{access view=guest}**Access to the full article is free, but requires you to register. Registration is simple and quick - all we need is your name and a valid e-mail address. We appreciate your interest in bridges.**{/access} {access view=!guest} That his discovery sounds so simple does not undermine its importance. When Princeton University awarded Gödel an honorary doctorate, the citation read, "His revolutionary analysis . . . has shaken the foundations of our understanding . . . of the human mind." Perhaps more significantly, Einstein, in his twilight years, chose to come to work largely, as he put it, to "have the privilege of walking home with Gödel."

And while they walked, they spoke. About truth.

"What is truth?" a Roman viceroy once asked. When I tell people that I am a mathematician, they often tell me (after reciting horror stories about their high school algebra teacher) that what they enjoyed most about math was its

*certainty*: In math, everything is either right or wrong, true or false, and you can always figure out which, if you're determined (or nerdy) enough.

Such was the attitude of the mathematical establishment at the turn of the last century. But consider this: "This sentence is false." Is it true or false? If it is true, then it's true that it's false, so it's false. But if it's false, then it is false that it's false, so it's true! (Someone get the Excedrin.)

Bertrand Russell, the renowned English aristocrat, philosopher, and mathematician, wrestled with this dilemma. The problem with "This sentence is false," he saw, was that the statement was talking about itself. If one could avoid mathematical "sentences" that talked about themselves, one could avoid such problematic mathematical brain-twisters - or paradoxes - as this.

Or so Russell hoped. He spent ten years of his life trying to lay the foundations of mathematics, brick by brick, so that no brick rested on itself. He did this by writing the mammoth three-volume treatise,

*Principia Mathematica*. (He was so careful to avoid paradoxes that he didn't even take arithmetic for granted. It isn't until the second volume that he proves 1+1=2.)

After he finished the

*Principia*, Russell wrote that his brain didn't work quite the same as before. (Note to Lord Russell: If you're going to ruin your brain, there are a lot more fun ways to do it than with mathematics.)

Enter Gödel. In 1931, at the tender age of 25, Gödel showed that Russell, the grand old man of mathematics, had wasted his time. No matter how carefully you build your house of math, there must always be a brick left out; your house must always be incomplete; there must always remain truths, even in arithmetic, about which we must forever remain ignorant.

How did Gödel do this? As with "This sentence is false," he created a mathematical statement that talked about itself. It's clear how a sentence in

*English*can talk about itself; but how can mathematics? By encoding English as mathematics. Suppose you assign numbers to the letters of the alphabet: 1 for A, 2 for B, 3 for C. Then you could either hail a cab or hail a 312. Mathematics doesn't use the letters of the alphabet; it uses symbols like +, ≠, and ∞. But you could encode these as well, and then describe mathematical statements like "1+1=2" as single code numbers.

What Gödel did was create a mathematical statement that said, "The statement with code number 42 cannot be proved." But - and this is the clever bit - he did this in such a way that the statement with code number 42 was that very same statement. In short, it was really saying, "

*This sentence*cannot be proved." So if it could be proved, it would be false; and if you can prove anything false, you can prove 1=0. But if it couldn't be proved, then it would be true. So, contrary to what we were all told in school, there would indeed be statements in math that could not be proved right or wrong, but which would in fact be right - you just couldn't

*prove*it!

That was Gödel's First Incompleteness Theorem. His second was that no matter how perfect your reasoning, or how precisely you lay your bricks, you can never exclude the possibility that someone, somewhere, some time, might use your system of mathematics to show that 1=0.

At some point, Gödel went mad. Fearful of being poisoned, he would have his wife, a former cabaret dancer, test his food. And when she was no longer there, he succumbed to malnutrition. Along with inventing "proofs" for the existence of God, Gödel, a refugee from Hitler's Austria, also discovered a way that the United States could, legally, descend from a democracy into a fascist dictatorship. But the roots of Gödel's illness went back many years. "You know, Gödel has really gone completely crazy," Einstein once told an acquaintance. The acquaintance was perplexed, so Einstein explained: "He voted for Eisenhower."

The Incompleteness Theorems revolutionized mathematics, and inspired men like John von Neumann, who created Game Theory (made famous by

*A Beautiful Mind*), and Alan Turing, who built some of the first electronic computers (helping to save the Allies from the Nazis while he was at it). Philosophers posited that Gödel's theorems showed us that the human mind has some special quality that cannot be mimicked by a computer: We human beings can see that "the Gödel sentence" is true, although a machine can not. Sir Roger Penrose, the English physicist, has even suggested that Gödel's theorems may help us discover a new physics that will explain the mystery of consciousness itself.

So happy birthday, Kurt Gödel. The last 100 years would have been incomplete without your genius.

*****

Professor Jonathan David Farley is a mathematician and Science Fellow at Stanford University's Center for International Security and Cooperation. *Seed Magazine has named him one of "15 people who have shaped the global conversation about science in 2005."*

Contact Information

Professor Jonathan David Farley

Center for International Security and Cooperation

Stanford University

616 Serra Street

Stanford, California 94305

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