2014-09-23

Mathematical incompletness and life

A friend told me once that we are living in an Universe full of unprovable truths.

He was referring to the Gödel's incompleteness theorems. Without being very formal, this says that there are many mathematical propositions that might be true but we cannot prove them using a computer program or any predictive way of checking, no matter how huge amount of time we can spend. We can find the proof for some, by luck, but for most of them we cannot find either a proof or disproof. In fact this "unprovable" propositions are way more many than the provable propositions. The provable propositions are like the set of natural numbers "N" and the unprovable propositions are like the set of real numbers (R).

Example of unprovable truths can be, for example: "this is the best possible compression of this string" (see Kolmogorov complexity). If we can find a better compressor we can say this assertion is false, however we cannot say, usually, that there is no better compressor that could be found. Even if we actually found the best possible compression, we can never prove it is the best (except some very simple cases).

The search for a better compression is an infinite search usually, there is no algorithm that can check all possible compressions and find the best one. This issue is related to the undecidability of Turing halting problem.

What has this to do with life?
One could argue that a meaning of life is to search for such better optimizations for some problem(s). Like we can search for a better file compression, we can search for a better theory that explains the Universe laws in the most simple way. It is not a proof, but I would argue that this is a problem that cannot find a clear answer in any given huge amount of time.

This approach is appealing because has some special properties that we prefer to assign for a meaning of life:

"What is after reaching the goal of life?"
It is a bit depressing to think that there is a goal in life and you can actually reach it. What is after, death?

Actually, solving such undecidable problems is proved to not have a definitive answer, after any huge amount of time we would spend. You can solve some of them after a while, but some others will continue to give better answers. It's an infinite improvement that is proven to always find a better answer after the last best know answer. Seems like something that could motivate people for a very long time, no?

The free will
In a deterministic Universe you could determine the future decisions knowing a snapshot of the Universe and the laws of Physics. However, improving the answer of such undecidable problems does not provide shortcuts. The complexity of simulating the outcome of such research has around the same complexity as letting the Universe to run it by itself.

So a supernatural being (if exists) could not calculate what humans could find about such questions; at least not easier that letting humans to do that work. Such supernatural being could do his own work on the same problem, but there is no guaranty that more processing power will find the prove faster that this "handful of dust".

This seems even more motivating: the race is not to find something that is already known, there is always something that cannot be known even for a being that creates an Universe. This is the same as we cannot know all the truths when we create a simple mathematical axiomatic system.

Of course, any human exploration of the truths space is finite in nature, and feasible to be simulate by an almighty being. However, betting on an unprovable "truth" that is future used/developed by the humanity is not something having a clear outcome that could be simulated.

Bottom line
This is just too speculative, even for me. The concepts that I presented might not be even rigorous from the mathematical point of view. I don't even advocate this view on human life. However, I think it might be an interesting starting point to dream about an answer for the fundamental philosophical questions of life.

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