University of Virginia Library

INFINITY

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.

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):

[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).

III. MATHEMATICS

A famous Greek encounter with infinity is the
“puzzles” (logoi) about motion by Zeno of Elea, about
the middle of the fifth century B.C. Best known is the
conundrum about “Achilles and the Turtle.” It main-
tains, against all experience, that in a race between
a quick-footed Achilles and a slow-moving Turtle, if
the Turtle has any head start at all then Achilles cannot
overtake him, ever. In fact, by the time Achilles has
reached the Turtle's starting point the latter has moved
on by a certain distance; when Achilles has covered
that distance, the Turtle has again gained a novel
distance, etc. This gives rise to an unending sequence
of distances; and the puzzle maintains that Achilles
cannot exhaust the sum of the distances and come
abreast with the Turtle (Ross, Aristotle's Physics, Intro-
duction; also A. Edel, Aristotle's Theory...).

The puzzles have an enduring appeal; but their role
in stimulating Greek rationality cannot be easily
gauged, because the Greek documentation of them is
very sparse and hesitant. The puzzles were transmitted
only by Aristotle, not in his Metaphysica, which is
Aristotle's work in basic philosophy, but only in the
Physica, and only in the second half of the latter, which
deals with problems of motion, and not with concep-
tions and principles of physics in their generality as
does the first half. Furthermore, in classical antiquity
the puzzles are never alluded to in mathematical con-
texts, and there is no kind of evidence or even allusion
that would link professional mathematicians with them.

In a broad sense, in classical antiquity the conception
of infinity belonged to physics and natural philosophy,
but not to mathematics proper; that is, to the area of
knowledge with which a department of mathematics

is entrusted today. Nobody in antiquity would have
expected Archimedes to give a lecture “On Infinity”
to an academic audience, or to his engineering staff
at the Syracuse Ministry of Defence. Also, no ancient
commentator would have said that Anaxagoras (fifth
century B.C.) had introduced a mathematical aspect of
infinity, as is sometimes asserted today (e.g., in Revue
de Synthèse, pp. 18-19).

Furthermore, such Greek efforts by mathematics
proper as, from our retrospect, did bear on infinity,
were—again from our retrospect—greatly hampered
in their eventual outcome by a congenital limitation
of Greek mathematics at its root (Bochner, The Role
of Mathematics..., pp. 48-58). As evidenced by
developments since around A.D. 1600, mathematics, if
it is to be truly successful, has to be basically opera-
tional. Greek constructive thinking however, in math-
ematics and also in general, was basically only idea-
tional. By this we mean that, on the whole, the Greeks
only formed abstractions of the first order, that is ide-
alizations, whereas mathematics demands also abstrac-
tions of higher order, that is abstractions from abstrac-
tions, abstractions from abstractions from abstractions,
etc. We are not underestimating Greek ideations as
such. Some of them are among the choicest Greek
achievements ever. For instance, Aristotle's distinction
between potential infinity and actual infinity was a
pure ideation, yet unsurpassed in originality and im-
perishable in its importance. However, as Aristotle
conceived it, and generations of followers knew it, this
distinction was not fitted into operational syllogisms,
and was therefore unexploitable. Because of this even
front-rank philosophers, especially after the Renais-
sance, mistook this distinction for a tiresome scholas-
ticism, until, at last, late Victorian mathematics began
to assimilate it into its operational texture.

In the seventeenth and eighteenth centuries, mathe-
matics was so fascinated with its newly developing raw
operational skills, that, in its ebullience, it hid from
itself the necessity of attending to some basic con-
ceptual (ideative) subtleties, mostly involving infinity,
the discovery and pursuit of which had been a hallmark
of the mathematics of the Greeks. Only in the nine-
teenth century did mathematics sober down, and finally
turn its attention to certain conceptualizations and
delicate ideations towards which the Greeks, in their
precociousness, were oriented from the first. But even
with its vastly superior operational skills, modern
mathematics had to spend the whole nineteenth cen-
tury to really overtake the Greeks in these matters.

This raises the problem, a very difficult one, of
determining the role of the Middle Ages as an interme-
diary between Greek precociousness and modern ex-
pertise. In the realm of mathematical infinity the
thirteenth and fourteenth centuries were rather active.
But studies thus far have not determined whether, as
maintained in the voluminous work of Pierre Duhem
(ibid., p. 117), a spark from the late Middle Ages leapt
across the Renaissance to ignite the scientific revolution
which centered in the seventeenth century, or whether
this revolution was self-igniting, as implied in well-
reasoned books of Anneliese Maier. And they also have
not determined what, in this area of knowledge, the
contribution of the Arabic tributary to the Western
mainstream actually was.

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:

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:

(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.

