[FOM] RE: Real Numbers
sean.stidd@juno.com
sean.stidd at juno.com
Fri May 9 10:25:20 EDT 2003
--- John Pais <pais at kinetigram.com> wrote:
"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."
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Professors Pais and Makarov are correct to note that the conception of the reals given by the theory of ordered fields is more central to mathematical practice than the conception provided by the Cauchy sequence or Dedekind cut constructions of them. But good standard textbook expositions, including the excellent one Pais provides quotations from, do not provide an explanation of the use of the definite article in this context.
"The" impllies that there is a single collection of objects or structure which we are to identify with the real numbers; this is inconsistent with the claim that real numbers are "any consistent isomorphic copy" of anything, which implies a multiplicity, or rather allows for one, which possibility is in fact realized in the case of the reals by way of our different constructions of them. There is a surface contradiction here which at least some philosophically minded persons would like to see resolved, even after getting through courses in analysis.
The basic options for its resolution would seem to be:
1. Say that the reals are some particular collection of objects, which are characterized up to isomorphism by the ordered field definition. In this case one has a second choice between
1a. The view that the reals ARE Dedekind cuts, Cauchy sequences, etc. This is the position of some set-theoretic foundationalists: reals are defined in terms of rationals are defined in terms of the integers, naturals, etc. which in turn are sets. Of course if you like a different kind of foundation than ZFC you may also hold 1a but with a different collection of objects as your basis. Virtues: one can clearly specify which things one is talking about in terms of a theory which has been shown able to duplicate the vast majority or all of ordinary mathematics so far. Flaws: such identifications seem essentially arbitrary, for reasons similar to those Benacerraf brought up, at every level: which sets are the natural numbers? which constructions in the rationals are the reals? etc. Also, there is the pedagogical objection that Pais raises, that they lead the student to focus on features of the reals irrelevant to the working analyst's understanding of them.
1b. The view that the reals are a particular collection of objects, not essentially reducible to some more fundamental sort of object, to which various constructions in other object-domains are isomorphic. This seems to be the most natural way to interpret the working mathematician's language, but it is not consistent with the IDENTIFICATION of the reals with "any consistent isomorphic copy" of those objects. If one accepts 1b one should rather say that the Dedekind cut/Cauchy sequence extensions of the rationals produce an object-domain which is isomorphic to the reals, but which is not identical to them. Virtues: This view seems most responsive to the way mathematicians normally talk about their subject-matter. Flaws: Set theory (or your other favorite foundation) loses some of its luster as the one true theory which characterizes the only, unique objects that mathematics is about. Note however that accepting 1a does not entail that set theory loses one kind of central importance to mathematics; it just becomes a lingua franca rather than a lingua characterica.
2. Structuralists by contrast have a technique for accepting the use of "the" and the "any consistent isomorphic copy" together in the way that Pais does, by claiming that the real numbers are to be identified with the structure-up-to-isomorphism that all these different specifications of object domains share. On this view all the various set-theoretic and analytic defintions pick out real number systems, which are real number systems by virtue of having the real number structure; and it is that structure itself to which the phrase 'the real numbers' refers. This in turn implies that there aren't any object-domains which essentially are the reals. What 'the real numbers' are is a particular structure up to isomorphism, and anything that has that structure can serve as an exemplar of it. Virtues: Preserves the 'intuitive' philosophy of mathematics of many working mathematicians that they are both talking about unique objects (and thus entitled to the definite article) and that the only thing that matters about those objects is their structure up to isomorphism. Flaws: Status of structuralist foundations unclear; creates ugly, counterintuitive cross-theoretic identifications.
3. Various 'anti-realist' alternatives.
Pais and Makarov seem to want to reject 1a but Pais' summary seems to me to waffle between 1b and 2, and therefore not to count as a 'clear' answer to at least one kind of philosophical question that arises in this context. It is the best way to give the student-analyst the materials they need to go on, probably, but that does not mean that it gives complete understanding of what is being talked about and how we should think about it.
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