Subject: [HM] Weierstrass, Boussinesq, Cauchy...
From: Thierry Guitard (guitard@attglobal.net)
Date: Sat Sep 02 2000 - 18:57:58 EDT
Actually, the "correspondance" of Saint Venant-Boussinesq is in the
library of Institute of France, not in the Archives, I had read it
and have the cited letter by Z. under my eyes ( as I made xerox
copies of the main letters ) and as I explained at Gottingen and at
Paris ( seminaire ulm ) from 1991 to 1994, using these unpublished
letters , the significance of the polemic with Bertrand and the role
played by saint venant-Boussinesq on the question of singular
solutions can only be deconstructed in the Cauchy's legacy ! As
Bertrand didn't saw Peano phenomenum but two explicit solutions , to
compare to Cauchy lectures in 1824 or in the Prague dissertation
where he used the continuity of the solutions as a function of
initial data as a proof of unicity to prove that the solution is
unique ( so it is the analytical one )
Generally speaking, the philosophical and epistemological record of
historians on determinism and Laplace daemon, did not properly
understand the importance of the construction and the unicity in the
existence proof : what is a "trajectoire" ?
Perhaps newton understood it against the opinion of scholars as De
Gandt who believed that the first solution of the two body problem,
was by Bernouilli...but if we read some sample of physics course,
nowadays, ?
And even the locus classicus of Laplacean determinism is not
contextualized : Laplace had from 1871 a proof of unicity of the
solution of the differential equation who was taught by Lagrange or
Poisson, founded on a hidden lemma ( if all the derivative of any order
are null...) Dahan as Israel etc...in their books said there was no
proof before Cauchy of existence and unicity of functions as solution
of Differential equations after other affirmed it : it is patently
false as
1st ) D'Alembert gave a "proof" ( and if he gave an "argument" who
could be read nowadays with "esprit critique", he posed the question
! to be found in his published papers ! )
2nd) Cauchy was the first to prove the so called
Lipshitz-Cauchy-Picard theorem as he refuted the Laplace proof of
unicity ! and why did you made a new proof of a very well known
result in the cursus ? because the Laplacean was at his eyes faulty :
he had a counterexample...from 1816 ! the sequence of derivates equal
to zero , the Taylor Formula ( so called by Lagrange after he
discovered singular solutions in Taylor 1715 ) etc... suffices to
read the " Calcul des fonctions" or Poisson's 1806 ;
en passant, Poisson in his review of Cauchy 1814 made a remark
about the link he saw between the improper integrals used by Cauchy
and a problem of mechanic equilibrium, from his mechanics lectures ,
giving to Cauchy a program of research he worked in 1815-16 ( the
"annee sans ete" with a chaotic weather caused by a volcano ) and
well documented by himself in a mathematical description of his
sequence of papers in Turin , 1830 . Deakin paper in AHES on the
prehistory of dynamical system is a good introduction to the modern
point of view ( and Boussinesq reading of Poisson is documented ) but
didn't discussed the historical context : singular soltions in the
sense of Euler, Laplace and Lagrange with a historical bifurcation :
differential algebra ( Ritt, ideal theory and the error on the
radical ideal, Kolchin ) and theory of enveloppes ( Thom, Arnold,
Bruce, Demidov etc... ) who never meet again ( problems are not
always solved but forgotten ? and why ? not a whiggish history )
Cauchy knew very well his own history even it was not the standard
rewrote.
As for Boussinesq proper understanding,there was
1st) the polemic with Laplace and Poisson circa 1820 ( the lectures
on singular solutions censored in 1824 and published by Moigno in
1842 ; a oral transmission of Cauchy's lectures as the correpondance
between Darboux and Houel proved explicitly but as the dismissing of
the so-called Blanchet criterium ( Osgood or Tamarkine nowadays ) by
Darboux gave a striking fact, a Lagrangean weltanshung on this
question until late in the century ( Peano...)