V. THE INFINITELY LARGE

A true departure from Greek precedents was the
manner in which mathematics of the nineteenth cen-
tury set out, in earnest, to deal with infinity—especially
with the infinitely large—by confrontation and actu-
alization. One such development, which we will briefly

sketch in the next section, was inaugurated in geome-
try, that is in the theory of space structure; so-called
“open” spaces were boldly “closed off” by addition
of ideally conceived “infinitely distant” points that
were operationally created for such purposes. Inter-
nally these were important events which affected the
course of mathematics profoundly, even if philosophers
did not become aware of them; but externally the
dominant and spectacular development was Georg
Cantor's creation of the theory of sets and of transfinite
numbers. It had a wide appeal, and an enduring effect,
outside of professional mathematics too. Cantor's work
was not only a creation, it was a movement. As of a
sudden, infinity ceased to be an object of frequently
aimless and barren ideational speculations, and it be-
came a datum of refreshingly efficient operational
manipulations and syllogizations. The movement
brought to the fore novel thought patterns in and out
of mathematics, and it helped to create the tautness
of syntax in and out of analytical philosophy. Also our
present-day “New Mathematics,” which—at any rate
in the United States—is being introduced on all levels
of pre-college schooling, is a delayed response to a
permanent challenge which has been emanating from
Cantor's theory from the first.

But before these Victorian achievements, that is, in
the overlong stretch of time from the early Church
Fathers to the early nineteenth century, and even
during the ages of the scientific revolution and of the
Enlightenment, mathematical developments regarding
infinity were, on the whole, excruciatingly slow.
Newton, Leibniz, Euler, Lagrange, or even Carl Fried-
rich Gauss, would not have been able to express satis-
factorily, in words of theirs, when an infinite series is
convergent and when not. As we have already stated,
John Wallis introduced in 1656 our present-day symbol
“∞” for infinitely large, and he began to operate with
it as if it were one more mathematical symbol. This
can be done, to an extent. But, from our retrospect,
for about 150 years the operations with the symbol
were amateurishly and scandalously unrigorous. How-
ever, long before that, in the great mathematical works
of Euclid, Archimedes, and Apollonius, of the third
century B.C., there were well-conceived convergence
processes, which, within their own settings, were han-
dled competently and maturely. It must be quickly
added however, that this mature Greek mathematics
did not have the internal strength to survive, but was
lost from sight in the obscurity of a general decline
of Hellenism, whereas the mathematics of the seven-
teenth and eighteenth centuries, however beset with
shortcomings of rigor, has been marching from strength
to greater strength without a break.

It had been a tenet of Aristotle that there cannot
be anything that is infinite in actuality, meaning “that
no form of infinite exists, as a given simultaneously
existing whole” (Ross, Aristotle, p. 87). But 22 centuries
later, Georg Cantor retorted, boastfully, that his find-
ings clearly controverted the tenet. Cantor also ad-
duced illustrious predecessors of his, notably Saint
Augustine, who had anticipated the actual infinity of
his, even as it applies to natural numbers (Cantor,
Gesammelte..., pp. 401-04 and other passages).
These statements of Cantor are misleading, and we will
briefly state in what way.

On the face of it, Cantor was right in affirming that
there is an anticipation of the first transfinite cardinal
number in Saint Augustine's De civitate Dei, especially
in the chapter entitled “Against those who assert that
things that are infinite cannot be comprehended by the
knowledge of God” (Book 12, Ch. 18). However, this
anticipation and the others which Cantor adduces,
were ideations only, and were made and remained at
a considerable distance from mathematics proper. But
Cantor's theory of sets was produced in a spirit of truly
“abstract” mathematics; it quickly moved into the
central area of operational mathematics and has re-
mained there ever since. Within theological and philo-
sophical contexts, actual infinity, however exalted, is
hierarchically subordinate to a supreme absolute of
which it is an attribute. But in set theory, infinity,
although a property of an aggregate, is nevertheless
mathematically autonomous and hierarchically su-
preme; like all primary mathematical data it is self-
created and self-creating within the realm of mathe-
matical imagery and modality.

In some of his writings Cantor reflects on the nature,
mission, and intellectual foundation of his theories, and
these reflections create the impression that Cantor's
prime intellectual motivation was an urge to examine
searchingly Aristotle's contention that infinity can exist
at best only potentially, and never actually. But Can-
tor's mathematical work itself, if one omits his self-
reflections, suggests a different kind of motivation, a
much more prosaic one. It suggests that Cantor's theory
evolved out of his preoccupation with an everyday
problem of working mathematics, namely with Rie-
mann's uniqueness problem for trigonometric series.
Some of Riemann's work, for instance his momentous
study of space structure, is clearly allied to philosophy.
But the problem of technical mathematics which at-
tracted Cantor's attention was not at all of this kind.
There was nothing in it to stimulate an Ernst Cassirer,
Bertrand Russell, A. N. Whitehead, or even Charles
S. Peirce or Gottlob Frege. Also, the nature of the
mathematical problem was such, that Cantor was led
Sto conceive ordinal numbers first, cardinal numbers
next, and general aggregates last (Cantor, Gesammelte
..., p. 102, editor's note 2). But in a later systematic
recapitulation (ibid., pp. 282-356), which is “philo-

sophically” arranged, the order of concepts is reversed.

