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theorems that he produced extend our knowledge in profound and important
ways. His philosophical views on mathematical realism and the nature of
our faculties of intuition seem to me to be comparatively thin. To be sure,
one can imagine that had his health been better and his life been longer,
he might have produced more striking and compelling theorems to fill out
the informal views. But this is exactly my point: absent the mathematical
analysis, it is hard to say what these views amount to.
But I have come to realize that this way of separating Gödel s mathe-
matical work from his philosophical views is misleading. For, what is most
striking about Gödel s mathematical work is the extent to which it is firmly
rooted in philosophical inquiry. We never find Gödel making up mathe-
matical puzzles just for the sake of solving them, or developing a body of
mathematical techniques just for the sake of doing so. Rather, he viewed
mathematical logic as a sustained reflection on the nature of mathematical
knowledge, providing a powerful means of addressing core epistemological
issues. Gödel kept his focus on fundamental questions, and had the remark-
able ability to to advance our philosophical understanding with concrete and
deeply satisfying answers.
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We must know, we will know. A four-minute excerpt from the speech was later
broadcast by radio. The text of the excerpt and a translation by James T. Smith can be
found online, together with a link to an audio recording of the broadcast:
http://math.sfsu.edu/smith/Documents/HilbertRadio/HilbertRadio.pdf
The final pronouncement is also Hilbert s epitaph; see [16].
16
When one considers the history of science and philosophy in broad terms,
it becomes clear that the sharp separation between the two disciplines that
we see today is a recent development, and an unfortunate one. In contrast,
Gödel saw mathematics and philosophy as partners, rather than opponents,
working together in the pursuit of knowledge. This conception of logic, I be-
lieve, is Gödel s most important legacy to the metamathematical tradition,
and one we should be thankful for.
References
[1] Mark van Atten. On Gödel s awareness of Skolem s Helsinki lecture.
History and Philosophy of Logic, 26:321 326, 2005.
[2] Mark van Atten and Juliette Kennedy. On the Philosophical Develop-
ment of Kurt Gödel. Bulletin of Symbolic Logic, 9:425 476, 2003.
[3] Jeremy Avigad and Erich H. Reck. Clarifying the nature of the infi-
nite : the development of metamathematics and proof theory. Techni-
cal Report CMU-PHIL-120, Carnegie Mellon University, 2001.
[4] Steve Awodey and André W. Carus. How Carnap could have replied
to Gödel. In Steve Awodey and Carsten Klein, editors, Carnap brought
home: The view from Jena, Open Court, Chicago, 2004, pages 203 223.
[5] Steve Awodey and Erich H. Reck. Completeness and categoricity, part
I: 19th century axiomatics to 20th century metalogic. History and Phi-
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[6] L. E. J. Brouwer. Intuitionism and formalism, 1912. An English transla-
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inger, ed., Classics of Mathematics, Prentice Hall, Englewood, 1982,
pages 734 740.
[7] John W. Dawson, Jr. Logical dilemmas: The life and work of Kurt
Gödel. A K Peters Ltd., Wellesley, MA, 2005.
[8] William Ewald, editor. From Kant to Hilbert: A Source Book in the
Foundations of Mathematics. Clarendon Press, Oxford, 1996. Volumes
1 and 2.
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[9] Solomon Feferman. Kurt Gödel: Conviction and caution. Philosophia
Naturalis, 21: 546 562, 1984. Reprinted in Solomon Feferman, In the
Light of Logic, Oxford University Press, New York, 1998, pages 150
164.
[10] Michael Friedman. A parting of the ways: Carnap, Cassirer, and Hei-
degger. Open Court, Chicago, 2000.
[11] Kurt Gödel. Collected works. Oxford University Press, New York,
1986 2003. Edited by Solomon Feferman et al. Volumes I V.
[12] Jean van Heijenoort. From Frege to Gödel: A sourcebook in mathemat-
ical logic, 1879-1931. Harvard University Press, 1967.
[13] David Hilbert. Probleme der Grundlegung der Mathematik. Mathema-
tische Annalen, 102:1 9, 1929.
[14] Paolo Mancosu, editor. From Brouwer to Hilbert: The debate on the
foundations of mathematics in the 1920 s. Oxford University Press,
Oxford, 1998.
[15] Frank Plumpton Ramsey. Mathematical logic. Mathematical Gazette,
13:185 194, 1926. Reprinted in F. P. Ramsey, Philosophical papers,
Cambridge University Press, Cambridge, 1990, pages 225-244.
[16] Constance Reid. Hilbert. Springer, Berlin, 1970.
[17] Wilfried Sieg. Hilbert s programs: 1917 1922. Bulletin of Symbolic
Logic, 5:1 44, 1999.
[18] Hao Wang. From mathematics to philosophy. Routledge & Kegan Paul,
London, 1974.
[19] Hao Wang. A logical journey: From Gödel to philosophy. MIT Press,
Cambridge, MA, 1996.
[20] Hermann Weyl. David Hilbert and his mathematical work. Bulletin of
the American Mathematical Society, pages 612 654, 1944.
[21] Richard Zach. Hilbert s program. In the Stanford encyclopedia of phi-
losophy, http://plato.stanford.edu/entries/hilbert-program/.
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