10
RICE UNIVERSITY STUDIES
space, together with the corresponding definition of continuous (function or)
mapping in terms of neighborhoods; the rudiments of the fact that for a
system of differential and other functional equations the solution usually
depends continuously on initial conditions and other parameters; the
rudiments of the hypothesis that the mathematical laws of physics are
constant in space and time, or at most vary continuously; the rudiments
of the law that biological species evolve continuously; etc.
After Leibniz, among philosophers and mathematician-philosophers,
the greatest proponent of a universal law of continuity was C. S. Peirce,
whom we have already mentioned above. Like some other 19th century
philosophers before him (J. F. Herbarth, G. T. Fechner), he spoke not of
continuity but of synechism, and he did not acknowledge any indebtedness
to Leibniz. Peirce was familiar with the work of Georg Cantor and with the
methodology of working mathematics, and through the length of his
philosophical career he endeavored to find a conception and principle of
continuity that would apply to mathematics and ontology both. In this he
utterly failed, as he was bound to, because no ontology worthy of its name
is a mere “extension” of mathematics, and because in mathematics con-
tinuity may vary with the context and purpose, even freely so, whereas in
ontology proper this freedom is greatly curtailed if it is present at all.
Piecewise Analytic Functions. In the 18th century, working mathe-
maticians of the stature of Euler, d’Alembert, and Lagrange were trying
to find out what functions are or ought to be, and how and when they are
“given,” and mathematicians even began to classify functions, somewhat
ingenuously at times. Somehow their findings were uncertain, ambiguous,
and inconclusive, so much so that even historical accounts of them do not
quite agree with each other, or, at any rate, try to be as circumspect as
they can. There is a reason for this. In the 18th century “there was a near-
perfect, richly yielding, fusion of mathematics and mechanics” [4, p. 7],
so that a mathematical function was not only an object of mathematics, but,
by equal priority, also an object of mechanics, and thus had to satisfy
needs and expectations of both in equal measure. For instance, it seems that
in the thinking of Lagrange an analytic function was, in equal parts,
a function likely to occur in mathematical analysis and a function likely to
occur in a typical situation of his Mécanique analytique. Now a rather
simple situation arises if one throws a ball against a wall from which it
bounces back. The coordinates of the ball, as functions of time, cease to
be analytic at the time point of impact but are analytic in the adjoining time
intervals. Thus, Lagrange would have had difficulty in firmly deciding