A thought crossed my mind today about Gödel’s incompleteness theorem that I wanted to make a note of. Recently, I came across the theorem again recently when I saw a video espousing Roger Penrose’s stance on quantum conciousness. I was trying to understand what the theorem meant, and what Penrose’s idea was.Today, I came up with an analogy, and wanted to note it down as an aid to memory. It may be an incomplete (pun intended) analogy in itself, so I may expand it later, the more I learn.
Gödel’s theorem states that given a set of rules (axioms) that underlie a mathematical structure, there are some truths in that structure that cannot be proved from those rules. This is strange because it means there are some things that may be true, that are also unprovable.
The Gödel theorem achieves it goal by using a self-referential formula:
$$
G_T \;\equiv\; \neg \mathrm{Prov}_T(\left\langle G_T \right\rangle)
$$
Here we state that the sentence, $G_T$, is exactly equal to the statement that $G_T$ is not provable.[^1] A contradiction. This means that there are some sentences that can be constructed from our rules that are unprovable, but true! The actual Gödel formula seems fairly contrived. Something that seems purely theoretical. What use is a self contradictory statement in mathematics ? But in fact, more examples have been found in the wild, and they are not so abstract. So, we must try to comprehend this curio and this can be achieved through analogy.
The analogy is to think of a series of islands in the sea. Most of the islands are connected by bridges; rules that connect the sentences. However, some islands form a discrete archipelago; between them the rules that reinforce their group structure. These islands exist, their sentences are true, but there is no path from our islands to them.
Penrose’s argument is that something to be provable by a computer, it can only get there through bridges. Our robot (computer) is roaming around exploring everything, with a certain myopia, only able to see the bridges in front of it.
Humans on the other hand can see the far-off island; the one not connected to ours. Gödel showed this by the very act of understanding his incompleteness theorem; it is the very epitome of the far-off island.
Penrose theorises that the ability of us to see the island means we have some thought beyond computation. As we are in the physical world then this must be based on something in reality that is not computable. Penrose surmises that this is the collapse of the wave function. Something that can be seen as a mathematical trick, but something that is a break in the Schrödinger equation. An equation that is used to evolve a quantum state. He doesn’t claim to know where this happens in the brain, but has proffered a hypothesis.
However, a counterargument could be, is there a meta-system of logic. Something that exists outside our system of bridges; a series of underwater tunnels perhaps. Is is possible that there exists a higher order set of rules, or a transformation of our mathematical structure that allows the islands to be connected.
[^1] This is reminiscent of the halting problem, and Turing may have been inspired by this. Briefly, a program H decides if a program P stops. Then a new program J is created that when passed H, does the opposite. J called on itself then creates a paradox; it cannot decide itself.