We have dwelt on this, because, in our view, the
actual infinity as conceived by Cantor, is entirely
different from the actual infinity as conceived by Aris-
totle, so that there is no conflict between Aristotle's
denial and Cantor's affirmation of its existence. In
support of this view we observe as follows: according
to Cantor (Gesammelte..., pp. 174-75), Aristotle had
to deny the existence of an actual infinity, simply
because Aristotle was not intellectually equipped to
countenance the fact that if n is a finite number and
α a transfinite number, then α “annihilates” n, in the
sense that

Cantor observes that, contrary to what Aristotle may
have thought, this is a true and important fact, and
he derides Aristotle for not grasping it but finding
something incongruous in it. Cantor elaborates on this
fact by further noting that if α is a number of ordinal
type, and if the order of the addends n and α is in-
verted, then n is not annihilated, because, in fact

Also, Cantor interprets all this to imply—in all serious-
ness—that if a finite number has the temerity of placing
itself in front of an infinite ordinal number α then it
suffers annihilation, but if it has the prudence of rang-
ing itself in the rear of an infinite ordinal α then its
existence is mercifully spared.

This bizarre interpretation, however alluring for its
boldness, must not be allowed to detract from the fact
that Aristotle himself, in the given context (Physica,
Book 3, Ch. 5; 204b 12-20), to which Cantor refers
(he actually refers not to this passage in the Physica,
but to a less “authoritative” near-duplication of it in
Metaphysica, Book II, Ch. 10), speaks not of number
(arithmos), or even magnitude (megethos), but of
“body” (soma, σω̃μα ), which he expressly specifies to
be an elementary constituent of matter, like fire or air.
Aristotle asserts that such a body cannot be infinite,
because if it were, then the addition (or subtraction)
of a finite amount would not affect the sum total. This
assertion, whatever its merit, is a statement about
physics or natural philosophy, and not, as Cantor mis-
leadingly presents it, a statement about technical
mathematics. One can easily formulate a statement
which would sound very similar to the assertion of
Aristotle, and which a present-day physicist might
accept, or, at any rate, not find unreasonable. Thus,
a present-day physicist might reason that it is incon-
gruous to assume that the total energy of the universe
is infinite. In fact, if it were infinite, the addition or
subtraction of a finite amount of energy would not
change the total amount of energy, and the law of the
conservation of energy—if our physicist generally sub-
scribes to it—would become pointless when applied
to such a universe as a whole. It is true that nowadays
the law of conservation of energy, although adhered
to in laboratory physics, is not always observed in
cosmology. Thus in present-day cosmological models
with “continual creation of matter” the total energy
is nonfinite and the law of the conservation of energy
is not enforced. But the infinity involved in these
models leans more towards Aristotle's potentiality than
Cantor's actuality, and is certainly not as fully “actual”
as in Cantor.

VI. THE INFINITELY DISTANT

The standard perspective of the visual arts, which
was created in the sixteenth century, features a
“vanishing point.” This is a concrete specific point in
the total mimetic tableau, yet, in a sense, it represents
an infinitely distant point in an underlying geometry
(Christian Wiener, Introduction; E. Panofsky, Albrecht
Dürer). Mathematics since then, and especially in the
nineteenth century, has introduced various mathe-
matical constructs with infinitely distant points in
them, and we will briefly report on some of them.

There were no such tangible developments before
the Renaissance. Aristotle, in his Poetics and elsewhere,
speaks of the art of painting, but not of vanishing points
or other infinitely distant points in geometry. In antiq-
uity altogether, only later antiquity had some adum-
brations (Panofsky, “Die Perspective”...). In medieval
architecture, Gothic arches and spires would “vanish”
into the upper reaches of the aether; but they would
stay there and not converge towards concrete specific
points in the total tableau.

