Dieudonne on Dedekind

[John Baldwin]

To complement Hendrik's remark.  I had the unfortunate task my first year
of teaching of
being required to teach a course for future teachers that developed the
reals via Cauchy sequences.
The silver lining is that I never made that mistake again.
John T. Baldwin
Professor Emeritus
Department of Mathematics, Statistics,
and Computer Science M/C 249
jbaldwin at uic.edu
851 S. Morgan
Chicago IL
60607


On Tue, Aug 10, 2021 at 7:24 PM Hendrik Boom <hendrik at topoi.pooq.com> wrote:

> On Mon, Aug 09, 2021 at 03:43:43PM -0400, Colin McLarty 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.
>
> Both my wife and I took "Basic Concepts of Mathematical Analysis"
> in the first year of the honours mathematics program at
> the university of Manitoba in th 1960's.
>
> But we entered the program at different times, and got it taught in
> different ways.
>
> I got the axiomatic definition of the reals as a coplete ordered field.
>
> She got the construction of the reals out of the integers.  I don't know
> whether the reals were made out of Cauchy sequences or out of Dedekind
> cuts.
>
> The course I got was clear and comprehensible.
> The course she got turned her away from mathematics.
>
> I don't know if this is the kind of reason Dieudonné considered the
> approach useless ...
>
> -- hendrik
>
> >
> > This is from "Devons-nous enseigner les " mathématiques modernes " ?"
> Jean
> > A. Dieudonné, Bulletin de l’APMEP n° 292 de février 1974  on line at
> >
> >
> https://nam04.safelinks.protection.outlook.com/?url=http%3A%2F%2Fmichel.delord.free.fr%2Fdieudonne-1974.html&data=04%7C01%7Cjbaldwin%40groute.uic.edu%7Ca8490c392cab472f297908d95c5e58e9%7Ce202cd477a564baa99e3e3b71a7c77dd%7C0%7C0%7C637642382571975151%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=cv%2BopHTOlzULoQcAssj%2BoMnC%2Bd1NWwyrfhSZUBQpvZU%3D&reserved=0
> >
> > 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:
> > >
> https://nam04.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.philosophy.utoronto.ca%2Fdirectory%2Fjames-robert-brown%2F&data=04%7C01%7Cjbaldwin%40groute.uic.edu%7Ca8490c392cab472f297908d95c5e58e9%7Ce202cd477a564baa99e3e3b71a7c77dd%7C0%7C0%7C637642382571975151%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=M%2FasCWdoAUFWE8%2BUtVwoEahlEFbcijHNWjs9InF2PI8%3D&reserved=0
> > > Home page:
> https://nam04.safelinks.protection.outlook.com/?url=http%3A%2F%2Fwww.chass.utoronto.ca%2F~jrbrown%2Findex.html&data=04%7C01%7Cjbaldwin%40groute.uic.edu%7Ca8490c392cab472f297908d95c5e58e9%7Ce202cd477a564baa99e3e3b71a7c77dd%7C0%7C0%7C637642382571975151%7CUnknown%7CTWFpbGZsb3d8eyJWIjoiMC4wLjAwMDAiLCJQIjoiV2luMzIiLCJBTiI6Ik1haWwiLCJXVCI6Mn0%3D%7C1000&sdata=EKuosBE%2BOXc00heLggYmPX1sB8LWmQg%2FjOefR4UNkmc%3D&reserved=0
> > >
> > >
> > >
> > >
> > >
> > >
>
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