Dieudonne on Dedekind

[JOSEPH SHIPMAN]

I still don’t get it. Everyone knows the “infinite decimal expansion” definition coming in, and assuming the student is competent enough to understand the .99999... .00000 issue, the least upper bound property takes one easy class period to establish to everyone’s satisfaction (you construct the LUB one decimal place at a time), and the Dedekind construction can be easily seen to be the same except using all rationals except rationals with power-of-10 denominator.

Cauchy sequences are a whole other level and I get that you reserve them for advanced classes, but why is Dedekind a problem?

— JS

Sent from my iPhone

> On Aug 10, 2021, at 11:52 PM, John Baldwin <jbaldwin at uic.edu> wrote:
> 
> 
> 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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