[FOM] natural language and the F of M

[Hartley Slater]

Harvey Friedman writes (FOM Digest Vol 4, Issue 15):

>  >On Bunt's formal work, his 'Ensemble Theory', as I said, *includes*
>>Set Theory as just a special case, so his treatment of mass terms is
>>*additional to* his treatment of count terms, which means that, from
>>his point of view, there is *more to* mathematics than what can be
>>included in Set Theory.
>
>I would like to see you indicate enough of the theory to justify that
>claim: That there is "more to" mathematics than what can be included
>in Set Theory.

Please read Bunt's book if a justification for the above claim is 
wanted.  In addition to 225 pages on linguistics and semantics, Bunt 
has 70 pages on his formal Ensemble Theory, of which the last 25 
concern the relations between Ensemble Theory, Set Theory and 
Mereology.  I have no intention of giving any more than the broad 
outlines of the many close details which are involved.  Ensemble 
Theory has 10 axioms, and there is a further axiom to separate out 
discrete ensembles; discrete ensembles are proved to be the same as 
ZF sets (p277f); Leonard and Goodman are quoted as saying 'domains of 
discourse often cannot be modelled adequately in set-theoretical 
terms', 'The ordinary logistic defines no relations between 
individuals except identity and diversity' (p289); and by defining an 
'individual ensemble' as a non-empty ensemble that has no members, 
Bunt shows (p194f) that such individuals are the same as LLG 
individuals (where the first 'L' refers to Lesniewski).


>For example, class theory goes beyond set theory in certain senses.
>But the claim that class theory shows that there is "more to"
>mathematics than what can be included in Set Theory is unconvincing.
>I would be surprised if the situation with regard to mass terms is
>any different.


But, in the previously quoted passage, I very deliberately inserted 
'from his point of view', since, given there is a connection between 
stuff and proper classes, I myself would also prefer to see Bunt's 
total Ensemble Theory as more a Set-and-Class Theory.  That is a very 
rough characterisation, however, and the detail of the relation 
between stuff and proper classes means a lot more conceptual 
re-arrangements are required (see below).


>  >So Friedman might find Bunt's book of considerable interest (I am
>>surprised he has not already read Lavine's).
>
>I didn't say that I haven't read Lavine's. I have, and I even own it.
>Sometimes I write for the benefit of the FOM list, and not just
>myself.

My interest in Lavine's book was merely that it defends a von Neumann 
type axiomatisation of Set Theory, claiming it is 'natural'.  The 
topic was natural language and the foundations of Mathematics, and my 
point was that there was further material which supported as natural 
a division between sets and other things, showing, as above, that 
there is a relation between proper classes and stuff.  Why I should 
be expected to 'sketch' Lavine's book in the process defeats me, 
especially when, I suspect, not only Friedman but also a great many 
other members of the list have read (or, at least, know enough about) 
the book in question - and Lavine's principal interest lies 
elsewhere, in developing some of Mycielski's work.

>  >My own contribution has been to point out that, by replacing 'being
>>a member of a proper class' with 'being part of a mereological
>>totality' one can do justice to mass terms while producing a
>>combined theory with some resemblance to the von Neumann-style
>>axiomatisation of traditional Set Theory.  It also gives an
>>immediate and natural way out of Russell's Paradox, and the like.
>
>The claim that it gives a new way out of Russell's Paradox above and
>beyond what normal class theory does is surprising.

The claim was that the way out is 'immediate and natural'.  But it is 
also new.  There is s structural similarity between "if 'F' does not 
determine a proper class, then there is a set of Fs" and "if 'F' is a 
count term, then there is a set of Fs", so, to that depth of 
analysis, the resolution of Russell's Paradox and the like might not 
seem too novel. But starting things from the count/mass distinction 
has more radical consequences.  It means, for instance, that sets can 
no longer be taken as the foundation of a theory of number, since 
that theory is needed first, to make the count/mass distinction on 
which sets are based.   Number must come in instead simply through 
the definitions of the numerical quantifiers, as in David Bostock's 
work ('Logic and Arithmetic' Vols I and II, O.U.P 1974, and 1979 - 
*which everybody interested in the F of M should have read*).  Then, 
with '(nx)Fx' as 'there are exactly n Fs', 'F' is count iff 
(En)(nx)Fx, and it is then that there is a set of Fs.

It is the possiblity that ~(En)(nx)Fx which has been forgotten since 
Frege - along with a fair number of other crucial things.  If list 
members want a fuller story see, for a start, chapters 9 and 10 in my 
recent book 'Logic Reformed', available via

http://www.peterlang.com/all/index.cfm?vSuche=vSuche&vDom=3&vRub=3060

As a bonus, readers can also find a fuller story, in chapters 2 and 
3, about the nominaliser 'that' which I showed, in FOM Digest Vol 4 
Issue 10, enables a natural resolution of The Liar and related 
paradoxes - and, in chapter 8, an introduction to the parallel, 
predicate nominalisation which clears up Frege's problem with the 
concept Horse.  There is certainly more there than is included in 
traditional Set Theory and Logic, no matter what there is in Bunt's 
book.
-- 
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