How Kurt Godel Changed Math Forever

When Godel proved his Incompleteness Theorem in 1931 he turned an entire philosophy of mathematics on its head and forever set out of reach a sort of holy grail that mathematicians had be searching for ever since Euclid's Geometry.

Mathematics, it was thought, should be based on some bedrock--some incontravertible truth. And, in time we should be able to develop a system of mathematics so perfect as to be able to prove or disprove all postulates put before it.

At the time, Russell and Whitehead, two preeminent mathematicians of the time were working on Principia Mathematica--an attempt to base all of mathematics on the "bedrock" of pure logic.





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But Godel's incompleteness theorem stopped that effort. In its tracks!

In effect what it proves (using mathematics) is this: It is impossible to have a self consistent system of mathematics capable of proving or disproving any postulate set before it.

A perfect example of this would be Euclidean and Non-Euclidean Geometry. Euclidean geometry says that the shortest distance between two points is a line. Non-Euclidean Geometry shows this rather to be a curve. Sum the interior angles of a triangle in Euclidean Geometry and you get exactly 180 degrees. In Non-Euclidean Geometry the sum may be less or more than 180 degrees.

If "There" is a unification of all systems of math, Godel was able to prove that you can't get there from here! In fact, you can't get "there" period. Ever.

No kidding! Godel and Einstein were at Princeton together. They spent time together nearly every day, and walked home together each afternoon. Godel's grasp of mathematics was far beyond Einstein's and Einstein knew it.

Godel and Einstein were close friends. Einstein valued Godel's input on his ideas.





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This I'm afraid is beyond the scope of my ability to explain, however I can refer you to A World Without Time: The Forgotten Legacy of Godel and Einstein that can do a much better job than I. I own it. It's a fascinating read.

In short, Godel was able to quantify a number of different uses and notions of time which we "think" we have some handle on. Godel was able to show that each meaning we make use of is simply an illusion.

I believe, like the author, that Godel was way before his time on this one (pun intended). We're just not quite ready yet.

Fascinating stuff and a very deep discussion. I am looking at this point for Godel's original papers on the topic. Unfortunately, Godel was German and alas I only took one semester of German in college!





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