Subject: [HM] Bolzano, Cauchy, Epsilon, Delta (4/5)
From: Walter Felscher (walter.felscher@uni-tuebingen.de)
Date: Wed Feb 02 2000 - 06:02:40 EST
6. Bestiarium infinitesimale
In the preceding sections, Bolzano's formulations required no
interpretation to fit into today's terminology. Cauchy, on the other
hand, wrote about quantit/ees infiniment petits which do not appear
in today's terminology. He did so, however, only after introducing
limits in a terminology needing no interpretation either, and the
quantit/ees infiniment petites he defined as a special case of
limits. Thus together with the term limit also the term quantit/ee
infiniment petite received an interpretation in today's terminology
which, in the following, I shall call the standard interpretation.
Cauchy thus used the quantit/ees infiniment petites as a (rigorous)
fa,con de parler, a device permitting to formally keep a connection
with the past that had conceived infinitesimals as numbers from a lower
class of Archimedicity.
Abraham Robinson embedded the real numbers R into a non-Archimedean
field S in which all L-sentences true in R hold as well. [There L is a
language with names for all real numbers, predicate symbols for all
sets and relations of and between real numbers, and with function symbols
for all real functions. This metamathematical connection between R
and S can be expressed by a simpler looking 'solution set' condition
as found in J.Keisler's textbook.] It then becomes possible to extend the
notions of 'standard' analysis from R to a 'non-standard' analysis on S
where now the presence of infinitesimal numbers, i.e. numbers below
all 1/n with natural n , permits to phrase certain arguments in a new
and rigorous way which Leibniz and his contemporaries had had to leave
in vagueness.
In his book "Nonstandard Analysis" of 1966 Robinson quoted Cauchy's
definitions on pp.269-270 and then continued
We gather from the above passages that infinitely small quantities
are fundamental in Cauchy' approach to Analysis. However, these
quantities are not numbers but variables, or rather, states of
variables whose limit is zero. ...
Whatever the precise picture of an infinitely small quantity may
have been in Cauchy's mind, we may examine his subsequent definitions
and see what they amount to if we interpret the infinitely small
and infinitely large quantities mentioned in them in the sense of
Non-standard Analysis. For the notion of continuity, Cauchy's
definition may thus be interpreted as stating that for f(x) defined
in the interval a<x<b , f(x) is continuous in that interval if, for
infinitesimal a [alpha], the difference f(x+a)-f(x) is always (toujours)
infinitesimal. If now we interpret 'always' as meaning 'for all standard
x ' then we obtain ordinary continuity in the interval, but if by
'always' we mean 'for all x ' then we obtain uniform continuity.
With this last paragraph Robinson introduces what I choose to call
the Robinson interpretation:
(a) where Cauchy writes about quantit/ees infiniment petites,
assume them as infinitesimal numbers in the sense of Non-standard
Analysis and
(b) read Cauchy's following developments under this assumption,
together with assuming today's (or Weierstrasz') standard knowledge
about distinctions such as usual versus uniform continuity etc.
This interpretation resulted in beautiful mathematical insights. In
particular, certain erroneous statements of Cauchy's (on series of
continuous functions, on continuity in several variables ...) in this
way can be read as correct statements in Non-standard Analysis. In
this connection there are two informative articles by John P.Cleave:
Cauchy, Convergence and Continuity. British J.Phil.Sci. 22 (1971) 27-37
The concept of 'variable' in nineteenth century analysis. British J.
Phil.Sci. 30 (1979) 266-278
But it must be emphasized that Robinsons's interpretation is in no
way an explanation of the historical content of Cauchy's writings,
and it is clear from Robinson's word's above that it does not purport
to be one. Cauchy wrote explicitly that his quantit/ees infiniment
petites have valeurs num/eriques, real numbers :
"Lorsque les valeurs num/eriques successives d'une m^eme variable ... ".
Of course, the non-Archimedean, infinitesimal numbers of Pascal, say,
were still in the back of Cauchy's and his contemporaries' mind. But
for his quantit/ees infiniment petites Cauchy purposefully stipulated
that they were assignments to variables the values of which were real
numbers. There is no mention of infinitesimals anywhere in Cauchy's
writings.
Yet as the candle draws the moth, there appear from time to time
articles the authors of which disregard Cauchy's text and try to
persuade themselves that the Robinson interpretation explains what
Cauchy "actually meant". Of them I mention Gordon Fisher's
Cauchy and the infinitely small. Historia Math. 5 (1978) 313-331
where the possibility that the Robinson interpretation be historically
correct is repeatedly asserted as based on the (mistaken) reading that
Cauchy does _not_ expressly exclude infinitesimal values for his
quantit/ees infiniment petites:
"When the successive numerical values of the same variable decrease
indefinitely, in such a way as to fall below any given number, this
variable becomes what one calls an _infinitely small_ (un infiniment
petit), or an infinitely small quantity. A variable of this kind
has zero for limit." The first sentence _could_ mean that when the
numerical (i.e.absolute) values of a variable decrease in such a
way as to be less than any positive number, then the variable takes
on infinitesimal values. (p. 316 bottom)
We have already observed that Cauchy's definition of a variable
does not exclude infinitesimal values (p. 318, para 3 )
... might be taken as ... in which a [alpha] is not permitted to
take on non-zero infinitesimal values. But there is no necessity to
make such restriction, nor does Cauchy say that this would be
desirable or even possible (p.319, para 2 )
Into this connection there also belong the various articles and books
by D.Laugwitz in which, avoiding Non-standard Analysis as a detracting
invention of logicians, he develops a 'mathematics of the infinitesimal'
of his own making, and uses it to skillfully interpret the mathematics
of the times from Euler to Cauchy.
And so we have had a glance at the bestiarium infinitesimale.
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