Abstract
This note is a brief comment on Feferman’s Gödel, Nagel, Minds, and Machines. It emphasizes the need to expand proof theory and use its formal tools for the analysis of the informal proofs of mathematical practice. Natural formalization is seen as one important step toward providing what Feferman called for, namely, “an informative, systematic account at a theoretical level of how the mathematical mind works that squares with experience”.
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Notes
- 1.
Feferman’s remarks are based on the correspondence between Gödel and Nagel – beginning with a letter from Gödel on 25 February 1957 and ending with a short note again from Gödel on 29 August 1957 – and the Introductory Note to that correspondence by Parsons and Sieg; all of this can be found in volume V of Gödel’s Collected Works, pp.135–154.
- 2.
In a quite different way, Gödel and Turing tried to straddle this divide as far as the extending development of mathematics is concerned; that is discussed in my (2013).
- 3.
The beginnings of such investigations go back to the 1970s when human- and machine-oriented approaches to theorem proving were contrasted. More recently, Gowers and collaborators have pursued automatic theorem proving in a radically human-oriented way; see (Gowers [3]). In [8] I have discussed my own perspective on the automated, but heuristically informed search for conceptually structured proofs.
References
Gentzen, G.: Die Widerspruchsfreiheit der reinen Zahlentheorie. Mathematische Annalen 112, pp. 493–565 (1936)
Gödel, K.: Some basic theorems on the foundations of mathematics and their implications; delivered as the 25\(^{th}\) Gibbs Lecture on 26 December 1951; published in volume III of Gödel’s Collected Works, pp. 304–323. Oxford University Press (1995)
Gowers, T.: Interview. Not. Am. Math. Soc. 63(9), pp. 1026–1028 (2016)
Lukas, J.R.: 1959 Minds, machines and Gödel; given in 1959 to the Oxford Philosophical Society; published in Philosophy XXXVI, pp. 112–127 (1961)
Nagel, E., Newman, J.R.: Gödel’s Proof. Scientific American CXCIV, pp. 71–86 (1956)
Nagel, E., Newman, J.R.: Gödel’s Proof. New York University Press, New York (1958)
Penrose, R.: The Emperor’s New Mind: Concerning Computers, Minds, and the Laws of Physics. Oxford University Press, Oxford (1989)
Sieg, W.: Searching for proofs (and uncovering capacities of the mathematical mind). In: Feferman, S., Sieg, W. (eds.) Proofs, Categories and Computations – Essays in Honor of Grigori Mints, pp. 189–215. College publication (2010)
Sieg, W.: Gödel’s philosophical challenge (to Turing). In: Copeland, B.J., Posy, C.J., Shagrir, O. (eds.) Computability – Turing, Gödel, Church, and Beyond, pp. 183–202. MIT Press (2013)
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Sieg, W. (2017). A Brief Note on Gödel, Nagel, Minds, and Machines. In: Jäger, G., Sieg, W. (eds) Feferman on Foundations. Outstanding Contributions to Logic, vol 13. Springer, Cham. https://doi.org/10.1007/978-3-319-63334-3_18
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