Tuesday, December 12, 2006

Gödel’s Platonism

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.


Anonymous said...

Hi, I came across your blog and post on Goedel today as I have an Google alert on Relativism, my own personal pet project (against it, that is: http://millennium-notes.blogspot.com/). I first came across Goedel when I read Science before Science by Anthony Rizzi. Great stuff. I am an admiror of philosophy, but I really don't want to get into it too much because of all the crackpot post-modernism around (before you know it you get actually to buy that stuff!). I like your blog a lot and made a link to my own. Should you have objections, I'm looking forward hearing from you. If you can reciprocate, so much the better. All the best! Cassandra.

Steve said...

Thank you for your nice comment. I look forward to looking at your blog also. - Steve

Vincenzo said...

I have read that Godel Not only is a Platonist But belive even in GOd , in a personal GOD! He said that his idea for God is not meataphorical and "sive nature"(like spinoza and einstein)But he belived in a personal God in the sign of leibniz. as a matter of fact he deeply focus on leibniz works.

Steve said...

Right, that's interesting. I believe he admired Leibniz very much and developed an ontological argument for the existence of God in the tradition of Leibniz. A personal God doesn't follow from Platonism, but clearly Godel's views on God went beyond the Platonism.

Anonymous said...


I wonder if the ``G\"{o}del's ontological argument'' is a formalized version of an existing argument versus a full blown proof of the existence of God.

Steve said...

I'm not sure of the answer although I think he thought his version was an improvement over other arguments.

Marcus said...

Hello Steve,

I have long been interested in Gödel's mathematical Platonism. As I understand it, from reading his papers sometime ago as an undergrad, he reasons for going for mathematical Platonism was because he saw an analogy between how sensory perceptions and mathematical intuition. Sensory intuition leads to glimpses of physical objects, while likewise mathematical intuition leads to glimpses of mathimatical objects (i.e. natural numbers). Nevertheless, intuitions and sense perceptions need to be corrected by theories. Physics corrects sensory perception and mathematics corrects mathematical intuitions.

What do you think are some of the advantages and disadvantages of Göodel's account?

Steve said...

(With the caveat that I don’t much about Gödel’s thinking.)

I would think this is the advantage: if we could agree that experience endows humans with limited intuitive glimpses into the abstract realm, then Gödel had a valuable insight that building mathematical knowledge is a process whereby humans, alone and in collaboration, make converging progress toward truth. As you point out, this us analogous to the how science proceeds from the raw material of empirical observation.

The only disadvantage I see is the one shared generically by rationalists and Platonists, which is that they are seen by others as lacking a good account of how we have any contact at all with the abstract realm.

(I think there can be an answer to this question which makes use of the fact that all events in the natural world are actualizations of possibilities, and therefore by participating in the world’s events we are also in some kind of contact with an abstract realm of possibilities. This idea, which seems consistent with quantum mechanics, would be the basis of our rational intuition. The next step would be to assert that mathematical truths are ones which we determine to be true in all possible worlds, utilizing the insights from this ongoing acquaintance with abstract possibilia.)

LF said...

We know only the abstract possibilia within abstract systems, not actual things/events. It appears that almost all animals reptiles birds insects etc have or at some point in their existences have 2 eyes a nose and a mouth- usually in a head type area. All events in the natural world may be actualizations of possibilities and like Darwin, one finds ones self down the road to a priori land. No "all possible worlds" are possible.