2019-07-01 concepts

# Underdetermination, Gödel's incompleteness theorems

Hey all -

I was going to apologize for the last newsletter getting all serious and philosophical - but then I remembered that the Muse did it, and everything was okay again.

## + mental models #

* Underdetermination

Recently I watched a documentary on Netflix called “Behind the Curve” (2018), which chronicled the lives of people who believe the Earth is flat. There is a lot you can learn from this movie - about communities, tribes, truth and more - but today I just want to talk about a problem called underdetermination (also known as the Duhem-Quine thesis).

In the film, you have a couple of people who are scientifically-oriented. They’re adamant the Earth is flat, and they set up experiments to prove it. One experiment goes like this:

Hold out your left thumb, bend it at a 90-degree angle, and notice how the top is flat. That’s our flat earth.

Then, hold out your right thumb and notice how the top is curved. That’s our round earth.

On your left thumb - the flat earth - imagine standing on one side and shining a light to the other. The light should reach the other side without issue since nothing is in the way - the surface is flat.

On your right thumb - the round earth - imagine standing on one side and shining a light to the other. The light should not reach the other side because the curvature of your thumb gets in the way. In order to see the light, one side must elevate itself to overcome the curvature in the middle.

Well that’s essentially the experiment these flat Earth conspiracists ran[1]. If you shine a light at someone a few miles away, do they see it, or do you have to elevate yourself?

Perhaps unsurprisingly, the result demonstrated the Earth is indeed round.

Or did it?

As the film ends, we hear a few reasons why it did not definitively prove the Earth is round. One person speculated the light was hitting weeds, so the test was invalid. Another said that when they later tried to replicate the experiment, they didn’t get the same result. Their conclusion: “There was just nothing we could walk away from and say for sure was decided.”

This of course is not the first time in history people stubbornly held fast to a prior theory:

When Newton’s celestial mechanics failed to correctly predict the orbit of Uranus, scientists at the time did not simply abandon the theory but protected it from refutation by instead challenging the background assumption that the solar system contained only seven planets.[2]

There’s a problem here. Either (a) classical Newtonian mechanics isn’t the whole story, or (b) the assumption that the solar system contains only seven planets isn’t the whole story. (It turns out neither were!)

But there’s another problem here. A big one. Namely, science doesn’t prove things. It only gives us a reasonable amount of confidence that things are true. Said otherwise, science always underdetermines the truth value of any theory.

Think about it: you can gather more and more scientific evidence to support your theory, but you will never gather all of it. And until you do, the theory is quite simply not proven for every possible piece of evidence. This is also known as “the problem of induction.”

So science will never give us definitive proof. But certainly we can prove some things are true, right? Axiomatically true even, by logical necessity!

“This statement is true” is absolutely, positively, provably true - right?

Well, not quite.

* Gödel’s incompleteness theorems

In 1931, a logician named Kurt Gödel discovered something which shocked the mathematical community[3]. In what became known as his Incompleteness Theorems, Gödel demonstrated that you can’t possibly invent a formal, logical system which can prove the truth of all of its own statements.

In other words, “this statement is true” - or even “this statement is false” - cannot be shown to be either true or false using the logic of the system. These self-referential statements break the logic of a system, condemning it to a logical purgatory where true and false have no meaning.

It’s as though pure logic, when used against itself, reaches a universal speed limit of paradox. Some things can simply never be known to be true or false from within the system, and as a result, there will always be “irreducible uncertainty in the world.”

One thing I always wondered is why yogis and mystics always speak in paradox. They’re absolutely, positively incoherent. But now I wonder if it’s because paradox is the only thing we’ll ever know.