University of Virginia Library

CONTINUITY AND
DISCONTINUITY IN NATURE
AND KNOWLEDGE

I. INTRODUCTION

Continuity is a key conception in the history of ideas
in many fields. A form of continuity, under one name
or another, is frequently attributed to processes or
developments in mathematics, science, philosophy,
history, and theology. In mathematics proper and its
direct applications, especially to “exact” science, con-
tinuity is nowadays rigorously defined and unequivo-
cally fixed. Also, in mathematical contexts continuity
is sharply distinguished from neighboring concepts like
uniformity, steadiness, constancy, etc., all of which
have, in mathematical contexts, definitions of their
own. But, by reason of its general nature, continuity
is a sprawling concept. Outside of mathematics, it is
ambiguously conceived and loosely applied, and merg-
ers and fusions with neighboring concepts are really
unavoidable.

Thus, the historical books of the Old Testament are
a vast dissertation on the Lord's unceasing concern for
his “Chosen People”; and a theologian may pronounce
with almost no difference of meaning that the Lord
perseveres in this concern with “continuity,” or
“steadiness,” “constancy,” “uniformity,” etc.

In general history and theology imprecisions as to
the meaning of continuity may be enriching rather than
disturbing, but in descriptive science, in which impre-
cisions also occur, they may become outright embar-
rassing. A leading instance of such an embarrassment
is the case of the hypothesis of uniformitarianism. The
hypothesis asserts, for geology and biology, that there
has been a certain continuity of evolution since the
formation of the earth. But no satisfactory definition
of this would-be continuity has been agreed upon, or
is in sight. (See section V.)

Imprecisions as to the meaning of continuity may
border on the very threshold of “exact science.” A
notable case is Isaac Newton's description of his abso-
lute time, which runs as follows:

[F. Cajori trans., p. 6].

Newton makes it clear that this pronouncement,
which is made in an elucidating scholium only, is not
meant to be a primary operational definition of abso-
lute time but only a supplementary background de-
scription of it. The decisive ingredient of this descrip-
tion is the word “equably” (which does not occur in
the immediately following description of “absolute
space”), and it is meant to suggest that the flow of
time is somehow intrinsically continuous and uniform.
But this suggestion is tenuous and fugacious, and
Newton's sentence about the nature of his time cannot
be made into a truly informative definition.

Two thousand years before the Principia, Aristotle,
in his great essay on time (Physica, Book 4, Chs. 10-14),

performed much better than Newton. Aristotle's time
is hierarchically anterior to events in it and even to
our awareness of it; and Aristotle is much more per-
suasive than Newton in his expostulations that time
is intrinsically continuous, and that this continuity is
a preexistent standard by which to assess, for any
“movement”—which, in Aristotle, stands for a general
process in nature, more or less—whether it is continu-
ous, discrete, or constant (Bochner, Ch. 4). Further-
more, already many centuries before Aristotle, a lin-
guistic bond between time and continuity had been
clearly present in Homer, and we are going to describe
it briefly.

The Greek word for our “abstract” noun “continu-
ity,” as standardized by Aristotle, is the adverbial form
to synechés (τὸ συνεχές), and the cognate verb syn-
echein means literally “to hang, or hold together.” It
so happens that the Latin root of the English word
continuity also means literally “hanging, or holding
together”; but works on Indo-European linguistics do
not assert that the Greek and Latin stem words for
continuity had a common root in Sanskrit.

