University of Virginia Library

I. A SURVEY

The Old Testament exulted in the omnipotence of
the Creator, but it did not initiate problems about the
unboundedness of His power or the infinity of His
creation. The Hebrew of the Bible did not have a word
for “infinity” in general. It only had words about par-
ticular aspects of infinity, and leading among these was
the word 'olam. It designated eternity, that is infinity
in time, without reference to spatiality. Post-biblically,
however, the word began to acquire traits of spatiality,
ever more so, and it may have given to the present-day
Arabic word 'alam its meaning of “world,” “cosmos,”
“universe” (Encyclopedia of Islam, New Ed. [1960],
1, 349).

Greek literary works, in poetry and prose, were less
theocratic than the Old Testament. But from the first
there was in them an awareness of immensity and even
unboundedness in the cosmos, and Greek rationality
showed very early a disposition to examine the mean-
ing of infinity in its complexity.

The standard Greek word for infinity was apeiron
(ἄπειρον ; probable etymology: a = non, peiras or per-
as = limit, bound). Close cognates to it occurred in
Homer, and the word itself had a considerable literary
cachet, in poetry and prose, letters and science. The
word occurs in Hesiod and Pindar, in literal fragments
of most pre-Socratic philosophers, and in reports about
Pythagorean statements which seem verbally proxi-
mate to original utterances of theirs (Bochner, “The
Size of the Universe...,” sec. II).

Thales of Miletus, the father of Greek rational phi-
losophy, is not yet credited with memorable pro-
nouncements about apeiron, not even indirectly; but
Thales is the only pre-Socratic who is not so credited,
whereas his younger compatriot Anaximander already
is, even emphatically so (ibid., sec. III). After Anaxi-
mander, and up to and including Aristotle, each and
every philosopher dealt with infinity, openly or dis-
guisedly; and many of them had a good deal to say
about it. This in itself sets off Thales from all other
philosophers, and it justifies the shrewd observation in
Diogenes Laërtius that it was Anaximander, and not
Thales, with whom (Greek) speculative philosophy
truly began (Diogenes Laërtius, Book 1, Ch. 13).

It seems likely that Anaximander composed a book
which Anaximander himself, or others after him, called
“On Nature,” and that it included a chapter on apeiron,
perhaps at the head of the book. Apparently because
of this, late classical antiquity (many centuries after
Anaximander) formed a consensus that Anaximander
had been a one-man creator of the problem of infinity
in classical Greek thought, and that this had been his
central achievement. This however is a doubtful thesis.

There is nothing in Plato and Aristotle to suggest that
the problem, or problems, of infinity arose at a fixed
stage of the philosophical past at the initiative of a
specific philosopher. Plato never mentions Anaxi-
mander or ever alludes to him. Aristotle does mention
him, but relatively rarely, and somehow very guardedly
(Kirk and Raven, p. 108), and without singling him out
for a special link with the problem of infinity. In fact,
apeiron occurs in all eight books of Aristotle's Physica;
and, by Aristotle's express design, the major part of
Book 3, namely chapters 4 through 9, is a concise
systematic essay about apeiron. Yet, within this essay,
Anaximander is mentioned only once, along with other
pre-Socratics, and, within the essay, the total reference
to him is as follows:

(Physica, 203b 12-14, Oxford
translation).

Whatever late antiquity may have thought or said,
from reading Aristotle one gains the impression that
the study of infinity, in its various facets, had been from
the first an all-Hellenic enterprise, in which virtually
everybody had participated; and so indeed it had been.

Oswald Spengler, in his ambitious work The Decline
of the West, which was published in German in 1918,
made much of the thesis, to which, in the end, even
professional historians such as P. Kucharski and B.
Rochot subscribed, that unlike the Middle Ages and
the Renaissance, classical antiquity, before the onset
and diffusion of Hellenism, did not find it congenial
to abandon itself to the mystique of infinity, but was
aspiring to control and suppress infinity rather than
to contemplate and savor it. To this we wish to point
out that even during the Renaissance and after, leading
scientists like Copernicus, Kepler, Newton, and others,
were approaching problems of infinity with caution
and reserve, and in no wise abandoned themselves to
a mystique of the infinite. Furthermore, cosmology in
the twentieth century is also circumspect when admit-
ting infinity into its context.

