Subject: [HM] Bolzano, Cauchy, Epsilon, Delta (2/5)
From: Walter Felscher (walter.felscher@uni-tuebingen.de)
Date: Wed Feb 02 2000 - 06:00:44 EST
4. Cauchy
4A. Cauchy on variables and their limits
Today's mathematical textbooks still use the word 'variable', but they
do not define a mathematical object named by this word. A 'variable'
today is a linguistic object, a letter employed to denote, and it is
only implicitly that the student learns to speak about that. This
conforms to the tendency to view mathematics as dealing with
concepts only and to disregard connections with the language used to
speak about them. In modern mathematics, this elimination of
linguistic features has been carried out with remarkable (and
sometimes regrettable) success.
Augustin Louis Cauchy (21.8.1789 - 23.5.1857) set out to fulfill
d'Alembert's program in his two textbooks of analysis :
Cours d'Analyse de l'/Ecole Royale Polytechnique , Paris 1821
(reprinted in Oevres Compl\etes, s/er.2 , vol.3 )
R/esum/e des Le,cons donn/ees a l'/Ecole Royale Polytechnique sur
l'Calcul Infinit/esimal , Paris 1823
(reprinted in Oevres Compl\etes, s/er.2 , vol.4 )
In both books, Cauchy begins by attempting to give a mathematical
description of what is (spoken about as if it were a mental
experiment and in this vein then) conceived as a limiting process. He
explains on page 4 of 1821 (Oevres 3 , p.19) the term "variable
quantity"
On nomme quantit/e variable celle que l'on consid\ere comme devant
recevoir successivement plusieurs valeurs diff/erentes les unes
apr\es des autres. On d/esigne une semblable quantit/e par une
lettre prise parmi les derni\eres de l'alphabet. ...
He proceeds to explain limits of such assignments:
Lorsque les valeurs successivement attribu/ees \a une m^eme
variable s'approchent ind/efiniment d'une valuer fixe, de mani\ere
\a finir par en diff/erer aussi petit que l'on voudre, cette
derni\ere est appel/ee la limite de toutes les autres. ...
In particular, there are variables with assignments which have the
limit zero:
Lorsque les valeurs num/eriques successives d'une m^eme variable
d/ecroissant ind/efiniment, de mani\ere \a s'abaisser au-dessous
de tout nombre donn/e, cette variable devient ce qu'on nomme un
infiniment petit ou une quantit/e infiniment petite. Une variable
de cette esp\ece a z/ero pour limite.
The same definitions appear in 1823 (Oevres 4 , p.16). In Cauchy's
further text then, when these definitions are referred to, the phrase
"aussi petit que l'on voudre" usually is expressed by saying that the
difference, between the valeurs of an assignment to the variable, and
the valeur fixe, can be made smaller than any given positive number.
4B. Comments
A variable for Cauchy then _is not_ a letter, but a concept: the
concept of a form to be filled by _assignments_ A of values. The
domains L of such assignments A remain unspecified; apparently
they shall at least carry an order (such that one can speak of
valeurs les unes apr\es des autres). In particular, assignments _may_
be conceived as sequences defined on w (omega), but they may also be
more general assignments, e.g. the identity map on a whole interval
[the reader may recall the modern notion of a Moore-Smith sequence].
The ranges of assignments are not specified either, but it appears
clear from Cauchy's words that they shall be num/eriques, i.e. consist
of real numbers.
[Note 1 : Cauchy says that values are assigned to a variable, he
does not use the substantivated notion of an assignment. Nor does
he distinguish notationally between (1) a variable, for which he
may write x , (2) an assignment, and (3) the values of that assignment:
quite often x then also stands for a value assigned.]
