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