THE RISE OF FUNCTIONS
“holding together,” or “interlocking.” The word occurs already in Homer
in both a spatial and a temporal sense, and in its temporal sense, that is,
when referring to the flow of time, its meaning is already semi-figurative,
foreshadowing the connotations of meaning of today.
Centuries later the word appears, even profusely, as a technical term in
Aristotle’s Physics, in the kind of technical meaning which it might have in a
scientific or philosophical context of today. The meaning of synechés in
Aristotle might not be exactly the same as the meaning of continuous is
today, but in a cursory reading of the Physics the translation of synechés by
continuous is good enough.
As against this, there is the remarkable fact, which cannot be sufficiently
stressed or overinterpreted, that Greek mathematics proper, that is, the
Greek mathematics as it is known from the works of Euclid, Archimedes,
Apollonius, Heron, Ptolemy, etc., never, but never, states, asserts, suggests,
or negates that something in mathematics is synechés in a technical mean-
ing of the term, nor does it ever take recourse to an obvious verbal equivalent
of it.
In the Physics, however, synechés, when used technically, occurs in a
manner which would be recognizably mathematical nowadays. When
occurring there, it is intended to describe, in rather involved thought patterns
of Aristotle, the essential mathematical feature of the linear continuum
(— co, ∞) of today, namely its “completeness” in the sense of Dedekind
and Cantor. Aristotle has great difficulties separating denseness from
completeness, but even professional mathematicians in the 17th and 18th
centuries might have had such difficulties too.
This is all the continuity that Aristotle is aware of. He never mentions
“topological” continuity, that is, continuity of a function or of a “mapping”
of any kind, except that he is aware of the fact (which he labors most re-
petitively) that in a uniform motion x = ct the continuity structures of the
spatial continuum {x} and the temporal continuum {t} are isomorphic.
The absence of topological continuity from Western thought lasted very
long. In fact, topological continuity is discernible for the first time only in
Leibniz, not in straight working mathematics, but in many expostulations
of something which Leibniz called a Law of Continuity (lex continui). This
Law was not really a hypothesis or principle of the metaphysics of Leibniz,
but rather a leitmotif of it. Among other things the Law asserted, or only
implied, that the data and features of the universe are all continuous, whether
asserted individually, in mutual correlation, or in functional dependence.
Thus, anybody so disposed may detect in Leibniz insights of the following
kind: the rudiments of a conception of space as a Hausdorff neighborhood