🚀 go-pugleaf

In article <298@mtnmath.UUCP> paul@mtnmath.UUCP (Paul Budnik) writes:
[...]
>Interpretations exist for quantum mechanics because the theory is not
>formalizable. One cannot define a model for a quantum mechanical system
>in ZFC. That is the standard criteria for whether a mathematical model
>has a rigorous definition. If that were possible there would be no
>ambiguitys about what quantum mechanics means in a mathematical sense
>and there would be no need for interpretations. 

I think this is somewhat misleading. I don't see what ZFC particularly
has to do with it, nor "formalizability". The ambiguities in the
interpretation lie mainly in what the terms, as formalized (which can
be done in a variety of ways), should be taken to mean. It is not so
much a mathematical or theoretical problem as a philosophical problem.

>It is not simply a matter of being extremely difficult to define such
>a model. It would take some major change to ZFC to allow one to rigorously
>define a random sequence. 

Are you saying that probabilistic theories in physics are inherently
dubious?

It is not clear that there even exists such a notion of "random
sequence" as you seem to think would be required. (There are, of
course, definitions of "random sequence", just not of the sort meant
in this context.) People can perfectly well formalize probability
theory, however.

Probabilistic models, on one interpretation, tell us how thoroughly to
expect certain events to occur, how much confidence to place in events
occurring.  It is not clear that one ought somehow to be able to
convert such a theory into one which gives a black-and-white criterion
for when a sequence of events does or does not conform to
expectations- for whether the sequence is "a random sequence" or
isn't. Confirmation is a matter of degrees.

>                    A change that I do not think any mathematicians
>foresee happening. 

Yet I don't think it has persuaded any of us to abandon probabilistic
models.

>                    It would also be difficult to rigorously define
>what happens when an observation is made and make this consistent with
>a rigorous definition of special relativity.

Perhaps so. My impression is that something in this direction has been
done, however, in specifying conditions which have to hold between the
operators associated with observables, observed at a space-like
separation, to prevent observation from being usable as an FTL
communications channel.

Keith Ramsay
ramsay@unixg.ubc.ca