Why we can't find a theory of everything

We must know, we will know

In this post I argue that we will never find a theory of everything, something that fully explains and links together all known physical phenomena. My argument is based by looking on other much simpler systems than the physical world.

David Hilbert believed that pure mathematics was black and white and absolutely clear. He and a lot of other mathematics set a goal to create a formalization of mathematics that would eliminate all the problems, especially the paradoxes that Bertrand Russell found in set theory. For around 30 years Hilbert et.al. worked very hard on this problem.

The thing they didn't know was that the problem they tried to solve was unsolvable. In 1931 Kurt Gödel proved that their efforts were a waste of time and they would either end up in an incomplete system or a system that includes contradictions. Kurt Gödel's findings shocked the mathematics world, especially since his proof was based in elementary number theory, in arithmetic. Gödel's incompleteness theorems are hard to explain, but the book Gödel, Escher, Bach does an effort on explaining them. I don't want to explain how he proves it, but the thing to remember is that he proves that formalization of elementary number theory would end up in an incomplete system or a system that includes contradictions.

Alan Turing proved in 1936 that the halting problem was unsolvable for the turing machine (which basically is an abstraction of a computer). The halting problem is a simple decision problem that is stated in an following way:

given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input.

There are MANY unsolvable problems - unsolvable problems are unsolvable no matter how much time, space or speed you have. Rice's theorem states that any non-trivial statement about a program is undecidable.

You can read more about the history in Historical Introduction - A Century of Controversy Over the Foundations of Mathematics.

The incompleteness theorem and the halting problem are related

Wikipedia's page on the halting problem explains how Gödel's problem can be reduced to the halting problem:

The weaker form of the theorem can be proved from the undecidability of the halting problem as follows. Assume that we have a consistent and complete axiomatization of all true first-order logic statements about natural numbers. Then we can build an algorithm that enumerates all these statements. This means that there is an algorithm N(n) that, given a natural number n, computes a true first-order logic statement about natural numbers such that, for all the true statements, there is at least one n such that N(n) yields that statement. Now suppose we want to decide if the algorithm with representation a halts on input i. We know that this statement can be expressed with a first-order logic statement, say H(a, i). Since the axiomatization is complete it follows that either there is an n such that N(n) = H(a, i) or there is an n' such that N(n') = ¬ H(a, i). So if we iterate over all n until we either find H(a, i) or its negation, we will always halt. This means that this gives us an algorithm to decide the halting problem. Since we know that there cannot be such an algorithm, it follows that the assumption that there is a consistent and complete axiomatization of all true first-order logic statements about natural numbers must be false.

It's much worse than that...

Gregory Chaitin postulates that mathematics have a much bigger problem than the one Gödel found:

In other words, it's not just that Hilbert was a little bit wrong. It's not just that the normal notion of pure mathematics is a little bit wrong, that there are a few small holes, that there are a few degenerate cases like ``This statement is unprovable''. It's not that way! It's much, much worse than that! There are extreme cases where mathematical truth has no structure at all, where it's maximally unknowable, where it's completely accidental, where you have mathematical truths that are like coin tosses, they're true by accident, they're true for no reason.

quote from Historical Introduction - A Century of Controversy Over the Foundations of Mathematics

Where does this leave us?

Mathematics was believed to be pure - to be ruled by simple rules. But as it can be read above there's something very fishy about mathematics and it's that the system is not pure and we can't find a set of axioms (basic facts) that defines mathematics!

How can a relative simple system like mathematics be unclear? How come we can't find a "theory of everything" for mathematics? How come mathematics seems to be ruled by randomness, by controlled chaos?

Today we try hard to find a theory of everything in physics, something that fully explains and links together all known physical phenomena. But I believe that this isn't possible. The thing to remember is that the physical world is much more complex than the mathematical world - - and it's also ruled by randomness, like those found in quantum physics. I.e. how can we find a theory for everything for the physical world when we can't find a "theory of everything" for a simple system like arithmetic?

I have some ideas on these matters that I am currently thinking about. I can't present it right now thought as my theory needs more thinking and more structure.

All models are wrong. Some models are useful.
- George Box