Re-Writing Descartes,
By Re-doing Mathematics
(and Relationships. and Self-Revision)...

Story Evolution

A reflection triggered by Grobstein's Writing Descartes ...


9 July 2004

With gratitude to the middleman/conversation starter....

René! WAKE UP!

(Sorry to shout, but THIS one's IMPORTANT....):

there's some new information "just" come down the pike--well, okay: I just picked it up in Jim Marshall's presentation in the Information Group yesterday morning, but I understand it's been circulating for @ least 40 years...

information you didn't have when you set out to find the un-doubtable, and which
--having it--you might never have set out to find.

During your morning meditations in bed, when (according to your biographer, Rob Wozniak), you were

struck by the sharp contrast between the certainty of mathematics and the controversial nature of philosophy, and came to believe that the sciences could be made to yield results as certain as those of mathematics ,

here's what you didn't know:

that in the 1960's, an American mathematician named Gregory Chaitin would begin working on a proof that the very structure of arithmetic is random. Sure, lots of equations (even some *diophantine* equations) are solveable, but their solutions may not generalizable; there may be no rule, no abstraction, to be drawn from the solution. What this means is that there are some patterns with no (further) explanation; what they are is simply what they are. To mathematicians obsessed with finding the explanations (for instance) for the properties of certain elusive functions, Chaitin says, "What we know now is good enough." I'm quite amused by the thought that, in doing so, he is following the pragmatists, saying that we cannot know everything, but we do "know enough"--and so calling a halt to a particular line of inquiry.

Now, Chaitin doesn't at all "give up" on math: he counsels instead against the (futile) search for certain foundations, and advises more work in experimental arithmetic. What he ended up with has been called the "ultimate in undecidability theorems": a claim that there are forms of mathematical information that are "not compressible"(i.e., that "there's nothing smaller than the information itself"). His resulting argument to "give up on completeness," to acknowledge that there are numbers, and number sequences, that are unknowable in terms of other things (=non-compressible), that some digits "stand on their own, independent of any others" (and so their appearance in a sequence is unpredictable, i.e. random) places uncertainty at the heart of mathematics...

Now (full disclosure) I checked out of doing math with h.s. Algebra II...but last fall, I was one of the organizers of a series of interdisciplinary conversations at Bryn Mawr exploring What Counts? Measuring Ourselves and the World. The punch line of that series, as I heard it enumerated repeatedly (though surely others were picking up other patterns) was that human beings have a long history, going back @ least to the Pythagoreans, of using numbers as a way of seeking fixed invariants in a universe that is full of change. In that series, I heard us repeatedly opposing the 'approximation of value' that numbers represent against a qualitative mode of working that attends more fully to the complexities of whatever (say: student performance) is being measured. But we didn't talk then about Chaitin's work--and learning about it for this first time this morning, I realized that the opposition (I heard us) set up in our series was another of those false binaries I've talked about elsewhere: the relationships between numbers no more form a formal axiomatic system than does...

The Human Brain.
The Universe.

In the "What Counts?" series, I heard us calling attention to the formal limits of number systems, but what Chaitin (just belatedly) taught me is that "foundationally" mathematics is NOT so limited, not so closed, not so "certain." And I'm (belatedly) now realizing that my childhood boredom (fear?) of mathematics was entirely displaced...it's just as uncertain, unpredictable (and thus: so surprising. and so interesting) as the rest of life. And I need to learn more about it...because it does have the ability to generate...

in the absence of certainty,

something new.

This has, for me, reverberations far beyond arithmetic.

For instance, the observation that the tyranny of the interpersonal is every bit as much of a danger as that of the intrapersonal...could now, in the language of Chaitin, be re-written as

"the danger is making a formal axiomatic system (=closed, w/ rules one can then predictably rely on) out of either"

...actually? out of anything, as per above:

the universe.
the self.

The July 9, 2004 Chronicle has a piece written by a college liberal arts prof about her current training in a psychoanalytic center. She concludes: "Freud famously referred to pyschoanalysis as 'the talking cure,' a description that my training institute would happily endorse. According to [the scholar-analyst] Adam Phillips, however, 'psychoanalysis...doesn't cure people so much as show them what it is about thesmelves that is incurable.'"

Seems to me a good/sharp/shorthand for what Anneliese mentioned: In my experience, the ability to think does not always suffice to change myself....insight (alone?) is generally acknowledged [in psychology] as rather limited, in therapeutic terms.

Reconception doesn't easily translate into revision.

So, René...is the earth (back/down there--where ARE you, anyhow?) shaking yet?

So...how far are we getting (and how do we measure the distance?) in revising you?

See the on-line forum for continuing conversation and to leave your own thoughts.

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