Dear Arnaud,
the post you pasted in your recent message to HM was not the one referred
to by Steve Maurer. My post was in fact submitted on February 9, 1997.
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Date: Sun, 9 Feb 1997 16:34:09 -0500
To: Stephen B Maurer <smaurer1@swarthmore.edu>
From: Julio Gonzalez Cabillon <jgc@adinet.com.uy>
Subject: Re: Implicit function theorem
Cc: math-history-list@maa.org
Dear Stephen,
At 10:39 PM 04/02/1997 -0500, you wrote:
> Here's a question I hope will move this group in some fresh directions.
>
> Who is responsible for the Implicit Function Theorem?
>
> I mean the theorem that says, roughly,
>
> if f(x,y) is a suitably differentiable function, with x and y from
> some normed spaces, then the set of values of y for which f(x,y)=0
> is locally a function of x, and differentiable.
>
> I'm sure the founders of calculus used this theorem (e.g., they
> assumed if x^2 y^3 + xy + e^x y^5 = 0, then y could be differentiated
> as a function of x), but when did someone decide there was something
> that needed to be proved, and who proved it?
>
> I expect the answer is: several people, perhaps independently at first,
> and later in more and more general situations. But I have never seen
> the historical progression laid out, let alone a single name attached
> to this theorem. Yet it is one of the great and crucial theorems of
> analysis. It is one of the main tools we have for assuring that a
> function exists even when we have no way to get a formula for it. Once
> mathematicians freed themselves for thinking that a solution required
> a formula, the importance of this result was surely clear to all.
>
> Steve Maurer
>
>
1. cf. page 155 of Leibniz' _Mathematische Schriften_, edited by Gerhardt,
vol. I, Berlin-Halle (Ascher-Schmidt), 1850.
2. cf. page 241 of Euler's "Institutiones Calculi Differentialis",
Academia Imperialis Scientiarum Petropolitana, 1755.
As Walter Felscher has pointed out, the Theory of the Implicit Function
in a modern setup is due to Ulisse Dini (cf. e.g.: "Lezioni di analisi
infinitesimale", vol 1, pp. 197 ff., Pisa 1907).
Regards,
julio
* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Some years ago I studied this topic from a historical standpoint, and
I tried to trace its evolution. I found that Dini's "Lezioni di analisi
infinitesimale" is especially rewarding. The chapter in question is:
Funcioni implicite di una o piu variabili indipendenti [pp 197-241]
It would be worth pointing out that Dini's text corresponds basically
to the lectures he delivered at the University of Pisa during 1876-78.
First.
It may convincingly be argued that the fact that Dini's presentation of
such a theory in a textbook does not say much about whether it is his own
harvest. None the less, I am fairly convinced that most of what appears
on pages 197 (onwards!) belongs to him. As far as I recall, both Leibniz
and Euler only work algorithmically (modern language) in the sense
described above "I'm sure the founders of calculus used this theorem (e.g.,
they assumed if x^2 y^3 + xy + e^x y^5 = 0, then y could be differentiated
as a function of x)".
Second.
It may also be argued that the statement "the modern set-up is due to
Ulisse Dini" might be misleading or even mistaken, since it has been
"claimed that almost all published proofs until circa 20 years ago were
flawed". Yes. But we may disagree with this (over-)statement, despite
we basically do understand the idea behind the remark. Right now, my
question is:
do you agree with the fact that the proofs of the implicit function
theorem we used to teach in c. 1980 were basically flawed?... I don't
think so. But as always I am ready and happy to learn more.
Un fuerte abrazo,
Julio GC