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Return to Kurt Godel Caricature |
The Undecidable Man
Without doubt Kurt's big claim to fame were his "undecidability" theorems (there's actually two of them) about how some mathematical statements have no proof, but are nonetheless true. Of course, that's not what he proved. He showed that if you have a sufficiently complex formal mathematical system which has no contradictions you can create a formula which is true but can't be proven within the system.
Sci-fi fans, on the other hand, might prefer his equations where the universe rotates in time. What you have then is a cosmology where the future eventually returns to the past. How about that! Kurt proved time travel is possible!
Alas, others, a bit more mundane, simply point out that this "forward to the past" universe is only one possible solution to the equation, and it may have no real meaning. In fact, there is another solution to Kurt's cosmology where time proceeds forward normally, and the universe physically rotates back to it's origin. That's probably the more meaningful solution.
But it's Kurt's undecidability theorems are what really caught the public's imagination (or for the minority of the public who even knows who Kurt is). The first paper is available in a cheap English paperback translation. But it's is pretty hard going. Instead the interested might try the small book "Godel's Proof" by James Newman and Ernest Nagel (New York University Press, 2002). This goes in enough detail so you can see what Kurt did. Strangely enough, if you have a decent mathematical backgroung and start reading Kurt's paper, not at the beginning, but at the definitions, it's easy enough to see where and how he's heading.Essentially Kurt showed with symbolic logic you can create a valid expressions - called a "well-formed formula" - which can be directly related to the English sentence "This sentence is not provable". The "this" refers to the sentence itself. So the sentence asserts its on non-provability. So if the sentence is true, it's not provable and if it's not true, you can prove a false statement - which means your system is inconsistent and worthless.
The kicker, though, is you can reason - but "outside the system" - that the sentence is true. For instance, suppose the sentence is provable. But it says it isn't! Then you can prove a false statement and you have what is known as an inconsistent system. But if your system is consistent, then the statement can't be proven. But that's just what the sentence says and if the system is consistent it's true but not provable. Which is what the sentence says. Ergo, it's true.
At this point you may find yourself scratching your head and wondering if what Kurt's trying to pull a fast one. Or at least wondering if the theorem is really what it's cracked up to be. After all what good is a mathematical (or logical) expression that simply asserts it can't be proven? At this point we need to admit there seems to be a bit of a schism between philosophers and logicians with the mathematicians at this point. Although no one seems to question the technical correctness of Kurt's proof, from time to time mathematicians will point out that it really didn't change math much as people think. Many hard-nosed mathematicians proceed through their careers with scarcely a glance at what Kurt did, thank you. Certainly he did not say there are great profound truths that you can't prove and certainly not (as some people maintain) that the Bible and US Constitution fall under the balliwick of Kurt's theorem.
After Kurt, proving the non-provable became quite the thing to do. Mathematician and early computer scientist Alan Turing showed that there's no point in trying to program a computer that will automatically solve everything. It can't be done. And for the curious, there are fairly simple proofs that true mathematical statements exist but which have no proof. You can read one of the bonehead examples as part of another essay by clicking here. Alan's proof - which is reasonably easy to understand is here
Kurt, like many famous mathematicians, had a mystical streak. Using modal logic, a logic that includes the concept of necessity and possibility, he worked out an ontological proof for the existence of God. The proof is quite short and not particularly difficult to understand. But it does require some new and not quite self evident axioms and Kurt also had to invent a completely new property for God (and everyone else who had it) called positivity. Once more empiricists snort that all this is simply reverse engineering. Find the answer you want, and work backwards until you come up with a system and axioms which are not self evident (and possibly nonsensical) that give you what you're trying to prove.
Kurt was born in what is now the Czech Republic and grew up in Vienna. Finding himself a citizen of a Nazi regime after the Anschluss, he left Austria and became an American citizen. When he was studying for his citizenship exam, he was aghast when his quick grasp of logic allowed him to deduced that the Constitution would permit American democracy to be legally and easily transformed into totalitarianism. That caused him a bit of trouble when the examining judge asked him why he had wished to become an American citizen. Kurt said he wanted to live in a free society, and his own country had been transformed into a dictatorship. "Which of course," the judge said, "can't happen here." Kurt leapt up and cried, "Oh, but you're wrong!" The judge, though, was not perturbed and Kurt got his citizenship.
What did Kurt mean? No one knows exactly, but since the US Constitution (with some exceptions as stated in Article V) puts no limits on the type or number of Amendments, it is completely possible to alter it so we could have a supreme monarch, no freedom of speech, and continual surveillance of our private lives. Plus anything else you want, really. So Kurt's conclusion was a simple matter of logic. You just wonder why the Judge thought otherwise.
Although a logician of first order, toward the end of his life Kurt somehow deduced someone was poisoning his food. He refused to eat and in 1978 died of starvation.
References
On Formally Undecidable Propositions of Principia Mathematica and Related Systems, Kurt Godel, Dover (1992). An English translation in a paperback. But be warned! This is tough going!
"On Formally Undecidable Propositions of Principia Mathematica and Related Systems" is also online. Since Kurt's symbolism is now somewhat out of date, you can try the annotated version of his paper at http://www.research.ibm.com/people/ h/hirzel/papers/canon00-goedel.pdf. This helps explain things when Kurt splashes up a set of symbols and says what you have is a recursive formula ".. as the reader may readily see". But the paper is still tough going.
Godel's Proof, Ernest Nagel, James Newman (Forward by Douglas Hofstadter), New York University Press (2002). A simplified treatment, but still goes through the steps of what Kurt did. This might be a bit much for the completely mathematically challenged, but the average American who knows how to operate a TV remote should be able to plow through it.
Who Got Einstein's Office? Eccentricity and Genius at the Institute for Advanced Study, Edward Regis, Perseus Books (1987). The first chapter is about Kurt and arguably is the best chapter of the book.
Godel's Theorem - An Incomplete Guide to Its Use and Abuse, Torkel Franzén, A K Peters, (2005). Very nice and readable book making sense of and also showing the nonsense in what people say about Godel's theorem. It reminds people that Godel's theorems are about formal mathematical systems not the U. S. Constitution, not the Bible.
"Does Godel's Theorem Matter to Mathematicians", Gina Kolata, Science Vol. 218, pp. 779-780 (1982). A problem with Godel's theorem is the statement he created seems kind of a logicians trick and isn't a "real" mathematical statement. This paper shows some examples that are real but for one reason or another can't be proven - again it must be stressed - within their system.
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