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Platonism and Mathematical Intuition in Kurt Gödel's Thought

Published online by Cambridge University Press:  15 January 2014

Charles Parsons
Charles Parsons*
Affiliation:
Department of Philosophy, Emerson Hall, Harvard University, Cambridge, Massachusetts 02138.E-mail: [email protected]
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Extract

The best known and most widely discussed aspect of Kurt Gödel's philosophy of mathematics is undoubtedly his robust realism or platonism about mathematical objects and mathematical knowledge. This has scandalized many philosophers but probably has done so less in recent years than earlier. Bertrand Russell's report in his autobiography of one or more encounters with Gödel is well known:

Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians might hope to meet it hereafter.

On this Gödel commented:

Concerning my “unadulterated” Platonism, it is no more unadulterated than Russell's own in 1921 when in the Introduction to Mathematical Philosophy … he said, “Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” At that time evidently Russell had met the “not” even in this world, but later on under the infuence of Wittgenstein he chose to overlook it.

One of the tasks I shall undertake here is to say something about what Gödel's platonism is and why he held it.

A feature of Gödel's view is the manner in which he connects it with a strong conception of mathematical intuition, strong in the sense that it appears to be a basic epistemological factor in knowledge of highly abstract mathematics, in particular higher set theory. Other defenders of intuition in the foundations of mathematics, such as Brouwer and the traditional intuitionists, have a much more modest conception of what mathematical intuition will accomplish.

Type
Research Article
Information
Bulletin of Symbolic Logic , Volume 1 , Issue 1 , March 1995 , pp. 44 - 74
Copyright
Copyright © Association for Symbolic Logic 1995

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References

REFERENCES

Writings of Gödel

The bibliographical references to Gödel's writings are those used by the editors in the Collected Works. The latter are cited as CW with a volume number, but Gödel's published writings are cited in original pagination (given in the margins of CWI and II), his unpublished writings by page number in CW III, if applicable, manuscript page otherwise. The following are the items cited:

Gödel, [CW I] Collected Works, Volume I: Publications 1929-1936, Feferman, Solomon (editor-in chief), Dawson, John W. Jr., Kleene, Stephen C., Moore, Gregory H., Solovay, Robert M., and van Heijenoort, Jean (editors), Oxford University Press, New York, 1986.Google Scholar
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Gödel, [1946] Remarks before the Princeton bicentennial conference on problems in mathematics, in Davis, [6], pp. 84–88, reprinted in CW II, pages 150–153.Google Scholar
Gödel, [*1946/9] Some observations about the relation of the theory of relativity and Kantian philosophy, versions B2 and C1, in CW III, pages 230-246 and 247–259.Google Scholar
Gödel, [1947] What is Cantor's continuum problem?, American Mathematical Monthly, vol. 54 (1947), pp. 515–525, reprinted in CW II, pages 176-187.Google Scholar
Gödel, [1949a] A remark about the relationship between relativity theory and idealistic philosophy, in Schilpp, [21], pp. 555–562, reprinted in CW II, pages 202–207.Google Scholar
Gödel, [*1951] Some basic theorems on the foundations of mathematics and their implications, Josiah Willard Gibbs Lecture, American Mathematical Society, 1951, in CW III, pages 304–323.Google Scholar
Gödel, [*1953/9] Is mathematics syntax of language?, versions III and V, in CW III, pages 334–356 and 356–362.Google Scholar
Gödel, [1958] Über eine bisher noch nicht benützte Erweiterung desfiniten Standpunktes, Dialectica, vol. 12 (1958), pp. 280–287, reprinted with a translation by Stefan Bauer-Mengelberg and Jean van Heijenoort in CW II, pages 240–251.Google Scholar
Gödel, [*1961/?] The modern development of the foundations of mathematics in the light of philosophy, transcription of German shorthand draft, with translation by Köhler, Eckehart and Wang, Hao, revised by Dawson, John, Parsons, Charles, and Craig, William, in CW III, pages 374–377.Google Scholar
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