But the Renaissance produced perspective; and it
also began to create novel theories of vision (V.
Ronchi). Furthermore, since around A.D. 1600 mathe-
matics began to construct, concretely, infinitely distant
points, and in the first construction, an implicit one,
the Euclidean plane E2 was “closed off” in all direc-
tions by the addition of a point at infinity on each ray
emanating from a fixed point. That is, E2 was viewed
(as already in De rerum natura of Lucretius) as an
“open” disk of infinite radius; it was made, geometri-
cally, into a “closed” disk by the addition of a “hoop”
of infinite radius around it. This construction was not
performed explicitly or intentionally, but was implied
in the following assumption. By Euclid's own definition,
two straight lines are parallel if, being in the same
plane, and being produced indefinitely in both direc-
tions, they do not meet one another in either direction
(T. L. Heath, I, 190). Now, around 1600 some mathe-
maticians began to assume, as a matter of course, that

Euclid's definition is equivalent to the description that
two straight lines in the same plane are parallel, if,
after being produced indefinitely, they meet at two
infinitely distant points at both ends of the configura-
tion (and only there). To assume this is, from our pres-
ent retrospect, equivalent to assuming that there is
around E2 the kind of hoop that we have described.

The same mathematicians soon began to sense, in
their own manner, that to close off E2 in this fashion
is neither intellectually original nor operationally
profitable. They began to “experiment” with other
procedures for closing off E2. These “experiments”
were a part, even a significant part, of the sustained
efforts to erect the doctrines of descriptive and projec-
tive geometries, and they were satisfactorily completed
in the course of the nineteenth century only.

The outcome was as follows. It is pertinent to install
our hoop around E2, but this is only a first step. The
total hoop is too wide, that is, not sufficiently restric-
tive, and it is necessary to “reduce” it in size by
“identifying” or “matching” various points of it with
each other.

First and foremost, it is very appropriate to “iden-
tify” all points of the hoop with each other, that is
to “constrict” the hoop to a single point. By the addi-
tion of this single point, the plane E2 becomes “sealed
off” as infinity, and the resulting two-dimensional
figure is topologically a spherical surface S2. Con-
versely, if one starts out with an S2, say with an ideally
smoothed-out surface of our earth, and removes one
point, say the North Pole, then the remaining surface
can be “spread out” topologically onto the E2. Such
a spreading out is done in cartography by means of
the so-called stereographic projection. This projection
of a punctured sphere S2 on the Euclidean E2 is not
only topological, that is bi-continuous, but also con-
formal, that is angle-preserving; and this was already
known to the astronomer and geographer Ptolemy in
the second century A.D. in his Geography (M. R. Cohen
and I. E. Drabkin, pp. 169-79).

The one-point completion which we have just de-
scribed can be performed for the Euclidean (or rather
Cartesian) space En of any dimension n, and the result
is the n-dimensional sphere Sn. Topologically there is
no difference between various dimensions, but alge-
braically there is. First, for n = 2, the plane E2 can
be viewed as the space of the complex numbers
z = x + iy, and the added point at infinity can be
interpreted as a complex number ∞, for which, sym-
bolically,

This interpretation is commonly attributed to C. F.
Gauss (1777-1855). Next, for n = 4, En can be inter-
preted as the space of quaternions a + ib + jc + kd,
which were created by William Rowan Hamilton
(1805-65), and the point at infinity can be interpreted
as a quaternion ∞ for which (°) holds. This can still
be done for E8, if it be viewed as a space of so-called
Cayley numbers (= pairs of quaternions), but no other
such cases of so-called “real division algebras” are
known (N. Steenrod, pp. 105-15; M. T. Greenberg, p.
87). As regards quaternions it is worth recording, as
a phenomenon in the history of ideas, that around 1900
there was an international organization of partisans
who believed that quaternions were one of the most
potent operational tools which the twentieth century
was about to inherit from the preceding one; the orga-
nization has been long extinct.

After the spheres, the next important spaces which
arise from En by a suitable addition of points at infinity
are so-called projective spaces; we will speak only of
“real” projective spaces, and denote them by Pn. (Other
projective spaces are those over complex numbers,
quaternions, or Cayley numbers; see Steenrod, Green-
berg, loc. cit.) For each dimension n, Pn arises from
En if one identifies each infinitely distant point of the
“hoop” around En with its antipodal point, that is, if
for each straight line through the origin of En the two
infinite points at the opposite ends of it are identified
(that is “glued together”). The resulting space is a
closed manifold (without any boundary), and it is the
carrier of the so-called elliptic non-Euclidean geometry
of F. Klein (S. M. Coxeter, p. 13). Klein's purpose in
devising his geometry was to remove a “blemish” from
the spherical (non-Euclidean) geometry of B. Riemann.
In Riemann's geometry any two “straight” (i.e.,
geodesic) lines intersect in precisely two points,
whereas in Klein's variant on it they intersect in pre-
cisely one point only.

The Pn, that is the real projective spaces, have a
remarkable property: for even dimensions n they are
nonorientable, but for odd dimensions orientable. A
space is orientable, if a tornado (or any other spinning
top), when moving along any closed path, returns to
its starting point with the same sense of gyration with
which it started, and it is nonorientable if along some
closed path the sense of gyration is reversed. In the
case of a Pn with an even-dimensional n the sense of
gyration is reversed each time the path “crosses” in-
finity. In particular, the space P2, that is the space of
two dimensional elliptic geometry, is not orientable,
but P3 is. Thus, in P2 a fully mobile society cannot
distinguish between right- and left-handed screws, but
in P3 it can.

