Infinite loops

Watching that MIT series on Godel, Escher, Bach has been really good for drawing the essence of the problem together. I notice that “This is a lie” is acknowledged as an existing problem (so I know it hasn’t been solved yet).

A note to the section on recursion. There is no such thing as recursion because the computer really implements a stack and a loop (I realise I’ve always implemented fractals like this! not recursively). That is to say that there is no “calling of itself” rather a dumping of new parameters onto a memory stack FIFO style and then running the code again taking the top parameters off the stack.

This is my argument that there is no such thing as self-reference. We can approximate it by feeding output into input so that the system reprocesses its own state – this ends up with a dynamic system which can negative feedback to a single equilibrium or a group of equilibria which it oscillates through or it can become chaotic where the path of the system (it is not a Markov chain) never repeats a value and so no matter how long the chain input never equals output. This is an infinite loop, but not self-reference.

Self-reference is where the current state depends upon itself – same time and place. In a loop, current state depends upon previous state – it is a path not a single value so the self is actually the path. A self-referential loop would have its whole loop-state depend upon the whole path.

Escher’s hand drawing hand picture is a good example of “self-reference”. For one hand to exist it must have drawn the other hand so that the other hand can draw it. I first met this with the thought that maybe God the creator of “all” things might have created himself – it’s an idea that doesn’t settle well. With Escher’s hands we are left hoping they are not at the same state of evolution – if one started the whole process (the chicken/egg problem) then it is not-self referential. But, we know the awful point is to ask how can they ever have got started. At what point was the hand fully formed enough to hold the pen to start drawing the other hand. It is a system which truly depends upon itself.

My quest is to find a simple proof that this is impossible. This way all attempts to explain existence (and self) upon itself fail. Explanation must always then demand more than what is present in the system (what was called God – what I would call the God principal) : complete understanding is then impossible: essential self dependence upon non-self is implied and interdependence is necessary. These simple ideas are the bed rock of moral behaviour suddenly gain a firm footing so it is not a trivial project.

p.s. So Godel’s theorems depend upon Godel numbering: the ordering of sentences of a calculus so that they can be referred to by their position in the set of possible sentences. One presumes then that Godel’s theorem itself has a Godel Number in Godel’s calculus. That is to say: not only is there a sentence which says “I am unprovable” which has a Godel number, but that Godel’s whole theorem has a Godel Number. Now I don’t think it is possible for the theorem which outlines how to generate Godel’s numbers to have a Godel number itself! That goes right back to the origin of this enquiry and the realisation that “the theorem” is just symbols which are as useful to the wind as autumns leaves. It takes a mind with a culture of logic to start playing games with these symbols and animate them (a la Wittgenstein). A string of symbols does not create Godel numbers and so can’t provide its own Godel number. Next stop…. fully understanding Godel Numbering (should have done that at college!! still I somehow got a 1st for my logic module)