Abstract
It is argued that to a greater or lesser extent, all mathematical knowledge is empirical.
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Notes
- 1.
This essay is based on a paper with the same title (but without the subtitle) read at a conference celebrating Harvey Friedman’s 60th birthday. Much of the text is taken verbatim from that paper.
- 2.
Feferman et al. (2003), vol. IV, pp. 502–505.
- 3.
Feferman et al. (2003), vol. III, p. 312.
- 4.
Feferman et al. (2003), vol. V, p. 204.
- 5.
Feferman et al. (2003), vol. III, p. 313.
- 6.
Feferman et al. (2003), vol. III, p. 50.
- 7.
A proof did not appear until the next millennium! See Green and Tao (2008).
- 8.
This was a celebrated result because the unsolvability of Hilbert’s tenth problem was an immediate corollary (Davis et al. 1996).
- 9.
Post (1994) p. 295.
- 10.
van Heijenoort (1967) p. 371.
- 11.
This discussion, including the quotations, is based on Paolo Mancosu’s wonderful monograph (Mancosu 1996).
- 12.
The method of exhaustion typically required one to have the answer at hand, whereas with indivisibles the answer could be computed.
- 13.
Emphasis is in the original.
- 14.
Of course this conventional use of the Greek letter \(\pi \) has nothing to do with the number \(\pi = 3.14159\ldots .\)
- 15.
Although Euler obtained this infinite product by using a doubtful analogy, Weierstrass eventually obtained it quite rigorously as a special case of a general theorem about complex analytic functions.
- 16.
That there are infinitely many primes has been proved in many different ways. The first proof is already in Euclid. It is very simple and elegant. To show that given any finite collection of primes, \(p_1,p_2,\ldots , p_n\), there is another prime not in that collection, one multiplies together all the primes in that collection and adds 1:
$$\begin{aligned} N = p_1p_2\ldots p_n +1 \end{aligned}$$Since none of \(p_1,p_2,\ldots , p_n\) are divisors of N, either N is itself a prime different from any of them or it is divisible by a prime different from them.
- 17.
A similar conclusion is obtained heuristically in Hardy and Wright (1960) pp. 371–372 without probabilistic considerations.
- 18.
Mancosu (1996) p. 172.
- 19.
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.
- 20.
Number theorists seem to regard the use of Grothendiek universes as a mere convenience. See McLarty (2010) for a careful discussion.
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Davis, M. (2018). Pragmatic Platonism. In: Hellman, G., Cook, R. (eds) Hilary Putnam on Logic and Mathematics. Outstanding Contributions to Logic, vol 9. Springer, Cham. https://doi.org/10.1007/978-3-319-96274-0_10
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