[FOM] real numbers

[Hartley Slater]

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