Dieudonne on Dedekind

[JOSEPH SHIPMAN]

That raises a more general issue.

In high school we learn of the reals as decimal expansions with the obvious equivalence for .99999... .00000..., the least upper bound property is stated and used, and no one complains because it’s sort of obvious how you construct the least upper bound one decimal place at a time.

Dedekind says “a real is a nonempty downward closed set of rationals with nonempty complement and the obvious equivalence if there is a top element”, and it’s not hard to make it intuitively clear that if you replace rationals with “rationals with denominator a power of 10” it works exactly the same as the previous definition involving infinite decimals.

In both cases, the point of the definition is simply to show that a complete ordered field exists and that you can do calculations with it to as much precision as you need.

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?

— 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/ 
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>> 
>> 
>> 
>> 
>> 
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