2nd) after the Darboux-Catalan exchange about the same intrigate
problem ( reinventing duality as in the Monge transformation so
called Legendre , but first by use of "polaire reciproque" ) , a
first polemic on the generality of the singular solution as enveloppe
( Darboux's paradoxe ), one century before Thom's program of solution
of the problem in the sense of Lagrange-Darboux-Thom, not of
Euler-Poisson-Ritt in Cartan Seminar in 1971 ( solved by Demidov , a
student of Arnold only ten years ago, as he established the tool : the
normal forms of the germs ),
a new polemic on singular solutions after the one in 1821-1824, the
1871 papers in Bulletin and CRAS, the Boussinesq paper rejected by
Bertrand :
3rd) the problem of determinism inherited from Cauchy work on what is
a solution of a differential equation...a problem who was the origin
on real versus complex analysis as Barbeau gave a good picture in a
HM paper ( non analytic but infinitely differentiable functions in
the real sense : as you can read, in Cauchy's paper in 1821, it was
the problem of unicity ( so you must have a concept of solution in
modern sense ...and as Volkert deconstructed it , the monsters had
their origin ( even in Bolzano who read Ampere's proof of mean value
thorem ) in the simple problem : if y'=0, prove that y is a
"constant"
the Henstock-Kurzweil integral gave us the tools to deconstruct the
Riemann integral as Cauchy solved only differential equation and
taught improper integrals to study singular solutions ( and case of
non unicity ) in scale of infinity ( the origin of du bois reymond
work ( Fisher ) is Cauchy's chapter on scale of infinitesimals used
in his so-called osgoodean criterium ); the Work of Volkert in AHES
proved my history , even if he didn't saw the link between the
determinism and the mean value theorem in his two papers in AHES on
monsters in the sense of Weierstrass or Levy ( who knew Perrin ), the
Bolzano function, and the proof of the mean value theorem by Binet,
Ampere etc..
The letter of Boussinesq to his "dear master" was from one who
elected Saint Venant to be his master in philosophy and choose to be
his "pupil" in mathematics also . It was my archaic use of the word
"student" perhaps you prefer "pupil" as Go Sei Gen was a pupil of
Kitani: I knew very well Picard's "eloge" and the nachlass of
Boussinesq...and some other reference as Marie Joe Nye or Hacking ,
and also the book of Paul on French thomists ( the Cauchy club held
by Mansion, Gilbert ...) not in the sense of the Thom of Largeault in
our recent contemporary context.( and even the Catholic Encyclopedia )
As for W., the main problem is to understand he discovered the
approximation as Boussinesq exemplified it in studying the "lissage"
of initial "arbitrary" conditions ( a continuous function ) in an
analytical one in the steady state of the solutions in two variables
of "equation de la chaleur dans une barre ": it was a physical
problem who gave W. the idea of his proof using a kernel as Laplace
invented it circa 1810 ( and Fourier borrowed it )...An account in
Barbeau show the heritage in the problem of what is analycity in
several variables...
I didn't understand the question "Was it in your book on fractals?" ;
there is several papers of B.M. on the subject ( SIAM Rev 1968 ) ,
well known, ...and other monsters constructed by iteration as the
Kieswetter's ( I made a biblio long ago for a dissertation of one of
my students ) and on this subject, the recent literature is important
( see the American Monthly ) : the link with the work of Besicovich
is explained in Fractal functions, fractal surfaces and Wavelets by P
Massopust
Accidentally, some HMer wrote on Argand and Legendre : the proof of
Legendre in his theory of numbers ( 06 ou 08 )( and as in his
exercices de calcul integral, reading it is a journey in a foreign
country ) didn't give credit to Argand booklet, but as it could be
found in reading the text, there was a new idea : the use of Newton
method ( so the Argand descent was translated analytically ) a new
notion : module ( at this time there was no absolute value for a
real, W. invented it several years after...a historical significant
order indeed ), and it was a step before Cauchy's...( two
publications on this proof after one translating Gauss thesis, in the
second, Legendre was credited )and before Cauchy's concept of arctan
( the break of 1017 ) and the modern trigonometry ( problem of
continuity of primitives and constants of integration, again...the
Cauchy's "idee fixe")
the notion of "argument" was the next step , read Cauchy 1821, as
said Borel one hundred year after ( and he read his definition of
order of infinitesimals as DBR or Hardy, a theory put by Bourbaki in
exercises !
hoping to help , with a question : what is the history of implicit
function theorem ?
