[FOM] natural language and the Fof M

[Hartley Slater]

Harvey Friedman has taken Dean Buckner to task, asking for natural 
language studies to be related more closely to issues in the 
foundations of mathematics (FOM Digest Vol 4, Issue 10).  Buckner 
must answer for himself, of course, but there is one issue I have 
been engaged with, which bears very closely on this matter (and see 
my 'Aggregate Theory versus Set Theory' shortly to appear in 
Erkenntnis)

As Michael Dummett forcefully pointed out in 'Frege: Philosophy of 
Mathematics' Duckworth, London, 1991, p94 (c.f. Mary Tiles, 'The 
Philosophy of Set Theory'  Blackwell, Oxford 1989, p151, Crispin 
Wright 'Frege's Conception of Numbers as Objects', Aberdeen U.P. 
Aberdeen, 1983, p3, Bob Hale and Crispin Wright 'The Reason's Proper 
Study' Clarendon, Oxford, 2001, pp315, 385, 415), Frege presumed that 
the concept of number was applicable to all concepts whatever, and 
so, in particular, provided no formalisation of the difference 
between count nouns and mass terms.  Very few thinkers since Frege 
have been man enough to tackle this lacuna in Frege's logic, but 
Harry Bunt has studied mass terms a good deal recently, and as a 
result has replaced Set Theory with a more comprehensive 'Ensemble 
Theory', which includes Set Theory as just a special case ('Mass 
Terms and Model Theoretic Semantics' C.U.P. Cambridge 1985).

One thing Bunt misses, though, relates closely to the argument 
between von Neumann-Bernays-type and Zermelo-Fraenkel-type 
axiomatisations of Set Theory.  Shaughan Lavine, for instance, has 
produced an axiomatisation equivalent to von Neumann's 'which seemed 
to me to be more natural and conceptually based than the usual ones - 
though I doubt it is the last word on the subject' ('Understanding 
the Infinite', Harvard U.P. Cambridge MA 1994, p320, see also 
pp141-53).  Maybe the following is a relevant further word on the 
naturalness of a fundamental distinction between sets and other 
things.  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.







-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html