Doing Mathematics Differently

Gregory Chaitin in Inference:

Between the physics of the very small (particle or high energy physics) and the physics of the very large (cosmology) lie complex systems like us. In this essay, my subject is conceptual complexity, the complexity of ideas. The main application of these concepts is in meta-mathematics, where one deals with incompleteness and the limits of pure thought.

Mathematics contains a great deal of complexity. In this area, as in so many others, Gottfried Leibniz put his finger on the basic issue.

He did so before anyone else.

Hermann Weyl was both a mathematician and a mathematical physicist. Weyl wrote on mathematics, general relativity, and quantum mechanics, as well as on art and philosophy. His smaller book on philosophy is entitled The Open World. It is made up of lectures Weyl gave in 1932 at Yale University.² In the philosophy of science, according to Weyl, complexity is essential in understanding the concept of a law of nature. If laws of nature may be arbitrarily complex, he argued, the very concept of a law becomes vacuous. What difference would remain between complex phenomena and the laws meant to explain them if the laws meant to explain them were as complex as the phenomena they are meant to explain?

Laws of nature must be simple.

A good point.

But Leibniz made the same point three centuries before.

More here.