[FOM] constructive cauchy
S. S. Kutateladze
sskut at math.nsc.ru
Fri Nov 9 23:26:49 EST 2007
11/10/2007, you wrote to me:
Gabriel Stolzenberg> It is said that Cauchy, in his argument in Cours d'Analyse about
Gabriel Stolzenberg> a limit of continuous functions being continuous, say, on [0,1],
Gabriel Stolzenberg> failed to see that the convergence must be
Gabriel Stolzenberg> uniform....
Gabriel Stolzenberg> Nonstandard analysis readings of this argument go back at least
Gabriel Stolzenberg> to John Cleave in around 1970. Here I wish to give a very sketchy
Gabriel Stolzenberg> first draft of a constructive reading of it.
A. Robinson in his book of 1961 explained the background of the matter
within nonstandard analysis. Cauchy described the functions under study as follows:
``An infinitely small increment given to the variable
produces an infinitely small increment of the function
itself.'' This yields uniform continuity rather than continuity as
envisaged by nonstandard analysis. Analogously, pointwise convergence at all
(standard and nonstandard) points of a compact interval amounts to uniform convergence.
Cauchy was a brilliant mind who looked at the entities of
analysis in a fashion closer to Leibniz than Newton. He felt the difference
between ``assignable'' and ``nonassignable'' numbers which was
neglected in the epsilon-delta technique but reconstructed by Robinson.
Gabriel Stolzenberg> In reading Cauchy's argument more carefully than I have here, it
Gabriel Stolzenberg> may be relevant to note that he allegedly accepted Abel's criticism
Gabriel Stolzenberg> of it. If he did, what was his understanding of where and how he had
Gabriel Stolzenberg> gone wrong?
Cauchy was ``intimidated'' and felt ashamed of his would-be ``error.''
He tried to suggest a better proof but published practically
the same ``new'' demonstration.
As mentioned by Henrik Kragh Sorensen, PhD, Department of Mathematics,
Agder University College, Norway, the relevant references are as
follows:
Cauchy, A.-L. (1821). Cours d'Analyse de l'Ecole Royale Polytechnique.
Premi`ere partie. Analyse Algebrique. Paris: L'Imprimerie Ro\-yale.
Photographic reproduction in (Bottazzini 1990).
Cauchy, A.-L. (1853). Note sur les series convergentes dont les divers
termes sont des fonctions continues d'une variable reelle ou imaginaire,
entre des limites donnees. In OEuvres Compl`etes d'Augustin
Cauchy, Volume 12 of 1st series, pp. 30-36. Paris: Gauthier-
Villars. Read to the Academie des Sciences on March 14, 1853
and originally published Comptes Rendus vol. XXXVI p. 454.}
Cauchy wrote in the second article:
``By the way, it is easy to see how one should modify the
announcement of the theorem so that it is true without
exception. [$\dots$]
Suppose now, that by attributing to $n$ a sufficiently great
value, one can make---for all values of $x$ between the
given limits---the module of the expression (3)
$[s_{n'}-s_n]$
(whatever $n'$ might be) and therefore the module of $r_n$
less than any number $\varepsilon$ however small one might like. [$\dots$]
Theorem I. If the different terms of the series
$$
(1)\quad u_0, u_1, u_2, \dots, u_n, u_{n+1}, \dots %\?
$$
are functions of a real variable $x$ and continuous with respect
to this function between given limits, and if the sum
$$
u_n +u_{n+1} + \dots + u_{n'-1}
$$
becomes infinitely small for infinitely great integral values
of $n$ and $n' > n$, the series {\rm(1)} is convergent, and the
sum $s$ will be a continuous function of $x$ between the given
limits.''
In fact, Cauchy repeated his original argument he intended to "correct."
Cauchy was a great master of analysis, the inventor of complex analysis.
He had a proper vision of the matters of convergence and continuity,
but his vision differed from the dogmata of the prevailing paradigm of
his critics.
Cauchy deserves a better fate than to be ridiculed
for the errors he had never committed. The whole story is a typical
illustration of the lack of understanding which results from
neglecting the dependence of the standards of rigor on time.
---------------------------------------------
Sobolev Institute of Mathematics
Novosibirsk State University
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