ONT Re: Changing Membership
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JA = Jon Awbrey
LO = Leonid Ototsky
MW = Matthew West
re. http://suo.ieee.org/email/msg08094.html
cf. http://suo.ieee.org/email/msg08093.html
cf. http://suo.ieee.org/email/msg08076.html
LO: Jon,
LO: Very important notes !
LO: Wednesday, March 13, 2002, 10:10:24 AM, JA wrote:
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.
JA: 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.
JA: 1. Sets are mathematical objects.
JA: 2. Elements of sets are mathematical objects.
JA: 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".
JA: 3. Mathematical objects are not physical objects.
JA: 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.
JA: 4. Ergo, sets and their elements are not physical objects.
JA: 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.
JA: No news there.
JA: 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.
JA: 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.
JA: 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.
JA: Anyways, this is kind of tentative,
and may of course change with time.
LO: There are some additions from "The duality principle ..." point of view.
They differ a "subject area" from a "classsification field". The first
one is "not closed" class in principle. The last is a "good set" when
the proper "primary" identifications from real objects to "minimal"
taxons are made already! (This is another very impotant theme).
LO: The minimal taxons "substitute" real objects in any model.
It is important to differ "taxonomical" properties from more
deep "diagnostic" properties. A value of a taxonomical property
may have a complex connection with them.
LO: What about to take this into account in the SUO ?
Leonid
=====================================================
Leonid Ototsky,
http://www.mgn.ru/~ototsky/ototskyhome.html
Chief Specialist of the Computer Center,
Magnitogorsk Iron&Steel Works (MMK)
http://www.mmk.ru
Russia
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Leonid,
Thank you for your remarks and references.
Maybe it would help to give some literature
citations and especially e-links to material
on the "duality principle" point of view, as
it sounds like this might also link up with
many other active perspectives in the SUO
project, John Sowa's use of lattices, the
Chateau d'IFF use of Galois tunneling, and
Jean-Luc Delatre's advert to Chris Hillman's
ideas about Formal Concept Analysis, plus the
efforts I have been making to mine, to smelter,
and to refine the raw materials of Charles Peirce's
oratory ore on Extension x Comprehension = Information.
One of the problems that we have here is that most of these
ideas are very familiar, or at least vaguely familiar, from
the common parlance of people who work in mathematics, both
"pure" and applied in their different ways, and also people
who use mathematical modeling in other applied disciplines.
Yet if you ask for references to these practical principles,
they say "common sense", or "folklore", or "it's obvious",
or "gosh, I never thought about it out loud before".
So it would help to start documenting this stuff
for the beneficial use of others, and, of course,
to begin hashing out the different interpretations
in a somewhat more explicit and formalized fashion.
I am interested in the use of mathematical systems to
describe physical systems, including those that hope to
be called "intelligent dynamical systems", present company
hopefully included. This work is carried out in a holistic
vein or paradigm, in the sense that it involves the making
of "formal" or "morphic" comparisons from whole system to
whole system, without excessive fixation on the assumed
elements, putative individuals, or potentially singular
local configurations, though of course these can play
their roles. The Big Question is, to be answered
in practical terms:
How does it makes sense to talk about the relation
between physical systems and mathematical systems?
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?
That is the brand of knowledge that dearly needs
to become embodied in our ontological theories.
Jon Awbrey
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