Abstract
This chapter intends to examine the widespread assumption, which has been uncritically accepted, that Zermelo simply adopted Hilbert’s axiomatic method in his axiomatization of set theory. What is essential in that shared axiomatic method? And, exactly when was it established? By philosophical reflection on these questions, we are to uncover how Zermelo’s thought and Hilbert’s thought on the axiomatic method were developed interacting each other. As a consequence, we will note the possibility that Zermelo, in his early as well as late thought, had views about the axiomatic method entirely different from that of Hilbert. Such a result must have far-reaching implications to the history of set theory and the axiomatic method, thereby to the philosophy of mathematics in general.
This chapter was originally published as Park (2008) in Korean.
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- 1.
Further, Muller claims that, at least in mathematics, Wittgenstein’s ‘social’ conception of meaning is harmonious with Hilbert’s ‘rational’ conception of implicit definability: “because accepted axioms of some branch of mathematics, together with the logical deduction-rules, govern the rigorous use of the primitive notions as they are actually used by the community of mathematicians in this branch” (Muller 2004, 429).
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Rowe (2000, p. 76): “While Hilbert failed to respond to Frege’s detailed critique, his silence should not be taken as a sign that he failed to comprehend the point of Frege’s criticisms”.
- 3.
Peckhaus (1994, p. 99): “Above all, it is remarkable that in this passage Hilbert upholds the traditional view that axioms are propositions, since back in 1899, in a famous controversy with Frege over the “Grundlagen der Geometrie”, Hilbert already advanced the “modern” notion that the word “axiom” does not denote a proposition but rather a propositional function”.
- 4.
In fact, Resnik even evaluates Frege as follows: “In any case, Frege was the first to give written evidence of an adequate grasp of the use of implicit definitions” (Ibid.).
- 5.
See Rowe (2000, p. 82): “Yet while retreating on the foundational front, Hilbert widened his interest in axiomatic dramatically during the twelve-year period from 1905 to 1917. This period marked at once the pinnacle of achievement for Göttingen mathematics as well as for “Hilbert’s school”.
- 6.
Moore also points out the same point as follows: “Although in 1900 Hilbert was aware of various paradoxes of set theory, he did not show any concern that set theory, or logic, was threatened”. Moore (2002, 47).
- 7.
Moore also gives us an interesting report in the same vein. According to the lecture Notes in Summer, 1905, even though Hilbert suggested the use of the axiomatic method, he did not suggest the axiomatization of set theory (Moore 2002, 49).
- 8.
Rowe (2000, p. 82). Rowe conjectures that for that very reason, even though Hilbert supported Zermelo strongly, he did not go out and play any active role in the controversy of Zermelo’s axiom of choice. As we already pointed out that Hilbert’s silence is a kind of mystery, We should welcome Rowe’s conjecture, whether it is right or wrong.
- 9.
Of course, one might treat this as a minority opinion. However, we should note that Moore’s interpretation has been accepted by significant number of scholars including Hallett, Peckhaus, and Kanamori.
- 10.
Kanamori (2004, 499f) discusses Zermelo’s reductionism as follows: “Zermelo pioneered the reduction of mathematical concepts and arguments to set-theoretic concepts and arguments from axioms, based on sets doing the work of mathematical objects”. Zermelo (1909) wrote at the beginning: “… for me, every theorem stated about finite numbers is nothing other than a theorem about finite sets”. See also Hallett (1984, p. 244f).
- 11.
In fact, everywhere in Taylor’s article, those that can be counted as the methodological differences are pointed out as the difference between Hilbert and Zermelo. For example, see Taylor (1993, 537). This point will be further discussed in what follows.
- 12.
Zermelo’s own manuscript and a note taken by Kurt Grelling are preserved (Ebbinghaus 2007, p. 97).
- 13.
For example, see Moore (1982).
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Park, W. (2018). Zermelo and the Axiomatic Method. In: Philosophy's Loss of Logic to Mathematics. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 43. Springer, Cham. https://doi.org/10.1007/978-3-319-95147-8_5
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