Metaphysical Musings

This is the text of a letter I sent to a physicist at ETH Zurich back in the mid 1990s. It probably isn't anything new, but one of those concepts that people present routinely, only to be rejected.

Over time I came to believe that it is, in essence, what Einstein, Podolski, and Rosen had tried to explain, just to have it turned in to a fairly straightforward paradox by the physics community - Who were unwilling, or unable to consider metatheoretical incompleteness in their field.

CONJECTURE:

The Tautology - The Heisenberg Uncertainty Principle is universally true is an undecidable property over the set of particles in the universe.

Applying the tools of Naive set theory to the experiment viewed as an endomorphic system, it appears to be impossible to produce a logical proof of the Heisenberg Uncertainty Principle using generalised transfinite induction. This is not, unfortunately, sufficient to prove the conjecture.

WHAT I BELIEVE CAN BE PROVEN:

1) There exist statements in particle physics that cannot be proven by applying the rules of inference to the set of particles in the universe.

2) Particle physics is not monomorphic.

Statement (2) follows directly from statement (1) - with the unusual result that there can be more than one logically consistent, and mutually incompatible theorys of the universe ( endomorphic and exomorphic ? ). The set theorist would immediately recognize that the difficulty results from the fact that the cardinality of countably infinite sets of objects ( particles in the universe ) corresponds to that of the natural numbers. In fact, it is possible to construct a rudimentary, albeit nearly trivial, particle theory using a generalized form of the peano axioms.

Once it has been established that one can support at least part of particle theory based on the peano axioms it becomes clear that the allowable structures ( groups , rings , fields, etc. - in the sense of P. Lorenzen ) on this subset of particle theory correspond to precisely those allowable for the set of natural numbers. Consequently, the incompleteness theorem of Kurt Godel can be applied directly. Structure theory turns out to be a very powerful tool for relating mathematics to it's applications. I find it to be more tractable then Category theory as well. In addition, as algebra and specifically Group theory return to the mainstream in modern physics, the tools of Naive set theory and mathematical logic will become increasingly relevant.

Philosophically, mathematicians have accepted for years that the incompleteness of Number Theory is not a flaw in mathematics, it is an intrinsic limitation associated with the way human beings perceive transfinite sets of discrete objects There was a time when mathematicians considered the theory of numbers to be a divine "blueprint" for intellectual reason, just as many physicists consider the theory of physics to be a potentially completeable blueprint for reality. Physicists will eventually accept, as did mathematicians, that even if their field is divinely inspired, it is not divinely executed. Physics is, after all, a product of the combined experience of human beings, and consequently, constrained by the limitations of the human consciousness.

Recently there has been a movement in the United States towards the view that, for philosophical reasons, the correct approach to physics is to model the observer as part of the system. In essence, the endomorphic model is the correct one. This opens up the whole question of provability in self-referential systems.

I am of the position that the question of decidability and provability in particle theory are consequences of the syntactic structure of particle theory itself. This is a very slightly broader interpretation than would result from the position that paradoxes are intrinsic to self-referential systems.

At a very basic level I suspect that particle physics will arrive at the same conclusion that number theory did sixty years ago. Provability and decidability are closely related concepts (I am hesitant to apply the over-used term "dual"). In mathematics, if you attempt to produce an axiomatic foundation in which the theory of the natural numbers is both unique and complete, one discovers that there are statements that are not recursively decidable. If you build a structure in which all statements are decidable, you sacrifice uniqueness and provability.

I suspect, heuristically, that any attempt by physics to put particle theory in a box will result in two probable outcomes. Either one will discover that the box is of unbounded volume - not all statements are recursively decidable, or if the box is of bounded dimension one cannot "plug up" all the holes - there will exist unprovable statements and uniqueness is lost.

I want to emphasize that this all seems horribly deep - It really isn't!

Why couldn't one create a model for the universe in which the observer is part of the system (endomorphic), and a model for the universe in which the observer is external (exomorphic), and does not influence his/her surroundings? Is it really that implausible that by doing so, you could create two incompatible theories that fit observation? Why not?

There really isn't anything spooky about this. Of course, the argument always deteriorates in to debates about multiple realities, and such. As far as I am concerned, there is only one reality - The one we live in. Any phenomenon that is fundamentally unobservable, directly, or indirectly, was never part of reality to begin with.