Kurt Gödel

I may not be much of a mathematician, but I have a favorite mathematician: Kurt Gödel.

At the beginning of the 20th Century, Bertrand Russell started an ambitious project. They would begin from basic principles of logic (truth, falsity and sets), and gradually derive the whole of modern mathematics. The work began to be published as a series of volumes named Principia Mathematica. After 379 pages, they’d gotten as far as conclusively proving that 1+1=2, so it looked as if they were well on the way to deriving integers, fractions, and so on. It seemed plausible that all of mathematics might one day be constructed from first principles, with every field able to trace its roots all the way back to the same base axioms.

Kurt Gödel destroyed that hope. In 1931, his first incompleteness theorem proved that any system capable of expressing statements about elementary arithmetic on the natural numbers — statements like “1+1=2” — could either be consistent, or it could be complete, but it could not be both. So either the Principia series would eventually contradict itself (giving different answers to the same problem depending on the route you used to get to the answer), or it would be impossible to prove some true statements about natural numbers using the methods described in the books.

There are, of course, formal systems which do not include rigorous definitions of the natural numbers; I understand that if you use only real number arithmetic and not natural numbers, you can have completeness and consistency; you just can’t prove that 1+1=2. (This fits my personal crackpot view that real numbers are purely a mathematical concept with no real world validity, inferior to natural numbers in every way bar cardinality.) Mathematics would continue — but without the hope of a single set of volumes deriving all of mathematics from scratch.

These days there are many fields of mathematics, each with their own rules and conventions and well-known results. A mathematician working in one area may not even be able to understand a current piece of research in another area. This fragmented edifice is, I think, partly a result of Gödel’s work. And no matter how perfect a given field of mathematics may be, the enterprise as a whole is stuck with imperfection. That may not matter as far as getting the job done, but it’s philosophically important.

But mostly I love the incompleteness result not for the result itself, but because it was so subversively constructed. Gödel proved that a complete unified theory of mathematics which included natural numbers could not have a solid foundation — using the very same mathematics. The proof is simple enough for a lowly computer scientist to follow, at least in outline: it involves using numbers to encode logical statements, so that meta-statements about the system itself can be introduced into the system. Self-referentiality of the “This statement is false” kind is then used to produce an insoluble paradox. Subversion, meta-statements, paradox — what’s not to love?

(I also find the continuum hypothesis quite beguiling to ponder. Gödel apparently thought it was false, but demonstrated that you wouldn’t be able to disprove it using conventional set theory. Paul Cohen later finished the joke by proving that you can’t prove it using conventional set theory either.)

Gödel is also the person who first proved that you could have closed timelike curves in the spacetime of general relativity, raising the possibility of time travel in an Einsteinian universe — at least in “Groundhog Day” style loops.

So today, on Groundhog Day and the anniversary of the death of Bertrand Russell, let’s hear it for Kurt Gödel, wrecker of the dreams of 20th Century mathematicians.