[FOM] Real Numbers
Hartley Slater
slaterbh at cyllene.uwa.edu.au
Sat May 3 01:24:41 EDT 2003
Lucas Wiman (FOM Digest Vol 5 Issue 4) has got me wrong, in a number
of ways. He says
>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.
>....... 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.
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. In my posting about the Cauchy
definition (FOM Digest Vol 4 Issue 35) I was just speaking about the
Cauchy definition (see again 'Axiomatic Set Theory' by Suppes,
specifically p181). And I have no objection to that definition other
than the fact that it *alone* does not produce items which are
comparable to rationals. I argued that a category mistake is
involved in thinking classes of Cauchy sequences *themselves* are
comparable to the rationals. Specifically, it is the rational reals
which can be less than, equal to or greater than other reals, and a
rational real is not in the same category as a rational - so one
cannot, for instance, have,: '2=[<2,2,2,...>]' (in Suppes'
symbolism, '<...>' forms a sequence, and '[...]' the associated
equivalence class). If further argument about the impossibility of
this identity is required, consider that unless the rational number 2
was defined differently (as indeed it is in Suppes, see p110), one
could not go on to define the infinite sequence within the rational
real in terms of it, as in Suppes' definition 56, p182.
>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.
No, it's not a matter of some pre-axiomatic grasp of a concept. The
Dedekind material from Grattan-Guinness, in FOM Digest Vol 5, Issue
2, was useful because it paralleled my point, using a different
definition of the reals. Dedekind appreciated that to get something
comparable to the rationals, in connection with his cuts, you could
not *just* use the cuts. You had to have something 'corresponding
to' the cuts as well; since even a rational number was 'not identical
with the cut generated by it'. The full quotation showed people
around his time did not like his assumption of some 'creative power'
to get these other entities, and indeed there is one place where he
backed down on that, as Grattan-Guinness records (p224 again):
>in a letter to Lipschitz [Dedekind] remarked parenthetically that if
>one does not wish to introduce numbers in his sense, "I have nothing
>against it; the theorem I prove [on completeness] then reads: the
>system of all cuts in the discontinuous domain of rational numbers
>forms a continuous manifold"
Russell's identification of cuts with numbers, and Dedekind's
proposed creation of 'corresponding' numbers, would both have enabled
the theorem to be upgraded to "the system of all numbers which
consist in, or correspond to, cuts in the discontinuous domain of
rational numbers forms a continuous manifold". But Dedekind backed
down to the safety of just saying something about cuts (and so,
apparently, he was not committed to an extension of the number
system).
To make my point again, by means of another example:
No equivalence class is comparable to any rational number, since they
are in different categories. Suppes, for one, makes an associated
category mistake when giving a proof which tries to show that, for
instance, every real number is representable as a decimal. Maybe it
is so *representable*, but his generalised proof, for any integer
radix r greater than 1, assumes, for a start (p190), that, given a
real number x, there is an integer a such that ar is equal to or less
than xr, making a the largest integer equal to or less than x. Here
'x' cannot be an equivalence class of Cauchy sequences of rationals,
i.e. something like '[<...>]', otherwise it would not be in the right
category. So the proof Suppes provides is faulty.
Sandy Lemberg writes (in this issue?):
>Your view of Q<R as fundamentally different from Q in and of itself, and
>that identification of the two Q's is a "category mistake", misses the
>point. It is all pervasive in mathematics to make such identifications,
>again because it is the structure and not any characteristics of its
>elements or "points" which is of interest. Thus, such identifications are
>not "category mistakes" at all, but rather commonplace mathematical
>practice.
So cuts are equivalence classes? It would be a mistake to identify
them. Following on from the above, there are three relevant
structures, one containing equivalence classes, one containing cuts,
and one containing rational numbers. Certainly the first two are
isomorphic. Dedekind's point was that the latter two were not
provably isomorphic without a further assumption, since one needs
elements in the third corresponding to irrational cuts. The category
mistake he pointed out consisted in identifying elements in the
second structure with elements in the third. The category mistake I
pointed out repeated his in the case of the first and third
structures.
--
Barry Hartley Slater
Honorary Senior Research Fellow
Philosophy, School of Humanities
University of Western Australia
35 Stirling Highway
Crawley WA 6009, Australia
Ph: (08) 9380 1246 (W), 9386 4812 (H)
Fax: (08) 9380 1057
Url: http://www.arts.uwa.edu.au/PhilosWWW/Staff/slater.html
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