[FOM] real numbers

[Hartley Slater]

Arnon Avron now says (FOM Digest Vol 5 Issue 37):

>Here it is you who dont distinguish between numbers and points
>It is generally taken for granted (not by me!) that
>there  is a correspondence between the two (although
>no *textbook* I know actually PROVE that this correspondence exists
>and has the required properties), but they are not identical.
>A number is a number (whatever this means). A point on
>a line is a point on a line.

Apart from attributing to me the very sort of error I have been 
labouring to eradicate, this is excellent!  At last, someone who can 
distinguish a correspondence from an identity - though, yes, even 
correspondences seem to be rarely PROVED, as I piointed out with 
regard to Suppes' attempt to derive the decimal corresponding to an 
arbitrary equivalence class of Cauchy sequences of rationals in FOM 
Digest Vol 5 Issue 5.

I presented, before that, in FOM Digest Vol 5 Issue 2, a quote from 
Grattan-Guinness showing that Dedekind also appreciated there was an 
further process needed to get from his irrational cuts to any 
corresponding numbers.  The length of the diagonal in a unit square 
is a certain number of those given units.  So that number is not any 
associated set, whether a Dedekind cut, or an equivalence class of 
Cauchy sequences of rationals.  For no length is a cut, or an 
equivalence class, of inches (say).  Russell, as Grattan-Guinness 
also pointed out, was nevertheless prepared to identify the number 
with the cut, so how he squared his fudging of the difference with 
his desire to make the foundations of mathematics rigorous must be 
still be weighing on his conscience.  It is presumably because of 
Russell's influence that people more generally have come to be quite 
happy with the idea that numbers are sets of different kinds.
-- 
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