[FOM] Real Numbers

[Hartley Slater]

Lucas Wiman (FOM Digest Vol 4 Issue 20) is working with a lot of 
out-of-date assumptions:

>Sure.  Complementation just has no meaning for numbers.  Numbers are 
>not sets, though they can be.

I am glad you agree that using von Neumann ordinals, or any other 
series of sets to represent the natural numbers is a category 
mistake. But while Benacerraf pointed out that, because of the 
multiplicity of such possible series, no one series could be the 
natural numbers, it is not, by contrast, that the natural numbers 
*can* be this series, or that series.  They cannot be *any* series of 
sets: the theory of number must preceed the theory of sets, because 
one needs the concept of number to decide which specific predicates 
determine sets - the count predicates.  So the natural numbers are 
not sets period.  You are presuming that the Fregean assumption that 
all predicates are count is correct.  But it isn't.

>I think that this problem of abstraction descends to the deepest 
>levels of mathematics, which is what makes your point about your 
>greengrocer totally irrelevant.  Your greengrocer has essentially no 
>understanding of the real numbers, and a fairly rudimentary 
>understanding of the natural numbers.  Work in f.o.m. has shown us 
>just how vague and indeterminate our intuitions can be about 
>seemingly clear notions like the real numbers, a subset of the 
>natural numbers, and even some basic number-theoretic assertions. 
>Should we really care about the intuitions of grocers?

Following on from the above, one thing which clearly makes my 
greengrocer very relevant, is that he asks not only 'How many Xs?', 
but also 'How much X?'.   I emphasised before (FOM Digest Vol 5 Issue 
17) that his familiarity with the first question means he knows that 
the natural numbers are second order predicates - and knows this 
without using 'intuition', or any theory.  But it is my greengrocer's 
familiarity with the second question that Wiman is now primarily 
forgetting - along with Frege.  For there is no plural in the second 
expression, and that is very significant, since it means that the 'X' 
there is not a count, but a mass term.  I spelt out some while ago 
how that makes Set Theory inapplicable to the case, and what must 
replace it, in a series of postings 'natural language and the F of 
M'.  This is a foundational matter that Wiman has yet to get to grips 
with. See (again), for a start, on the history of the point, Michael 
Dummett's 'Frege: Philosophy of Mathematics', Duckworth, London 1991, 
p94, but also, on mathematics which puts Set Theory aside, Harry 
Bunt's 'Mass Terms and Model Theoretic Semantics', C.U.P. Cambrtidge 
1985 passim.
-- 
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