Nineteenth-century mathematics has created many
other completions of En which have become the sub-

stance of the theory of Riemann surfaces and of alge-
braic geometry. Twentieth-century mathematics has
produced a one-point “compactification” (P. Alexan-
droff, “Über die Metrisation...”), which has spread
into all of general topology, and a theory of prime-
ends (C. Carathéodory, “Über die Begrenzung...”),
which in one form or another is of consequence in con-
formal mapping, potential theory, probability theory,
and even group theory.

In the nineteenth century, while mathematics was
tightening the looseness-at-infinity of Euclidean struc-
ture, French painting was loosening the tightness-at-
inifinity of perspective structure. The French movement
is already discernible in Dominique Ingres, but the
acknowledged leader of it was Paul Cézanne. Cézanne
was not an “anarchist,” wanting only to “overthrow”
classical perspective without caring what to put in its
place, but analysts find it difficult to say what it was
that he was aspiring to replace perspective by. We
once suggested, for the comprehension of Cézanne, an
analogy to developments in mechanics (Bochner, The
Role of Mathematics..., pp. 191-201), and in the
present context we wish to point out, in another vein,
that Cézanne was trying to loosen up the traditional
perspective by permitting several vanishing points
instead of one (E. Loran, Cézanne's Composition), and
by giving to lines of composition considerable freedom
in their mode of convergence towards their vanishing
points (M. Schapiro, Paul Cézanne). This particular
suggestion may be off the mark, but the problem of
a parallelism between nineteenth-century develop-
ments in geometry and in the arts does exist.

VII. THE COMPLETE AND THE PERFECT

Nonscientific aspects of infinity are usually broad and
elusive and mottled with ambiguities and polarities.
One of the worst offenders was Benedict Spinoza,
however much he presumed to articulate his thoughts
more geometrico. In fact, the term “infinite” stands in
Spinoza for such terms as “unique,” “incomparable,”
“homonymous,” “indeterminate,” “incomprehensible,”
“ineffable,” “indefinable,” “unknowable,” and many
other similar terms (Wolfson,... Spinoza, I, 138).
What is worse, Spinoza justified this license of his by
reference to Aristotle's dictum that “the infinite so far
as infinite is unknown” (ibid., I, 139), which Aristotle
certainly would not have allowed to be exploited in
this way.

But even when intended to be much more coherent,
the conception of infinity in a nonscientific context,
especially in theology, need not refer to the magnitude
of quantitative elements like space, time, matter, etc.,
but it may refer to the intensity of qualitative attributes
like power, being, intellect, justice, goodness, grace,
etc. There are large-scale philosophical settings, in
which infinity, under this or an equivalent name, does
not magnify, or even emphasize, the outward extent
of something quantifiable, but expresses a degree of
completeness and perfection of something structurable.

Because of all that, philosopher-theologians who
strive for clarity of thought and exposition are having
great difficulties with them. Thus, Saint Thomas
Aquinas, in a discourse on the existence and nature
of God in the entering part of his Summa theologiae,
compares and confronts the completeness and perfec-
tion in God with the infinite and limitless in Him. In
a “typically Thomistic” sequence of arguments and
counterarguments, completeness and infinity are al-
ternately identified and contrasted, as if they were
synonyms and antonyms in one; and, although Aquinas
very much strives for clarity, it would be difficult to
state in a few sharply worded declaratory statements,
what the outcome of the discourse actually is (Saint
Thomas Aquinas, Summa theologiae, Vol. II).

Completeness in philosophy is even harder to define
than infinity in philosophy, and the relation between
the two is recondite and elusive. The problem of this
relation was already known to the Greeks. As a prob-
lem of cognition it was created by Parmenides, and
then clearly formulated by Aristotle, but as a problem
of “systematic” theology it came to the fore only in
the second half of Hellenism, beginning recognizably
with Philo of Alexandria, and coming to a first culmi-
nation in the Enneads of Plotinus. From our retrospect,
the “One” (τὸ ἕν) of Plotinus was a fusion between a
divinely intuited completeness and a metaphysically
perceived infinity. Books V and VI of the Enneads are
full of evidence for this, and we note, for instance, that
a recent study of Plotinus summarizes the passage VI,
8.11, of the Enneads thus:

(A. H. Armstrong, Plotinus, p. 63).

Also, a study of Plotinus of very recent date has the
following important summary:

(J. M. Rist, p. 25).

