In parts of Quantum Mechanics where you seem unable to dig deeper into Reality by experiment – eg the peculiar behavior of streams of single photons in a ‘two slit experiment’ or observations of quantum co-relations that can hold *instantly* between sub-atomic particles irrespective of distance separating them – you are stymied! You are stuck with observed properties of Reality that you cannot analyze by experiment in terms of cause and effect at a lower structural level. No one to date has been able to come up with an experiment that yields unexpected results in these ‘weird’ parts of quantum mechanics. Therefore, any attempt to theorize beyond the experiments is nonsense. All the models we have derive the same, anticipated set of testable consequences too – and these models all look strange but in different ways. They are fairy tales that are consistent with Schrodinger’s and Heisenberg’s equations and Bell’s Inequality and prior experimental data. They suggest nothing useful we did not know before.
Our only hope right now seems to be the fact that both great 20th century efforts at a theoretical synthesis, Quantum Theory and General Relativity, are incomplete. Each ignores most of the part of Reality the other deals with. General Relativity in its sphere is as powerful as Quantum Theory is in its. Each theory makes correct predictions to twelve to fourteen decimal places! By taking the theories themselves as *data* and looking at them side by side – or better yet superimposed where possible, perhaps a generalization can be made in terms of hitherto unexplored *relations* between these theories. The current popular candidate for such a meta-theory is some version of String Theory. I hope the theoretical physicists can make a synthesis in terms of one of their String Theories – or even something else.
There is another, hopefully faint, but dismal possibility. We are a proper subset of Reality. Therefore we cannot contain a physical model of Reality complete in all its details, since the assumption that we could, would lead to an infinite regress – a contradiction – an instance of the well known paradox of the architect who attempts to draw a complete current picture of the room he is working in. But can we even express the fundamental *laws* governing the functioning of all of Reality from within the proper subset of Reality we find ourselves in? This too may be impossible for reasons suggested by the following analogy.
Within the Logic of Principia Mathematica (PM) you cannot consistently utter certain statements about that Logic – statements that might be true if uttered outside the Logic. Godel showed this by constructing a sentence that, if it were derivable from the axioms of PM, would be false and render the Logic inconsistent. Yet Godel’s sentence was a true statement about the logic when viewed in an external metalanguage. It was a sentence, S, that uttered that a sentence of such and such syntactic structure could not be derived from the axioms and S was an instance of that very structure! Thus if S were derivable, it would be false and the logic inconsistent. If underivable it would be external to the Logic and true, assuming the Logic was otherwise consistent. Also, Tarski showed adding the predicate “True” to the predicative calculus would render that calculus inconsistent. Such a truth predicate would in a sense be an attempt to say too much about the semantics of the predicate calculus from within that calculus.
If the process of modeling Reality at the most fundamental level is analogous to an attempt at modeling the semantics of a logic within that logic, then the problem of constructing a consistent ‘theory of everything’ in physics may be an insolvable one. It might amount to an attempt to say too much about all of Reality from within a proper subset of Reality. The apparent strangeness of the quantum world, might then be interpretable as results of limits to our knowledge that are inherent in the structure of Reality, because we would be trying to describe the semantics of all of Reality from within a proper subset of Reality in which certain concepts were lacking. Another description, not inconsistent with this, would be that the fundamental laws of Reality might be too rich to express in terms of any of the formalisms possible in our subset of it. An appropriate formalism may look inconsistent within our subset – much as the notion of closure under subtraction would look inconsistent within a subset of the integers consisting only of the positive integers. Some pairs of positive integers (eg 7 – 9 = -2) would simply have no difference representable within the subset since it excludes the negative integers.
I hope these pessimistic conjectures turn out to be irrelevant or flawed. First, they assume a certain situation in a formal logic is analogous at a deep level to a particular situation in Reality. Second, they assume that Reality can be usefully modeled as having a syntax and semantics and that the weird part of Quantum Mechanics corresponds in some sense to the semantics of Reality while ordinary experimental descriptions are in the same sense syntactic ones. The applicability of both of these assumptions to Quantum Mechanics is open to serious doubt. I certainly hope the part of Quantum Mechanics considered here does not turn out to
be: ‘A Wall Through Which No Door May Pass.’