There are four possibilities:
- choice function
If you have a Turing machine which replicates itself at every decision point in order to explore all possibilities, this is what mathematicians call non-determinism. Unfortunately for Copenhagen advocates, this is precisely what Everett's Many-Worlds theory does and it is understood to be perfectly deterministic. The result of a computation by a Turing Machine that replicates itself is not "an unknown and undetermined machine" among the set of machines that exist at that point in time, rather the result is the set of all the Turing machines that exist at that point in time. That set is well-defined.
This is the mathematical concept that has the most uses in physics. Stephen Hawking claimed that black holes are non-unitary (singular) and simultaneously claimed that Einstein was wrong so the two must be related, right? Not so. Setting aside the fact that the non-unitarity of the universe is hotly contested, since every law of physics is unitary, singularity doesn't have any of the qualitative properties ascribed to non-determinism. The outcome of multiplying a matrix by a singular matrix is very well-defined; the outcome of multiplying a matrix by a "non-deterministic" matrix is not supposed to be well-defined. But there seems to be a way to rescue the concept if you consider non-determinism to be the inverse of a singular matrix. Now we're getting close to non-determinism. Unfortunately, there are two interpretations of taking the inverse of a singular matrix. 1) you get the set of all matrices which multiplied with that matrix give you some identity, or 2) you get absolutely nothing. #1 gets you back to Branching and #2 clearly contradicts reality (the result of any allegedly non-deterministic experiment is always something).Choice function
A choice function is a function that "selects" an element from a set. If you have a set with ten elements then there are ten possible choice functions on it. Choice functions are the only way to modify the Copenhagen Interpretation so as to make it intelligible without making it an entirely different theory (ie, without making it into Many-Worlds). Unfortunately, it also immediately disproves the resultant theory.
Philosophy of science explains that its purpose is to explain everything we perceive around us in as concise and formal a manner as possible. So as it stands, the Copenhagen Interpretation is incomplete because it fails to explain everything. In fact, it explains almost nothing of what we perceive.
The Copenhagen Interpretation doesn't explain how you get from a particle in state A at time t=0 to that particle in state B at time t=1 and the underlying quantum mechanical equations (which are fully deterministic since "non-deterministic math" is an incoherent concept) only tell you that the particle will evolve from state A at time t=0 to states B, C, D, and E at time t=1 (there's a story in here about how Copenhagenites abuse the mathematical concept of probability if someone wants to see me rant about physicists). So in order to complete the Copenhagen Interpretation you need to add a choice function to it that selects which state the particle will be in at time t=1.
The problem is this. A complete theory of physics must explain all perceptions and all physical objects it defines. So the choice function that you add to the Copenhagen Interpretation must provide information on state changes of 10**70 particles (the estimated number of particles in the universe) for every time interval during which a state change can occur. And that time interval is short; if one were feeling uncharitable, one would choose Planck time (10^-43 seconds). And this is over the entire lifetime of the universe. If the universe has an open geometry then this means that the choice function must encode an infinite amount of information. But let's be charitable and assume that the choice function chosen contains only 10^100 bits of information.
Now here is where the Copenhagen Interpretation dies. The complexity of the complete 'Copenhagen + choice function A' theory is greater than "God did it". From a formal point of view, there is nothing wrong with the theory "God created the universe" where you define;
- 'the universe' = 'everything you perceive', and
- 'God' = 'a powerful entity that would want to create the universe'.
The only thing that's wrong with this theory is that it's too complex since 'the universe' must contain an exhaustive enumeration of every bit of perception you have ever and will ever experience. And yet, it's simpler than the Copenhagen Interpretation.Non-mathematics
By that I mean only that 'non-determinism' is an undefined concept. Not the well-defined concept "undefined" but an undefined, null, meaningless concept. Per the above paragraph, this violates the philosophy of science and makes the Copenhagen Interpretation into incoherent nonsense.
Hardline apologists for the Copenhagen Interpretation will claim "you can't explain everything" but how would they know when they've entirely given up on the endeavour?
Previously published on wiki wiki web.