Long before that, Aristotle devoted a chapter of his
Physica (Book 3, Ch. 6) to an express comparison be-
tween completeness and infinity, as he saw it. Aristotle
presents a thesis that infinity is directly and unmistaka-
bly opposed to “the Complete and the Whole” (τέλειον
καὶ ὅλον), and his central statement runs as follows:

(206b 34-207a 1, Oxford translation).

His definition then is as follows:

(ibid., 207a 7-11).

(ibid., 207a 13-14).

Immediately following this passage, Aristotle makes
respectful mention of Parmenides, and deservedly so.
The great ontological poem of Parmenides clearly
outlines a certain feature of completeness, as an attri-
bute of something that is, ambivalently, an ontological
absolute and a cosmological universe. Ontologically
this universe was made of pure being and thought
itself, and there has been nothing like it since then
(W. K. C. Guthrie, A History..., Vol. 2; Untersteiner,
Parmenide...; L. Tarán, Parmenides...). And yet,
as we have tried to demonstrate in another context,
the Parmenidean completeness was so rich in allusions
that it even allows a measure of mathematization in
terms of today, more so than Aristotle's interpretation
of this completeness would (Bochner, “The Size of the
Universe...,” sec. V).

The Parmenidean being and thought, as constituents
of the universe, were conceived very tightly. In the
course of many centuries after Parmenides, they were
loosened up and gradually transformed into the
Hellenistic “One” and “Logos,” which were conceived
more diffusely, and less controversially. Also, in the
course of these and later centuries, the Parmenidean
universe, with its attribute of completeness, was overtly
theologized, mainly Christianized.

Aristotle took it for granted that the ontological
universe of Parmenides, in addition to being complete,
was also finite, and Parmenides did indeed so envisage
it, more or less. But what was a vision in Parmenides
was turned into a compulsion by Aristotle. That is,
Aristotle maintained, and made into a major proposi-
tion, that the Parmenidean universe could not be other
than finite, because, for Aristotle, completeness some-
how had to be anti-infinite automatically.

With this proposition Aristotle may have over-
reached himself. Mathematics has introduced, entirely
from its own spontaneity, and under various names,
several versions of completeness, any of which is remi-
niscent of the notion of Parmenides, and, on the whole,
finiteness is not implied automatically. On the contrary,
the completeness of Parmenides can be mathematically
so formalized, that a universe becomes complete if it
is so very infinite that no kind of magnification of it
is possible (Bochner, loc. cit.). But mathematizations
of the conception of completeness are of relatively
recent origin, and it would not be meaningful to pursue
the comparison between mathematical and philo-
sophical versions of the conception beyond a certain
point.

BIBLIOGRAPHY

The only general history of infinity is the book of Jonas
Cohen; a supplement to it, heavily oriented toward theol-
ogy, is the essay of Anton Antweiler. Of considerable inter-
est is a collection of articles in the 1954 volume of the Revue
de Synthèse.

A comprehensive study on infinity in Greek antiquity is
the work of Mondolfo. The author is a staunch defender
of the thesis that Greek thought had fully the same attitude
towards infinity as modern thought. About infinity in the
Old Testament see the books of C. von Orelli, Thorleif
Boman, and James Barr. Occasionally one encounters the
view that, in a true sense, infinity was originally as much
a Hebraic intuition as a Greek one, and perhaps even more
so. Such a view is implied in the books just cited, and it
was expressly stated in Revue de Synthèse, p. 53 (remark
by M. Serouya).

Infinity in the pre-Socratics is competently dealt with in
the recent work of Guthrie. Infinity in all of Greek philoso-
phy, Hellenic and Hellenistic, is also fully dealt with in
the great Victorian standard work of Eduard Zeller. It is
still very good on infinity in Plotinus, and also in Philo,
in spite of recent special studies on the two, especially on
Plotinus. In the case of Philo, it is not easy to locate infinity
specifically in his work, and even in Wolfson's detailed study
of Philo there is very little direct reference to it.

About infinity in medieval philosophy, European and
Arabic, and in subsequent philosophy up to and including
Spinoza, there is a wealth of material in Wolfson's two-
volume work on Spinoza. All of volume I is very pertinent,
and not only the parts dealing expressly with infinity, like
Chapter V, part III (Definition of the term “Infinite”), and
Chapter VIII (Infinity of Extension). The latter chapter is
of special interest for the genesis of Descartes' view on the
nature of infinity of his extension (or étendue); see section
II above.

About infinity in scientist-philosophers, or cosmologists,
or astronomers from Nicholas of Cusa to Newton and Leib-
niz there is the informative work of Koyré, which features
a judicious selection of verbatim excerpts, all in English.
There are also recent books about the relevance of infinity
to nonscientific general philosophy, such as the books of
Bernardete, Welte, and Heimsoeth.