[Note 2 :Speaking in today's terminology, every half-line (z,->)
contains a cofinal subset consisting of natural numbers above a
certain number N . The situations dealing with limits of
assignments A which Cauchy considers then ca n be subsumed in the
following cases. If a is finite and
(c1) if L is a cofinal subset of a half-line (z,->) with ascending order
then given e there is y in L such that l>y implies |a-A(l)| < e ,
(c2) if L is an open interval (0,z) with descending order
then given e there is d in L such that l<d implies |a-A(l)| < e ,
and if a is infinite and
(c3) if L is a cofinal subset of a half-line (z ,->) with ascending order
then given n there is y in L such that l>y implies n < A(l) ,
(c4) if L is an open interval (0,z) with descending order
then given n there is d in L such that l<d implies n < A(l) . ]
A variable with an assigment converging to zero is said "to become" a
"quantit/e infiniment petite". Since already a variable was introduced
as a "quantit/e variable" by Cauchy, the word "quantit/e" here appears
to have a more abstract meaning than that of number or of magnitude in
the geometric continuum. [Of course, it is left open whether a quantit/e
variable, with an assigment converging to zero, actually _is_ or only
_becomes_ a quantit/e infiniment petite.] Thus a quantit/e infiniment
petite is of quite a different species than are numbers (the valeurs
num/eriques of variables) or, equivalently, geometric magnitudes.
Cauchy's purpose when introducing the quantit/ees infiniment petites
is expressed in 1823 , p.9 :
Mon but pricipal a /et/e de concilier la rigeur, dont je m'/etais
fait une loi dans mon Cours d'analyse, avec le simplicit/e qui
r/esulte de la cond/eration directe des quantit/es infiniment petites.
It rests on the observation that a number c is the limit of the assignment
of values j to the variable x if, and only if, 0 is the limit of the
assigment of j-c to the variable x-c , i.e. this variable under that
assignment becomes infiniment petite. Based on it, statements about
limits can be translated into statements about quantit/ees infiniment
petites.
As an informative example, let me quote from 1823 , Oevres 4 , p.18 :
Cela pos/e, si la variable y est exprim/ee en fonction de la
variable x par la /equation
(1) y = f(x) ,
D_y [D_: capital delta], ou l'accroissement de y correspondent \a
l'accroissement D_x de la variable x , sera d/etermin/e par la
formule
(3) y + D_y = f(x + D_x) .
[p.19] ... Il est bon d'observer que, des /equations (1) et (2)
r/eunies, on conclut
(5) D_y = f(x + D_x) - f(x) .
Soient maintenant h et i deux quantit/es distinctes, la premi\ere
finie, la seconde infiniment petite, et a = i/h [a: alpha] le
rapport infiniment petit de ces deux quantit/es. Si l'on attribue
\a D_x la valeur finie h , la valeur de D_y , donn/ee par
l'equation (5), deviendra ce qu'on appelle la diff/erence de la
fonction f(x), et sera ordinairement une quantit/e finie. Si, au
contraire, l'on attribue \a D_x une valeur infiniment petite, si
l'on fait par example
D_x = i = a.h [a: alpha] ,
la valeur de D_y , savoir
f(x+i) - f(x) ou f(x + a.h) - f(x)
sera ordinairement une quantit/e infiniment petite. C'est ce que
l'on verfiera ais/ement \a l'/egard des fonctions
A^x , ...
auxquelles correspondent les diff/erences
A^x+i - A^x = (A^i - 1).A^x , ...
dont chacune renferme un facteur A^i - 1 ou ... qui converge
ind/efiniment avec i vers zero.
So Cauchy here considers a quantit/e infiniment petite i and a quantit/e
finite h (of which we are not certain whether it shall be understood
as a positive number or as a variable with an assignment converging to
something different than zero). Cauchy then writes a = i/h which is
(a variable with) an assigment having as values the quotients of the
values of i and the value(s) of h (there is the obvious notational
confusion which comes from denoting both the variable i and its
values by the same letter). If now D_x is i = a.h then D_y is
f(x+i)-f(x) . This he illustrates by the exponential function
f(x)=A^x where
D_y : A^x+i - A^x = (A^i - 1).A^x .
Thus here D_y is a quantit/e infiniment petite, presented by the
quantit/e infiniment petite B with values A^i - 1 and the constant
number A^x .
In particular, the above example shows that the three quantit/ees
infiniment petites i , D_y and B have as values ordinary real numbers
(and all converge to zero). There are no 'infinitesimal', non-Archimedean
numbers ever used by Cauchy for his quantit/ees infiniment petites.
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