Kurt Gödel

Lost innocence

This coming Saturday marks the birthday of the mathematician Kurt Gödel. Born in what was then Austria, on April 28 1906, Gödel died in Princeton, New Jersey on January 14 1978, having developed a paranoia that he was being poisoned and, as a result, starving himself to death (an altogether odd end for one of the greatest logicians the world has ever known).

Two years ago, when Time magazine conducted a poll to determine the 20 most influential thinkers of the 20th century, Gdel was one of just two mathematicians in the score. He came in ninth, ahead of astronomer Edwin Hubble, physicist Enrico Fermi, philosopher and economist John Maynard Keynes, transistor inventor William Shockley, biochemists James Watson and Francis Crick who discovered the double helix pattern of DNA, polio vaccine developer Jonas Salk, and world wide web creator Tim Berners-Lee. (Albert Einstein ended up as Time's person of the century.)

Gödel is best known for his discovery, in 1931, of the famous Gödel Incompleteness Theorem. In everyday terms, this says that no matter how hard you try, you will never be able to reduce all of mathematics to the application of fixed rules. Regardless of how many rules and procedures you write down, there will always be some true facts that you can't prove.

Some scientists, most notably the Oxford mathematician and physicist Sir Roger Penrose, have used Gödel's Incompleteness Theorem to argue that the human brain does not operate like a computer, and in particular that artificial intelligence is not achievable. According to Penrose's interpretation of Gödel's theorem, mathematics has an element that is completely creative.

The above summary of the Incompleteness Theorem is a bit, dare I say, incomplete. Here's what Gödel really did.

Because mathematics deals with abstractions, you cannot establish mathematical facts by making observations in the world or by performing experiments the way other scientists do. You have to use proofs -- logically sound arguments. Those arguments have to start somewhere. So, you start off by writing down some initial set of basic assumptions, known as axioms.

The axioms are supposed to be so simple that their truth is beyond question. Things like, two straight lines are either parallel or else meet in exactly one point, or when you add two numbers it does not matter which one goes first. (The first of these two examples turned out to be less "obviously true" than first appeared, but that's another story.)

Once you've written down the axioms, to decide whether some given statement is true or false, you try to prove it from the axioms.

This way of doing mathematics was introduced by the ancient Greeks 2,500 years ago, and worked well throughout history. It was always assumed that the only thing that could prevent you from being able to decide the truth or falsity of any mathematical question that came up was if you'd overlooked one or more basic assumptions to put into your list of axioms. And that was easy to fix: just add in the missing axiom(s).

Gdel shattered this belief, and changed forever our understanding of mathematics. His Incompleteness Theorem says that you can never find enough axioms. No matter how carefully you try to make sure you have written down all the basic assumptions, there will always be some questions that you can't answer. Mathematical knowledge is destined to remain forever incomplete.

In fact, the situation is even worse. Gödel went on to show that one of the questions you cannot answer is whether your chosen set of axioms is consistent or not. You can never be sure that, in writing down your axioms, you haven't made a mistake and introduced some subtle conflict.

Until Gödel's result, everyone assumed that, unlike all the other sciences, mathe matics offered absolute certainty, and one path to perfect, sure knowledge. To some extent, that's still true. Pythagoras's theorem about right-triangles is as true today as it was when it was first proved 3,000 years ago, and it will remain true forever. Once a mathematician has proved something, it's truth is permanent. (Unlike the situation in any other science, where theories are constantly being modified or overturned.) However, what Gdel discovered was that this kind of certainty does not extend to all questions of mathematics. Some remain beyond proof. If you think such a statement is true, you simply have to take it as an act of faith.

In that sense, Gödel's discovery can be regarded as the end of the age of innocence in the field of mathematics.

Keith Devlin, Thursday April 26, 2001 Guardian Newspapers Limited