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#### IV. MATHEMATICS

In mathematics, continuity is an all-pervading concept.
Topology is a relatively recent major division of math-
ematics, and in the half-century 1890-1940 it was a
vast exercise in continuity from a novel comprehensive
approach. Also, this novel pursuit of continuity sup-
plemented but did not supersede the study of continu-
ity in “analysis,” in which, knowingly or not, it had
been a central conception since the fifth century B.C.

It will suit our purposes to distinguish, and keep
apart two aspects of continuity in mathematics.

Aspect (1). Continuity of linear ordering. This aspect
of continuity is suggested by, and is embodied in the
intrinsic continuity structure of the so-called linear
continuum of real numbers – ∞ < t < ∞.

Aspect (2). Continuity of a function y = f(x). The
simplest, and still very important case of a continuous
function f(x) arises if x and y are both real numbers,
and in this case a function is “equivalent” with an
ordinary graph or chart on ordinary graph paper. In
the general case, a function y = f(x) is a “mapping”
from any topological space X:(x) to any other topo-
logical space Y:(y).

Aspect (1) was envisioned by the Greeks, and they
worked long and hard at elucidating it. Aspect (2)
however eluded them. The Greeks had fleeting chance
encounters with it, but they were not inspired to focus
on it in any manner. This failure of the Greeks to
recognize aspect (2) far outweighed their ability to
identify aspect (1). By a purely scientific assessment,
this failure greatly contributed to the eventual decline
of Greek mathematics in its own phase.

Even in the recognition of aspect (1) the Greeks had
two blind spots. Firstly, Greek mathematics never

498

created the real numbers themselves. When the Greeks
formed the product of two quantities that were repre-
sented by lengths then, conceptually, the product had
to be represented by an area. Descartes may have been
the first to state expressly, as he did with some emphasis
at the very beginning of his La Géométrie (1637), that
the product may also be represented by a length. The
Greek substitute for our concept of real numbers was
their quasi-concept of magnitude (μέγεθοσ; megethos),
and the corresponding elementary “arithmetic” was
the Greek theory of proportions, as presented in
Euclid's Elements, Book 5.

The Greek magnitudes were a “substitute” for posi-
tive
real numbers only; and we view it as a second
blind spot of the Greeks that they did not even intro-
duce a magnitude of value 0 (= zero), which, by con-
tinuity, would be the limiting case of magnitudes of
decreasing (positive) values. Thus, Greek mathematics
never had the thrill of conceiving that two coincident
lines form an angle of value 0; and Greek physics of
locomotion, as expounded in Aristotle's Physica, Books
5-8, always viewed “rest” (ἢρεμία) as a “contrary to
motion” (κίνησις) and never as a motion with velocity 0.

In fact, the first outright criticism of this Aristotelian
view is to be found only in Leibniz. It is strongly
implied in his pronouncement that “the law of bodies
at rest is, so to speak, only a special case of the general
rule for bodies in motion, the law of equality a special
case of inequality, the law for the rectilinear a sub-
species of the law for the curvilinear” (H. Weyl, p.
161).

This pronouncement of Leibniz was part of a uni-
versal lex continui (“Law of Continuity”) which runs
through his entire metaphysics and science. Leibniz
did not present the law in a systematic study of its
own, but he frequently reverted to it, presenting some
of its aspects each time. Leibniz recognized, reflec-
tively, the importance of functions for mathematics.
He coined the name “function” in 1694, and, what is
decisive, he was well aware of our aspect (2) of con-
tinuity (Bochner, pp. 216-23). But he did not “create”
functions in mathematics. As rightly emphasized by
Oswald Spengler, the concept of function began to stir
in the late fourteenth century, and its emergence con-
stituted a remarkable difference between ancient and
post-medieval mathematics. Also, as early as 1604, that
is 90 years before Leibniz coined the name, Luca
Valerio had de facto introduced a rather general class
of (continuous) functions f(x) to a purpose, and had
operated with them competently in the spirit of the
mathematics then evolving. However, it was Leibniz
who was the first to assert, more or less, that functions
and functional dependencies in nature are usually con-
tinuous. Thus he states the maxim of cognition that
“when the essential determinations of one being ap-
proximate those of another, as a consequence, all the
properties of the former should also gradually approxi-
mate those of the latter”
(Wiener, p. 187).

It is not easy to state the direct effect of Leibniz'
Law of Continuity on the growth of mathematics and
physics. In working mathematics, the meaning and role
of continuity unfolded excruciatingly slowly in the
course of the eighteenth and nineteenth centuries,
through cumulative work of Lagrange, Laplace,
Cauchy, Dirichlet, Riemann, Hankel, P. du Bois Rey-
mond, Georg Cantor, and others, without any manifest
reference to the metaphysically conceived lex continui
or Leibniz. Of course, the “Law” of Leibniz may have
been burrowing deep inside the texture of our intellec-
tual history, thus affecting the course of mathematics.
But to establish this in specific detail would be very
difficult.

Mathematics of the nineteenth century elucidated
basic facts about both aspects of continuity, for real
numbers, and for real and complex numbers. These
facts about continuity were intimately connected with
facts about infinity, especially about the infinitely small.
The efforts to elucidate these two sets of facts, severally
and connectedly, had begun with early Pythagoreans
and Zeno of Elea, and it took twenty-four centuries
to bring them to fruition.

The twentieth century greatly widened the scene of
continuity, especially of its aspect (2), by extending the
conception of continuity from functions from and to
real (and complex) numbers to functions from and to
general point-sets, that is general aggregates of mathe-
matical elements. In fact, a numerical function y = f(x)
is continuous if it transforms “nearby” numbers x into
“nearby” numbers y. Therefore, in order to apply the
notion of continuity to a function y = f(x) from a
general point-set X:(x) to a general point-set Y:(y) it
suffices to know what is meant by the statement that
points of X or points of Y are “sufficiently near” each
other. Now, in the twentieth century this has been
achieved by the introduction of a so-called topological
structure on a general point-set. For any given topo-
logical structure it is meaningful to say when two
points of the set are “near” each other, and when the
two point-sets X and Y are each endowed with a topo-
logical structure of its own it thus becomes meaningful
to say when a function y = f(x) is continuous. The
conception of a topological structure opened new
vistas, and it has become involved in most of the math-
ematics of today.