Archive for July, 2019

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

Leibniz and Computing

If you are interested in the history of the philosophical/logical/poetical dimensions of computing, you will be interested in Leibniz, the greater contemporary of Newton who independently created calculus. Martin Davis’s astonishing history of just this dimension of the history of computing, titled Engines of Logic, provides fascinating insight into the life and work of Leibniz and its relation with the history of computing. Leibniz is often thought of as the granddaddy of computing.

You will also be interested in Godel’s work. The incompleteness theorems. For these suggested an answer in 1930 to a question posed at the turn of the century by Hilbert, a question that is important in the development of the theory of computation. And Turing used Godel’s work in his 1933 paper on the ‘decision problem’ to answer the question and, almost incidentally, introduce the notion of the Turing machine, the theoretical model of a computing device that is still used today.

I picked up an excellent book: Gottfried Wilhelm Leibniz: Philososophical Writings edited by G.H.R. Parkinson. It’s actually readable and understandable, neither of which is true of another translation I have of this work by Leibniz.

Anyway, what I wanted to point out tonight is that in the brilliant introduction by Parkinson to this volume, he says that Leibniz thought of what he called the “principle of sufficient reason” as one of his guiding lights.

“Leibniz said many times that reasoning is based on two great principles–that of identity or contradiction, and that of sufficient reason….The principle of sufficient reason says that every truth can be proved…”

Now, it is just this which Godel demonstrates is false. Godel proved the necessary existence, in sufficiently powerful formal systems, of what he called “undecidable propositions”, namely ones that are true but not provable. These are ideas that are true for no reason; they’re definitely true but definitely not provable. That makes them true for no reason.

It’s the necessary existence of these sorts of propositions that complicates the entire structure of human knowledge. Leibniz’s principle of sufficient reason is what he needs to lay the foundation for the possibility of machines that reason as far and wide as it is possible to reason, verily and forsooth into a perfectly known and understood universe.

If everything that is true is provable, and if devising proofs can be reduced to a mechanical procedure, then there is no impediment, except possibly time and an infinity of theorems, to a machine generating all the proofs of all the theorems.

Mathematics becomes, in such a world, something that we can leave to a machine.

But Godel showed that not everything that is true is provable (his example was a lot like the proposition “This proposition is not provable.”). If not everything that’s true is provable, then knowledge, perforce, must always be incomplete. Everything that’s true can’t be completely exhausted via a theorem-proving machine.

Anyway, it’s interesting that Leibniz’s “principle of sufficient reason” back in the 17th century, is so strongly related to Godel’s work.