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Dictionary of the History of Ideas | ||

*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

*n*ε,, or, to

*n*ε > 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.

Dictionary of the History of Ideas | ||