FOM: Category Thry & Topos Thory:An Interpretation
Robert Tragesser
RTragesser at compuserve.com
Mon Jan 19 04:38:18 EST 1998
I suggest an interpretation of
Category Theory and Topos Theory, while not bolstering
the claims made on their behalf to function as "global
foundations", nevertheless suggests they have an
absolute significance (g.ph.i.??), Category Theory
for what it is, Topos Theory for what it points toward.
My proposed interpretation of Category Theory
derives from its very interesting applications by the
theoretical biologist, Robert Rosen (e.g., Life Iteself).
Those of us
who didn't have a clue to the significance of abstractly
presented k-forms until they read Wheeler et al will appre-
ciate the possibility of of finding an important dimension
of significance for a mathematical theory which had only
internal applications through seeing how it is importantly
applied outside pure mathematics.
Since I'm not sure what the interest is, or how
much needs to be said, or what problems people might have,
I'll give the ideas. . .details laterin answer to problems. . .
Category Theory. See this as a powerful and sensitive
theory/logic of wholes and parts, reading "A f-arrow B"
as "A is an f-part of B". Organisms grade off from very
well integrated functional (and not just structural) wholes
to very poorly integrated such wholes. Organisms are stretched
in time in various senses (e.g., life cycles, chains of
life cycles). There are different diachronic and synchronic
levels of organization and functions, and interactions among
them which typically cannot be organized into a hierarchy. There
are (inevitably) structural/functional problems in each phase
of an organism it is to "its" advantage to solve at a later phase,
and there will typically be several "choices" of solution
"possible".
Category theory is a powerful tool for modelling all of
these phenomena, e.g., Rosen is able to distinguish nonmechanical
"anticipatory" systems from mechanical systems. How well?
How definitively? This is being looked into. But Category Theory
represents in any case a decided advance over the "systems logic"
brought to bear by Norbert Wiener in "Newtonian and Bergsonian
Time".
History: Aristotle's progress beyond Plato was to recognize
whole-part structures as giving a broader (and so more fundamental)
range of logical entities than "class" alone (to speak over simply)
provides. But the tradition really did not follow this out.(Barry Smith,
Peter Simons. . .) Biology has in recent years crossed a threshold
which makes theoretical biology not just a phantasy of life
scientists who suffer from physics envy) and one can speculate
reasonably
that the first fruitful, powerful theories will deeply involve
Category Theory as the most powerful logical engine of parts & wholes.
TOPOS THEORY.
ZFC-Set have the cache of being cobbled together
out of ubiquitous phenomena, collections, combinations,
classes, meaning of general terms,. . . insofar as it
makes sense to ask, How many? Along one axis (there are
others of course), ZFC represents to the notion: Collection
for which it is meaningful to ask (in the sense of having
a definite answer) How many?, and sets themselves are "units",
sets are such that there can be sets of sets (understanding
that a set is a collection for which How many? has
a definite answer).
It is an easy remark that there are "collections"
for which it doesn't make sense to unambiguously ask How many?
Swarms, Bose-Einstein sets. . ."collections" where
the units composing them vary (swarms, packs) or where
their members are not units (e.g., they are partial, or they
are otherwise not sharply distingushed).
The principal mathematical example is Brouwer's
(emergent) set theory.
ZFC-Set is, then, tremendously short-sighted.
What we should like, then, is a "Set Theory" which
embraces are .logically articulable and
mathematically sigfnificant
varieties of "collections"
(and not just those for which How many? is well-defined.)
Now such "collections" may be (non-trivially)
regarded as various sorts of wholes, so we should expect
Category Theory to play a role in representing the different
sorts of "collections". One can see topos theory
as being a step in the right direction toward such a uniform
theory of all possible logically articulable, mathematically
potent categories of "collection". (Here I am drawing on --
and pushing further -- Goldblatt's philosophical motivation
for topos theory.)
Remarks: (1)BOOLE I see topos theory as a step toward restoring a
possibility dropped by Boole in his original paper "Mathematical
analysis of logic" -- that there are other logics, of categories
of "class".
(2)BROUWER It is odd -- indicative perhaps of a failure of
fluidity in logical imagination -- that mathematics cling so
powerfully to the real number continuum. As Brouwer showed us,
there are other powerful analyses/representations of continua.
Shouldn't one rather lean toward a logic/mathematics in which
there are fundamentally different varieties of continuity?
(N.B., as Hobson elaborately remarks, our intuitions
of continuity can quite support incompatible analyses/representations).
rbrt tragesser
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