Bobby Azarian writes:

Before Kurt Gödel, logicians and mathematicians believed that all statements about numbers — and reality more generally — were either true or false, and that there must be a rule-based way of determining which category a specific statement belonged to. According to this logic, mathematical proof is the true source of knowledge.

The Pythagorean theorem, for example, is a mathematical conjecture that is true: It has been proved formally, and in more ways than one. With many theorems, it may be extremely difficult to find proof, but if it is true, it must have a proof — and if it is false, then it should be impossible to prove with the fundamental axioms and the rules of inference of the formal mathematical system.

At least, that was the assumption made by leading mathematicians of the early 20th century like David Hilbert, and later Bertrand Russell and Alfred North Whitehead, who attempted to design an ultimate formal system that could, in theory, prove or disprove any conceivable mathematical theorem. Meanwhile, scientists and philosophers at that time were trying to demystify the mind by showing that human reasoning was the product of purely algorithmic processes. If we could somehow access the exact steps that brains were following to ascertain something, they argued, we would find that they were using strict rules of logic.

A brain, then, was nothing more than a squishy Turing machine — a simple device operating on reasonably simple rules that could compute the solution to any problem solvable with computation, given enough time and memory. This would mean that all the mystery and magic associated with conscious thought could be boiled down to logical operations, or rule-based symbol manipulation. The mind would be no more mysterious than a computer — everything it did would be determinable, definable and understood mathematically. It was a pretty sensible stance at the time.

But Gödel, an eccentric Austrian logician, disproved that view even before Alan Turing invented his abstract machine, in a quite roundabout and loopy way. In 1931, Gödel published his famous incompleteness theorem, as it became known, which called into question the power of mathematics to explain all of reality — along with the hypothesis that the mind works like a formal system, or a mathematical machine.

With a clever use of paradox, Gödel would destroy the idea that truth is equivalent to mathematical proof. Taking inspiration from an old Greek logic statement involving self-reference called the “liar’s paradox,” he constructed a proposition about number theory using a ridiculously complex coding scheme that has become known as Gödel numbering. Although the theorem is virtually impossible to understand for anyone without an advanced degree in mathematics, we can comprehend it by translating it into similar statements in common language. [Continue reading…]