Now, the verb synechein and the adverb synechés
occur already in Homer, but on different levels of
abstraction. The verb occurs in Iliad 4, 133, in the
expression: “The golden clasps of the belt were held
together,” in which its meaning is quite concrete. But
the adverb, which is used twice, is both times used
in a semi-abstract meaning, namely in the meaning of:
continually (in time). The first occurrence is in Iliad
12, 25-26, thus: “Zeus made it rain continually”; and
the second occurrence is in Odyssey 9, 74, thus: “There
for two nights and two days we lay continually.” Also,
in the second passage the adverb synechés is reinforced,
seemingly redundantly, by the adverb aiei (αἰεί) which
means: always, ever, eternally. The Odyssey thus
adumbrated a tripartite bond between continuity, time,
and eternity, and this bond has been variously contem-
plated and exalted in general philosophy and theology
since. This bond is nowhere stronger than in the Old
Testament, but the extant canon of the Old Testament
does not have a word whose functions would corre-
spond to those of synechés.

Post-Hellenistically, this bond is also verified by the
order in which the cognates of our English word con-
tinuity have come into use. According to the entries
in the Oxford English Dictionary, our word continual
(in time) was the first to emerge. It occurs already
around A.D. 1340, in the phrase: “great exercise of body
and continual travail of the spirit,” in one of the so-
called English Prose Treatises of the hermit Richard
Rolle of Hampole (1290-1349). But for all other cog-
nates of continuity the same dictionary quotes only
from Chaucer, who wrote about half a century after
Hampole, or from later sources, even much later ones.
Thus, according to this dictionary, our adjective con-
tinuous gained currency in the seventeenth century
only.

Continuity has many shadings of meaning and
therefore also many antonyms. The leading antonym
to “continuous” is “discrete”; other ones are: saltatory,
sudden, intermittent, indivisible, atomic, particulate,
and even monadic. A monad however, being a kind
of synonym for unity and one-ness, may suggest both
continuity and discreteness, at one and the same time.
The monad of Leibniz, as presented in his Monadology,
is apparently of such a kind; that is, it also suggests
continuity, even if it is an irreducible ultimate unit,
not only of physical structure but also of consciousness,
cognition, and metaphysical coherence. In the thinking
of Leibniz this simultaneousness is grounded in the
all-pervading lex continui, which maintains that “all”
basic constituents of the universe are somehow contin-
uous, be they physical or metaphysical, elemental or
rational.

Long before Leibniz, the outlines of a lex continui,
and an involvement of unity with continuity, were
already present in the great ontological poem of Par-
menides (sixth century B.C.) from which we quote two
passages (L. Tarán, p. 85).

(frag.
8, lines 22-26).

In sum, the Parmenidean “Being” is one, yet contin-
uous; homogeneous, that is continuously distributed,
yet indivisible; ungenerated, and imperishable, and
atemporal. Such bold accumulations of divers attributes
in one have been occurring in Western philosophy ever
since. And Western science has been harboring self-
dualities and near-inconsistencies, many of which affect
continuity, from early Pythagoreans until our very day.

Our present-day intellectual discomfort, if any, over
the contrast between the continuous and the discrete
is an inheritance from the nineteenth century. Through
the length of the nineteenth century there was a wide-
spread predilection for continuity in all areas of
knowledge, in mathematics, physics, earth-and-life sci-
ence, general philosophy, and even in historiography.
This predilection manifested itself in a tendency to
subsume and subordinate the discrete under the con-
tinuous, even when the presence of the discrete was
freely and fully acknowledged. In the twentieth cen-

tury the Victorian outlooks on continuity have been
modified and reoriented, some gradually and some
vehemently; and the discrete has come into its rightful
own, reaching a high-point in quantum mechanics.

II. PHYSICAL SCIENCE

The nineteenth century created the great doctrines
of molecular chemistry, thermodynamics, and statis-
tical mechanics, all of which take full cognizance of
the fact that physical matter consists of discrete parti-
cles. Yet, in an overall sense, “Physical Science” of the
nineteenth century gave decisive preference to “field”
theory over “particle” theory, that is to continuity over
discreteness.

