Notre Dame Journal of Formal Logic

Cardinality, Counting, and Equinumerosity

Richard G. Heck

Abstract

Frege, famously, held that there is a close connection between our concept of cardinal number and the notion of one-one correspondence, a connection enshrined in Hume's Principle. Husserl, and later Parsons, objected that there is no such close connection, that our most primitive conception of cardinality arises from our grasp of the practice of counting. Some empirical work on children's development of a concept of number has sometimes been thought to point in the same direction. I argue, however, that Frege was close to right, that our concept of cardinal number is closely connected with a notion like that of one-one correspondence, a more primitive notion we might call just as many.

Article information

Source
Notre Dame J. Formal Logic Volume 41, Number 3 (2000), 187-209.

Dates
First available: 26 November 2002

Permanent link to this document
http://projecteuclid.org/euclid.ndjfl/1038336841

Digital Object Identifier
doi:10.1305/ndjfl/1038336841

Mathematical Reviews number (MathSciNet)
MR1943492

Zentralblatt MATH identifier
1009.03009

Subjects
Primary: 03A05: Philosophical and critical {For philosophy of mathematics, see also 00A30}

Keywords
Frege logicism counting arithmetic

Citation

Heck, Richard G. Cardinality, Counting, and Equinumerosity. Notre Dame Journal of Formal Logic 41 (2000), no. 3, 187--209. doi:10.1305/ndjfl/1038336841. http://projecteuclid.org/euclid.ndjfl/1038336841.


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References

  • [1] Boolos, G., Logic, Logic, and Logic, Harvard University Press, Cambridge, 1998.
  • [2] Burge, T., "Frege on knowing the foundation", Mind, vol. 107 (1998), pp. 305--47.
  • [3] Burge, T., "Frege on apriority", pp. 11--42 in New Essays on the A Priori, edited by P. Boghossian and C. Peacocke, The Clarendon Press, Oxford, 2000.
  • [4] Carey, S., "Continuity and discontinuity in cognitive development", pp. 101--29 in An Invitation to Cognitive Science: Thinking, edited by G. Smith and D. Osherson, The MIT Press, Cambridge, 2d edition, 1995.
  • [5] Demopoulos, W., "The philosophical basis of our knowledge of number", Noûs, vol. 32 (1998), pp. 481--503.
  • [6] Dummett, M., Frege and Other Philosophers, The Clarendon Press, New York, 1991.
  • [7] Frege, G., Grundgesetze der Arithmetik, H. Pohle, Jena, 1903.
  • [8] Frege, G., The Foundations of Arithmetic, Northwestern University Press, Evanston, 1980. Translated by J. L. Austin.
  • [9] Frege, G., "Review of E. G. Husserl, Philosophie der Arithmetik I", pp. 195--209 in Collected Papers on Mathematics, Logic, and Philosophy, edited by B. McGuiness, Basil Blackwell, Oxford, 1984.
  • [10] Gelman, R., and C. R. R. Galistel, The Child's Understanding of Number, Harvard University Press, Cambridge, 1978.
  • [11] Hale, B., "Grundlagen \S 64", pp. 91--116 in The Reason's Proper Study, edited by B. Hale and C. Wright, The Clarendon Press, Oxford, 2001.
  • [12] Heck, R. G., Jr., "Finitude and Hume's Principle", Journal of Philosophical Logic, vol. 26 (1997), pp. 589--617.
  • [13] Heck, R. G., Jr., Language, Thought, and Logic. Essays in honour of Michael Dummett, Oxford University Press, New York, 1997.
  • [14] Heck, R. G., Jr., "The finite and the infinite in Frege's Grundgesetze der Arithmetik", pp. 429--66 in The Philosophy of Mathematics Today (Munich, 1993), Oxford University Press, New York, 1998.
  • [15] Kripke, S., "De re beliefs about number", Whitehead Lectures given at Harvard, 1993.
  • [16] Parsons, C., "Intuition and number", pp. 141--57 in Mathematics and Mind (Amherst, 1991), edited by G. Alexander, Oxford University Press, New York, 1994.
  • [17] Parsons, C., "Frege's theory of number", pp. 182--207 in Frege's Philosophy of Mathematics, edited by W. Demopoulos, Harvard University Press, Cambridge, 1995. Originally published in Philosophy in America, Cornell University Press, Ithaca, 1965, pp. 180--203.
  • [18] Wright, C., Frege's Conception of Numbers as Objects, Aberdeen University Press, Aberdeen, 1983.
  • [19] Wright, C., "On the philosophical significance of Frege's Theorem", pp. 272--306 in The Reason's Proper Study, edited by B. Hale and C. Wright, The Clarendon Press, Oxford, 2001.