----- Original Message -----
From: "Brent Meeker" <[EMAIL PROTECTED]>
Sent: Friday, July 22, 2005 8:31 PM
Subject: Re: what relation do mathematical models have with reality?
On 22-Jul-05,Stephen P. King wrote:
Ok, I am rapidly loosing the connection that abstract models
have with the physical world, at least in the case of
computations. If there is no constraint on what we can
conjecture, other than what is required by one's choice of logic
and set theory, what relation do mathematical models have with
Is this not as obvious as it appears?
Here's my $0.02. We can only base our knowledge on our experience
and we don't experience *reality*, we just have certain
experiences and we create a model that describes them and
predicts them. Using this model to predict or describe usually
involves some calculations and interpretation of the calculation
in terms of the model. The relation of the model to reality, if
it's a good one, is it gives us the right answer, i.e. it
predicts accurately. Their are other criteria for a good model
too, such as fitting in with other models we have; but prediction
is the main standard. So in my view, mathematics and theorems
about computer science are just models too, albeit more abstract
ones. Persis Diaconsis says, "Statistics is just the physics of
numbers." I have a similar view of all mathematics, e.g.
arithmetic is just the physics of counting.
Ok, I would agree completely with you if we are using Kant's definition
of *reality*- Dasein: existence in itself, but I was trying to be point out
that we must have some kind of connection between the abstract and the
One thing that I hope we all can agree upon about *reality* is that what
ever it is, its properties are invariant with respect to transformations
from one point of view to any other. It is this trait that makes it
"independent", but the problems with realism seem to arise when we consider
whether or not this *reality* has some set of properties to the exclusion of
any others independent of some observational context.
QM demands that we not treat objects as having some sharp set of
properties independent of context and thus the main source of
counterintuitive aspects that make QM so difficult to deal with when we
approach the subject of Realism. OTOH, we have to come up with an
explanation of how it is that our individual experiences of a world seem to
be confined to sharp valuations and the appearance of property definiteness.
Everett and others gave us the solution to this conundrum with the MWI. Any
given object has eigenstates (?) that have eigenvalues (?) that are sharp
and definite relative to some other set of eigenstates, but as a whole a
state/wave function is a superposition of all possible.
So, what does this mean? We are to take the a priori and context
independent aspect of *reality* as not having any one set of sharp and
definite properties, it has a superposition of all possible. The trick is to
figure out a reason why we have one basis and not some other, one
partitioning of the eigenstates and not some other.
What does this have to do with mathematics and models? If we are going
to create/discover models of what we can all agree is sharp and definite-
our physical world, we must be sure that our models agree with each other.
This, of course, assumes that there is some connection between abstract and
concrete aspect of *reality*.