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Dictionary of the History of Ideas | ||

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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

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.

Dictionary of the History of Ideas | ||