Thus, in the beginning of the nineteenth century,
Thomas Young and A. Fresnel rendered a decision,
which was then long adhered to, that light is composed
of waves and not of corpuscles. Secondly, D. S. Poisson,
C. F. Gauss, and G. Green created field conceptions
of “potential,” with parallel mathematical properties,
for electrostatics, magnetostatics, and gravitation.
Thirdly, Maxwell's electrodynamics, as corroborated by
Heinrich Hertz, was a field theory, and it is frequently
viewed as the representative field theory of the cen-
tury. Fourthly, mathematical physicists from Cauchy
to Kirchhoff created and standardized the theory of
mechanics of continuous media, which assumes that
mechanical matter is distributed continuously, with a
finite density point-by-point. This mathematical as-
sumption runs counter to the indisputable hypothesis
of basic physics and chemistry that matter is atomistic,
or molecular; but, operationally, the theory of contin-
uous media has been overwhelmingly successful and
it is simply indispensable for many contexts. Finally,
in statistical phenomena and theories, nineteenth-
century physics was led to operate almost exclusively
with the Gaussian law of probability, and this law
represents continuous distribution par excellence.

Furthermore, within fluid mechanics, which is a part
of the theory of continuous media, Helmholtz created
a major theory of vortices. Lord Kelvin, his contem-
porary, was so impressed with it, that he hastily dashed
off a theory of “vortex-atoms,” in which individual
atoms, even in their singleness, were made into contin-
uously spread-out vortices à la Helmholtz. This attempt
of Kelvin was so ill-advised that present-day physics
has all but “suppressed” the memory of it (R. H. Silli-
man). Lastly, Helmholtz fully shared the general pre-
sumption that acoustics is a theory of waves, that is,
a field theory, although it was he who pioneered in
the discovery (for which he is justly famous), that,
physiologically, there are discrete “tone atoms,” that
is tones which the ear cannot “resolve” by physically
subdividing them.

Physics of the twentieth century changed all this.
It did not give up nineteenth-century insights but it
refocused them. All field-like constructs of the preced-
ing century were fully retained, and even enlarged and
added to; but they were all balanced or complemented
by the introduction of appropriate particle-like con-
structs of a dual kind. Thus, electric fields were bal-
anced by electrons, light waves by photons, sound
waves by phonons, and even gravitational fields were
balanced by would-be duals which are hopefully called
gravitons; conversely, and most importantly, all ele-
mentary particles of matter were balanced by undula-
tory counterparts, the so-called de Broglie waves. In
the final outcome, the nineteenth-century link between
physics and continuity was nowise weakened, but it
was balanced by an equally viable link between physics
and discreteness, and the whole structure of physics
has been brought to rest on a duality between the
continuous and the discrete. Except for sporadic and
disjointed anticipations in philosophy of science (M.
Jammer, p. 241), general philosophy of the late Victo-
rian and even Edwardian era was unprepared and ill-
equipped to cope with the novel postures in basic
physics, and only very slowly is general philosophy
accommodating itself to the stubborn fact that the
duality principle in physics is here to stay.

A peculiar adumbration of our present-day duality
between the continuous and the discrete may be dis-
cerned in the outlooks of the first atomists Leucippus
and Democritus (fifth century B.C.). They recognized
from the first that an atomic hypothesis does not only
assert that physical matter is “granulated,” that is, built
up of particles which in a suitable sense are indecom-
posable, but also that these particles “interact” with
each other across the “void” that separates them from
one another. They interact unceasingly, and for the
most part “invisibly.” It is the mode and manner of
these interactions which constitute the structure of
matter, mainly in its microscopic properties, but also
in its macroscopic attributes. Democritus saw this more
clearly than most participants in the seventeenth-
century “Revival of Atomism” (R. H. Kargon, 1966),
and even chemists of the nineteenth century may have
been lagging behind Democritus in this crucial insight.
Democritus may have also known, in thought patterns
of his, that even if an atomic theory intends to be
“philosophical” rather than “physical” (van Melsen),
it still has to establish its “legitimacy” by offering a
context of physical explanations of some degree of
novelty. Giordano Bruno, for instance, did not know
this at all. He offered various atomistic and monad-

ological statements, but they served no purpose in
physics (K. Lasswitz, I, 391-92).

