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
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.