Abstract
It is argued that to a greater or less extent, all mathematical knowledge is empirical.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
See [5], vol IV, pp. 502–505.
- 2.
See [5] vol. III, p. 312.
- 3.
See [5] vol. V, p. 204.
- 4.
See [5], vol. III, p. 313.
- 5.
See [5] vol. III, p. 50.
- 6.
See [4] pp. 331–339.
- 7.
See [9] p. 295.
- 8.
See [10] p. 371.
- 9.
This discussion, including the quotations, is based on Paolo Mancosu’s wonderful monograph [7].
- 10.
The method of exhaustion typically required one to have the answer at hand, whereas with indivisibles the answer could be computed.
- 11.
See [7] p. 172.
- 12.
Because otherwise the consistency of ZF would be provable in ZF contradicting Gödel’s second incompleteness theorem. For that matter the set \(V_{\omega 2}\) cannot be proved to exist from the Zermelo axioms alone; in ZF its existence follows using Replacement.
- 13.
Number theorists regard the use of Grothendiek universes as a mere convenience. See [8] for a careful discussion.
References
Barrow, J. D., & Tipler, F. J. (1986). The anthropic cosmological principle. Oxford: Oxford University Press.
Boole, G. (1865). A treatise on differential equations. London: Macmillan and Co.
Davis, M. (2005). What did Gödel believe and when did he believe it? Bulletin of Symbolic Logic, 11, 194–206.
Davis, M., Matiyasevich, Yu., & Robinson, J. (1976). Hilbert’s tenth problem. Diophantine equations: Positive aspects of a negative solution. In Proceedings of Symposia in Pure Mathematics: Positive Aspects of a Negative Solution (Vol. XXVIII, pp. 323–378).
Feferman, S., et al. (1986–2003). Kurt Gödel Collected Works (Vols. I–V). Oxford: Oxford University Press.
Frege, G. (1892). Rezension von: Georg Cantor. Zum Lehre vom Transfiniten. Zeitschrift für Philosophie und philosophische Kritik, new series, 100, 269–272.
Mancosu, P. (1996). Philosophy of mathematics & mathematical practice in the seventeenth century. Oxford: Oxford University Press.
McLarty, C. (2010). What does it take to prove Fermat’s last theorem? Grothendiek and the logic of number theory. Bulletin of Symbolic Logic, 16, 359–377.
Post, E. L. (1944). Recursively enumerable sets of positive integers and their decision problems. Bulletin of the American Mathematical Society, 50, 284–316. Reprinted: M. Davis (Ed.), The undecidable Raven Press, New York 1965; Dover, New York 2004. Reprinted:: M. Davis (Ed.), Solvability, provability, definability: The collected works of Emil L. Post, Birkhäuser 1994.
van Heijenoort, J. (Ed.) (1967). From Frege to Gödel: A source book in mathematical logic, 1879–1931. Cambridge: Harvard University Press.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Davis, M. (2016). Pragmatic Platonism. In: Omodeo, E., Policriti, A. (eds) Martin Davis on Computability, Computational Logic, and Mathematical Foundations. Outstanding Contributions to Logic, vol 10. Springer, Cham. https://doi.org/10.1007/978-3-319-41842-1_14
Download citation
DOI: https://doi.org/10.1007/978-3-319-41842-1_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-41841-4
Online ISBN: 978-3-319-41842-1
eBook Packages: Religion and PhilosophyPhilosophy and Religion (R0)