Dictionary of the History of Ideas Studies of Selected Pivotal Ideas |

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VII. | INFINITY |

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Dictionary of the History of Ideas | ||

#### 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.

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:

Further, they identify it [the infinite] with the Divine, for

it is “deathless and imperishable”, as Anaximander says with

the majority of physicists

(*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

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

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):

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 *per*ception 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

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).

de Synthèse,

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

*n*ε,, or, to

*n*ε > 1, and the last relation can be

verbalized thus:

If ε is any positive real number, then on adding it to itself

sufficiently often, the resulting number will exceed the

number 1.

The Greeks did not have our real numbers; but if

we nevertheless superimpose them on the mathematics

of Archimedes, then the statement just verbalized be-

comes a particular case of the so-called “Postulate of

Archimedes,” which, for our purposes, may be stated

thus:

If *a* and *b* are any two magnitudes of the same kind (that

is if both are, say, lengths, areas, or volumes), then on adding

*a* to itself sufficiently often, the resulting magnitude will

exceed *b;* that is *na > b,* for some *n.*

(E. J. Dijksterhuis,

*Archimedes,* pp. 146-47 has the wording of the postulate

in original Greek, an English translation of his own, and

a comparison of this translation with various others).

The Greek theory of proportion was a “substitute”

for our present-day theory of the linear continuum for

real numbers, and the infinitely small is involved in

interlocking properties of denseness and completeness

of this continuum (see Appendix to this section). Our

real numbers are a universal quantitative “yardstick”

by which to measure any scalar physical magnitude,

like length, area, volume, time, energy, temperature,

etc. The Greeks, most regrettably, did not introduce

real numbers; that is they did not operationally abstract

the idea of a real number from the idea of a general

magnitude. Instead, Euclid's Book 5 laboriously estab-

lishes properties of a linear continuum for a magnitude

(μέγεθος, *megethos*) in general. If the Greeks had been

inspired to introduce our field of real numbers and to

give to the positive numbers the status of magnitudes,

then their theory of proportions would have applied

to the latter too, and their theory of proportions thus

completed would have resembled an avant-garde the-

ory of twentieth-century mathematics.

Within the context of Zeno's puzzles, Aristotle was

also analyzing the infinitely small as a constituent of

the linear continuum which “measures” length and

time. He did so not by the method of circumvention,

which the professional mathematicians of his time were

developing into an expert procedure, but by a reasoned

confrontation *à la* Georg Cantor, which may have been

characteristic of philosophers of his time. In logical

detail Aristotle's reasoning is not always satisfactory,

but he was right in his overall thesis that if length and

time are quantitatively determined by a suitable com-

mon linear continuum, then the puzzles lose their

force. In fact, in present-day mathematical mechanics,

locomotion is operationally represented by a mathe-

matical function *x* = φ(*t*) from the time variable *t* to

the length variable *x,* as defined in working mathe

matics; in such a setup Zeno's paradoxes do not even

arise. It is not at all a part of a physicist's professional

knowledge, or even of his background equipment, to

be aware of the fact that such puzzles were ever con-

ceived.

The “method of exhaustion” is a Greek anticipation

of the integral calculus. In the works of Archimedes,

the syllogistic maturity of the method is equal to that

of the Riemann-Darboux integral in a present-day

graduate text, but in operational efficiency the method

was made obsolete by the first textbook on the integral

calculus from around A.D. 1700 (C. B. Boyer, p. 278).

However the method also embodied the postulate of

Archimedes, and this postulate has an enhanced stand-

ing today. An innovation came about in the late nine-

teenth century when G. Veronese (*Grundzüge,* 1894)

and D. Hilbert (*Grundlagen,* 1899) transformed the

“postulate” into an “axiom,” that is into an axiomatic

hypothesis which may or may not be adjoined to suita-

ble sets of axioms, in geometry, analysis, or algebra.

This gives rise to various non-Archimedian possibilities

and settings, some of which are of interest and even

of importance.

