Abstract
Bernays has not drawn scholarly attention that he deserves. Only quite recently, the reevaluation of his philosophy, including the projects of editing, translating, and reissuing his writings, has just started. As a part of this renaissance of Bernays studies, this chapter tries to distinguish carefully between Hilbert’s and Bernays’ views regarding the axiomatic method. We shall highlight the fact that Hilbert was so proud of his own axiomatic method on textual evidence. Bernays’ estimation of the place of Hilbert’s achievements in the history of the axiomatic method will be scrutinized. Encouraged by the fact that there are big differences between the early middle Bernays and the later Bernays in this matter, we shall contrast them vividly. The most salient difference between Hilbert and Bernays will shown to be found in the problem of the uniformity of the axiomatic method. In the same vein, we will discuss the later Bernays’ criticism of Carnap, for Carnap’s project of philosophy of science in the late 1950s seems to be a continuation and an extension of Hilbert’s faith in the uniformity of the axiomatic method.
An earlier version was published in Korean as Park (2011).
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Corry claims that the expression “assistant” is somewhat misleading (Corry 2004, p. 70).
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According to Specker (1979), which is a very brief biography of Bernays, in 1912 Bernays received a doctoral degree for his study on the theory of numbers under the supervision of Landau at Göttingen. And, in the same year he acquired a license for professorship with his study on the theory of function under Zermelo at the University of Zürich. On Hilbert’s invitation, he moved to Göttingen, and received once more the license for professorship with his work on logic. But, when he was an undergraduate student at Berlin, he learned physics from Plank at Berlin, and from Born at Göttingen. On the other hand, he learned philosophy from Riehl, Stumpf, and Cassirer at Berlin, and from Nelson at at Göttingen. Under the influence of Lelson, who was one of the center figures of neo-Friesianism, Bernays wrote a paper on moral philosophies of Sidgwick and Kant in the 1910’s. Bernays’ background in physics, there are useful and important pieces of information in Corry (2004, pp. 295–296). Bernays was deeply involved in mathematical physics and the foundational issues related to it, publishing Bernays (1913). Also, he lectured on gas theory up to 1920 at Göttingen. We cannot simply bypass Corry’s observation that there is an interesting parallel between Zermelo and Bernays in that both showed a transition of interest from mathematical physics to foundations of mathematics.
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I have in mind the Collected works of Gödel, Zermelo, Carnap, Hilbert-Bernays project, and Bernays project.
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Parsons (2006) takes 1945 as the starting point for the later Bernays. That seems to be a very safe choice that allows at least 6 years from the period of collaboration with Hilbert. However, such a choice could be a too safe one, since there is virtually an unanimous agreement that the true author of Hilbert and Bernays (1934, 1939) was Bernays. We need to trace the beginning of the period of the later Bernays back to sometime after 1934 when Bernays was expelled to Zürich by Nazis, especially when he was attracted to Ginseth’s thought. For the purpose of this paper, this problem may be ignored.
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For example, we have a report of very suggestive episode in Specker (1979, p. 383).
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The interpretation that counts securing the autonomy of mathematics as the central motivation for Hilbert’s thought has been recently emphasized by Franks (2009). It seems important that Bernays mentioned this point.
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Hallett also emphasizes the following points. Contrary to the standard view about the axiomatic system before Hilbert, axioms are no longer true in Hilbert. Furthermore, they are not even judgments that can be either true or false. Even the entire axiom system does not express the truth (Hallett 1995, p. 137). Hallett thinks that such a disagreement over whether axioms can have truth values ultimately originates from the disagreement over the status of axioms. In case we assume that axioms are true, since they must be truths about the primitive concepts that appear in the axioms, a universal logic that can determine the primitive concepts would be pursued. On the other hand, if we reject the assumption that axioms are true, there would be no need for invoking primitive concepts before developing a theory (Hallett 1995, p. 141).
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I discuss the problem of implicit definition in Park (2008). See also Chun (2008), Choi (2007). It is Schlick who made this problem famous, and Friedman discusses it in several writings (see, especially Friedman 1992, 1999). However, interestingly enough, Majer underestimates this problem as subsidiary (Majer 2001, 2002). See also Park (2012). We may continue our interest by turning to Frege/Hilbert controversy. See Choi (2009). For the position of Bernays himself, we need to read Bernays (1942). Usually, based on this review, we say that Frege/Hilbert controversy has not been concluded once and for all.
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For example, Majer claims that the widespread view that Hilbert separated himself from Euclid’s axiomatic approach, and aimed at an entirely new axiomatics must be wrong (Majer 2006, p. 158).
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It is interesting that there is Bernays (1922b) among his writings.
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Parsons also complains about Bernays’ essay style of writing, which lacks systematic approach. Further, he thinks that Bernays’ discretion in discussing different positions and his loyalty to those he had close friendship tend to make his position as close to syncretism (Parsons 2006, p. 148).
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Above all, Parsons seems to discuss Bernays’ anti-foundationalist position by assuming that Hilbert’s studies in foundations of mathematics at least indirectly makes him a foundationalist. However, the recent study of Franks shows persuasively that there are notably anti-foundationalist elements in Hilbert himself (see Franks 2009, pp. 32–40). The problem of the critique of a priori knowledge is an old problem that provoked in-depth discussion for a long time within the Hilbert school. It shows much more complex modes than one might think. However, if we have in mind Nelson’s criticism of Hilbert, and Bernays’ rebuttal, there is no doubt that there are almost insurmountable difficulties in contrasting Hilbert and Bernays through this problem (see Bernays 1928). The case of structuralism no doubt shows affinities not only with Bernays but also with Hilbert. In fact, many mathematical structuralists of our time trace their views back to Hilbert. Thus, here again it is difficult to highlight the differences between Hilbert and Bernays.
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The prime examples of this project are Carnap (1956, 1958/1975, 1959/2000), and they were discussed by Carnap (1966/1974) in sufficiently detailed fashion. Carnap (1961) must be extremely important for understanding the relationship between Hilbert and Carnap. Friedman (1992) views Carnap’s later project in philosophy of science was already budding in Carnap (1934, 1939). See Friedman (2008, 2011) for how later Carnap expanded the idea of Ramsey sentence by the impetus from Hempel.
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Park, W. (2018). Between Bernays and Carnap. 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_7
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