[FOM] numbers and sets (again)

[Hartley Slater]

Martin Davis has written to me:
>I would encourage you to post explaining why you think that the 
>set-theoretic foundation for mathematics with which most advanced 
>mathematics textbooks begin, is no longer relevant.

That was because I expressed to him my disappointment at what seemed 
to me a shallow interest in foundational issues amonst members on the 
list, despite it being 'FOM'.  I had in mind here centrally FOM 
reactions to Benacceraf's points about 'what numbers cannot be'.  In 
discussions on the list earlier this year, which I took part in, 
these points, and similar ones about real numbers, were taken to be 
irrelevant by many other correspondents.

Thus it was once thought (i.e. before Benacceraf) that the theory of 
von Neumann ordinals (or any other collection of sets with 
successively different numbers of members) provided a foundation for 
the theory of number, and indeed that one could rightly say such 
things as '2={{{}}}', '2={{}, {{}}}'.  Those identities do not hold, 
however, because numbers are not sets: the items on the right have 
complements, for instance, and their connection with the number 2 is 
not that either *is* the number 2, but that one can appropriately 
count two things in connection with each of them.  Likewise with two 
lines '||'; or two of anything, for that matter; these have the same 
relation with the number 2, but even more clearly are not the number 
itself.  The theory of number, within set theory at least, has not 
moved on from such times, and you may remember when Benacceraf wrote 
- 1965.

I am not going to repeat the points I made in those discussions, 
since in my judgement they were fruitless.  But I copy below a brief 
portion of a recent Erkenntnis piece by myself on a related matter 
(see issue 59, 2003, 189-202, 'Aggregate Theory versus Set Theory', 
which discusses the BSL 2000 topic of 'New Axioms' and some of 
Maddy's work; the excerpt is from pp193-4).  Offprints of the whole 
article can be obtained from me, if a mailing address is provided.

Is Frege's cardboard the set of cards, or the set of suits.  The 
cardboard is 52 cards, and also 4 suits - indeed that observation was 
the basis for Frege's point that the stuff of the world has no 
inherent number properties. But if the cardboard was a set in the 
current mathematical sense it could not be both sets, since one set 
has 52 members, while the other has 4 members. Now the pack of cards 
is an ordinary physical object, which can be traded, held in the hand 
etc. so the pack of cards is just the cardboard. But where is the 
other set, if the cardboard is just one of the two? And not only the 
location of the other set is in question, of course, for what does 
the set of suits weigh, or cost, by contrast with the pack of cards, 
and is it as much fun to play with? We are approaching the reductio 
of the current position. For the cardboard is the pack of cards, but 
it cannot be one set without being the other, on grounds of symmetry; 
so it also has to be the set of suits. That is mysterious from the 
current mathematical point of view, for if the same object is each of 
two sets, as mathematically understood at the moment, then the two 
sets must have the same members. But the pack is clearly not four 
membered!
	The point to remember, to extricate oneself from this muddle, 
is that packs are, by definition, packs of cards, so the principle of 
division which identifies the members is brought in along with the 
collective term, once one describes the cardboard with it. That is 
fundamentally what makes the things seen an aspect of the whole, 
since what members are involved depends on the physical totality 
being described in a certain way. One can often describe the same 
totality with two collective nouns, each pointing to different 
divisions: thus a nation of people might be a federation of tribes. 
Normal collective nouns are species-specific in this way, to one 
extent or another, with prides being prides of lions, flocks being 
flocks of sheep, or birds, etc. But by using a non-species-specific 
term like 'set', this further descriptive element which makes evident 
the membership is removed. If one talks of the cardboard as a 'set', 
therefore, which set one has in mind is indeterminate until one 
specifies further what the set is a set of. Nevertheless the 
cardboard is both a set of cards and a set of suits. And that means 
that, when we do use the bare, non-species-specific term 'set', it 
just means 'totality'. The number of cards certainly differs from the 
number of suits, but still the totality of the cards remains the same 
as the totality of the suits. The differentiation comes in only by 
means of reference to the members of the set, which shows, as Maddy 
anticipated (Maddy 1990a, 60), that it is the plural terms for those 
members - 'the cards', 'the suits' - which are doing the work. We do 
not say 'the number of the pack of cards is 52' but 'the number of 
cards is 52', and even 'the number of packs of cards is 1', so a 
number is not a property of an individual thing, as Frege tried to 
make out, but some things.
	How does Aggregate Theory formalise these matters? It is 
convenient, first of all, to draw the count-mass distinction using 
the mereological relation 'proper part of' ('<'). Then 'P' is a mass 
term if and only if....
-- 
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, M207 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