[FOM] Real Numbers

[Lucas Wiman]

In two recent postings, Hartley Slater accuses not only me, but a large 
number of authors of set theory and real analysis textbooks of making a 
systematic category error regarding the definition of the real number by 
means of Cauchy sequences (or presumably Dedekind cuts, or any other set 
theoretic account of the reals).  As I understand him, this is 
essentially the same argument that Paul Benacerraf made in his famous 
paper "What Numbers Could Not Be."  In this paper, 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 believes this because a class of Cauchy 
sequences is not comparable with rational numbers, but only with an 
equivalence class generated by what he calls a rational real (like 1/2, 
1/2, 1/2,...).  A few points:

Why can't equivalence classes of Cauchy sequences sit in a line?  I can 
certainly imagine identifying real numbers with sequences of rationals 
converging to them (which do sit in a line), and hence why not look at 
the real as being the same as the class of all of them?  There is a 
quite attractive duality here.  We certainly do something like this in 
projective geometry--one can think of a point (possibly ideal) as being 
the same as all lines which contain it.  This leads to a beautiful 
point-line duality, which allows us interchange point and line in 
theorems of projective geometry.

When mathematicians (myself included) say equal or the same, we often 
mean isomorphic.  Philosophers frequently miss this point.  
Mathematicians don't care about these sorts of "category errors" 
precisely because they are not errors.  Thus when people talk about 
getting the same permutation group (on n elements) no matter what 
n-membered set is used, they don't mean that all these groups are equal 
in the philosophical sense.  Rather they mean that the group 
theoretically relevant aspects of them are equal.  So when speaking 
about group theory, why say they're different?  Much the same thing 
happens here.  The differing axiomatizations of the reals become clearer 
when viewed in this light.  An axiomatization for the reals in terms of 
infinite strings of binary numbers shows the computational aspects of 
the reals.  Cantor's axiomatization (above) shows the sequential 
completeness, Dedekind's shows the least upper bound axiom.

Gian-Carlo Rota, David Sharp and Robert Sokolowski wrote an interesting 
article on this topic (reprinted in Rota's book Indiscrete Thoughts).  
Here is a quote from the introduction:  "The items of mathematics, such 
as the real line, the triangle, sets, and the natural numbs, share the 
property of retaining their identity while receiving axiomatic 
presentations which may vary radically.  Mathematicians has axiomatized 
the real line as a one-dimensional continuum, as a complete Archimedean 
ordered field, as a real closed field, or as a system of binary decimals 
on which arithmetical operations are performed in a certain way.  Each 
of these axiomatizations is tacitly understood by mathematicians as an 
axiomatization of the *same* real line.  That is, the mathematical item 
thereby axiomatized is presumed to the *same* in each case, and an 
identity is not questioned."  Near the end:  "Mathematicians are still 
discovering new axiomatizations of the real line.  Such a wealth of 
axiomatizations shows that to the mathematician there is only *one* real 
line.  ...  The need for further axiomatizations is motivated by the 
discovery of further properties of the real line.  The mathematician 
wants to find out what *else* the real line can be.  He wants ever more 
perspectives on the same real line."  Rota et al go on to define a 
pre-axiomatic grasp of a subject, which allows the mathematician to see 
mathematical objects defined in different ways as the same.  On most 
points I agree with the conclusions of this article, and I highly 
recommend it to anyone interested in the topic of mathematical identity.

So, yes, I agree with Slater.  The real numbers are not equivalence 
classes of Cauchy sequences.  They can be, but they can also be Dedekind 
cuts, binary numbers, real closed fields, one dimensional continua, or a 
number of other things.

Regarding Slater's commentary on Dedekind, the creative power postulated 
by Dedekind seems to be very similar to Rota et al's pre-axiomatic grasp 
of a concept, though of course the language is a bit different.

- Lucas Wiman