Infinity in mathematics is accounted for in any general
history of mathematics, but especially in Boyer's The History
of the Calculus. For the history of Zeno's paradoxes the

main account, with full references, is the article in nine
parts, commencing in 1915, by F. Cajori in the American
Mathematical Monthly. The references are carried to 1936
in the lengthy introduction to Ross's edition, with commen-
tary, of Aristotle's Physics. To judge by an incidental remark
in Cajori's account, the first outright association of the
paradoxes with mathematics is documented only from the
seventeenth century A.D., in the work of Gregory of St.
Vincent.

For the roots and rise of Georg Cantor's set theory there
is much material in Cantor's Collected Works which have
been edited by Ernst Zermelo. The principal memoirs of
Cantor were translated into English, with introduction and
notes, by P. E. B. Jourdain. There is a lack of studies on
how the emergence of Cantor's set theory fits into the
history of ideas; there is, for instance, no special study on
how it reflects itself in the philosophical system of Charles
S. Peirce (cf. Collected Papers of Charles S. Peirce, ed. C.
Hartshorne and Paul Weiss, Cambridge, Mass. [1933], Vol.
IV).

The following works are additional references for the study
of infinity. Paul Alexandroff, “Über die Metrisation der im
kleinen kompakten topologischen Räume,” Mathematische
Annalen, 99 (1924), 294-307. Anton Antweiler, Unendlich,
Eine Untersuchung zur metaphysischen Weisheit Gottes auf
Grund der Mathematik, Philosophie, Theologie (Freiburg im
Breisgau, 1935). Saint Thomas Aquinas, Summa theologiae,
Latin text and English trans. by Blackfriars (London and
New York, 1962), Vol. II. Aristotle, Physica, trans. R. P.
Hardie and R. K. Gaye in the Oxford translation of Aris-
totle's works under the general editorship of W. D. Ross,
Vol. 2 (Oxford, 1930). See also W. D. Ross, below. A. H.
Armstrong, Plotinus (New York, 1962). James Barr, Biblical
Words for Time (London, 1961). José A. Bernardete, Infinity,
an Essay in Metaphysics (Oxford, 1964). Salomon Bochner,
The Role of Mathematics in the Rise of Science (Princeton,
1966); idem, “The Size of the Universe in Greek Thought,”
Scientia, 103 (1968), 510-30. Hermann Bondi, Cosmology,
2nd ed. (Cambridge, 1960). Thorleif Boman, Das hebräische
Denken im Vergleich mit dem Griechischen, 4th ed. (Göttin-
gen, 1965); 3rd ed. trans. as Hebrew Thought Compared With
Greek Thought (Philadelphia, 1961). Carl B. Boyer, The
History of the Calculus (New York, 1959). F. Cajori, “The
History of Zeno's Arguments on Motion,” American Mathe-
matical Monthly, 12 (1915), 1-6, 39-47, 77-82, 109-15,
143-49, 179-86, 215-20, 253-58, 292-97; idem, Sir Isaac
Newton's Mathematical Principles of Natural Philosophy and
His System of the World, trans. Andrew Motte (1729), re-
vised by F. Cajori (Berkeley, 1934; many reprints); cited
as Principia. Georg Cantor, Gesammelte Abhandlungen
mathematischen und philosophischen Inhalts, ed. Ernst
Zermelo (Berlin, 1932). C. Carathéodory, “Über die Beg-
renzung einfach zusammenhängender Gebiete,” Mathe-
matische Annalen, 73 (1913), 343-70. Morris R. Cohen and
I. E. Drabkin, A Source Book in Greek Science (New York,
1948). Jonas Cohen, Geschichte der Unendlichkeitsproblems
im abendländischen Denken bis Kant (Leipzig, 1869). James
H. Coleman, Modern Theories of the Universe (New York,
1963). H. S. M. Coxeter, Non-Euclidean Geometry (Toronto,
1957). E. J. Dijksterhuis, Archimedes (New York, 1957).
Diogenes Laërtius, Lives of Eminent Philosophers, 2 vols.
(London and Cambridge, Mass., 1925). Abraham Edel, Aris-
totle's Theory of the Infinite (New York, 1934). George
Gamow, The Creation of the Universe (New York, 1952).
Marvin T. Greenberg, Lectures on Algebraic Topology (New
York, 1967). Gregory of St. Vincent, Opus geometricum
quadratura circuli et sectionum coní (Antwerp, 1647).
W. K. C. Guthrie, A History of Greek Philosophy, Vols. 1 and
2 (Cambridge, 1962 and 1965). T. L. Heath, The Thirteen
Books of Euclid's Elements (Cambridge, 1908); idem, History
of Greek Mathematics, 2 vols. (Oxford, 1921). Heinz Heim-
soeth, Diesechs grossen themen der abendländischen Meta-
physik und der Ausgang des Mittelalters, 3rd ed. (Stuttgart,
1954). Werner Heisenberg, Physics and Philosophy, the
Revolution in Modern Science (New York, 1958). David
Hilbert, Grundlagen der Geometrie (Leipzig, 1899), many
editions and translations. P. E. B. Jourdain, Contributions
to the Founding of the Theory of Transfinite Numbers
(Chicago and London, 1915). Immanuel Kant, Critique of
Pure Reason, trans. Norman Kemp Smith (London, 1929).
G. S. Kirk and J. E. Raven, The Presocratic Philosophers
(Cambridge, 1957). Alexandre Koyré, From the Closed World
to the Infinite Universe (Baltimore, 1957). P. Kucharski,
“L'idée de l'infini en Grèce,” Revue de Synthèse, 34 (1954),
5-20. Earle Loran, Cézanne's Composition, 2nd ed. (Berke-
ley, 1944). Anneliese Maier, DieVorläufer Galileis im 14.
Jahrhundert (Rome, 1949); idem, Zwei Grundprobleme der
Scholastischen Naturphilosophie (Rome, 1951); idem,
Zwischen Philosophie und Mechanik (Rome, 1958); idem,
Metaphysische Hintergründe der Spätscholastischen Natur-
philosophie (Rome, 1955). Rodolfo Mondolfo, L'infinito nel
pensiero dell'Antiquità classica (Florence, 1965). Isaac
Newton, see Cajori, above. C. von Orelli, Diehebräischen
Synonyma der Zeit und Ewigkeit, genetisch und sprachver-
gleichlich dargestellt (Leipzig, 1871). Erwin Panofsky,
Albrecht Dürer (Princeton, 1945); idem, “Die Perspective
als 'Symbolische Form,'” in Vorträge der Bibliothek Warburg
(1924-25); the latter is reprinted in Panofsky's Aufsätze zu
Grundfragen der Kunstwissenschaft (Berlin, 1964). W. Pauli,
ed., Niels Bohr and the Development of Physics (New York,
1955). Charles S. Peirce, Collected Papers of Charles S.
Peirce, ed. C. Hartshorne and Paul Weiss, 6 vols. (Cam-
bridge, Mass., 1933), Vol. IV. Revue de Synthèse (Centre
international de Synthèse), 34, New Series (1954). J. M. Rist,
Plotinus: The Road to Reality (Cambridge, 1967). B. Rochot,
“L'infini Cartésien,” Revue de Synthèse, 34 (1954), 35-54.
Vasco Ronchi, The Science of Vision (New York, 1957).
W. D. Ross, ed., Aristotle's Physics, A revised text with intro-
duction and commentary (Oxford, 1936); idem, Aristotle, a
complete exposition of his works and thought (Cleveland,
1959). Meyer Schapiro, Paul Cézanne (New York, 1952). A.
Schoenfliess, “Projective Geometrie,” Encyclopädie der
mathematischen Wissenschaften, Vol. III, Leipzig, 1898-),
Abt. 5. Basil Schonland, The Atomists (1830-1933) (Oxford,
1968). Oswald Spengler, The Decline of the West, trans. C. F.
Atkinson, 2 vols. (New York, 1926-28). Norman Steenrod,
Topology of Fibre Bundles (Princeton, 1965). Leonardo
Tarán, Parmenides, A Text with Translation, Commentary,