-------------------------- Message d'origine --------------------------
De : Martin Zerner
Envoye : vendredi 1 septembre 2000 12:38
A : historia-matematica@chasque.apc.org
Objet : Re: [HM] Weierstrass
Benoit Mandelbrot wrote:
>
> First things first: I am a faithful reader of HM and am immensely
> grateful to Julio Gonzalez Cabillon for undertaking this task.
> The internet begat many sins but also made narrowcasting workable
> and created new invisible colleges of great value. This is one and
> I regret that time is too short for me to contribute regularly.
>
And it gives an opportunity to talk once more, a long time after we
met in the countryside of Aix! I am specially interested in this
question of "strange" functions and devoted some work to it. However
this discussion brought me much information I did not know.
I remember having read somewhere that Jean Perrin, in his book "Les
Atomes" (1912), remarked that, in view of the discontinuity of matter,
singular functions might be more relevant than regular ones in the
study of some physical phenomena. Was it in your book on fractals?
> Secondly, the Weierstrass function W was largely forgotten in the
> broad community of mathematicians (as witnessed by previous
> correspondence) and J.R. Newman's "World of Mathematics" has a
> graph of a different function.
This means that at that point at least Emile Borel, a close friend of
Perrin was aware.
However in the 1870s, the examples of Hankel were much more widely
known than Weierstrass', in spite of the fact that the latter is much
stronger.
See:
Darboux G. 1875 Memoire sur les fonctions discontinues Annales
E.N.S. (2) 4 p.57-112 (1875)
and
Dini U. 1878 Fondamenti per la teorica delle funzioni di variabili
reali Pise : Nistri
One point I do not find in this discussion is what I thought to be
first publication of Weierstrass' example by Dubois-Raymond in 1875,
with the permission of W. and due credit to him. I think the reference
should be in:
Dugac P. 1973 Elements d'analyse de Karl Weierstrass Arch. for
Hist. of Exact Sc. 10 p. 41-176.
One mathematician at least paid very soon attention to the
implications of these examples to physics, be it in a way which seems
strange to us.
He feared that the existence of non differentiable functions would
induce engineers to drop the study of mathematical analysis and
mathematicians to give up any interest in applications. As a remedy
he proved a weaker form of Weierstrass approximation theorem, where
polynomials are replaced by m times differentiable functions.
The proof should be as an appendix in:
Boussinesq J. 1878 Conciliation du veritable Determinisme mecanique
avec l'Existence de la Vie et de la Liberte morale. Precede d'un
rapport de M. Paul Janet a l'Academie des Sciences morales et
politiques. Gauthier-Villars, Paris
This he explains in a 1878 letter to Saint Venant (this correspondence
is kept in the library of the Institut de France).
Incidentally, Thierry Guitard wrote on this list that Boussinesq was
a student of Saint Venant. This is true inasmuch as he considered him
as his master, but he never studied under him. See my article:
M. Zerner Origine et reception des articles de Boussinesq sur le
determinisme, Contra los Titanes de la Rutina / Contre les Titans de
la Routine (sous la direction de S. Garma, D. Flament et V. Navarro),
ed. Comunidad de Madrid 1994, p. 319-333 and the references therein.
Later, interest in the Weierstrass function is testified by the fact
that it is in the second edition (1893) of the first volume of Jordan's
"Traite d'analyse". I do not remember whether it also is in the appendix
of volume 3 of the first edition (1887). Characteristic of his interests,
Jordan proves that the function is nowhere differentiable but also that
it is not of bounded variation. The function can also be found in
Goursat's "Cours d'Analyse mathematique", the most widely used textbook
on the subject in France in the first half of this century.
Martin Zerner
11bis rue Charbonnel
F75013 Paris
(33) 1 53 62 03 49
zerner@paris7.jussieu.fr
This archive was generated by hypermail 2b28 : Sat Sep 02 2000 - 19:45:45 EDT