ONT Re: Changing Membership
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JA = Jon Awbrey
SR = Seth Russell
Yes, I worried that it might be read that way,
but I could not find the proper dispensation
of emphases to forestall that interpretation,
so I just left it flat, but that is also why
I rephrased the whole question all over again:
JA: How does it makes sense to talk about the relation
between physical systems and mathematical systems?
JA: That is to say, what is the optimal way to understand
the way that mathematical models are used to describe
practical realities among the communities of practice
that use them all the time to achieve these purposes?
So let me try again.
How should we formalize our statements about
the relationships between physical systems
and mathematical systems in such a way
that active experienced users of this
knowledge will think that we know
what they are talking about?
Because that is definitely a task for the future.
SR: The structures of a mathematical systems can represent the structures
of physical systems. Just like the neural structures in your brain can
represent the self same physical system. We say that the representation
of the physical system by the mathematical system is more or less accurate
(true if you like) just in the proportion that the mathematical system can
predict events of the physical system. Embedded in that paragraph is my
definition of represent. You seem to use the word 'associated' for the
same relationship.
You mean the "representation is structure-preservation" type of definition?
SR: Just as we ~forget~ that our awareness's are entailed by the neural
representations in our brain and hence gives us the illusion that our
awareness walks among real physical objects, so a mathematical system
can ~forget~ that it's structures are mere representations and function
as if they were the real things. So it can say that a real physical
object is a member of a set ... and make no apologies for its confusion.
Okay.
SR: The confusion of the map for the territory is when the actual mathematical
system is confused with the physical system. That does not happen unless
it makes a category error. Frequently parts of a mathematical system are
used to represent other parts of the mathematical system. I call that
reification and do not consider such reification to be any kind of
confusion; rather it is the mathematical system turning its eye
(as it were) on itself ... one neural analogy of that could be
introspection.
Alright, up to the business about "reification",
where I suspect that we use the word diversely.
So we are just talking about category theory,
and morphisms from a space X to a space Y,
with the slight Catch-22 that X is really
too much an unknown to be formalized in
its own right, but only in the "forms"
of one or another choice for Y, which
is what I call quasi-category theory.
Jon Awbrey
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