Dieudonne on Dedekind

[Alexander M Lemberg]

>From "Foundations of Modern Analysis" (p 16 of Elsevier edition):

"The material in this chapter is completely classical; the main
difference
with most treatments of the real numbers is that their properties are
here
derived from a certain number of statements taken as axioms, whereas in
fact these statements can be proved as consequences of the axioms of set
theory (or of the axioms of natural integers, together with some part of
set
theory, allowing one to perform the classical constructions of the
"Dedekind
cuts" or the "Cantor fundamental sequences"). These proofs have great
logical interest, and historically they helped a great deal in clarifying
the
classical (and somewhat nebulous) concept of the "continuum". But they
have no bearing whatsoever on analysis, and it has not been thought
necessary
to burden the student with them; the interested reader may find them in
practically any book on analysis; for a particularly lucid and neat
description,
see Landau [16]."

Sandy

On Mon, 9 Aug 2021 15:43:43 -0400 Colin McLarty <colin.mclarty at case.edu>
writes:
> He may have said this.  But what I know he said is that Dedekind 
> reals were
> useless, compared to the axiomatic definition of the reals as a 
> complete
> ordered field, for students before the "troisieme cycle 
> d'university."   I
> am not sure exactly what that meant in France in 1974, but it seems 
> to have
> meant something like graduate students.
> 
> This is from "Devons-nous enseigner les " mathématiques modernes " 
> ?" Jean
> A. Dieudonné, Bulletin de l’APMEP n° 292 de février 1974  on 
> line at
> 
>  http://michel.delord.free.fr/dieudonne-1974.html
> 
> Ces applications, cependant, sont bien au-dessus du niveau de 
> l'étudiant
> > avant le troisième cycle des universités, et je partage 
> l'opinion de Thom
> > que les "coupures" traditionnelles de Dedekind ou les façons 
> analogues de
> > "définir" des nombres réels sont parfaitement inutiles et même 
> nuisibles à 
> > ce niveau. ...  Mais je pense qu'il ne peut être que profitable 
> Ã 
> > l'étudiant de posséder une liste précise des propriétés 
> fondamentales des
> > nombres réels qu'il utilisera constamment en analyse et c'est ce 
> que l'on
> > appelle un "système d'axiomes des nombres réels"
> 
> 
> Colin
> 
> 
> On Mon, Aug 9, 2021 at 2:52 PM James Robert Brown 
> <jrbrown at chass.utoronto.ca>
> wrote:
> 
> >
> > Several years ago I read something by Jean Dieudonne where he said
> > (perhaps only in passing) that he wanted to reject Dedekind’s 
> theory of
> > real numbers because it was fruitless; it was not generating any 
> new
> > research.
> >
> > Does anyone know of an article by him (or anyone else) along this 
> line?
> >
> > Many thanks,
> >
> > Jim
> >
> >
> > *************************
> > James Robert Brown
> > Professor Emeritus
> > Department of Philosophy
> > University of Toronto
> > Toronto  M5R 2M8
> > Canada
> > Home: 519-439-2889, Cell: 519-854-0131
> > Philosophy Dept. page:
> > http://www.philosophy.utoronto.ca/directory/james-robert-brown/
> > Home page: http://www.chass.utoronto.ca/~jrbrown/index.html
> >
> >
> >
> >
> >
> >


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