# [FOM] real numbers

Hartley Slater slaterbh at cyllene.uwa.edu.au
Mon May 12 03:21:42 EDT 2003

```John Pais (FOM Digest Vol 5 Issue 15) comments on my last posting:

>  > ............  For instance, I can ask my greengrocer for 2 1/2 pounds
>>  of cabbage, but he would be non-plussed if I asked for the complement
>>  of 2 1/2 pounds of cabbage, or for members of 2 1/2 pounds of cabbage.
>>
>>  How does my greengrocer enter the sophisticated realm of debates on
>>  the foundations of mathematics?  Because he clearly knows better than
>>  many theorists in this area what sort of thing, in the first place, a
>>  number is: it has to number things.
>
>You say: "it has to number things". Why do you think so?

I was talking about the rationals, as the example, and the previous
discussion made clear.  In connection with the more elementary case
of the integers the point can be made most briefly: the point is that
one may put set-theoretic expressions in place of 'S', but not in
place of 'n' in '(nx)(x is in S)'  where (1x)Fx iff (Ex)(y)(Fy iff
y=x), and ([n+1]x)Fx iff (Ex)(Fx.(ny)(Fy.~(y=x))).  For more, see,
for instance, my reply to Friedman under 'natural language and the F
of M' in FOM Digest Vol 4 Issue 16.  Miguel Lerma adds:

>In the realm of Mathematics there are no cabbages, nor apples, nor bricks,
>nor planets, nor anything "material" at all. Mathematics is concerned only
>with generalities that do not depend on peculiarities of any objects from
>the real world - in other words, "2 cabbages plus 2 cabbages is 4 cabbages"
>is just a property of cabbages, "2 plus 2 is 4" is Mathematics.  Mathematics
>starts where all references to the real world end.

But the '4' in '2+2=4' is the same '4' as in '(4x)(x is one of the
cabbages)', which means that the above grammatical point still holds.
So why did I refer to my greengrocer?  That is because, while the
idea that numbers are second-order properties has been fully
discussed in the literature, and academic professionalism requires
all interested parties to have read books like those by Bostock,
still, *in that context*, this idea remains a theory too easily seen
to be debatable.  How is the debate to be settled? By 'committment'
to this in opposition to all the other theories, or by some nebulous
appeal to 'intuition'?  My greengrocer doesn't use his intuition to
know that he is regularly asking questions of the form 'How many
Xs?', and 'How much X?'; and he hasn't a clue about any theory of
numerical quantification.

But fitting in with the local tradesmen is only one of the
'desiderata' preceeding any settled theory.  Didn't Frege have
arguments, for instance, about the priority of numerical statements
like 'Nx.Fx = n'?  From (nx)Fx, however, there follows (Em)(mx)Fx, by
existential generalisation, and so ([em(mx)Fx]y)Fy, (where 'e' is
epsilon, and the epsilon term reads 'the number of Fs' ) by the
epsilon definition of the quantifiers; it follows therefore, by
uniqueness, that em(mx)Fx=n.  Yet the reverse entailment does not
hold, because of the possibility of mass terms - then 'the number of
Fs' does not gain its reference from its sense, since it must be
non-attributive, and so semantically arbitrary, which means that
em(mx)Fx=n can be (accidentially) true without (Em)(mx)Fx.  The
quantificational form (nx)Fx, therefore, is the logically prior
expression.  Pais continued:

>However, I just don't see the point in digging ones heels in, ignoring current
>mathematical practice and its purposes, and maintaining essentially
>(merely) that
>mathematicians aren't using certain words correctly, e.g. 'number'
>or 'the'. Who
>should determine how, and whether or not, the concept of 'number'
>should change and
>evolve, mathematicians or greengrocers?

But clearly I am all for the current concept of number evolving
(amongst theorists), specifically into one where it is possible that
~(En)(nx)Fx.  That will not only enable mass terms to be
accommodated: it will show in a different way that the concept of
number preceeds the concept of a set, since sets can only be formed
using count terms (see again the above reference).  And I have given
many further external references in my postings to counter any
impression that getting the formation rules right is all that needs
to be done.  That is only the start (though it is the only start),
and fuller mathematical results always do follow.  For example,
setting Tarski right about the grammatical category of what is true
leads to the sort of specific results in protothetic I provided in
FOM Digest Vol 4 Issue 10, under 'consistency and completeness in
natural language'.

--
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

```