It is true that since the Renaissance many philoso-
phers—as distinct from philosophizing scientists—were
disposed to opt for untrammeled infinity in their find-
ings, and Giordano Bruno (1548-1600) was a leader
among them. But even this disposition may have been
a stage of a development that reached back into classi-
cal antiquity, into Hellenism at any rate. Long before
Bruno infinity of space was vigorously advocated, from
an anti-Aristotelian stance à la Bruno, by the Jewish
philosopher Hasdai Crescas (1340-1411) and this
apparently created a fashion (Wolfson, Crescas'...,
pp. 35-36). Crescas was apparently a late product of
Judeo-Arabic scholasticism, which in its turn was de
rived from Islamic philosophy that had been in bloom
in the tenth and eleventh centuries A.D.; there are
historical appraisals that Islamic philosophy in its turn
had been an off-shoot of general Hellenism (R. Walzer,
passim).

If Greek natural philosophers of the Hellenic period
were indeed wary of infinity, then this was due largely
to the fact that they had precociously discovered how
difficult it is to comprehend infinity in its conceptual
ramifications and not because they had an innate hesi-
tancy to be intimate with it. As is evident from various
works of Aristotle (Physica, De caelo, De generatione
et corruptione, etc.), the Greeks had by then created
a host of problems about infinity that are familiar to
us from cosmology, physics, and natural philosophy;
then as today significant problems about infinity were
correlated with problems about continuity, motion,
matter, genesis of the universe, etc.

For instance, on one occasion Aristotle suggests that
the belief in infinity is derived from five sources: (i)
from the infinity of time, (ii) from the divisibility of
magnitude, (iii) from the fact that the perpetuity of
generation and destruction in nature can be maintained
only if there is an infinite source to draw on, (iv) from
the fact that anything limited has to be limited by
something else, and finally, (v) from the fact that there
is no limit to our power of thinking that would inhibit
the mental attribution of infinity to numbers, to mag-
nitudes, or to what is outside the heavens. (Physica,
203b 15-25; our paraphrase is adapted from W. D.
Ross, Aristotle's Physics, p. 363.) These aspects of in-
finity are timeless; they might have been envisioned,
spontaneously, by Aquinas, Descartes, Leibniz, Kant,
or Herbart. There is most certainly nothing “ancient”
or “antiquarian” about them, and there is nothing in
them to suggest that Aristotle had any kind of innate
hesitancy to face infinity when meeting it.

Furthermore, certain special problems about infinity
which are generally presumed to be “typically medie-
val” were formulated in later antiquity and had roots
in the classical period itself. Thus, Philo of Alexandria
(first century A.D.) and Plotinus (third century A.D.) have
fashioned lasting problems about infinity which are
theological, in the sense that the infinity involved is
a leading attribute of divinity. The Middle Ages them-
selves knew that problems of this kind reached back
at least to Plato, and they may have reached back even
to Xenophanes (late sixth century B.C.); except that only
later antiquity loosened them out of the matrix of
“natural philosophy” within which they had come into
being first.

After the Middle Ages this theologically oriented
infinity of Hellenistic provenance gave rise to a
“secular” infinity in general philosophy, notably so in
the many philosophical systems that were burgeoning

during the scientific revolution which extended over
the sixteenth, seventeenth, and eighteenth centuries
and centered in the seventeenth century. We note that
the transition from theological to ontological infinity
was a natural development which was not “revolu-
tionary” in itself. Logically it does not matter much
whether an infinity is a leading attribute of a theolog-
ically conceived divinity, or of some secular absolute
with a commanding standing in the realm of cognition
and morals, and being and belief. Paradigmatically it
was the same infinity, whether the absolute, of which
it was an attribute, was the rationality of Descartes,
the logicality of Leibniz, the morality of Spinoza, the
sensuality of Hobbes, the empiricism of Locke, or the
idealism of Berkeley.

In the second half of the eighteenth century Im-
manuel Kant made a very curious use of infinity in
his Critique of Pure Reason. The fame of the work rests
on its early chapters, in which Kant posits and ex-
pounds his celebrated thesis that space and time are
a priori absolutes of a certain kind, namely that they
are not objectively real, but only subjectively ideal in
a peculiarly Kantian sense, which Kant himself calls,
“aesthetical.” After expounding this thesis Kant dwells
at length on other matters, but in the second half of
the treatise, namely in the long section called “The
Antinomy of Pure Reason” (trans. N. K. Smith, pp.
384-484), he returns to the thesis and undertakes to
fully demonstrate that neither space nor time can be
objectively real. Kant reasons after the manner of a
medieval schoolman, namely by having resort to old-
fashioned antinomies. On the presumption that space
(or time) is objectively real, Kant presents a thesis that
it then would have to be finite, and an antithesis that
it then would have to be infinite; from which it follows,
according to Kant, that space (or time) cannot be
objectively real but must be aesthetically ideal.

