In the early twentieth century, the mathematical community embarked on a grand quest to understand all of mathematics. This quest is entertainingly documented (with some poetic license) in the comic book Logicomix: An epic search for truth. The driving conceit of the leaders of the quest, the Vienna circle, was that—just as Euclid came up with five axioms to explain all of geometry—the same could be done for all of mathematics.
The crash came when a young mathematician, Kurt Gödel (1906–1978) discovered that a variant of the Liar’s Paradox, “Is a liar who says he’s a liar telling the truth?”, proves that mathematics cannot be reduced to a fixed set of axioms. His demonstration became known as Gödel’s first incompleteness theorem. The theorem states that an axiomatic system cannot be both complete (enumerate all mathematical truths) and consistent (only prove true statements) at the same time.
Gödel’s discovery brought back a sense of wonder to mathematics and to the rest of human knowledge. His incompleteness theorem underlies the fact that human investigation can never exhaust all that can be known. Every discovery builds a path to a new discovery. C. S. Lewis coined the phrase “further up and further in” to capture the never-ending adventure of the search for truth in his book, The Last Battle.
Gödel’s theorem has been used to argue that the human mind is limitless. If the mind is capable of knowing an unlimited number of truths, it cannot be limited by the finite material world at least. Philosopher J. R. Lucas in particular is known for using Gödel’s theorem to argue that the mind cannot be reduced to a mechanism.
Arguments like these have been widely criticized. Critics argue that the problem posed by Gödel’s theorem can be easily circumvented by saying the mind is inconsistent. Obviously, not everything we believe is true and the fact we can believe both false and true things means that we are not limited by Gödel’s theorem.
Yet it seems strange to think that the human ability to be wrong leads us to being right. Indeed, in 1984 a famous mathematician named Leonid Levin (right) proved that the critics’ solution is unsuccessful, as he explains on his website: Adding randomness into the equation only makes the problem pointed out by Gödel worse.
This result has broad implications that go well beyond mathematics. The fundamental implication is that nothing within math, science, and technology can create information. Yet information is all around us. This problem arises in many areas: evolution, artificial intelligence, economics, and physics. All of these fields investigate various kinds of information yet researchers re flummoxed when trying to explain the origin of the information. For example, it is possible to prove that evolutionary algorithms are incapable of generating information.
On the other hand, the human mind seems to be fundamentally creative. Our everyday acts are filled with intentional decisions between alternatives. We cannot even be conscious without creativity seeping out of our thoughts. Thus, it should not be surprising that Gödel’s theorem shows that the human mind is profoundly beyond the reach of our current theories about reality.
More on Kurt Gödel’s (and others’) profoundly revealing math:
Faith is the most fundamental of the mathematical tools: An early twentieth century clash of giants showed that even mathematics depends on some unprovable assumptions. (Daniel Andrés Díaz-Pachón)
The mind can’t be just a computer. Gödel demonstrated that fact and Turing tried to live with it. (Analysis)
Things exist that are unknowable. A tutorial on Chaitin’s number (Robert J. Marks)
Could one single machine invent everything? The king’s perpetual innovation machine was all ready to roll but then a skeptic butted in. (Eric Holloway)