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Saturday, September 17, 2005

continuation on Goldstein

In my last post on Goldstein’s book, Incompleteness, I said that Goldstein tried to present Goedel’s two theorems from the perspective of Goedel’s own sense of what these theorems ultimately proved. That is, what they proved meta-mathematically. Goldstein is interested in the divergence between the popular image of what Goedel was up to and the fact that Goedle’s incompleteness proof, by insinuating a moment of absolute uncertainty into any formalization of arithmetic, seems to point to the insufficiency of conventionalism, not to affirm it.

The uncertainty, you will remember, goes like this: in any formally consistent language adequate for number theory, a., there will be one proposition that one can generate from the axioms of the system the truth or falsity of which can’t be decided by those rules, and b., that the consistency of the system can’t be proved within the system.

Now, Goldstein’s major point is to show why Goedel might take his theorems as evidence that there are real ideal objects. In other words, that at least one proposition in number theory must be either true or false without the system being able to determine its truth or falsity with its own resources begs the question of what the truthmaker, here, is.

However, Goldstein subverts her point a bit by admitting that Goedel’s view of the meaning of his work was conflicted. In public, he liked to claim that the theorems pointed to the reality of mathematical ideal objects, insofar as we associate reality with what makes a proposition true. But in private, Goedel was less certain. Here is what Goedel said to his student, Hao Wang:

“Either the human mind surpasses all machines (to be precise it can decide more number theoretical questions than any machine) or else there exist number theoretical questions undecidable for the human mind.”

Goldstein asks herself what the second part of this disjunct means, and gives us a … well, a postmodern answer. That is, one that refers the conceptual question to the personality quirks of its inventor.

“I think that what he is considering here is the possibility that we are indeed machines – that is, that all of our thinking is mechanical, determined by hard-wired rules – but that we are under the delusion that we have access to unformalizable mathematical truth.”

As she says a few paragraphs later:
“This possibility – its being precisely the possibility that gave Goedel pause – is particularly interesting when we consider an aspect of Goedel’s opaque inner life that we have touched upon before: his own serious delusions.”

Well, I want to tickle Goldstein a bit here, but I’m not really interested in pursuing the path of delusion. Rather, I want to pursue the path of the excluded middle, which is of course the framing assumption here. There is, I believe, a term of art in Zen, “mu”. “Mu” is neither yes nor no. It is, in a sense, the bifurcating moment itself, Deleuze’s “inclusive disjunct.” Perhaps all Goedel’s theorems are about is that we don’t have a formal grasp of the logic of “mu”.

Unfortunately, this post has sailed away from the point I originally started out to make. That is, I wanted to point out one peculiarity, from the formalist p.o.v., about Goedel’s proof – and that is that it depends on the possibility of constructing Goedel numbers, which is, in turn, the most extreme expression of formalism, and the most resolutely anti-Platonist “moment” in a theorem that Goedel thinks shows us the Platonist structure of number theoretical truths. Should I go into this? Hmm, perhaps not.

Two essays you might want to check out on the Web. Paul Bernays essay on Platonism in Mathematics is here. Putnam’s defense of Wittgenstein’s comments about Goedel are here.

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