Abstract
Despite the limitations of Frege’s and Gentzen’s analysis of deduction, throughout the twentieth century mathematical logic has been extolled, and the importance of Aristotle’s logic downplayed. For example, Russell states that mathematical logic gives thought “wings. It has, in my opinion, introduced the same kind of advance into philosophy as Galileo introduced into physics.” On the contrary, Aristotle’s “logic put thought in fetters.” Thus “any person in the present day who wishes to learn logic will be wasting his time if he reads Aristotle.”
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- 1.
Russell (1999, 68–69).
- 2.
Ibid., 68.
- 3.
Russell (2004, 194).
- 4.
Hilbert (2004b, 563).
- 5.
Hilbert (1996a, 1093).
- 6.
Hilbert (2004b, 563).
- 7.
Barnes (2003, 21).
- 8.
Frege (1979, 12).
- 9.
See, for example, Cellucci (2007, 188–191).
- 10.
Frege (1984, 242).
- 11.
Hilbert (1967, 465).
- 12.
Gentzen (1969, 74).
- 13.
‘Interpretation’ here means ‘full interpretation’, namely, interpretation where the domain of n-ary relations is set of all n-ary relations on the domain of individuals.
- 14.
For details, see, for example, Cellucci (2007, 202–208).
- 15.
Hilbert (1983, 195).
- 16.
Hilbert (1967, 477).
- 17.
For further details on Hilbert’s conservation programme see, for example, Cellucci (2007, 43–47).
- 18.
Hilbert and Bernays (1968–1970, I, 44).
- 19.
- 20.
Hilbert (1967, 472).
- 21.
For further details on Hilbert’s consistency programme see, for example, Cellucci (2007, 47–48). On the equivalence between Hilbert’s conservation programme and consistency programme, see ibid., 48–49.
- 22.
Hilbert (1967, 471).
- 23.
Hilbert (1980, 39–40).
- 24.
Spinoza (2002, 19).
- 25.
Ibid., 20.
- 26.
Hilbert (1996b, 1113).
- 27.
Graham et al. (1994, 56).
- 28.
For details see, for example, Cellucci (2007, 192).
- 29.
Hintikka (2000, 44).
- 30.
Ibid.
- 31.
Ibid.
- 32.
Hintikka (1996, 95).
- 33.
‘Logical truth’ here means ‘statement true in all full interpretations’. For this version of the strong incompleteness theorem for second-order logic, see Robbin (2006, 163).
- 34.
Boghossian (2003, 248).
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Cellucci, C. (2013). The Limitations of Mathematical Logic. In: Rethinking Logic: Logic in Relation to Mathematics, Evolution, and Method. Logic, Argumentation & Reasoning, vol 1. Springer, Dordrecht. https://doi.org/10.1007/978-94-007-6091-2_12
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