[FOM] natural language and the Fof M: the main pt.

[Peter Apostoli]

HS:

The main point is that sets are, by definition, sets of
discrete elements, and so, once the difference between such elements
and stuff is available, we can obtain a natural foundation for a
difference between sets and other things.

Bunt proved 'A continuous ensemble has no atomic parts...[it] has no
members', adding 'this last result emphasises once again that
continuous ensembles are fundamentally different from sets' (Bunt
1985, p262).  He then defined discrete ensembles, 'An ensemble is
discrete iff it is equal to the merge of its atomic parts', which
meant that 'Discrete ensembles will be seen to be the antipodes of
continuous ensembles, and to be in every respect like ordinary sets'
(Bunt 1985, p263).  Continuous ensembles are then subject to the laws
of Mereology, while discrete ensembles are handled by the parallel,
but abstract, part-whole theory of singletons, as in David Lewis'
later work ('Parts of Classes' Blackwell Oxford, 1991). Bunt
developed his Set Theory in the Zermelo-Fraenkel manner, however, and
so missed the fact that his distinction shows, in a more elementary
way than with Zermelo-Fraenkel, how the naive Set Abstraction Scheme
is conditional.  Specifically, on Bunt's principles, he should say,
simply:
iff F is count, then there is a set x such that (y)(Fy iff y is in x).
That, amongst other things, relieves difficulties with paradoxical
predicates on the right through the possibility of the denial of the
stated precondition on the left.  But it also reveals there is a
close relation between proper classes and stuff.

Specifically, for instance, y is water not because it is a member of
a certain set, but because it is part of a certain aggregate (the
totality of all water), while the formal possibility of a non-set
equivalent has previously been taken to be a matter of y still being
a member, but a member of a different type of thing to a set - a
proper class.  The representation of stuff in terms of membership is
on account of the influence of late nineteenth century thinking about
number.

PA

Greetings from Spring-time Toronto.

A concept script may be given which correctly articulates - in
 the definition of an "admissible concept" (begriff) - the principle of
individuation characteristic of count nouns, so long as we already have
in hand a really sound (concretely realized) identity theory.

Frege was deeply aware
of the the lacunas you mention above in his concept script. He gave
expression
to it in his problem about JC, which stems from the formal incompleteness of
his identity theory.
After all, who needs "Hume's Principle" to decide the truth value of "x =
the number of F's"
or "JC = the number of F's" if you already have an adequate theory of "="?

Is it not striking that Bunt should offer Frege a way out, by denying the
the concepthood of problematic pluralities? There is indeed a
close relation between proper classes and stuff. Since the latter is central
to our
 *commonsense* conception of the world, Bunt's account of
aggregates of stuff as "continuous collections" is understandable. It
accords with an old
tradition [Kant, Aristotle] of identifying continuity of matter with
infinite divisibility or
 order theoretic density. But at least on certain understandings
(at once ancient and contemporary) the distinction between aggregates of
stuff and collections of discrete objects does not survive analysis at the
most fundamental level of reality. On this undertanding, the distincion
between
pluralites of discrete objects and "stuff" is relative, or one of level:
All collections are of discrete objects, but under graded indentity
(indiscernibility).
The phenomenal continuum - that of the world we expreience - is derived from
an underlying discrete ontology. E.g., you will find this idea in Milesian
atomism and
on all three roads to quantum gravity

(As observed last summer in Munich)
relative identity theory accommodates the essential aspects of Bunt's basic
framework as thus far described.
In this way Bunt, Lewis and and the tradition in set theory that takes
singleton
and inclusion ("parthood") as more basic than elementhood can be unified in
a rough setting.