[FOM] real numbers

[Hartley Slater]

John Pais (FOM Digest Vol 5 Issue 14) objects to my pointing out 
that, unlike rational numbers, Dedekind cuts have complements, and 
equivalence classes of Cauchy sequences of rational numbers have 
members:

>Mathematically, these observations seem peculiar and naive. They seize on
>unimportant peripheral features of some specific constructions, while ignoring
>(missing) the mathematical content and purpose of these constructions.
>Unfortunately, they seem to involve some sort of mirage that 
>attempts to locate
>rational (and real) numbers in one particular construction or another.

I'll readily grant that my remarks are obvious and trivial: the many 
people who have taken rational numbers to be Dedekind cuts or Cauchy 
equivalence classes have been making *egregious* mistakes.  But I do 
not have 'one particular construction' for the rationals in mind when 
I say these things: any structure which defines the rationals must 
satisfy certain desiderata, so all I need are some of those 
desiderata.  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.  It has been well said, 
'remember, you can convince anyone of anything so long as they are 
clever enough'.  It takes a clever person to miss what is patent, and 
missing what is patent has been endemic in logic and the foundations 
of mathematics for quite some while.

As readers of my postings may, by now, have realised, it is a full 
time job, pointing out the crashingly evident to those whose minds 
are elsewhere - even if it only takes two lines, in each case.  Some 
will remember me telling Tarski, 'There aren't two snows, Alfred, 
there are two patches of snow', and other peopIe still need to hear 
'There aren't zero snows, Gottlob and Crispin, there is merely no 
snow'.  I also had to tell Tarski that what is true is that snow is 
white; and others, more generally, that one mustn't fudge the 
use-mention distinction.  I have had occasion to tell readers about 
how Russell and Grelling missed the context sensitivity of reflexive 
pronouns; so that they, as a consequence, with many others, came to 
make the big mistake of confusing facts with properties.  There were 
other, similar banalities, as well.

Anyone going to the Joint Session of the Aristotelian Society and 
Mind Association in Belfast, July 18-21, or the LMPS do in Oviedo, 
August 7-13, can hear it all again.
-- 
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