January 27th, 2017
(written by lawrence krubner, however indented passages are often quotes). You can contact lawrence at: firstname.lastname@example.org
My strongest subject was always logic, even though I was terrible at most other forms of math. And though I have doubts about randomness, I certainly love inconsistency, which can be approached in a logical way. Every computer programmer who writes concurrent code has to deal with inconsistent data, and it would be great if logicians thought more deeply about how to formalize this.
With Gödel, at first there was a lot of shock. As a student in the late 1950s and early 1960s, I would read essays by Weyl, by John von Neumann, and by other mathematicians. Gödel really disturbed them. He took away their belief in the Platonic world of ideas, the principle that mathematical truth is black or white and provides absolute certainty.
But now, strangely enough, the mathematics community ignores Gödel incompleteness and goes on exactly as before, in what I would call a Hilbertian spirit, or following the Bourbaki tradition, the French school inspired by Hilbert. Formal axiomatic theories are still the official religion in the mathematics community.
As for the work on Ω, the logic community thinks it is crazy. They do not understand conceptual complexity, which is akin to entropy, and they have no feeling at all for randomness, which has a long history in physics, but not in logic. To them, these are very strange ideas. Logicians hate randomness.
That is precisely why they became logicians.
Gödel incompleteness is even unpopular among logicians. They are ambivalent. On the one hand, Gödel is the most famous logician ever. But, on the other hand, the incompleteness theorem says that logic is a failure. No logician would like the message from a course on logic to be that they should fire all the logic professors! Books on logic may talk about incompleteness, but they do so in the last chapter, and you never get to the last chapter when you give a course!
The ideas discussed in this essay—randomness and lack of structure—are very close to ideas from physics, such as entropy and disorder in statistical mechanics. To physicists, these ideas seem familiar and comfortable. What the work discussed here really shows is that pure mathematics may not be identical to theoretical physics, but it is not that different. Pure mathematicians like to think that they have absolute truth, and that we physicists—I consider myself an honorary physicist—use proofs that are heuristic and non-rigorous. But even in pure mathematics there are wonderful non-rigorous heuristic proofs of the kind that mathematical physicists feel comfortable with. Leonhard Euler used them all the time!12