[FOM] real numbers

[Hartley Slater]

Matt Insall writes (FOM Digest Vol 5 Issue 17):

>In one currently common presentation of the rational
>numbers in advanced courses, the rational numbers
>also have complements in the sense in which Dedekind
>Cuts have complements.

Indeed, if the number 5 were the same as the set {p|p<5} then not 
only would 'the complement of {p|p<5}' make sense, but also 'the 
complement of 5' would do so.  Back to the drawing board, Dedekind, 
and Peano!  Your axioms only generated arithmetic properties of the 
natural numbers!  Silly you!  You forgot entirely about the 
set-theoretic ones!  A similar point goes in the case Lucas Wiman 
asks me specifically to consider (FOM Digest Vol 5 Issue 18), namely 
the equation of complex numbers with [ordered] pairs of reals.  For 
the latter have complements, even if nothing about those complements 
has yet appeared in complex number theory.  No doubt on Argand 
diagrams there are the straight lines between ordered sets - so sets 
have geometric and spatial properties.  And is such a line the graph 
of a linear function?  No!  Lo!  It is the very function itself, 
since algebraic entities, we now realise, can be seen!  Wiman expands 
on this intoxicating point of view:

>Mathematicians do not use identity in the same way as philosophers. 
>I cannot emphasize this enough.  In the chain of maps I showed 
>above, no set was included in the following set, but they were 
>``pretty much" included in the following set.  They're the same in 
>every way that matters, and that's all that mathematicians mean when 
>they say ``equals."  No category error, no problem.  It seems to me 
>that Slater is objecting to what mathematicians commonly call ``an 
>abuse of notation."  This argument seems, therefore, silly and 
>trivial.

I answered the point about triviality in my previous posting (FOM 
Digest Vol 5 Issue 17).  As for 'sillyness', see above.  But if it 
was just a carefree matter of a change of notation, then not only 
would there be no category error in 'the complement of 5+3i', and the 
rest, there would also be no problem - no problem at all - about 
stating the rules which hold for the 'pretty much' and 'all that 
matters' notions of identity and equality that Wiman hypothesises. Or 
is it all so rubbery as to be quite lawless?  Either he must confess 
there are no rules at the bottom of the relevant mathematical 
practises, or he must now specify them, since they are evidently very 
much needed, in the sober and rigorous study we call 'The Foundations 
of Mathematics', to augment Leibniz' Law.  And when worrying about 
those rules, Wiman might like to address the following sort of 
problem:

If the rational r were 'identical', in any sense, with the associated 
Dedekind cut {p|p<r}, or equivalence class [<r,r,r,...>], then an 
infinite regress could be generated, by repeatedly substituting the 
R.H.S. into itself.  The definition of the cut, and the class 
*requires* a prior, separate definition of r, otherwise there would 
be a circularity.
-- 
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