II. NATURAL PHILOSOPHY
When the Greeks started out to take stock of the
physical and cosmological phenomena around them—
whether they were Ionians, poet philosophers, Pythag-
oreans, Eleatics, or pluralists—they quickly perceived,
in their own patterns of discernment, the difficulty of
separating the infinite and the indefinite. But the
Greeks did not allow themselves to become frustrated
over this. During the formative stages of their rational-
ity, even still in Plato, the Greeks reacted to this
difficulty by investing the word apeiron with both
meanings in one, and they added a range of interme-
diate and proximate meanings too. Thus, in the context
of a pre-Socratic philosopheme, and even still in Plato,
apeiron, when translated into a modern idiom, may
have to be rendered variously by: infinite, illimited,
unbounded; immense, vast; indefinite, undetermined;
even by: undefinable, undifferentiated. Furthermore,
in the meaning of: infinite, illimited, unbounded,
apeiron may refer to both bigness and smallness of size;
and—what is important—in its meaning of indefinite,
undetermined, undifferentiated, etc., apeiron may refer
not only to quantity but also to quality (in our sense),
even indistinguishably (Bochner, The Role of Mathe-
matics, Ch. 2).
A prominent ambiguity, to which we have referred
before (ibid.) occurs in a verbatim fragment of Xeno-
phanes (frag. B 28). In an excellent translation of W.
K. C. Guthrie it runs (emphasis added):
At our feet
We see this upper limit of the Earth
coterminous with air, but underneath
it stretches without limit
[es apeiron].
The apeiron in this fragment clearly refers only to what
is under the surface, and not also to what is above the
horizon; but what this apeiron actually means cannot
be stated. Commentators since the nineteenth century
have been debating whether it should be translated by
“infinite” or “indefinite.” We think that the point is
undecidable, and we have previously adduced testi-
mony from latest antiquity in support of this conclu-
sion.
We are not asserting that a Greek of the sixth or
fifth centuries B.C., when encountering the word
apeiron, had to go through a mental process of deciding
which of the various meanings, in our vocabulary, is
intended. The shade of meaning in our sense was usu-
ally manifest from the context; whatever ambiguities
presented themselves, were inherent in the objective
situation, rather than in the subjective verbalization.
Aristotle, in his usage and thinking, tends to take
apeiron in the meaning of “infinite in a quantitative
sense,” and in the second half of his Physica (Books
5-8), which deals with locomotion, terrestrial or orbital,
apeiron is taken almost exclusively in this sense. At any
rate, the second half of the Physica becomes as intel-
ligible as it can be made, if apeiron is taken in this
sense exclusively. But in the first half of the Physica,
which is a magnificent discourse on principles of
physics in their diversity, Aristotle is unable to keep
vestiges of the indefinite out of his apeiron, and even
tints of quality are shading the hue of quantity. In
keeping with this, Aristotle's report on the “puzzles”
of Zeno (see next section, III), in which apeiron has
to be quantitative, is presented by him in the second
half of the Physica, and only there; in the first half
of the Physica there is no mention of the puzzles at
all, not even in the connected essay about apeiron (see
section I, above), in which all aspects of the notion
are presumed to be mentioned.
All told the Greeks created a permanent theme of
cognition when, in their own thought patterns, they
interpreted the disparity between perception and con
ception as an imprecision between the indefinite and
the infinite. Also, our present-day polarity between the
nuclear indefinite of quantum theory and the opera-
tional infinite of mathematics proper is only the latest
in a succession of variations on this Greek motif.
A remarkable confirmation of this Greek insight
came in the twilight period between Middle Ages and
Renaissance. In fact, in the first half of the fifteenth
century Nicholas of Cusa broke a medieval stalemate
when he made bold to proclaim that the universe, in
its mathematical structure, is, in one sense, neither
finite nor infinite, and, in another sense, both finite and
infinite, that is, indefinite. A century later, in the first
half of the sixteenth century, Nicholas Copernicus took
upon himself to rearrange the architectonics of our
solar system, but about the size of the universe he
would only say, guardedly, that it is immense, whatever
that means (A. Koyré, Ch. III). It is true that in the
second half of the sixteenth century Giordano Bruno,
a much applauded philosopher, made the universe as
wide-open and all-infinite as it could conceivably be;
but Johannes Kepler, a scientists' scientist, countered,
with patience and cogency, and incomparably deeper
philosophical wisdom, that this would be an astrophys-
ical incongruity, and in the question of the overall size
of the universe Kepler ranged himself alongside Aris-
totle (A. Koyré, Ch. IV).
In the first half of the seventeenth century, René
Descartes, the modern paragon of right reason and
clear thinking, insisted that his extension (étendue),
which was his space of physical events, is by size
indefinite and not infinite; although in some of his
Méditations, when dealing with the existence of God
in general terms, Descartes imparts to God the attri-
bute of infinity in the common (philosophical) sense
(B. Rochot).
The Platonist Henry More, an intolerant follower
of Giordano Bruno, put Descartes under severe pres-
sure, philosophically and theologically, to change the
verdict into indefinite, but Descartes, to his immea-
surable credit, would not surrender (Koyré, Ch. V and
VI). And, in the second half of the seventeenth century
and afterwards, Isaac Newton, in all three editions of
his incomparable Principles of Natural Philosophy
(1687, 1713, 1726), when speaking of cosmic distances,
uses the Copernican term “immense” (for instance, F.
Cajori, ed., Principia, p. 596), but avoids saying
whether the size of the universe is finite or infinite,
or perhaps indefinite; although between the first and
second editions, in a written reply to a query from
the equally intolerant divine Robert Bentley, Newton
made some kind of “admission” that the universe might
be infinite (A. Koyré, pp. 178-89).
Even the aether of electrodynamics in the nineteenth
century, although it filled a Euclidean substratum of
infinite dimensions, had, by quality, a feature of in-
definiteness, or rather of indeterminacy, adhering to
it. By pedigree, this aether was a descendant of the
“subtle matter” (matière subtile) of Descartes, which
had been as indefinite as the étendue which it filled,
and it is possible that, by a long evolution, both had
inherited their indeterminacy from the original apeiron
of Anaximander, which may have been the first “subtle
matter” there ever was.
Finally we note that an imprecision between the
indefinitely small and the infinitely small intervenes
whenever a substance which is physically known to
be distributed discontinuously (granularly, molecularly,
atomistically, nuclearly) is mathematically assumed, for
the sake of manipulations, to be distributed continu-
ously. Without such simplifying assumptions there
would be no physics today, in any of its parts. It was
the forte of nineteenth-century physics that it excelled
in field theories, which are theories of continuous dis-
tribution of matter or energy, and that at the same
time, and in the same contexts, it was pioneering in
the search of “particles” like atoms, molecules, and
electrons (B. Schonland).