Aristotle, who lived about a life span after Democ-
ritus, was much concerned with the first atomists and
their doctrine. He was opposed to atomism, but not
because it assumed that matter consists of minimal
constituents. In this assumption, Aristotle might have
acquiesced. He was a biologist, and a very great one
too, and as such he had it in his thinking that an organic
tissue (like flesh, skin, bone, etc.) consists of “minimal”
parts; just as in modern biology a tissue is composed
of cells, which are ultimate units of life. What Aristotle
could not accept for himself was the crucial assertion
of atomism that, ordinarily, any two atoms are sepa-
rated from each other by a spatial vacuity, which
surrounds each of the atoms and extends between any
two of them. Aristotle simply “abhorred” a vacuum,
any vacuum, and he could do so with metaphysical
justification. Firstly, even in present-day biology, cells
are adjoined wall to wall, without biological interstices;
and secondly, in physics proper, Aristotle was a
thermodynamicist, and it is a fact of present-day phys-
ics, which Aristotle anticipated, that in a purely
thermodynamical system an absolute vacuum is not
allowed for. It is true that since the nineteenth century
this kind of thermodynamics has been deemed com-
patible with an atomic, or rather molecular structure
of the substances which compose the system (Bochner,
p. 160). However, this reconciliation of apparent op-
posites has been brought about by statistical mechanics;
but the basic attitudes of this doctrine were far beyond
the reach of Aristotle, and of antiquity in general.

In spite of his opposition to atomism, Aristotle had
a masterful grasp of the achievements of the Atomists,
as evidenced, for instance, by the following passage.

(Meta-
physica, 985b 4-10; trans. Kirk and Raven, pp. 406-07).

This remarkable statement could serve as a motto
for the polarity between particle and field and even
for the de Broglie duality between corpuscle and wave.
It is notable that Aristotle even “apologizes” for the
Atomists for expressing the polarity between full and
void in the quaint, and possibly misleading Parmenid-
ean contrast between “Being” and “Non-Being,” and
Aristotle is reassuring the reader that the “Non” in
“Non-Being” is only a façon de parler without any
negative intent or force. And Aristotle's casual obser
vation that “the two together are the material causes
of existing things” is an oracle for the ages, for our
age of physics, at any rate.

III. ATOMISM AND DISCONTINUITY
IN THEOLOGY

In the European West, atomism since Democritus
has been persistently associated with forms of atheism,
or at least with suspicions of it. But, as against this,
in the Islamic Middle East, in the tenth and eleventh
centuries A.D., when Islam's philosophy and theology
were at their height, Islamic theologians—most of
whom were of Persian extraction—based their ortho-
doxy, which was philosophically articulated, on a radical
form of atomism and discontinuity in nature. (For a
balanced recent account see M. Fakhry, Chs. 1 and
2.) From the Islamic approach it was the avowal of
continuity which represented atheism, and the avowal
of discontinuity which represented theism.

It is worth noting that a late Victorian scholar, a
leading one, finds “Mephistophelian humor” in the fact
that Islamic theists could embrace “atheistic” atomism.
The scholar concludes that this came about because
Aristotle had depicted Democritus so engagingly in-
stead of warning theists against seeking refuge with
him (L. Stein, pp. 331-32).

This Islamic doctrine, whatever its origin, was part
of the so-called Kalam. Its intent was not so much to
deny continuity as to deny causation, but it strongly
correlated the two. And it denied causation, because
any general law of causation would circumscribe, and
even inhibit God's freedom of intervention and thau-
maturgy. Thus, within this intellectual setting, the
physical atomism of the Kalam became a scientific
occasionalism of its philosophy.

