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Pragmatic Platonism

Mathematics and the Infinite
  • Martin Davis
Chapter
Part of the Outstanding Contributions to Logic book series (OCTR, volume 9)

Abstract

It is argued that to a greater or lesser extent, all mathematical knowledge is empirical.

References

  1. Barrow, J. D., & Tipler, F. J. (1986). The anthropic cosmological principle. Oxford: Oxford University Press.Google Scholar
  2. Boole, G. (1865). A treatise on differential equations. London: Macmillan and Co.Google Scholar
  3. Davis, M. (2005). What did Gödel believe and when did he believe It? Bulletin of Symbolic Logic, 11, 194–206.MathSciNetCrossRefGoogle Scholar
  4. Davis, M., Putnam, H., & Robinson, J. (1961). The decision problem for exponential diophantine equations. Annals of Mathematics, 74, 425–436. Reprinted in S. Feferman (Ed.). (1996). The collected works of Julia Robinson (pp. 77–88). American Mathematical Society.MathSciNetCrossRefGoogle Scholar
  5. Davis, M., Matiyasevich, Y., & Robinson, J. (1976). Hilbert’s tenth problem. Diophantine equations: positive aspects of a negative solution. In Proceedings of Symposia in Pure Mathematics, vol XXVIII: Positive Aspects of a Negative Solution (pp. 323–378). Reprinted in S. Feferman (Ed.). (1996). The collected works of Julia Robinson (pp. 269–378). American Mathematical Society.Google Scholar
  6. Feferman, S., et al. (1986–2003). Kurt Gödel collected works, volume I–V. Oxford: Oxford University Press.Google Scholar
  7. Frege, G. (1892). Rezension von: Georg Cantor. Zum Lehre vom Transfiniten. Zeitschrift fr Philosophie und philosophische Kritik, new series, 100, 269–272.Google Scholar
  8. Green, B., & Tao, T. (2008). The primes contain arbitrarily long arithmetic progressions. Annals of Mathematics, 167, 481–547.MathSciNetCrossRefGoogle Scholar
  9. Hardy, G. H., & Wright, E. M. (1960). An introduction to the theory of numbers (Fourth Edition ed.). Oxford: Clarendon Press.zbMATHGoogle Scholar
  10. Mancosu, P. (1996). Philosophy of mathematics and mathematical practice in the seventeenth century. Oxford: Oxford University Press.Google Scholar
  11. 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.MathSciNetCrossRefGoogle Scholar
  12. Post, E. L. (1944). Recursively enumerable sets of positive integers and their decision problems. Bulletin of the American Mathematical Society, 50, 284–316. Reprinted: Davis, M. (1965, 2004). The undecidable. New York: Raven Press; New York: Dover. Reprinted: E. L. Post, & M. Martin Davis (Ed.). (1994). Solvability, provability, definability: The collected works. Birkhäuser.Google Scholar
  13. Putnam, H. (1975). What is mathematical truth? Historia Mathematica, 2, 529–533.MathSciNetCrossRefGoogle Scholar
  14. Putman, H. (1995). Philosophy of mathematics: why nothing works. Works and life (pp. 499–511). Cambridge: Harvard University Press.Google Scholar
  15. Robinson, J. (1952). Existential definability in arithmetic. In: Transactions of the American Mathematical Society (Vol. 72, pp. 437–449). Reprinted in S. Feferman (Ed.). (1996). The collected works of Julia Robinson (pp. 47–59). American Mathematical Society.Google Scholar
  16. van Heijenoort, J. (Ed.). (1967). From Frege to Gödel: A source book in mathematical logic, 1879–1931. Cambridge: Harvard University Press.zbMATHGoogle Scholar
  17. Weyl, H. (1944). David Hilbert and his mathematical work. Bulletin of the American Mathematical Society, 50, 612–654.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUSA
  2. 2.BerkeleyUSA

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