George Szell, speaking about Glenn Gould once said, "that nut is a genius!" This was, if anything, even more true of Kurt Godel. How genius? When Einstein was elderly and working at the Institute for Advanced Studies, he said that the only reason he went to the office was "to have the privilege of walking home with Godel". Experts say that the two greatest logicians of all time were Aristotle and Godel, in reverse order. How nutty? When the Nazis took over in Germany, Godel went back to Europe from America in order to protect his `rights' as a privatdozent (roughly equivalent to adjunct professor) So nutty that, when he was dying, he convinced the hospital not to treat him for a condition his insurance did not cover. He was paranoid, convinced that people were conspiring to poison him. After his wife died, he essentially starved himself to death. He also showed something about math that shook it to its foundations.
More below the fold
If any branch of knowledge was thought to be certain, it was mathematics. In 300 BC, Euclid used Aristotelian logic to deduce the principles of geometry and other branches of math in what became the longest lasing textbook of all time: The Elements. 2100 years later, it turned out that geometry wasn't so certain, and mathematicians tried to do the same thing for arithmetic. Godel showed that this was impossible. He showed that there will always be true arithmetical statements that cannot be proven.
Aristotle codified the rules of logic by declaring certain patterns of argument to be syllogisms. A syllogism consists of two premises and a conclusion. For example:
All men are mortal.
Plf515 is a man.
therefore plf515 is mortal.
If you accept the premises, you have to accept the conclusion. The form of argument is valid, even if it is not correct. For example
All dogs weigh more than 200 pounds
Brutus is a dog
Therefore, Brutus weighs more than 200 pounds
The argument is valid, even though the premises (at least one of them) are false.
Euclid took some common notions and some postulates and derived much of geometry, and a good bit of other math, as well. This was thought not only to be valid, but true. That is, it was thought that his geometry was a description of the real world, and that no other geometry was even possible. This turned out not to be correct (although it took a long time for people to see that). What Euclid did was create an axiomatic, or formal, system. Two goals of such a system are consistency and completeness.
Completeness means that every meaningful statement in the system should be provable as either true or false. That is, if you come up with a formal system for arithmetic, then every statement about arithmetic should be capable of being proven or disproven. There should be no theorems that cannot be decided.
In mathematics, the word `consistency' essentially means `cannot yield contradictions'. When a set of rules applies to the real world, it must be consistent, because if one thing is true, the opposite cannot also be true. If my apartment has 2 bathrooms, it cannot have 3. When Euclidean geometry was thought to describe the real world, no one needed to worry about consistency. If a triangle's angles add up to 180 degrees, they cannot add up to more, or less.
But when people started to question whether geometry did, in fact, apply to the real world, consistency became a problem. A mathematical system which is inconsistent is useless. When geometry started to be questioned, mathematicians tried to make a formal system for arithmetic. Peano came up with a set of 5 axioms that he thought would be sufficient to derive arithmetic.
1. 1 is a natural number
2. Every natural number has a successor
3. 1 is not the successor of any natural number
4. No two natural numbers have the same successor.
5. If a property is true of 1, and holds for the successor of every natural number for which it holds, then it holds for all natural numbers. (This is the principle of mathematical induction).
Gottlob Frege built on this, and attempted to reduce all of arithmetic to logic. But, just as Frege's masterwork was going to press, Bertrand Russell found a problem in it, a paradox. His paradox is based on an ancient one, often called Epamenides Paradox. It exists in many forms, but the clearest may be the statement:
This statement is false
If this statement is true, then it is false. But if it is false, then it is true. Russell's paradox uses a similar theme, but full consideration of it would take us too far afield. You can read more about it here Wikipedia entry
In attempt to get rid of these paradoxes, Russell and Whitehead produced the monumental Principia Mathematica; three very long very dense volumes. Then, in 1930, Godel showed that, not only was Principia Mathematica flawed, but that the whole idea behind it was impossible. He showed no formal system that is powerful enough to derive arithmetic can be both complete and consistent. The details of how he did this are formidably complex, and well beyond my ability to understand, much less explain. But here is a very brief outline.
First, he figured out a way to turn every statement ABOUT numbers into a number.
Second, he used those numbers to make every syllogism about numbers into a number.
Third, he discovered an arithmetic relationship between the numbers for the premises and the numbers for the conclusion that held only for true arguments.
Finally, he took a version of this statement
This statement cannot be proven
and showed that it was true.
Sources: Most particularly, a chapter in C. C. Clawson's Mathematical Mysteries.
Also: Rebecca Goldstein's wonderful book: Incompleteness.
And, of course, I got hooked on this by Hofstadter's Godel Escher Bach (Escher is one of my favorite painters, and Bach is my favorite composer).