The nineteenth century was crowded with eminent
representatives of general philosophy—idealists, posi-
tivists, historicists, early existentialists—and, in large
works, there was much discourse on absolutes, and on
their attributes, with infinity among them. But nothing
that was said about infinity in works other than scien-
tific ones struck a new note and need be remembered.

In contrast to this, in the twentieth century, and
starting in the last decades of the nineteenth century,
the topic of infinity has become alive with innovations;
but these innovations, even when adopted and ex-
ploited by philosophers, came about primarily in
mathematics and fundamental physics and only sec-
ondarily in philosophy of any kind. Yet there continue
to be those philosophers who from an intellectual
devotion to religion, theology, or “non-scientific” phi-
losophy seek refuge in perennial problems about the
infinity of God and of “secular” absolutes; and there
continue to be books and discussions in which seem-
ingly timeless problems about infinity are thought
about, talked about, and written about almost as much
in the restless hours of today as they used to be in
the leisurely days of yesteryear (J. A. Bernardete; B.
Welte; H. Heimsoeth).

In present-day mathematics, infinity is an everyday
concept, widespread, matter-of-fact, operational, in-
dispensable. There is no mystique, uncertainty, or
ambiguity about it, except that certain foundational
verities about infinity are not demonstrable, but have
to be posited axiomatically, and thus taken “on faith.”
This organic assimilation of infinity to the general body
of mathematics has been a part of the total develop-
ment of analysis since the early nineteenth century.
Outstanding within the total development was Georg
Cantor's creation of the theory of sets and transfinite
numbers, between 1870 and 1890. It was a catalytic
event, and more. But a complete account has to name
a string of predecessors, like Cauchy, Abel, Bolzano,
Hankel, Weierstrass, and others.

Furthermore, the nineteenth century began and the
twentieth century completed a separation between
infinity in mathematics and infinity in physics, in spite
of the fact that, since the nineteenth century, physics,
more than ever, explains and interprets what mathe-
matics expresses and exposes (Bochner, The Role of
Mathematics..., especially the Introduction and Ch.
5). While infinity of mathematics has ceased to be
syllogistically different from other concepts, and oper-
ationally suspect, interpretative infinity of physics is
more problematic and intriguing than ever.

In fact, in rational physics and also cosmology,
whether in ancient or modern times, the infinite has
always been inseparable from the indefinite and even
the undetermined. Physics of the twentieth century has
greatly compounded the situation by creating stirring
hypotheses which may be viewed as novel conceptions
of the role of the indefinitely small and the indefinitely
large in the interpretation of physical events and phe-
nomena from the laboratory and the cosmos (W. K.
Heisenberg; W. Pauli). Thus, the law of Werner
Heisenberg (the uncertainty principle), which states
that for an elementary particle its position and mo-
mentum cannot be sharply determined simultaneously,
is a statement of unprecedented novelty, about the
indefinitely small in physics. Next, the law of Max Born
(statistical distribution of particles), which states that
for large assemblages of matter the density of distribu-
tion is a probability and not a certainty, straddles the
indefinitely small and indefinitely large. Finally, the
unsettling principle of de Broglie (duality of waves and
corpuscles), which states that every elementary particle

has, ambivalently, two realizations, a corpuscular and
an undulatory, can be interpreted to mean that even
the undifferentiated cannot be separated from the in-
definite and the infinite; in this interpretation it reaches
back to an uncanny insight of the Greeks, which was
perceived by them dimly but discernibly.

The indefinitely large also occurs in present-day
cosmology (J. H. Coleman; G. Gamow; H. Bondi).
Among cosmological models of the universe that are
presently under active study there are hardly any that
are as completely infinite as was the universe of
Giordano Bruno, which played a considerable role in
philosophy between 1700 and 1900. The models with
“continual creation” are nonfinite, but they are in-
definitely large rather than infinitely large (Bondi).
Even the “universe of telescopic depth,” which is
presumed to reach as far out into the galactic vastness
as the most powerful telescopes will at any time reach,
is indefinitely large, inasmuch as there is a “rim of the
universe” at which “galaxies fade into nothingness”
(Coleman, p. 63).

In sum, in our days, the philosophical conception
of infinity is back again in the matrix of “scientism”
(philosophy of nature) in which it was first set, molded,
and shaped in the sixth and fifth centuries B.C., in small
Greek communities of inexhaustible vistas.

In the sections to follow we will enlarge upon some
of the topics raised in this survey.