Dieudonne on Dedekind

[Colin McLarty]

>
>
> It’s not clear to me what Dieudonne’s complaint is, because any issues
> with Dedekind’s definition are equally applicable to the high school
> definition. How, and when, does he want the teaching to be different?
>


He wants undergrad students to know the real numbers have the least upper
bound property.  But he does not want them taught that somehow you begin
with sets of rational numbers and then derive the property for sets of
reals from that.  Note that historically Dedekind did base his account of
real numbers on the least upper bound principle, while he
explicitly refused to identify real numbers as cuts on the rationals.

And Dieudonne wants undergrads taught (a great deal, actually) about
convergent sequences and series of real numbers, obviously including the
Cauchy convergence criterion.  But he does not want them taught that real
numbers are defined by Cauchy sequences of rational numbers, with the
theory of Cauchy sequences of real numbers as a consequence of that.

Dieudonne wants the Dedekind cut construction of reals, and the Cauchy
sequence construction of them, taught to postgrad students who will make
use of them as cases of completions in topological ring theory.

Colin



> — JS
>
> Sent from my iPhone
>
> On Aug 9, 2021, at 8:58 PM, Colin McLarty <colin.mclarty at case.edu> wrote:
>
> 
> 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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