A famous account of this atomism is incorporated
in the Guide of the Perplexed (Maimonides, Part I, Chs.
71-75). As usual with Maimonides, his report is some-
what over-systematic, but the account seems very reli-
able and adequate. Now, according to this account,
the Mutakallemim—that is, the professors of Kalam—
atomized, or rather quantized (in the sense of our
quantum theory) everything: matter, space, time, and
motion.

Specifically they taught that the seemingly continu-
ous locomotion of a body is in fact not really continu-
ous but a succession of leaps between discretely placed
positions; and they apparently took it for granted that
there is a universal minimal distance between any two
positions. Also, what is important, a leap from position
A to position B consists of two interlocking subevents;
the original body in position A ceases to exist, and an
“identical” body comes into being in position B. This

sounds surprisingly like the leap of a Bohr electron,
when rotating around a proton, from one energy level
into a neighboring one; except that in the Kalam, the
second subevent follows on the first “occasionalisti-
cally,” that is by an act of God, and not “causally,”
that is by a law of nature.

Somewhat more occasionalistic, but still compatible
with our physics of today, was the insistence of the
Mutakallemim that if a white garment turns red by
being dipped into a red dye then it is wrong to say
that red pigment has been transferred from the dye
to the garment. Rather, by God's volition, an amount
of red pigment ceased to be in the dye, and a corre-
sponding amount of the pigment was created in the
garment.

Most alien to our thinking is the “Hypothesis of
Admissibility.” It apparently asserted that anything
which is “imaginable” is also possible. It is “imagin-
able” that man might be much larger in size than he
is now, and he might indeed so be; in fact, he might
be as large as a mountain. Fire usually goes upward,
but we can “imagine” it going downwards, and so
indeed it might go.

Even more striking than the atomistic pronounce-
ments, were the accompanying occasionalistic theses,
and the latter were displayed most dazzingly in the
work of the Iranian Muslim theologian al-Ghazali.
Nevertheless, they were leading Islamic philosophical
thought into a cul-de-sac, and it was very fortunate
for the nascent medieval civilization on the European
continent that the leading European schoolmen,
Muhammadan, Jewish, and Christian, were refusing to
be drawn into this blind alley. In the twelfth century,
the Spanish Jew Maimonides was opposed to the occa-
sionalistic doctrines of the Kalam, and so were also,
very systematically, his contemporary Averroës (a
Spanish Muslim), and, almost a century later, the Latin
schoolman Thomas Aquinas in his Summa contra Gen-
tiles, Book III.

It is regrettable, though, that this opposition to the
Islamic occasionalism also kept the West from becom-
ing generally acquainted with its scientific atomism.
Saint Thomas, for instance, has very little about it.
Almost a century after Aquinas, a Karaite schoolman,
Aaron ben Elijah of Nicodemia (1300-69), who stood
intellectually between West and East, made a last
major attempt to keep Islamic atomism alive, but to
no avail (Husik, Ch. 16).

It appears that the atomism of the Islam had been
greatly influenced by the atomisms of Democritus and
Epicurus, but it is not easy to say why the metaphysical
and religious evaluations were so divergent. It has been
suggested that Islamic philosophers were exposed to
Indian influences (S. Pines), and also that a primitive
atomism may have arisen within the Kalam indige-
nously (O. Pretzl). There are intimations that, from the
beginnings of Islamic thought there had been reflec-
tions, naive ones, on the concentration of space and
matter in elemental units. Also, the problem of the
differences between Islamic and Greek atomism is
compounded by the fact that there had been diver-
gences of philosophy even between Democritus and
Epicurus themselves.

It is reported that Democritus was of a serene dispo-
sition in his personal deportment. This serenity in
manners may have corresponded to a determinism in
scientific outlook which takes it for granted that, ordi-
narily, the physical constellation of today will deter-
mine the physical events of tomorrow. In the universe
of Democritus, atoms were unceasingly in motion, by
fixed laws and unchangeable rhythms. In the course
of their motions atoms would combine to form
“worlds”—which we may take to be solar systems, or
galaxies, in our experience—and the worlds could also
fall apart by dissolution of the combinations of atoms
which constitute them. Also, by their structure, the
worlds of Democritus were mostly (spiral) vortices, and
once upon a time the vortices emanated from some
kind of “turbulence,” that is, from some kind of
“primordial chaos” (Diogenes Laërtius).

