journal article
Sep 01, 1986
Making room for mathematicians in the philosophy of mathematics
Topics
No keywords indexed for this article. Browse by subject →
References
16
[1]
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.
[2]
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).
[3]
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.
[4]
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.
10.2307/2318997
[5]
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.
10.1016/0001-8708(79)90018-5
[6]
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.
10.2307/2321752
[7]
Gottlob Frege,The Foundations of Arithmetic, (Oxford: Basil Black-well, 1968), preface.
[8]
Jacque Hadamard,The Psychology of Invention in the Mathematical Field, (New York: Dover, 1954), 13.
[9]
A graphic representation of this increase is given in Davis and Hersh,op. cit., especially 29-30.
[10]
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).
[11]
I have done so in my paper, Gödel, Wittgenstein and the Nature of Mathematical Knowledge” in P. Asquith, ed.,PSA 1984, forthcoming.
10.1086/psaprocbienmeetp.1984.2.192520
[12]
Imre Lakatos,Proofs and Refutations, (Cambridge: Cambridge University Press, 1976).
10.1017/cbo9781139171472
[13]
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.
[14]
Hilary Putnam, “Mathematics Without Foundations”, reprinted in hisMathematics, Matter and Method, (Cambridge: Cambridge University Press, 1975), 45.
[15]
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.
10.1145/359104.359106
[16]
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.
Metrics
15
Citations
16
References
Details
- Published
- Sep 01, 1986
- Vol/Issue
- 8(3)
- Pages
- 44-50
- License
- View
Authors
Cite This Article
Thomas Tymoczko (1986). Making room for mathematicians in the philosophy of mathematics. The Mathematical Intelligencer, 8(3), 44-50. https://doi.org/10.1007/bf03025789
Related
You May Also Like
The elements of statistical learning: data mining, inference and prediction
James Franklin · 2005
1,084 citations