Subject: [HM] Bolzano, Cauchy, Epsilon, Delta (1/5)
From: Walter Felscher (walter.felscher@uni-tuebingen.de)
Date: Wed Feb 02 2000 - 05:59:16 EST
Bolzano, Cauchy, Epsilon, Delta
In this article, I describe how the notions of limit and continuity
were explained and used between 1817 and 1823 by Bolzano and Cauchy.
In a closing section I discuss the extent to which the technique of
epsilon and delta serves a tool to write finite proofs of statements
which, involving limits and continuity, refer to infinite processes.
1. Today's terminology
For more than one hundred years, the following definitions have come
into common use.
Let f be a function defined in a neighbourhood V = {x;|a-x|<q} of a
number a .
A number b is defined to be the " B-limit of f at a " if
for every e>0
there exists d>0 (with d<q)
for every y : if |a-y|<d then |b-f(y)|<e .
Let s be a sequence <x(i);i<w > of numbers x(i) indexed with the set
w (omega) of natural numbers. The sequence s is defined to " converge
to a " , and a then the " limit of s ", if
for every e>0
there exists n in w
for every i in w : if i>n then |a-x(i)|<e .
A number b is defined to be the " C-limit of f at a " if
for every sequence s = <x(i);i<w> with values in V
if s converges to a then fs = <f(x(i))|i<w> converges to b .
Lemma: b is B-limit of f at a if, and only if, b is C-limit of f at a .
[In this connection, an oberservation of Sierpi/nski 1916 should be
mentioned: the implication that a B-limit is a C-limit requires,
and actually is equivalent, to the following weak form of the Axiom
of Choice: for every countable family of non empty sets of real
numbers there exists a choice function.]
Let % be one of B or C . The function f is defined to be %-continuous
at a if f(a) is the %-limit of f at a .
Lemma: f is B-continuous at a if, and only if, f is C-continuous at a .
[Again, the statement that a B-continuous function is C-continuous
requires, and actually is equivalent, to that weak form of the Axiom
of Choice.]
Methodological remark. The B-notions require quantifiers ranging
over sets of numbers (namely e, d, y). The C-notions also require
such quantifiers (namely e, n, i), but in addition they require a
quantifier ranging over all sequences s . Sequences are notions
of a higher order (the 2nd ) than numbers (of 1st order). In so
far, the logical complexity of the C-notions is higher than that
of the B-notions. [For usual analysis, that fact is irrelevant
since the notion of function is of 2nd order as well.] In particular,
B-continuity directly connects local properties at a and at
f(a), while C-continuity makes use of an intermediate, additional
abstraction, that of sequences and their convergence (or their
limits): limiting processes at a are set into correspondence with
limiting processes at f(a).
Terminological remark. Not only are the above definitions in
common use, also the writing of the letters "epsilon" for e and
"delta" for d has become canonical throughout the literature
(even that printed otherwise with cyrillic letters). Thus the
B-notions are denoted with help of the letters epsilon and delta.
Convergence of a sequence, and so the C-limit, is denoted with
help of the letter epsilon and with a letter n or N . So the
C-notions do not involve to write a letter such as delta. To
work with notions employing epsilon sometimes is said to perform
_epsilontics_, while to speak about epsilon-delta techniques
indicates that also a letter (and then a quantifier) for delta is
used - i.e. that B-notions are considered.
2. D'Alembert's program
Jean-Baptist le Rond d'Alembert (17.11.1717 - 29.10.1783) was,
together with Euler and the brothers Bernoulli, one of the mathematicians
representing the heroic age of calculus. Together with Denis Diderot
he also edited the "Encyclop/edie ou Dictionnaire Raisonn/e des Sciences,
des Arts et des M/etiers". In its 9th volume of 1765 he wrote in the
article "Limite"
On dit qu'une grandeur est la limite d'une autre grandeur, quand
la seconde peut approcher de la premi\ere plus pr\es que d'une
grandeur donn/ee, si petite qu'on la puisse supposer, sans pourtant
que la grandeut qui approche, puisse jamais surpasser la grandeur
dont elle approche; en sorte que la diff/erence d'un pareille
quantit/e \a sa limite est absolument inassignable ...
