticized on many grounds. Among philosophers, Lakatos and Putnam have criticized the foundational program; mathematicians such as Davis, Hersh and Wilder have made similar claims. For a collection of some of the more pointed critiques, see Tymoczko,New Directions.
- Hilary Putnam, “Mathematics Without Foundations”, reprinted in hisMathematics, Matter and Method, (Cambridge: Cambridge University Press, 1975), 45.
- Richard De Millo, Richard Lipton and Alan Perlis, “Social Processes and Proofs of Theorems and Programs”,Communications of the ACM, Vol. 22, No. 5, (1979) 271–280, reprinted in Tymoczko,New Directions. CrossRef
- This echoes a point of Lesbesgue, “in the studies on the foundations and methods of mathematics, there must be a large place for psychology and even for aesthetics” from his “Les controverses sur la théorie des ensembles et la question des fondements”, in F. Gonseth (ed.),Les entretiens de Zurich, (Zurich: Leeman, 1941), above translation by Gregory Moore. For a fuller discussion of art and mathematics, see A. Borel, “Mathematics: Art and Science”,The Mathematical Intelligencer, Vol. 5, No. 4, 1983, 9-17.
- Making room for mathematicians in the philosophy of mathematics
The Mathematical Intelligencer
Volume 8, Issue 3 , pp 44-50
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- Thomas Tymoczko (1)
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- 1. Department of Philosophy Smith College Northampton, 01063, Mass.