and Critical Essays (Princeton, 1965). Mario Untersteiner,
Parmenide, Testimonianze e Frammenti (Florence, 1958). G.
Veronese, Grundzüge der Geometrie (Berlin, 1894); the orig-
inal edition in Italian is almost never quoted. Richard
Walzer, Greek into Arabic; Essays in Islamic Philosophy
(Oxford, 1962). Bernhard Welte, Im Spielfeld von Endlich-
keit und Unendlichkeit. Gedanken zur Deutung der men-
schlichen Daseins (Frankfurt-am-Main, 1967). Christian
Wiener, Lehrbuch der darstellenden Geometrie, 2 vols.
(Leipzig, 1884). Harry Austryn Wolfson, Crescas' Critique
of Aristotle, Problems of Aristotle's Physics in Jewish and
Arabic Philosophy (Cambridge, Mass., 1929); idem, The
Philosophy of Spinoza, Unfolding the Latent Processes of
His Reasoning, 2 vols. (Cambridge, Mass., 1934); idem, Philo:
Foundations of Religious Philosophy in Judaism, Christi-
anity, and Islam, 2 vols. (Cambridge, Mass., 1947). Eduard
Zeller, DiePhilosophie der Griechen in ihrer geschichtlichen
Entwicklung, 3 vols. (1844-52); the English translation ap-
peared in segments.

SALOMON BOCHNER

[See also Abstraction; Axiomatization; Continuity; Cos-
mology; Mathematical Rigor; Newton on Method; Number;
Rationality; Space; Time and Measurement.]