Abstract
Aufbau §176 “demonstrating” the non-constructability of the real (as a mind-transcendent) concept had §17 of Weyl’s 1926 book, Philosophie der Mathematik und Naturwissenschaften squarely in its sights. Weyl had argued that postulation of a real, external world is both necessary for natural science and that such an objective world can be constructed, but only in abstract mathematical symbols far removed (“distilled”) from immediately given content. This objective world is a “symbolic construction of exactly the same kind as that which Hilbert carries through in mathematics”. For Hilbert and Weyl, symbolic construction is the twentieth century manifestation of Kant’s regulative idea of unity of nature.
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Notes
- 1.
E.g., Carnap- -> (1937, 46, 148, 305) and Carnap (1939a, 184). Weyl- -> is not mentioned at all in Carnap’s encyclopedia monograph (1939b) on foundations logic and mathematics. The sole mention of Weyl in Carnap’s “Intellectual Autobiography” is a reference to the intuitionism of Brouwer- -> and Weyl (1963, 48).
- 2.
- 3.
Atiyah- -> (2002, 3).
- 4.
As pointed out by Michael Friedman- -> in his annotations to the translation - ->of Carnap (1922, 189–190).
- 5.
For details, - ->see - -> Brading and Ryckman (2008).
- 6.
Stein (1994, 637). - ->
- 7.
For details, - -> see Ryckman (2005, 83–84).
- 8.
Cf. Beller- -> (1999, particularly Ch. 9).
- 9.
Stadler- -> and Uebel- -> (eds.) (2012, 93).
- 10.
Russell- -> (1914, 110–111): “Now verifiability is by no means the same thing as truth; it is, in fact, something far more subjective and psychological. For a proposition to be verifiable, it is not enough that it should be true, but it must also be such as we can discover to be true. Thus verifiability depends upon our capacity for acquiring knowledge, and not only upon the objective truth.”
- 11.
Weyl (1946, 275).
- 12.
Sieg (1999) pinpoints this paper as the first published presentation of Hilbert- ->’s finitist program of metamathematics.
- 13.
“Thus all human cognition begins with intuitions, goes from there to concepts, and ends with ideas.”
- 14.
Hilbert (1922, - -> Ewald- -> translation, 1121).
- 15.
Weyl (1929, 157).
- 16.
Weyl (1925, - ->GA II, 540): “Without doubt, if mathematics is to remain a serious cultural concern, some sense must be connected to the Hilbertian formulae game. And I see only one possibility of attaching an independent intellectual significance (geistige Bedeutung) to it, including its transfinite components. In theoretical physics we have before us the great example of cognition of an entirely different imprint than the common intuitive or phenomenal knowledge that expresses purely what is given in intuition. While here every judgment has in intuition its own completely realizable meaning, this is by no means the case with the single propositions of theoretical physics. Rather there it stands that when a proposition is to be confronted with experience, only the system as a whole comes into question.”
- 17.
Weyl (1928, - -> Bauer-Mengelberg- -> and D. Føllesdal- -> translation, 484).
- 18.
Weyl- -> (1925, GA II, 542).
- 19.
Thanks to the conference participants for comments and to an anonymous reviewer for helpful suggestions; special thanks to Christian Damböck for organizing the Munich conference and to Friedrich Stadler- -> for institutional support.
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Ryckman, T. (2016). What Carnap Might Have Learned from Weyl. In: Damböck, C. (eds) Influences on the Aufbau. Vienna Circle Institute Yearbook, vol 18. Springer, Cham. https://doi.org/10.1007/978-3-319-21876-2_2
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