[FOM] real numbers

[Hartley Slater]

Just to hand in this morning's mail, I find, is an updated version of 
Bostock's work on the non-set-theoretic account of number I drew list 
members' attention to before, i.e. what John Pais called 'numerical 
philosophy'.  See 'Frege's unoffical arithmetic' by Agustin Rayo, JSL 
67.4 Dec 2002, 1623-1638, which contains references to other work in 
the same area by Harold Hodes.  Rayo explains why Frege came to see 
his account of numbers as objects as faulty, because of the problem 
with Basic Law V exposed by Russell, and how Frege's subsequent 
'unoffical proposal', amongst other things, avoids Benacceraf's 
problem altogether.

On a related matter, Mark Steiner says (FOM Digest Vol 5 Issue 28)

>However, I think that core mathematicians would have little sympathy for =
>the argument, however valid, that real numbers are not "really" Dedekind =
>cuts.  (I'm not taking sides here, just stating a sociological fact.)

The points I have made were that *the natural numbers* and *the 
rational numbers* were not Dedekind cuts, i.e. that 'r={p|p<r}' is 
false - that would make those numbers objects of a certain sort.  And 
the 'core mathematician' Suppes was quite clear that rational numbers 
are not the same as rational real numbers, as I also pointed out.  I 
am sure, nevertheless, that Steiner is right about the sociological 
fact that many would resolutely find no problem with 'r={p{p<r}'.  No 
one likes to admit they have made a mistake, especially not as big a 
mistake as Frege made.
-- 
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