Aristotle made the major pronouncement (*Physica,*

Book 3, Ch. 7) that a magnitude (*megethos*) may be-

come infinitely small only potentially, but not actually.

This is an insight in depth, and there are various possi-

bilities for translating this ideational pronouncement

from natural philosophy into a present-day statement

in operational mathematics. We adduce one such

statement: although every real number can be repre-

sented by a nonterminating decimal expansion, it is

generally not possible to find an actual formula for the

entire infinite expansion; but *potentially,* for any pre-

scribed real number, by virtue of knowing it, any

desired finite part of its decimal expansion can be

obtained.

*Appendix.* A linearly ordered set is termed *dense* if

between any two elements there is a third. It is termed

*complete* if for any “Dedekind Cut,” that is for any

division of the set into a lower and upper subset, (i)

either the lower subset has a maximal element, (ii) or

the upper subset has a minimal element, (iii) or both.

If the set is both dense and complete, possibility (iii)

cannot arise, so that either the lower subset has a

maximum, or the upper subset has a minimum. This

single element is then said to lie on the cut, or to be

determined by the cut.

*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

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-

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

*n*+ α = α.

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

*n*= α

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 *E*2 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, *E*2 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

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

*E*2 the kind of hoop that we have described.

The same mathematicians soon began to sense, in

their own manner, that to close off *E*2 in this fashion

is neither intellectually original nor operationally

profitable. They began to “experiment” with other

procedures for closing off *E*2. 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 *E*2, 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 *E*2 becomes “sealed

off” as infinity, and the resulting two-dimensional

figure is topologically a spherical surface *S*2. Con-

versely, if one starts out with an *S*2, 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 *E*2. Such

a spreading out is done in cartography by means of

the so-called stereographic projection. This projection

of a punctured sphere *S*2 on the Euclidean *E*2 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 *E*2 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 *E*8, 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 *P*2, that is the space of

two dimensional elliptic geometry, is not orientable,

but *P*3 is. Thus, in *P*2 a fully mobile society cannot

distinguish between right- and left-handed screws, but

in *P*3 it can.

Nineteenth-century mathematics has created many

other completions of *En* which have become the sub-

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:

The absolute transcendence of the One as unconditioned,

unlimited, Principle of all things: particular necessity of

eliminating all spatial ideas from our thoughts about Him

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

Also, a study of Plotinus of very recent date has the

following important summary:

discussion on the infinity of the Plotinian One, and from

it we learn much. The chief participants are now in basic

agreement that the One is infinite in itself as well as infinite

in power

(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:

to be. It is not what has nothing outside it that is infinite,

but what always has something outside it

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

His definition then is as follows:

a part outside what has already been taken. On the other

hand what has nothing outside it is complete and whole.

For thus we define the whole—that from which nothing

is wanting—as a whole man or a whole box

(ibid., 207a 7-11).

akin. Nothing is complete (teleion) which has no end (telos);

and the end is a limit

(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

parts, commencing in 1915, by F. Cajori in the

*American*

Mathematical Monthly.The references are carried to 1936

Mathematical Monthly.

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

Eine Untersuchung zur metaphysischen Weisheit Gottes auf

Grund der Mathematik, Philosophie, Theologie

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,

Words for Time

*Infinity,*

an Essay in Metaphysics(Oxford, 1964). Salomon Bochner,

an Essay in Metaphysics

*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-

Denken im Vergleich mit dem Griechischen,

gen, 1965); 3rd ed. trans. as

*Hebrew Thought Compared With*

Greek Thought(Philadelphia, 1961). Carl B. Boyer,

Greek Thought

*The*

History of the Calculus(New York, 1959). F. Cajori, “The

History of the Calculus

History of Zeno's Arguments on Motion,”

