[FOM] Real Numbers

[Hartley Slater]

Jaspar Stein writes (FOM Digest Vol 5 Issue 24):

>I may misunderstand your concept of 'count predicates', but anyway I
>fail to see the relevance of them to natural numbers. Surely +they+ are
>'count'? And surely there +are+ sets (like the von Neumann ordinals)
>that are count? So why forbid the usual equating of N and omega?

The relevance is that 'How many Xs?' only makes sense if 'X' is count 
- otherwise we ask 'How much X?' with 'X' a mass term.  Number words 
are not in the right category for themselves to be count or mass, 
since while one can say 'It is a pear' or 'It is broccoli', one 
cannot just say 'It is two'.  Using Frege's example: a pack is four 
suits, but also 52 cards, so it isn't in itself 4 or 52 - that is the 
whole point of the 'second order predicate' understanding, since a 
first-order predicate is required, like 'is a suit', before a 
numerical judgement can be made  Certainly one can count the number 
of *members* in a von Neumann ordinal, indeed it is that which 
demonstrates the isomorphism with the natural numbers, and gets 
people slackly thinking they *are* the natural numbers.  They are 
merely exemplars of different  numbers of things.  I am pleased to 
have support on this (yet again, painfully obvious point) from Sean 
Stidd, in the same FOM issue.

But John Pais has yet to be convinced:

>As Kunen indicates, this program of foundations is aware of, but not
>concerned with or interested in, philosophical quibbles regarding
>identity. For mathematicians, mathematical identity *is* isomorphism.

Unless Pais (unlike Wiman) can provide the rules for his 
non-Leibnizian notion of 'identity', he will have to agree that 
'numbers are sets' is an abuse of language.  But I am concerned with 
what numbers are, in the proper, exact, Leibnizian sense of 'are'. 
Set Theory might mirror certain results about numbers, but the 
foundations of mathematics should concern itself with *those 
numbers*, in the first place.

>In contrast to this 'numerical philosophy', with its strange sounding and
>spurious claims and characterizations such as the above, all most all
>mathematicians and logicians that do foundation of mathematics, start
>with set theory ab initio.

Not Bostock, whose 'Logic and Arithmetic' books I have referred you 
to before (FOM Digest Vol 4 Issue 16).  Bostock, for a start, takes 
the natural numbers to be second order quantifiers, as above.

>you declined to give us enough of a sketch of Bunt's work so that we
>could fairly evaluate whether or not it could both address the concerns
>of numerical philosophy and serve as a replacement foundation for
>mathematics such as that laid out in Kunen's book.

I pointed out in that ZF Set Theory was a proper part of Bunt's 
Ensemble Theory, so I did give you enough. But what you are calling 
'numerical philosophy' is the second-order account, found at length 
in Bostock.

>I don't plan on buying the book (I need more convincing), and I 
>think at this point in the
>discussion everyone would like to see details supporting for example your
>claim regarding: "mathematics which puts Set Theory aside".  It just
>seems that if Bunt's work (1985) offered a serious replacement foundation
>for mathematics, then it would be more well-known by now. So, though I
>remain skeptical, "my brain is open".

I doubt whether Bunt's book is now available outside a university 
library, or a professor's bookshelf.  And David Lewis, for one, knew 
about it in 1991: he acknowledged that Bunt's development of Set 
Theory prefigured his own, in 'Parts of Classes'.  One has to 
remember that Dummett only pointed out in the same year the 
inadequacy of Frege's logic in the mass term area.  And a time lag of 
100 years is quite an improvement on the two millenia it took 
Aristotelian Logic to acknowledge relational inferences.

-- 
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