David Bourget (Western Ontario)
David Chalmers (ANU, NYU)
Rafael De Clercq
Jack Alan Reynolds
Learn more about PhilPapers
In addition to this being the centenary of Kurt Gödel’s birth, January marked 75 years since the publication (1931) of his stunning incompleteness theorems. Though widely known in one form or another by practicing mathematicians, and generally thought to say something fundamental about the limits and potentialities of mathematical knowledge, the actual importance of these results for mathematics is little understood. Nor is this an isolated example among famous results. For example, not long ago, Philip Davis wrote me about what he calls The Paradox of Irrelevance: “There are many math problems that have achieved the cachet of tremendous significance, e.g. Fermat, 4 color, Kepler’s packing, Gödel, etc. Of Fermat, I have read: ‘the most famous math problem of all time.’ Of Gödel, I have read: ‘the most mathematically significant achievement of the 20th century.’ … Yet, these problems have engaged the attention of relatively few research mathematicians—even in pure math.” What accounts for this disconnect between fame and relevance? Before going into the question for Gödel’s theorems, it should be distinguished in one respect from the other examples mentioned, which in any case form quite a mixed bag. Namely, each of the Fermat, 4 color, and Kepler’s packing problems posed a stand-out challenge following extended efforts to settle them; meeting the challenge in each case required new ideas or approaches and intense work, obviously of different degrees. By contrast, Gödel’s theorems were simply unexpected, and their proofs, though requiring novel techniques, were not difficult on the scale of things. Setting that aside, my view of Gödel’s incompleteness theorems is that their relevance to mathematical logic (and its offspring in the theory of computation) is paramount; further, their philosophical relevance is significant, but in just what way is far from settled; and finally, their mathematical relevance outside of logic is very much unsubstantiated but is the object of ongoing, tantalizing efforts..
|Keywords||No keywords specified (fix it)|
|Categories||categorize this paper)|
Setup an account with your affiliations in order to access resources via your University's proxy server
Configure custom proxy (use this if your affiliation does not provide a proxy)
|Through your library||
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
Matthias Baaz (ed.) (2011). Kurt Gödel and the Foundations of Mathematics: Horizons of Truth. Cambridge University Press.
Paolo Mancosu (1999). Between Vienna and Berlin: The Immediate Reception of Godel's Incompleteness Theorems. History and Philosophy of Logic 20 (1):33-45.
Peter Smith (2013). An Introduction to Gödel's Theorems. Cambridge University Press.
Richard Tieszen (1994). Mathematical Realism and Gödel's Incompleteness Theorems. Philosophia Mathematica 2 (3):177-201.
Roman Murawski (1997). Gödel's Incompleteness Theorems and Computer Science. Foundations of Science 2 (1):123-135.
Raymond M. Smullyan (1992). Gödel's Incompleteness Theorems. Oxford University Press.
Panu Raatikainen (2005). On the Philosophical Relevance of Gödel's Incompleteness Theorems. Revue Internationale de Philosophie 59 (4):513-534.
Added to index2009-01-28
Total downloads31 ( #63,581 of 1,413,324 )
Recent downloads (6 months)1 ( #154,079 of 1,413,324 )
How can I increase my downloads?