[FOM] numbers and sets (again)
Hartley Slater
slaterbh at cyllene.uwa.edu.au
Thu Sep 18 00:09:14 EDT 2003
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
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