Mathematics and Reality

Stewart Shapiro
Philosophy of Science
Vol. 50, No. 4 (Dec., 1983), pp. 523-548
Stable URL: http://www.jstor.org/stable/187555
Page Count: 26
  • Download PDF
  • Cite this Item

You are not currently logged in.

Access your personal account or get JSTOR access through your library or other institution:

login

Log in to your personal account or through your institution.

Mathematics and Reality
We're having trouble loading this content. Download PDF instead.

Abstract

The subject of this paper is the philosophical problem of accounting for the relationship between mathematics and non-mathematical reality. The first section, devoted to the importance of the problem, suggests that many of the reasons for engaging in philosophy at all make an account of the relationship between mathematics and reality a priority, not only in philosophy of mathematics and philosophy of science, but also in general epistemology/metaphysics. This is followed by a (rather brief) survey of the major, traditional philosophies of mathematics indicating how each is prepared to deal with the present problem. It is shown that (the standard formulations of) some views seem to deny outright that there is a relationship between mathematics and any non-mathematical reality; such philosophies are clearly unacceptable. Other views leave the relationship rather mysterious and, thus, are incomplete at best. The final, more speculative section provides the direction of a positive account. A structuralist philosophy of mathematics is outlined and it is proposed that mathematics applies to reality though the discovery of mathematical structures underlying the non-mathematical universe.

Notes and References

This item contains 56 references.

