IV. THE INFINITELY SMALL
Relative to the infinitely small, Greek mathematics
attained two summit achievements: the theory of pro-
portions, as presented by Book 5 of Euclid's Elements,
and the method of exhaustion for the computation of
areas and volumes, as presented by the essay “On
Sphere and Cylinder” of Archimedes. Eudoxus of
Cnidos (408-355 B.C.), the greatest Greek mathe-
matician before Archimedes—and a star member of
Plato's Academy, who was even an expert on
“Hedonism and Ethical Purity”—had a share in both
achievements. But not a line of his writings, if any,
survives, and he is, in historical truth, only a name.
The durable outcome of these efforts was a syllogistic
procedure for the validation of mathematical limiting
processes. On the face of it, such a process requires
an infinity of steps, but the Greeks devised a procedure
by which the express introduction of infinity was cir-
cumvented. The Greeks never bestowed mathematical
legitimacy on an avowed conception of infinity, but
they created a circumlocution by which to avoid any
direct mention of it. Thus the word apeiron occurs
in Archimedes only nontechnically, and very rarely too.
In the nineteenth century, Georg Cantor and others,
but mainly Cantor, legitimized infinity directly, and
the world of thought has not been the same since. But
the Greek method of circumvention lives on too, as
vigorously and indispensably as ever; except that a
symbol for infinity—namely the symbol “x221E;” which
was introduced by John Wallis in 1656—has been
injected into the context, with remarkable conse-
quences. The symbol occurs, for instance, in the limit
relation
lim n薔蜴1 /n = 0,
which, notwithstanding its un-Archimedian appear-
ance, is purely Archimedian by its true meaning. In
fact, since 1/n decreases as n increases, the Cauchy
definition of this relation states that corresponding to
any positive number ε, however small, there exists an
integer n such that 1/n < ε. Now, this is equivalent to
1 <
nε,, or, to
nε > 1, and the last relation can be
verbalized thus:
If ε is any positive real number, then on adding it to itself
sufficiently often, the resulting number will exceed the
number 1.
The Greeks did not have our real numbers; but if
we nevertheless superimpose them on the mathematics
of Archimedes, then the statement just verbalized be-
comes a particular case of the so-called “Postulate of
Archimedes,” which, for our purposes, may be stated
thus:
If a and b are any two magnitudes of the same kind (that
is if both are, say, lengths, areas, or volumes), then on adding
a to itself sufficiently often, the resulting magnitude will
exceed b; that is na > b, for some n.
(E. J. Dijksterhuis,
Archimedes, pp. 146-47 has the wording of the postulate
in original Greek, an English translation of his own, and
a comparison of this translation with various others).
The Greek theory of proportion was a “substitute”
for our present-day theory of the linear continuum for
real numbers, and the infinitely small is involved in
interlocking properties of denseness and completeness
of this continuum (see Appendix to this section). Our
real numbers are a universal quantitative “yardstick”
by which to measure any scalar physical magnitude,
like length, area, volume, time, energy, temperature,
etc. The Greeks, most regrettably, did not introduce
real numbers; that is they did not operationally abstract
the idea of a real number from the idea of a general
magnitude. Instead, Euclid's Book 5 laboriously estab-
lishes properties of a linear continuum for a magnitude
(μέγεθος, megethos) in general. If the Greeks had been
inspired to introduce our field of real numbers and to
give to the positive numbers the status of magnitudes,
then their theory of proportions would have applied
to the latter too, and their theory of proportions thus
completed would have resembled an avant-garde the-
ory of twentieth-century mathematics.
Within the context of Zeno's puzzles, Aristotle was
also analyzing the infinitely small as a constituent of
the linear continuum which “measures” length and
time. He did so not by the method of circumvention,
which the professional mathematicians of his time were
developing into an expert procedure, but by a reasoned
confrontation à la Georg Cantor, which may have been
characteristic of philosophers of his time. In logical
detail Aristotle's reasoning is not always satisfactory,
but he was right in his overall thesis that if length and
time are quantitatively determined by a suitable com-
mon linear continuum, then the puzzles lose their
force. In fact, in present-day mathematical mechanics,
locomotion is operationally represented by a mathe-
matical function x = φ(t) from the time variable t to
the length variable x, as defined in working mathe
matics; in such a setup Zeno's paradoxes do not even
arise. It is not at all a part of a physicist's professional
knowledge, or even of his background equipment, to
be aware of the fact that such puzzles were ever con-
ceived.
The “method of exhaustion” is a Greek anticipation
of the integral calculus. In the works of Archimedes,
the syllogistic maturity of the method is equal to that
of the Riemann-Darboux integral in a present-day
graduate text, but in operational efficiency the method
was made obsolete by the first textbook on the integral
calculus from around A.D. 1700 (C. B. Boyer, p. 278).
However the method also embodied the postulate of
Archimedes, and this postulate has an enhanced stand-
ing today. An innovation came about in the late nine-
teenth century when G. Veronese (Grundzüge, 1894)
and D. Hilbert (Grundlagen, 1899) transformed the
“postulate” into an “axiom,” that is into an axiomatic
hypothesis which may or may not be adjoined to suita-
ble sets of axioms, in geometry, analysis, or algebra.
This gives rise to various non-Archimedian possibilities
and settings, some of which are of interest and even
of importance.
Aristotle made the major pronouncement (Physica,
Book 3, Ch. 7) that a magnitude (megethos) may be-
come infinitely small only potentially, but not actually.
This is an insight in depth, and there are various possi-
bilities for translating this ideational pronouncement
from natural philosophy into a present-day statement
in operational mathematics. We adduce one such
statement: although every real number can be repre-
sented by a nonterminating decimal expansion, it is
generally not possible to find an actual formula for the
entire infinite expansion; but potentially, for any pre-
scribed real number, by virtue of knowing it, any
desired finite part of its decimal expansion can be
obtained.
Appendix. A linearly ordered set is termed dense if
between any two elements there is a third. It is termed
complete if for any “Dedekind Cut,” that is for any
division of the set into a lower and upper subset, (i)
either the lower subset has a maximal element, (ii) or
the upper subset has a minimal element, (iii) or both.
If the set is both dense and complete, possibility (iii)
cannot arise, so that either the lower subset has a
maximum, or the upper subset has a minimum. This
single element is then said to lie on the cut, or to be
determined by the cut.
