What Gödel Discovered (2020)

In 1931, a 25-year-old Kurt Gödel wrote a proof that turned mathematics upside down. The implication was so astounding, and his proof so elegant that it was...kind of funny. I wanted to share his discovery with you. Fair warning though, I’m not a mathematician; I’m a programmer. This means my understanding is intuitive and not exact. Hopefully, that will come to our advantage since I have no choice but to avoid formality 🙂. Let’s get to it.

For the last 300 years, mathematicians and scientists alike made startling discoveries, which led to one great pattern. The pattern was unification: ideas that were previously thought to be disparate and different consistently turned out to be one and the same!

Newton kicked this off for physicists when he discovered that what kept us rooted on the Earth was also what kept the Earth dancing around the sun. People thought that heat was a special type of energy, but it turned out that it could be explained with mechanics. People thought that electricity, magnetism, and light were different, but Maxwell discovered they could be explained by an electromagnetic field.

Darwin did the same for biologists. It turned out that our chins, the beautiful feathers of birds, deer antlers, different flowers, male and female sexes, the reason you like sugar so much, the reason whales swim differently...could all be explained by natural selection.

Mathematicians waged a similar battle for unification. They wanted to find the “core” principles of mathematics, from which they could derive all true theories. This would unite logic, arithmetic, and so on, all under one simple umbrella. To get a sense of what this is about, consider this question: How do we know that 3 is smaller than 5? Or that 1 comes before 2? Is this a “core” principle that we take on faith (the formal name for this is called an “axiom”) or can this be derived from some even more core principle? Are numbers fundamental concepts, or can they be derived from something even more fundamental?

Mathematicians made great progress in this battle for core principles. For example, a gentleman called Frege discovered that he could craft a theory of sets, which could represent just about everything. For numbers, for example, he could do something like this:

Here, he represents 0 the empty set. 1 as the set which contains the set for 0. 2 as the set that contains the set for 1 and 0. From this he could set a principle to get the “next” number: just wrap all previous numbers in a set. Pretty cool! Frege was able to take that and prove arithmetic rules like “1 + 1”, “numbers are infinite”, etc.

This looked formidable and cool, but Bertrand Russell came along and broke the theory in one fell swoop. He used the rules that Frege laid out to make a valid but nonsensical statement. He proved something analogous to 1 + 1 = 3. This sounds innocuous; it was after-all just one statement. But nevertheless it was disastrous for a foundational theory of mathematics. If you could prove that 1 + 1 = 3, then you can’t really trust any true statement that results from this foundation.

This put mathematicians on a tail-spin. They even dubbed this period the “Foundational Crisis of Mathematics”. In an effort to solve this problem, a mathematician called Hilbert laid down some requirements for what a fundamental theory of mathematics had to look like. He said that this theory must be a new language, with a set of rules that satisfied two primary constraints: Completeness and Consistency.

Bertrand Russell worked together with Alfred North Whitehead to develop a theory of their own called Principia Mathematica. They started by writing a new language with a few simple rules. Their language was dense and the work laborious, but they kept on proving a whole bunch of different truths in mathematics. It certainly looked like Principia Mathematica could serve as the foundational theory for Mathematics. Until Gödel came along.

He proved that Principia Mathematica did indeed have true mathematical statements which could not be proven in the language. Principia Mathematica was incomplete. He showed that the entire enterprise behind Hilbert’s Program could never work. He did this by showing that if a language could represent numbers, then unprovable statements would necessarily pop up.