A proprement parler, la limite ne conincide jamais, ou ne devient
jamais /egale, a la quantit/e dont elle est la limite; mais celle-ci
s'en approche toujours deplus en plus et peut en diff/erer aussi
peut qu'on voudra ...
In its 4th volume 1754 of he wrote in the article "Diff/erentiel"
Celui-ci nous paro^it suffire pour faire entendre aux commen,cans la
vraie m/etaphysique du calcul diff/erentiel. Quand une fois on
l'aura bien comprise, on sentira que la supposition que l'on y
fait de quantit/es infiniment petites, n'est que pour abr\eger &
simplifier les raisonnemens; mais que dans le fond le calcul
diff/erentiel ne suppose point n/ecessairement l'existence de ces
quantit/es; que ce calcul ne consiste qu'a d/eterminer
alg/ebriquement la limite d'un rapport de laquelle on \a d/eja
l'expression en lignes , & \a /egaler ces deux limites, ce qui
fait trouver une des lignes que l'on cherche.
Il ne s'agit point, comme on le dit encore ordinairement, de
quantit/es infiniment petites dans le calcul diff/erentiel; il
s'agit uniquement de limites de quantit/es finites. Ainsi la
m/etaphysique & des quantit/es infiniment petites plus grandes ou
plus petites les unes que les autres, est totalement inutile au
calcul diff/erentiel. On ne se sert du terme d'infiniment petit,
que pour abr\eger les expressions. Nous ne dirons donc pas avec
bien des g/eometres qu'une quantit/e est infiniment petite, non
avant qu'elle s'/evano"uisse, non apr\es qu'elle est /evano"uie,
mais dans l'instant m^eme o\u elle s'/evano"uit; car que veut
dire une d/efinition si fausse, cent fois plus obscure que ce
qu'on veut d/efinir ? Nous dirons qu'il n'y a point dans le
calcul diff/erentiel de quantit/es infiniment petites.
Reading these words today, we may receive the impression that they
might as well have been written at the time of Weierstrass, of Cantor,
or even by a contemporary mathematician: all that matters in analysis
is the notion of limit, and there is no place at all to conceive
infinitesimals. Yet when they were written, the details of
how to mathematically work with limits were hardly worked out, and
what d'Alembert wrote here was less a description of the actual
state of affairs, but rather a program to be carried out in the
future. It _was_ carried out, with the decisive steps performed by
Bolzano and Cauchy.
3. Bolzano
Bernhard Bolzano (5.10.1781-18.12.1848 ) studied mathematics in
Prague with Stanislav Vydra and Franz-Josef Gerstner. At the same
time, he had studied theology, and in 1805 , only days before receiving
his doctorate in mathematics, he was ordained as a (secular) priest;
already in 1807 he obtained a chair in "Religionslehre" in Prague.
In this position, in 1817 Bolzano published a book of 60 pages with
the title
Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey
Werthen, die ein entgegengesetzes Resultat gewaehren, wenigstens
eine reelle Wurzel der Gleichung liege
which I have discussed in my article "Bolzano and continuity" here on
[HM]. There he gave a definition of continuity of a function f at an
argument x :
if x is such argument then the difference f(x+w)-f(x) [w: omega]
can be made smaller than any given quantity if only w is assumed as
small as wanted
and said article I have shown that whenever Bolzano uses his definition
in his proofs, this use consists precisely in the verification of
given e>0 , there is d>0 such that w<d implies |f(x+w)-f(x)| < e .
For example, in the case of particular a particular f Bolzano determines,
on p.58 of his book, for a given e (he uses the letter D ) the number
d as the smaller of two numbers w_1 [omega index 1 ] and D/S .
Thus Bolzano's notion of continuity is intended precisely as that of
B-continuity.
At first sight, Bolzano's concepts, presented with unambiguous perspicacity,
appear to have sprung from his head, as Pallas did spring from that
of Zeus.
Bolzano does not appear to have had contact with mathematicians apart
from his acquaintances in Prague. In consequence, his mathematical
work remained completely unknown and came to the notice of the
mathematical community only thirty years after his death. In
particular, Bolzano's writings had no influence upon the re-discovery
of B-continuity by Weierstrasz and others.
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