Godel and Philosophy

It’s inspiring when math/logic leaps from the empyrean to the inner life. We see such a leap in Godel’s work: he actually showed that there exist truths that are not provable, truths that are true for no reason, thereby bringing the quotidian in more fruitful relation with the empyrean.

That there are truths which are not provable is something that we had intuited and even acknowledged for a very long time. Some things are true but not simply hard to prove–they’re impossible to prove. Yet definitely true. Courts of law acknowledge that proof, in matters of law, must extend only to the elimination of reasonable doubt. Not to fully demonstrable absolute proof. Because it can be and often is impossible to prove beyond all doubt what is true.

Previous to Godel’s work, meta-mathematics, ie, reasoning about mathematics and logic, had had some dramatic results. Such as the revelation that relatively consistent non-Euclidean geometries were possible. This came as a bolt of lightning not only to mathematics but to philosophy. Because even the great Kant–as well as many another philosopher–had provided, when pressed for an example of the existence of a-priori truth, the parallel postulate or the logically equivalent notion that the sum of the angles of a triangle is 180 degrees. But neither of these ideas is true in non-Euclidean geometries. In spherical geometry, for example, there are no parallel lines, and the sum of the angles of a triangle can be as large as 270 degrees. And this rocked the idea that a-priori truth exists at all, cuz the prime exemplars had been axioms of Euclidean geometry.

Godel’s work in meta-mathematics–which is now simply called logic–was at least as lightning-boltish as non-Euclidean geometry had been in the 1800’s.

When I studied math as an undergraduate, the most beautiful, profound work I studied was that of Georg Cantor concerning set theory and, in particular, infinite sets. Cantor actually proved things worth knowing about the infinite. And he did so in some of the most beautiful proofs you will ever encounter. Stunning work. He developed something called “the diagonal argument”. I won’t go into it, but it’s really killer-diller–and Godel uses that argument in his own proofs! He draws on this profound work by Cantor in his own incredible proof.

And, in turn, in Turing’s paper that layed the groundwork for the computer age, he also uses Cantor’s diagonal argument–and acknowledges Godel’s work as having helped him on his road.

You see, the poetics of computer art has this rich philosophical, mathematical history among its parents. It has this in its genes. It’s important to computer art for very many reasons. But one of those reasons is to understand where it comes from.

“Oh! Blessed rage for order, pale Ramon,
The maker’s rage to order words of the sea,
Words of the fragrant portals, dimly-starred,
And of ourselves and of our origins,
In ghostlier demarcations, keener sounds.”
from The Idea of Order at Key West, Wallace Stevens

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