Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-25T16:59:17.209Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematics and Reality

Published online by Cambridge University Press:  01 April 2022

Stewart Shapiro
Stewart Shapiro*
Affiliation:
Department of Philosophy, Ohio State University of Newark
Get access Rights & Permissions [Opens in a new window]

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.

Type
Research Article
Information
Philosophy of Science , Volume 50 , Issue 4 , December 1983 , pp. 523 - 548
Copyright
Copyright © Philosophy of Science Association 1983

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

†

I would like to thank Mike Resnik, John Corcoran, Robert Kraut, Mark Steiner, and Bill Lycan for many useful comments on previous versions of this paper and for encouragement to continue the project. I would also like to thank, as groups, the members of the Kenyon Symposium, the Ohio State University Philosophy Colloquium, the Buffalo Logic Colloquium, and the Hebrew University Philosophy Colloquium for devoting a session to this project and for providing many thoughtful and useful comments.

References

Ayer, A. (1946), Language, Truth, and Logic. New York: Dover publications.Google Scholar
Barbut, M. (1970), “On the meaning of the word ‘structure’ in mathematics” in Lane, pp. 367–388.Google Scholar
Benacerraf, P. (1965), “What numbers could not be”, Philosophical Review 74: 47–73.CrossRefGoogle Scholar
Bourbaki, N. (1950), “The architecture of mathematics”, American Mathematical Monthly 57: 221–232.CrossRefGoogle Scholar
Budden, F. J. (1972), The Fascination of Groups. Cambridge, England: Cambridge University Press.Google Scholar
Cohen, P. (1971), “Comments on the foundations of set theory” in Scott, D. (ed.), Axiomatic Set Theory. Providence, Rhode Island: American Mathematical Society, pp. 9–15.CrossRefGoogle Scholar
Crowell, R. and Fox, R. (1963), Introduction to Knot Theory. Boston: Ginn and Company.Google Scholar
Curry, H. (1958), Outlines of a Formalist Philosophy of Mathematics. Amsterdam: North Holland Publishing Company.Google Scholar
Dedekind, R. (1888), “The nature and meaning of numbers” in R. Dedekind, Essays on the Theory of Numbers, Beman, W. W. (ed.) (1963). New York: Dover Press, pp. 31–115.Google Scholar
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.Google Scholar
Field, H. (1980), Science Without Numbers. Princeton: Princeton University Press.Google Scholar
Frege, G. (1903), Grundgesetze der Arithmetic, Volume 2.Google Scholar
Glymour, C. (1980), Theory and Evidence. Princeton: Princeton University Press.Google Scholar
Gödel, K. (1964), “What is Cantor's continuum problem”, in Benacerraf, P. and Putnam, H. (eds.), Philosophy of Mathematics. Englewood Cliffs, New Jersey: Prentice-Hall, pp. 258–273.Google Scholar
Goodman, Nelson (1972), Problems and Projects. Indianapolis: Bobbs-Merill.Google Scholar
Goodman, Nicolas D. (1979), “Mathematics as an objective science”, American Mathematical Monthly 88: 540–551.CrossRefGoogle Scholar
Harman, G. (1975), “Meaning and semantics”, in Semantics and Philosophy. New York: NYU Press, pp. 1–16.Google Scholar
Hempel, C. (1945), “On the nature of mathematical truth”, American Mathematical Monthly 52: 543–556.CrossRefGoogle Scholar
Heyting, A. (1956), Intuitionism, An Introduction. Amsterdam: North Holland Publishing Company.Google Scholar
Kraut, R. (1980), “Indiscernibility and ontology”, Synthese 44: 113–135.CrossRefGoogle Scholar
Lane, M. (ed.) (1970), Introduction to Structuralism. New York: Basic Books.Google Scholar
Maddy, P. (1980), “Perception and mathematical intuition”, The Philosophical Review 89: 163–196.CrossRefGoogle Scholar
Nutter, J. T. (1980), Structuralism in the philosophy of mathematics, Ph.D. Dissertation, SUNY/Buffalo.Google Scholar
Piaget, J. (1968), Le Structuralisme. Paris: Presses Universitaires de France.Google Scholar
Polya, G. (1954), Mathematics and Plausible Reasoning. Princeton: Princeton University Press.Google Scholar
Polya, G. (1977), Mathematical Methods in Science. Washington, D.C.: Mathematical Association of America.CrossRefGoogle Scholar
Putnam, H. (1971), Philosophy of Logic. New York: Harper Torchbooks.Google Scholar
Resnik, M. (1975), “Mathematical knowledge and pattern cognition”, Canadian Journal of Philosophy 5: 25–39.CrossRefGoogle Scholar
Resnik, M. (1980), Frege and The Philosophy of Mathematics. Ithaca, New York: Cornell University Press.Google Scholar
Resnik, M. (1981), “Mathematics as a science of patterns: Ontology and reference”, Nous 15: 529–550.CrossRefGoogle Scholar
Resnik, M. (1982), “Mathematics as a science of patterns: Epistemology”, Nous 16: 95–105.CrossRefGoogle Scholar
Robinson, A. (1965), “Formalism” in Bar-Hillel, Y. (ed.), Logic, Methodology and Philosophy of Science. Amsterdam: North Holland Publishing Company, pp. 228–246.Google Scholar
Shapiro, S. (1981), “Understanding Church's thesis”, Journal of Philosophical Logic 10: 353–365.CrossRefGoogle Scholar
Shapiro, S. (1983), “Conservativeness and incompleteness”, Journal of Philosophy 80: 521–531.CrossRefGoogle Scholar
Steiner, M. (1975), Mathematical Knowledge. Ithaca, New York: Cornell University Press.Google Scholar
Turing, A. (1936), “On computable numbers, with an application to the Entscheidungsproblem”, reprinted in Davis, M. (ed.), The Undecidable (1965). Hewlett, New York: The Raven Press, pp. 116–153.Google Scholar
Turnbull, R. (1978), “Knowledge of the forms in the later platonic dialogues”, Proceedings and Addresses of the American Philosophical Association 51: 735–758.CrossRefGoogle Scholar
Wilson, M. (1981), “The double standard in ontology”, Philosophical Studies 39: 409–427.CrossRefGoogle Scholar