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Wednesday, September 14, 2005

incompleteness

LI’s been reading Incompleteness, Rebecca Goldstein’s book on Kurt Goedel.

Goldstein’s book pursues an interesting philosophical argument and a feeble intellectual historical one. The latter consists of lumping together disparate currents (logical positivism, subjectivism, social constructionism, formalism) under the rubric “postmodernism, ” and then claiming that the postmodern annexation of Goedel’s incompleteness theorem is philosophically suspect. Postmodern here is a shapeshifting label lifted straight out of the Saturday arts section of the New York Times, but with little real meaning outside of being a caricature for a kind of touchy feely relativism that Goldstein evidently dislikes. Ourselves, we dislike the term, partly because it so often functions just as it functions in Goldstein’s text, as a moving target under which is gathered a diffuse sensibility.

But if it does have a distinct intellectual historical meaning, we imagine that Lyotard hit on it: postmodernity is what is entailed by the collapse of all the great metanarratives of modernity; Marxism, progress, revolution, laissez faire capitalism. In this, it is rather like the End of History and other low rent apocalypses that popped up at the end of the Cold War.

Goldstein’s feeble intellectual history argument allows her to group together logical positivism and subjectivism – whatever the latter is – as variants of the same thing. We think that this is much too gross a reading of logical positivism, and indeed of modernism itself.

The more interesting argument is Goldstein’s defense of Goedel’s own conception of what he was up to: a vindication of the Platonist view of mathematics. Goldstein is obviously more comfortable with these issues, and she does a very nice job of untangling the misconceptions around the apparent paradoxes entailed by incompleteness, showing that they are paradoxes relative to a positivist and/or formalist view of mathematics. For Goedel, and for Goldstein, Goedel’s incompleteness theorems aren’t paradoxes, but capital evidences against the formalist or positivist view of mathematics.

Goldstein begins with a nice clarification of the Platonist position. Bertrand Russell famously tweaked Goedel by writing:

“Goedel turned out to be an unadultered Platonist, and apparently believe that an eternal “not” was laid up in heaven, where virtuous logician might hope to meet it hereafter.”

Goedel was understandably peeved by Russell’s joke. As Goedel pointed out, his own position was consistent with Russell’s statement, in 1919, that “logic is concerned with the real world just as truly as zoology…” Russell’s fall into the Dunciad quicksands of positivism was due, in Goedel’s opinion, to Wittgenstein’s malign influence.

Goldstein unpacks the meaning of Platonism by way of a nice example: Goldbach’s conjecture. As she observes, this conjecture has never been proven. Goldbach’s conjecture is that all even numbers greater than two are the sum of two primes. As Goldstein astutely remarks:

“The fact that Goldbach’s conjecture remains unproven means (at least according to the Platonist) that lurking out there beyond the point where mathematicians have checked there might be a counterexample… Then again… there may not be a counter-example: every even number may be the sum of two primes, without there being a formal way to prove that this is so. A Platonist asserts that there either is or isn’t a counter-example, irrespective of our having a proof one way or another.”

Like Schroedinger’s cat, which is either alive or dead, the Platonist thinks that the structure of reality is such that nothing can be real that is not either so or not-so: either the conjecture is right or wrong. (actually, Plato recognized doxa as being half real and half not – but let’s not mess up Platonism by referring to Plato). Nothing in nature would continence it being structurally indeterminate. Now, it is easy to see how the Platonist’s claim can get a bit confusing. To return to Russell’s joke, we like to think of the real in terms of crude correspondences of object to perception. We think that the real is what we encounter, or meet. Hence the comedy of the virtuous logician meeting some cartoon “not” in logical heaven. But the Platonist contends for the existence of abstract structures that simply are not encounterable by the senses. They are, rather, encountered by the intellection – by Reason. That encounter should count as real – that is to say, the mind has a specific reality as an organ that detects the suprasensible, and the suprasensible – abstract structures – exists as what can so be detected. And just as there can be false sensibles – for instance, the flying horse – that do not overthrow the structure of the sensible itself, so, too, there can be false supersensibles – the square circle – which do not overthrow the structure of the supersensible itself. In this way, the logical is on par with the zoological.

The Goldbach example cleverly creates a sense for the direction in which Goedel was going. Goldstein’s point is to drive a wedge between Goedel’s incompleteness theorems and the formalist assumptions about mathematics. For the formalists, and most notably Hilbert, mathematics is what is generated by some given set of axioms. The formalist conjecture is that these axioms entirely determine what is true about what is in the system – so that, according to the formalist, we will eventually understand why the successor of the successor of zero has the singular property of demarcating a property change in the natural numbers such that Goldbach’s conjecture is correct. As Goldstein points out, Wittgenstein’s language games rely on a similar sense of the power of conventions to determine the truth content of discourse. To use Davidson’s notion (and to abbreviate it a bit) coherence precedes correspondence.

Wittgenstein keeps popping up in Goldstein’s account as a sort of devil’s advocate. Wittgenstein referred to Goedel’s incompleteness theorems as “logische Kunststuecken” – logical tricks. Goldstein’s sympathy with Goedel moves her to dismiss Wittgenstein’s phrase as one deriving from the panic of seeing certain of his fundamental presuppositions collapse. We aren’t sure that is entirely right. Put in terms of the formalist vs. Platonist conception of mathematics, there is something odd about Goedel’s incompleteness theorems. We will dissert on this in another post.

3 comments:

kmort said...

Russell's joke points out the issue, really: logic does make use of variables and predicates that refer to objects which can be defined. DO numbers have to exist (subsist?) in some realm of "Reals" which has no relation to the world? I have tried to crack Goedels' theorem a few times (still working on 'er) and it always seems to me that a positivist answer--more or less that self-referentiality is either meaningless or verboten--is adequate; it is difficult to imagine scenarios where Goedel's theorem does actually lead to say a real-world program breaking down. Some truths in an axiomatic system may not be confirmable, but then the question is, via CS Peirce: what does it matter? Mathematical relations were derived from experience or perception, formulated, systematized, or...what? ghost worlds

Self-referentiality is verboten; and though I think Goedels theorem is not just to be dismissed there I think arguments, both inductive and analytical, whichcan refute it or at least attenuate its effect

roger said...

Hi Kmort!
I am going to tussle with these comments more in my second post about this. Just a short comment: I agree with you that any departure from the materialist framework (which I take to be that one in which the laws of thermodynamics hold) is probably not going to work. On the other hand, I don't think it is quite right to compare ghosts to Goedel's logical or mathematical structures. I think that even if one could prove there were no God or ghosts, the status claim that there are intellectual structures in the universe would be unaffected. And conversely.

Although much of Goldstein's book would annoy you, I'd still recommend it, especially if you are working on the significance of Goedel's incompleteness theorem.

Caleb Addy said...

Are there other forums/blogs that are more specific for this topic? I have not found one.