[FOM] real numbers

Hartley Slater slaterbh at cyllene.uwa.edu.au
Thu May 1 06:40:55 EDT 2003

Further to my point (FOM Digest, Vol 4, Issue 35) about the category 
mistake involved in conflating rationals with rational reals, I have 
come across some historical material on the parallel issue when reals 
are defined in terms of Dedekind cuts (see 'From the Calculus to Set 
Theory, 1630-1910', ed. I. Grattan-Guinness, Duckworth, London, 1980, 

>Many authors who adopted Dedekind's basic ideas preferred not to 
>follow him in defining the real numbers as creations of the mind 
>corresponding to cuts in the system of rational numbers.  In the 
>Principles of Mathematics, Bertrand Russell emphasised the advantage 
>of defining the real numbers simply as ...segments of the 
>rationals....  But Dedekind had his reasons...for defining the real 
>numbers as he did.  When Heinrich Weber expressed his opinion in a 
>letter to Dedekind that an irrational number should be taken to be 
>the cut, instead of something new which is created in the mind and 
>supposed to correspond to the cut, Dedekind replied "We have the 
>right to grant ourselves such a creative power, and besides it is 
>much more appropriate to proceed thus because of the similarity of 
>all numbers.  The rational numbers surely also produce cuts, but I 
>will certainly not give out the rational number as identical with 
>the cut generated by it; and also by introduction of the irrational 
>numbers, one will often speak of cut-phenomena with such 
>expressions, granting them such attributes, which applied to the 
>numbers themselves would sound quite strange".

If the reals are to be comparable to the rationals, i.e. to be less 
than, equal to or greater than them, as the case may be, then 
Dedekind was entirely right, since a cut cannot be less than a 
rational, even if all the members of that cut are less than it. 
Russell's definition ignores the category mistake in, for instance, 
'{x|x < 2} < 2' and '{x|xsquared < 2} < 2'.  There is therefore a 
further assumption involved in getting from the mathematics of cuts 
to the mathematics of the supposed items corresponding to cuts which 
are comparable to the rationals.  Dedekind gave himself the 'creative 
power' to generate the 'something new' certainly required - and it 
looks like he did so expressly to avoid a category mistake, as the 
last part of the above quote nicely illustrates.  But, of course, 
whether the assumption of the supposed items, or the granting of 
Dedekind's 'creative power' is defensible is a different, and very, 
very interesting question.

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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