In addition to this being the centenary of Kurt Godel'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, four-color, Kepler's packing, Godel, etc. Of Fermat, I have read: 'the most famous math problem of all time'. Of Godel, 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 Godel'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, four-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, Godel's theorems were simply unexpected, and their proofs, though requiring novel techniques, were not difficult on the scale of things.
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