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Re: Dear Nico B. (Not Benschop) & 5 BSM
Posted:
May 23, 2000 10:43 PM
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n_f_benschop@my-deja.com wrote:
> Pertti Lounesto <Pertti.Lounesto@hit.fi> wrote:
> > Jan Stevens wrote:
> > > Pertti Lounesto <Pertti.Lounesto@hit.fi> writes:
> > > >> > Jan Stevens wrote:
> > > >> >> Pertti Lounesto <lounesto@pop.hit.fi> writes:
> > > >> >>>
> > > >> >>> Dear Nicolas Bourbaki,
> > > >> >>>
> > > >> >>> I think that your Lemme 5, on page 151, of
> > > >> >>>
> > > >> >>> N. Bourbaki: Alg\`ebre, Chapitre 9, Formes
> > > >> >>>sesquilin\'eaires et formes quadratiques,
> > > >> >>>Hermann, Paris, 1959,
> > > >> >>>
> > > >> >>>does not hold. For a counterexample, consult
> > > >> >>>
> > > >> >>>http://www.hit.fi/~lounesto/counterexamples.htm.
> > >
> > > It would be really helpful for readers of Bourbaki if
> > > Lounesto pointed out which step of the proof in the original
> > > version is wrong. After all the case distinction dim E even
> > > or odd is made there.
> >
> > Bourbaki's mistake has been regarded significant enough
> > to comment by Deheuvels 1981, p. 355 and Moresi 1988, p. 621,
> > in R. Deheuvels: Formes quadratiques et groupes classiques,
> > Presses Universitaires de France, Paris, 1981, and
> >
> > R. Moresi: A remark on the Clifford group of a quadratic
> > form; pp. 621-626 in Stochastic Processes, Physics and
> > Geometry, Ascona/Locarno, 1988.
> >
> > As for locating the erroneous step of the proofs, I have never
> > commented on that, because the validity of a counterexample to
> > a theorem/lemma does not depend on its proof. [*]
>
> ...[*]... But would it not increase insight into the matter?
Yes.
> Which, after all, is the purpose of scientific / maths research!
Yes.
> That is: to learn from mistakes.
> Or in your words: to improve our cognitive structures ;-)
But, engaging in a discussion about error in a proof,
instead of error in a theorem, results easily into an
endless debate about possible interpretations and
evasions from the part of the mistake-maker, while
the mistake-maker might have had other goals than
those perceived by the error-detector. One can
avoid fruitless debates about possible interpretations
of the mistake-maker by focusing on the error in the
theorem, and giving a counterexample which satisfies
all the assumptions of the theorem without the
conclusions being valid.
For the art of falsifying theorems with counterexamples
see http://www.hit.fi/~lounesto/counterexamples.htm.
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