Dieudonne on Dedekind

Tue Aug 10 09:41:56 EDT 2021

It is worth mentioning though that, Dieudonne specifically meant Dedekind
cuts, and Cauchy sequences of rationals, have "have no bearing whatsoever
on *analysis*."  He did use them, and their analogues, in number theory.
Speaking of the whole arithmetization of analysis, from natural numbers to
real numbers, he said:

Moreover, this boring mathematics gave birth to at least two important
> ideas (another example of the phenomenon mentioned above). The first, of
> algebraic nature, is the symmetrization which leads from a "semi-group" to
> a group, the value of which has been measured very recently in the
> definition of "Grothendieck groups" in K-theory; the other, "completion",
> although topological in nature, is now used mostly in algebra where it
> serves as a powerful tool leading to the definition of p-adic numbers and
> similar "topological rings". These applications, however, are well above
> the pre-graduate student level of universities, and I share Thom's view
> that Dedekind's traditional "cuts" or analogous ways of "defining" real
> numbers are completely unnecessary and even harmful at this level.

Par ailleurs, ces mathématiques ennuyeuses ont donné le jour au moins à
> deux idées importantes (un autre exemple du phénomène mentionné ci-dessus).
> La première, de nature algébrique, est la symétrisation qui conduit d'un
> "semi-groupe" à un groupe, dont on a mesuré très récemment toute la valeur
> dans la définition des "groupes de Grothendieck" en K-théorie ; l'autre,
> "la complétion", bien que de nature topologique, est utilisée à présent
> surtout en algèbre où elle sert d'outil puissant conduisant à la définition
> des nombres p-adiques et des "anneaux topologiques" similaires. 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.

http://michel.delord.free.fr/dieudonne-1974.html

Colin

On Mon, Aug 9, 2021 at 9:21 PM Alexander M Lemberg <sandylemberg at juno.com>
wrote:

> 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
> > à
> > > 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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