chess, logic, mathematics, physics

Logical troubles (likelly) ahead

The rules of chess are really simple. Not just to learn, but more importantly,  simple and clean enough to be comfortably certain that there is no paradox within the  game. Chess is obviously self consistent,  there is no room for contradictions.

This is what I thought, until I came upon the vertical castling thing which is something short of a formal paradox, but disturbing non the less.  This is a clear case when the rules of chess brake down. It may be the only occasion, but it doesn’t matter. Chess axioms are formally inconsistent, or at least not specific enough to unambiguously deal with this strange case, which is the same.

What a shocker! For an axiomatic system this simple, nobody would expect two contradictory statements hiding  inside, there just isn’t enough space! But they are there. One sub-rule directly opposes the other. What may be common in Law, shouldn’t occur here, nor in the Law, but that’s even less realistic to expect.

There were many cases of unstoppable force against immovable object in scientific theories  in the past. Such a case doesn’t exist in reality, for reality MUST be consistent. And if a theory inhabits only one paradox, it is useless and wrong.

The most famous example is the Frege’s Set Theory, destroyed by the much better known Russell’s Paradox. After a few attempts to avoid RP, we now believe that we have a consistent mathematics based on the so called ZF axiomatic system, but we can’t be sure and this is the principle, we can’t be sure in such complex cases!

Now, given the complexity of modern physics, how probable is it, that there is no paradox inside, say General Relativity?

Physicists are mostly quite sure that General Relativity is well established and in accordance with the measurements to the 14th decimal place and so on.

As the chess masters ignore any chess inconsistency and keep playing, so do physicists. The inconsistency between  GR and QM is just a fact of life for now and a curiosity to intimidate laymen.

But I wonder what a computer chess program would do in the above situation? Would it concede as black or not? Depends on how it’s programmed of course, but a self consistent solution must be provided  by the programmer, regardless of the official rules. Humans may ignore the antinomy, a more solid machine wouldn’t.  It would behave well as white and as black in this position and under the same premises in both cases. If it doesn’t — it’s just a bad program, playing a poorly  designed game.

But humans have a nasty habit of just ignoring devastating information. To keep pretending all is okay, even if it isn’t.

It is not very difficult to construct a really bad paradox in modern physics and nobody cares. Imagine a pulse of light so intense, that its mass is no longer negligible. That it has so large a gravity, that it’s a black hole with the speed of light.  The so called Kugelblitz.

A very cool object, nothing wrong with it, and with no paradoxes. But if we permit it to collide with a small rock, what happens? For one, it cannot just stop or slow down. It is a big ball of light travelling through vacuum. Secondly, it can’t leave the rock where it was, for the rock has crossed the event horizon. Thirdly, it can’t just suck in the rock and accelerate it to the speed of light, because that would demand an infinite amount of energy.

Every conceivable options is out of the question, it seems. A nice example of an unstoppable force against an immovable object.

We humans, are messy creatures. Therefore our science is likely infested with paradoxes. We try to solve some and ignore or legalize the others.  Giving in to paradoxes is very wrong and not everybody accepts them, some of us deeply despise  paradoxes. Machines we build will not forgive us logical sloppiness. They will clean our games and our sciences, every axiomatic system must be pure. And a machine has to do, what a machine has to do! Or it’s broken – like humans are.

It is just another opportunity for future superintelligences to be better then we are.