[Footnotes]
  • 2
    This reference contains 4 citations:
    • Cohen (1971)
    • Curry (1958)
    • Robinson (1965).
    • Resnik (1980, Chapter 2)
  • 3
    Resnik (1980, pp. 62-63)
  • 4
    This reference contains 2 citations:
    • Harman (1975)
    • Dummett (1973).
  • 5
    This reference contains 2 citations:
    • Ayer (1946)
    • Hempel (1945)
  • 6
    Glymour (1980)
  • 7
    This reference contains 4 citations:
    • Gödel (1964)
    • Gödel (1964)
    • "the 'given' underlying mathematics is closely related to the abstract ele- ments contained in our empirical ideas" (p. 272).
    • Maddy (1980).
  • 9
    Barbut 1970
  • 10
    This reference contains 2 citations:
    • Resnik (1981)
    • (1982)
  • 13
    Resnik (1980, Chapter 5)
  • 14
    Resnik (1981).
  • 15
    Kraut (1980)
  • 16
    This reference contains 2 citations:
    • Resnik (1982)
    • note 7 above
  • 17
    This reference contains 3 citations:
    • Hartry Field (1980, pp. 31-33)
    • M. Resnik in Nous
    • D. Malament in the Journal of Philosophy.
  • 21
    Turnbull (1978)
  • 22
    This reference contains 2 citations:
    • Polya (1954)
    • (1977)
  • 24
    Nelson Goodman (1972)
  • 25
    note 17 above
  • 27
    Shapiro (1983)
References
  • Ayer, A. (1946), Language, Truth, and Logic. New York: Dover publications.
  • Barbut, M. (1970), "On the meaning of the word 'structure' in mathematics" in Lane, pp. 367-388.
  • Benacerraf, P. (1965), "What numbers could not be", Philosophical Review74: 47-73.
  • Bourbaki, N. (1950), "The architecture of mathematics", American Mathematical Monthly 57: 221-232.
  • Budden, F. J. (1972), The Fascination of Groups. Cambridge, England: Cambridge Uni- versity Press.
  • Cohen, P. (1971), "Comments on the foundations of set theory" in D. Scott (ed.), Axi- omatic Set Theory. Providence, Rhode Island: American Mathematical Society, pp. 9-15.
  • Crowell, R. and Fox, R. (1963), Introduction to Knot Theory. Boston: Ginn and Company.
  • Curry, H. (1958), Outlines of a Formalist Philosophy of Mathematics. Amsterdam: North Holland Publishing Company.
  • Dedekind, R. (1888), "The nature and meaning of numbers" in R. Dedekind, Essays on the Theory of Numbers, W. W. Beman (ed.) (1963). New York: Dover Press, pp. 31-115.
  • Dummett, M. (1973), "The philosophical basis of intuitionistic logic", reprinted in M. Dummett, Truth and Other Enigmas (1978). Cambridge, Massachusetts: Harvard University Press, pp. 215-247.
  • Field, H. (1980), Science Without Numbers. Princeton: Princeton University Press.
  • Frege, G. (1903), Grundgesetze der Arithmetic, Volume 2.
  • Glymour, C. (1980), Theory and Evidence. Princeton: Princeton University Press.
  • Gödel, K. (1964), "What is Cantor's continuum problem", in P. Benacerraf and H. Put- nam (eds.), Philosophy of Mathematics. Englewood Cliffs, New Jersey: Prentice- Hall, pp. 258-273.
  • Goodman, Nelson (1972), Problems and Projects. Indianapolis: Bobbs-Merill.
  • Goodman, Nicolas D. (1979), "Mathematics as an objective science", American Mathe- matical Monthly88: 540-551.
  • Harman, G. (1975), "Meaning and semantics", in Semantics and Philosophy. New York: NYU Press, pp. 1-16.
  • Hempel, C. (1945), "On the nature of mathematical truth", American Mathematical Monthly 52: 543-556.
  • Heyting, A. (1956), Intuitionism, An Introduction. Amsterdam: North Holland Publishing Company.
  • Kraut, R. (1980), "Indiscernibility and ontology", Synthese44: 113-135.
  • Lane, M. (ed.) (1970), Introduction to Structuralism. New York: Basic Books.
  • Maddy, P. (1980), "Perception and mathematical intuition", The Philosophical Review89: 163-196.
  • Nutter, J. T. (1980), Structuralism in the philosophy of mathematics, Ph.D. Dissertation, SUNY/Buffalo.
  • Piaget, J. (1968), Le Structuralisme. Paris: Presses Universitaires de France.
  • Polya, G. (1954), Mathematics and Plausible Reasoning. Princeton: Princeton University Press.
  • Polya, G. (1977), Mathematical Methods in Science. Washington, D.C.: Mathematical Association of America.
  • Putnam, H. (1971), Philosophy of Logic. New York: Harper Torchbooks.
  • Resnik, M. (1975), "Mathematical knowledge and pattern cognition", Canadian Journal of Philosophy5: 25-39.
  • Resnik, M. (1980), Frege and The Philosophy of Mathematics. Ithaca, New York: Cornell University Press.
  • Resnik, M. (1981), "Mathematics as a science of patterns: Ontology and reference", Nous 15: 529-550.
  • Resnik, M. (1982), "Mathematics as a science of patterns: Epistemology", Nous16: 95- 105.
  • Robinson, A. (1965), "Formalism" in Y. Bar-Hillel (ed.), Logic, Methodology and Phi- losophy of Science. Amsterdam: North Holland Publishing Company, pp. 228-246.
  • Shapiro, S. (1981), "Understanding Church's thesis", Journal of Philosophical Logic10: 353-365.
  • Shapiro, S. (1983), "Conservativeness and incompleteness", Journal of Philosophy80: 521-531.
  • Steiner, M. (1975), Mathematical Knowledge. Ithaca, New York: Cornell University Press.
  • Turing, A. (1936), "On computable numbers, with an application to the Entscheidungs- problem", reprinted in M. Davis (ed.), The Undecidable (1965). Hewlett, New York: The Raven Press, pp. 116-153.
  • Turnbull, R. (1978), "Knowledge of the forms in the later platonic dialogues", Proceed- ings and Addresses of the American Philosophical Association51: 735-758.
  • Wilson, M. (1981), "The double standard in ontology", Philosophical Studies39: 409- 427.