[FOM] RE: Real Numbers
pais at kinetigram.com
Thu May 8 11:21:21 EDT 2003
Victor Makarov wrote:
> In mathematics, every mathematical concept Z exists only in the context of a
> certain mathematical theory T (let us call this theory T as the host theory
> of the concept Z). For the most of mathematicians (not set-theorists) the
> host theory for the concept "real number" is the theory of the field R. For
> set-theorists the host theory is a set theory (ZFC, ZFA, NF, ... ). So
> asking. what real numbers are we should add "in what theory". And for the
> most of mathematicians the answer is well-known:
> real numbers are elements of the field R.
This posting is to emphasize and document the "well-known" part of Victor's
In contrast to what some philosophers think, it is not the case that "the sky is
falling" regarding mathematicians' conception of rational numbers, real numbers,
etc., and this is well-documented in mathematical practice and literature. In
particular, to see this one need only consult a standard text such as Walter
Rudin's, Principles of Mathematical Analysis (Third Edition), McGraw-Hill, 1976.
Here below are some excerpts:
"Experience has convinced me that it is pedagogically unsound (though logically
correct) to start off with the construction of the real numbers from the
rational ones. At the beginning, most students simply fail to appreciate the
need for doing this. Accordingly, the real number system is introduced as an
order field with the least-upper-bound property, and a few interesting
applications of this property are quickly made. However, Dedekind's construction
is not omitted. It is now in an Appendix to Chapter 1, where it may be studied
and enjoyed whenever the time seems ripe."
THE REAL FIELD
"We now state the *existence theorem* which is the core of this chapter.
1.19 Theorem There exists and ordered field R which has the least-upper-bound
Moreover, R contains Q as a subfield.
The members of R are called *real numbers*.
The proof of Theorem 1.19 is rather long and a bit tedious and is therefore
presented in an Appendix to Chap. 1."
"Theorem 1.19 will be proved in this appendix by constructing R from Q. We shall
divide the construction into several steps.
Step 1 The members of R will be certain subsets of Q called *cuts*.
"Step 8 We associate with each r in Q the set r* which consists of all p in Q
such that p < r...."
"Step 9 We saw in Step 8 that the replacement of the rational numbers [each
rational number] r by the corresponding 'rational cuts' r* in R preserves sums,
products, and order. This fact may be expressed by saying that the ordered
field Q is *isomorphic* to the ordered field Q* whose elements are rational
cuts. Of course, r* is by no means the same as r, but the properties we are
concerned with (arithmetic and order) are the same in the two fields.
*It is this identification of Q with Q* which allows us to regard Q as a
subfield of R.*
The second part of Theorem 1.19 is to be understood in terms of this
identification. Note that the same phenomenon occurs when real numbers are
regarded as a subfield of the complex field, and it also occurs at a much more
elementary level, when the integers are identified with a certain subset of Q.
It is a fact, which we will not prove here, that *any two ordered fields with
the least-upper-bound property are isomorphic*. The first part of Theorem 1.19
therefore characterizes the real field R completely."
End of exerpts.
I can't image a more elegant or pedagogically sound exposition, which clearly
explains the use of the definite article regarding *the rationals*, *the reals*,
etc. As Victor mentioned above, actual mathematical practice deals with the
theory of ordered fields, and an abstract conception of real numbers R as any
convenient isomorphic copy of a least-upper-bound complete ordered field, since
such a structure is unique up to isomorphism. This perspective is one of the
hallmarks of modern mathematics.
So, folks, there are no "category mistakes" involved here. On the other hand,
one may wonder what philosophy of mathematics, devoid of or remote from
mathematical practice, and its fundamental concepts such as morphism and
isomorphism is really all about.
E-mail: pais at kinetigram.com
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