[FOM] Real Numbers

[Lucas Wiman]

 >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.

Sure.  Complementation just has no meaning for numbers.  Numbers are not 
sets, though they can be.  Much as colors are not books, though a book 
can have color.  Your objection seems to me to be roughly like saying 
``You say that this book is red, but the book has pages.  Red does not 
have pages, so you were wrong in saying the book is red."  Of course red 
things can have pages, but number things can also have elements--it's 
just not a number thing to ask about.

Note that an abuse of notation is different from a change of notation.  
It often consists of saying that two things are the same when one knows 
that they ``really" aren't equal.  For example, in topology, one might 
tire of saying ``the topology of <R, T>", where R is the reals, and T is 
the set of all open sets of R in the Euclidean topology, and rather just 
say "the topology of R."  In reality, R has no topology (since a 
topology in most formalisms is an ordered pair of two kinds of sets), 
but it's convenient to speak of it as if it did, since it's clear what 
is actually meant.

 >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.

Well, a good start would be something along the lines of ``a complete 
ordered field," though even that is too restrictive.  In general, the 
notion of an isomorphism is the relevant one here, though there are 
always new kinds of isomorphisms to be found.  See my posting on Rota.

It is fairly rubbery.  To me, mathematics is about finding patterns, and 
those patterns are the fundamental ``unit" of identity.  If we already 
knew about precise conditions for finding such identities, I don't think 
mathematics would be that interesting.  I also don't think that 
specifying these sorts of precise laws is a part of the foundations of 
mathematics.  Certainly Brouwer and the pre-Dummett intuitionists agreed 
on this point, though they were highly interested in the foundations of 
mathematics.

 >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.

Yeah, OK.  What's your point?  The reals might be required a priori for 
some definitions of the rationals.  For example, one might introduce the 
reals as urelements (as many people have suggested), introduce axioms 
for a complete ordered field on them, and then say that the rationals 
are the reals which are in all subfields of the reals.  One needs the 
whole field of the reals to define them in that case.  I'm sure less 
natural examples could be produced, giving the rationals as equivalence 
classes of the reals or something along those lines.

I see my point about the complex numbers was too brief and not clearly 
made.  My point wasn't that sets can have geometric content, but that 
there is nothing objectionable to a set-theoretic treatment of complex 
numbers.  If you don't like that, you can pick your favorite Hilbert 
space (L^2, say) or something similarly abstract.  One has no a priori 
attachment to this or that formalism concerning the complexes or L^2, or 
whatever.  I think that this problem of abstraction descends to the 
deepest levels of mathematics, which is what makes your point about your 
greengrocer totally irrelevant.  Your greengrocer has essentially no 
understanding of the real numbers, and a fairly rudimentary 
understanding of the natural numbers.  Work in f.o.m. has shown us just 
how vague and indeterminate our intuitions can be about seemingly clear 
notions like the real numbers, a subset of the natural numbers, and even 
some basic number-theoretic assertions.  Should we really care about the 
intuitions of grocers?

- Lucas Wiman

P.S. I'm going to be out of town for the next 5 days, so don't expect a 
fast reply to criticisms of the above.