SUO: *Date 13 Mar 2002 -- Changing Membership
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MW = Matthew West
MW: Being a member of a set is not something that changes over time.
Possessing a property can change over time. This means that
property possession is different from set membership.
Matthew, & All,
Something about this sort of statement really sounds wrong to me.
I have wrestled with it for a couple of days now, and here is the
best that I can work out for the time being.
1. Sets are mathematical objects.
2. Elements of sets are mathematical objects.
If you cannot imagine what in the world such statements might mean,
please feel free to interpret them as idiomatic figures of speech,
affording the paraphrase that sets and their elements are objects
of signs of a sort that we call "mathematical". In most settings,
the descriptors "formal" or "logical" will convey the point just
as well as "mathematical". These adjectives are meant to impart
no more than the fact that the meanings of the signs in question
are determined solely by bodies of formal, logical, mathematical
expressions that we know as "theories".
3. Mathematical objects are not physical objects.
The reason I say this is because the meanings of physical signs,
that is, the sorts of signs that refer to physical objects and
physical phenomena, are not determined solely by theories, as
a large share of their meanings reside in the experiences of
these objects and phenomena themselves. Notice that there
is a difference between being determined by a law, which
we may not know well enough to be able to write down,
and being determined by a theory, at least, of the
sort whose finite axiom set has been written down.
4. Ergo, sets and their elements are not physical objects.
Now, I am perfectly well aware that we very often speak as
if physical objects could be sets, or the elements of sets,
but that just tells me that we very often speak loosely.
No news there.
The confusion arises, I guess, from the fact that we very often
use mathematical systems in the description of physical systems.
And so I would have to classify this as yet another instance of
uncritically and unreflectively projecting the mathematical map
onto the physical territory, and thus confounding both of them.
Aside from all of this, as we went through several times before,
it only coronates confusion to describe mathematical objects as
"timeless". The factor of time, as the conventional aspect and
the standardized parameter of a physical process, is simply not
a determinant in the definition of a mathematical object proper.
Nevertheless, we do use mathematical systems to describe
time-evolving physical systems. And when we do this, it
is perfectly possible for a function f : R -> X to give,
for every real time associated with a real number t in R,
the position of the test object in a physical space that
is parameterized by a mathematical space X. In this way,
then, in the only figure of speech that could make sense,
the "associate membership" of the test object in various
subsets of X is indeed something that changes over time.
Anyways, this is kind of tentative,
and may of course change with time.
Jon Awbrey
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