[FOM] Real Numbers

[Lucas Wiman]

Me (a while ago):

 >Benacerraf argues (among other things) that since any
 >set theoretic representation of the natural numbers includes some
 >properties which numbers do not have (like 3 being an element of 5),
 >numbers cannot be sets.  Slater argues along similar lines that
 >since equivalence classes of Cauchy sequences do not sit in a line,
 >and the real numbers do, that the reals cannot be classes of Cauchy
 >sequences.

Slater (a while ago):

 >No, my point does not resemble Benacerraf's.  I was not saying that,
 >since there are varying definitions of the reals, no one definition
 >can claim any special precedence.

Slater (more recently):

 >One categorical difference here is that the
 >equivalence class has members while the rational number does not.
 >There is nothing on the geometric line which corresponds to the
 >inside of an equivalence class, even if the appropriate equivalence
 >classes map onto all the points on that line.  Cauchy, for instance,
 >took infinitesimals to be members of [<0,0,0,...>], but they have no
 >decimal representation.  The rational number zero has no inside.

Benacerraf (in "What numbers could not be", pp. 286-87 in his and 
Putnam's anthology):

 >I propose to deny that all identities are meaningful, in particular to 
discard all questions of the form
 >['arithmetical expression'='non-arithmetical expression'] as senseless 
or 'unsemantical' ...  Identity
 >statements make sense only in contexts make sense only in contexts 
where there exist possible
 >individuating conditions.  If an expression of the form 'x=y' is to 
have a sense, it can be only in
 >contexts where it is clear that both x and y are of some kind or 
category C, and that it is the
 >conditions which individuate things as *as the same* C which are 
operative and determine its truth value.

This does seem very close to Slater's point about whether rationals are 
comparable to equivalence classes of Cauchy sequences, unless I'm 
seriously misreading both.  Slater says that it is senseless (a category 
error) to identify rational numbers with the rational reals 
corresponding to them, because they have elements, whereas the rationals 
do not.  Benacerraf says essentially the same thing: it's just 
meaningless to ask whether the rationals have elements

This is moving the debate up a few levels.  Natural numbers are often 
identified with the Von Neumann ordinals, integers with equivalence 
classes of pairs of natural numbers, rationals with equivalence classes 
of pairs of integers, reals with equivalences classes of Cauchy 
sequences of rationals, and the complex numbers are equated with pairs 
of reals.  Slater's point seems to be found at each stage of this tower 
of equivalence-classing.  At each point in this tower, there are 
injective maps as follows:
N-->Z-->Q-->R-->C
Each map preserves both the algebraic and the metrical structure of each 
object, so that the images of the natural numbers in the complex numbers 
are as good as the Von Neumann ordinals.  Hartley, are you bothered by 
the last map in this chain of maps?  Is it a category error to call the 
complex numbers pairs of real numbers?  Explain.

A few points:
(1)  Say we look at 2 as a Von Neumann ordinal.  Then the question ``Is 
5 an element of 2?" really asks the question becomes (when properly 
interpreted in terms of the interpretations it's natural to use) ``Is 
5>2?"  The answer is obvious.
(2) Say we look at the rational reals corresponding to the rationals m/n 
and p/q, with m not equal to p, and n not equal to q.  In Slater's 
notation, this becomes, [<m/n,...>] and [<p/q,...>].  When we ask ``Is 
<m/n,...> an element of [<p/q,...>]?", what we're really asking is 
``Does n/m=p/q?"  What matters is the interpretation of a given object 
in this or that situation.  That is the sense in which the reals can be 
both equivalence classes of Cauchy sequences or Dedekind cuts on the 
rationals.  They are both models of the same theory, or even a group of 
bi-interpretable theories.
(3) These sorts of structuralist claims are in no way a challenge to set 
theory.  The claim isn't that set theory fails to provide a foundation 
for mathematics (it succeeds wildly), but rather that it fails to 
explain mathematical practice.  I think the kind of structuralism 
implicit in model theory is a good approximation thereof.
(4)  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.

- Lucas Wiman