*American Mathe-*

matical Monthly,12 (1915), 1-6, 39-47, 77-82, 109-15,

matical Monthly,

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-

Newton's Mathematical Principles of Natural Philosophy and

His System of the World,

vised by F. Cajori (Berkeley, 1934; many reprints); cited

as

*Principia.*Georg Cantor,

*Gesammelte Abhandlungen*

mathematischen und philosophischen Inhalts,ed. Ernst

mathematischen und philosophischen Inhalts,

Zermelo (Berlin, 1932). C. Carathéodory, “Über die Beg-

renzung einfach zusammenhängender Gebiete,”

*Mathe-*

matische Annalen,73 (1913), 343-70. Morris R. Cohen and

matische Annalen,

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

im abendländischen Denken bis Kant

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

totle's Theory of the Infinite

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).

quadratura circuli et sectionum coní

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,

Books of Euclid's Elements

*History*

of Greek Mathematics,2 vols. (Oxford, 1921). Heinz Heim-

of Greek Mathematics,

soeth,

*Diesechs grossen themen der abendländischen Meta-*

physik und der Ausgang des Mittelalters,3rd ed. (Stuttgart,

physik und der Ausgang des Mittelalters,

1954). Werner Heisenberg,

*Physics and Philosophy, the*

Revolution in Modern Science(New York, 1958). David

Revolution in Modern Science

Hilbert,

*Grundlagen der Geometrie*(Leipzig, 1899), many

editions and translations. P. E. B. Jourdain,

*Contributions*

to the Founding of the Theory of Transfinite Numbers

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).

Pure Reason,

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,

to the Infinite Universe

“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,

Jahrhundert

*Zwei Grundprobleme der*

Scholastischen Naturphilosophie(Rome, 1951); idem,

Scholastischen Naturphilosophie

*Zwischen Philosophie und Mechanik*(Rome, 1958); idem,

*Metaphysische Hintergründe der Spätscholastischen Natur-*

philosophie(Rome, 1955). Rodolfo Mondolfo,

philosophie

*L'infinito nel*

pensiero dell'Antiquità classica(Florence, 1965). Isaac

pensiero dell'Antiquità classica

Newton, see Cajori, above. C. von Orelli,

*Diehebräischen*

Synonyma der Zeit und Ewigkeit, genetisch und sprachver-

gleichlich dargestellt(Leipzig, 1871). Erwin Panofsky,

Synonyma der Zeit und Ewigkeit, genetisch und sprachver-

gleichlich dargestellt

*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,

Grundfragen der Kunstwissenschaft

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-

Peirce,

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,

duction and commentary

*Aristotle, a*

complete exposition of his works and thought(Cleveland,

complete exposition of his works and thought

1959). Meyer Schapiro,

*Paul Cézanne*(New York, 1952). A.

Schoenfliess, “Projective Geometrie,”

*Encyclopädie der*

mathematischen Wissenschaften,Vol. III, Leipzig, 1898-),

mathematischen Wissenschaften,

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,*

*617*

*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

keit und Unendlichkeit. Gedanken zur Deutung der men-

schlichen Daseins

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,

of Aristotle, Problems of Aristotle's Physics in Jewish and

Arabic Philosophy

*The*

Philosophy of Spinoza, Unfolding the Latent Processes of

His Reasoning,2 vols. (Cambridge, Mass., 1934); idem,

Philosophy of Spinoza, Unfolding the Latent Processes of

His Reasoning,

*Philo:*

Foundations of Religious Philosophy in Judaism, Christi-

anity, and Islam,2 vols. (Cambridge, Mass., 1947). Eduard

Foundations of Religious Philosophy in Judaism, Christi-

anity, and Islam,

Zeller,

*DiePhilosophie der Griechen in ihrer geschichtlichen*

Entwicklung,3 vols. (1844-52); the English translation ap-

Entwicklung,

peared in segments.

SALOMON BOCHNER

[See also Abstraction; Axiomatization; Continuity; Cos-mology; Mathematical Rigor; Newton on Method; Number;

Rationality; Space; Time and Measurement.]

Dictionary of the History of Ideas | ||