I got around to reading Rebecca Goldstein’s brief and engaging book on Gödel -- Incompleteness: The Proof and Paradox of Kurt Gödel. The book centered on the irony that Gödel’s own philosophical interpretation of his work (which indeed may have driven his efforts to begin with) was in complete opposition to how it was most commonly interpreted by others.
Gödel was a Platonist, believing that the mind was able to make contact with absolute mathematical reality. Given that he was an attending member of the Vienna circle in the 1920’s, which was the locus of logical positivism, many assumed he was of like mind, believing there was no truth beyond what man could empirically discover. Gödel’s extreme reluctance to speak or write on his views helped make this misunderstanding possible. Indeed, the incompleteness theorems have often been co-opted by sloppy post-modernists (along with relativity theory and the uncertainty principle) in making the case for truth relativism. They would focus on the conclusion that we can’t construct formal systems (large enough to at least encompass arithmetic) that are both complete and provably consistent and treat this fact as revealing a limitation in our ability to reach absolute truth. Gödel believed the actual lesson was that the human mind can and does perceive truth beyond the capability of formal systems (equivalently, algorithmic computing machines).
[UPDATE 12 January 2012: For a review quite critical of Goldstein see Solomon Feferman's here. He says she has no basis for taking Godel's later platonism and suggesting it motivated his earlier seminal work.]
To digress a moment, I just about forgot I have an old post on Gödel. This blog is ably serving one of its functions -- an external memory module. In that post I noted the consensus of experts that while the incompleteness theorems may point toward philosophical conclusions (such as thinking the mind surpasses a computer), they don’t provide any proofs after you depart their formal setting. However, one philosophical stance I said they did appear to support was the notion of an ultimate limit on “objective” knowledge. (Note I also maintained that such an observation need not lead to thorough-going relativism.) Now, in reading more about Gödel’s own views, I’m not feeling confidant that assertion quite captures things.
In one of those happy coincidences, the recent update to the Online Papers in Philosophy blog maintained by Jonathan Ichikawa [UPDATE: this blog not longer exists] had a paper by eminent mathematical logician Solomon Feferman which examined one of Gödel’s rare talks (the 1951 “Gibbs” lecture). In the talk, Gödel presented the philosophical implications in terms of a disjunction thus: “Either…the human mind…infinitely surpasses the powers of any finite machine, or else there exist absolutely unsolvable Diophantine problems.” By the most reliable accounts, Gödel did indeed believe that the mind surpassed finite systems and therefore there were no (ideally) unsolvable problems, but he expressed a bit of caution in this talk by presenting the disjunction.
Feferman, in keeping with the usual deflationary mode of papers by experts on this topic, respectfully shows how the imprecision and complexity of these issues prevent one from reaching a logical proof of Gödel’s claim (or even ruling out that the disjuncts could both be true). Once again, the broad statements about the mind and its capabilities can’t be derived from the mathematical arguments which inspire them. Still, many interesting facets of these issues are illuminated in the discussion.
The Feferman paper included one (unpublished) Gödel quote which struck me as very insightful. Gödel (responding to something Turing had written) says: …”mind, in its use, is not static, but constantly developing, i.e., we understand abstract terms more and more precisely as we go on using them…though at each stage the number and precision of the abstract terms at our disposal may be finite, both…may converge toward infinity…” We are clearly finite and contingent creatures, and so the idea that we are in direct contact with Platonic truths is hard to support; but we do appear to have a gift of rationality which allows us to converge toward absolute truths. I see a connection here to the idea of modal rationalism, which I’ll explore in a future post.
One more note: that OPP update had another paper which touches on this topic, Philip Ebert’s “What Mathematical Knowledge could not be”. This is a nice survey of positions on the reality of mathematical objects; it doesn’t itself advance an argument.