All this sounds astonishingly “modern.” Primordial
turbulence, and spiral-shaped galaxies are giant-sized
discontinuities in nature, the account of which fills the
pages of any book on cosmogony of today; and it must
not be held against the first atomists that they did not
explain their provenance, because present-day cosmol-
ogy cannot explain it either (J. H. Oort, p. 20).

The system of Democritus was not “atheistic” in a
militant sense, but it was indifferent to divinity in a
passive sense. Since everything in nature and life was
presumed to follow predictably by laws and rhythms,
there was apparently no need, or rather no room, for
a Divine Providence that would affect the fate of man,
or the course of the world, by acts of willed interven-
tion and prodigy. Very much later though, mostly in
response to Islamic occasionalism, the counterargu-
ment was fashioned that it is noncontinuity and inde-
terminacy which bespeak the absence of divine Provi-
dence; and that it is continuity and causality in nature
which testify to a rule by Providence and perhaps even
to an original creation by a divine resolve.

While the system of Democritus has the mystique
of an incomparable classical creation, the atomic sys-
tem of Epicurus, over a century later, bears the mark
of an important but epigonic adaptation. It had a great
appeal though. But the appeal was not due to the
power of scientific inventiveness in Epicurus, who had
set “Epicureanism” in motion, but to the beauty of

Lucretius' De rerum natura in which it is poetically
enshrined. The latter work is not an essay in science
but a poet's sweeping vision of the Great Chain of
Being in its manifold manifestations; however, by some
irrationality of inspiration, which has been a puzzle
to many a poet and literary critic since, Lucretius
transported his vision through the rather amorphous
medium of Epicurus' system of knowledge, and thus
immortalized Epicurus' variant on atomism in the
process. Democritus had been a physicist, first and
foremost, and very genuinely so. Epicurus however was
first and foremost a moralist and a social critic, even
if he elected to transmit his philosophemes in a setting
of physical assumptions; and it was this humanism
which attracted Lucretius to him.

With regard to discontinuity in the universe Lucre-
tius avers, as did Democritus long before him, that,
by conjunction and disjunction of atoms, numerous
“galaxies” are formed and dissolved. He even alludes
to a primordial turbulence (nova tempestas), but, re-
grettably, not to vortices (P. Boyancé, p. 273). Lucre-
tius even seems to suggest, in words of his own—what
is apparently not in the extant reports about
Democritus—that the separate galaxies of the universe
are likely to be distributed throughout the universe
with a certain uniform frequency of occurrence (De
rerum natura, Book II, lines 1048-66; C. Bailey, 2,
964-65).

Lucretius also has the significant report—which most
regrettably does not occur in the extant remains of
Epicurus himself, but has also been confirmed by
Cicero, Plutarch, and others—that the atom of Epi-
curus was endowed with a so-called clinamen of his
invention. It was a small-scale swerving motion of the
atom, and Epicurus superimposed it on the large-scale
rectilinear motion that had been advocated by De-
mocritus. This clinamen was designed to temper the
basic determinism of physics by an element of inde-
terminism; and as a suggestion in physics it was a
remarkable adumbration of indeterminacies in the
physics of our day. But Epicurus, and his followers ever
since, went much too far in using it as a physical
justification for indeterminacies in the science of man,
namely as a justification for the freedom of human will
and for man's self-mastery, in a moral, social, and
theological sense.

Epicurus was adopted as the ancestral creator of the
nineteenth-century Marxist doctrine that certain fixed
assumptions in physics are an unfailing indicator of
certain fixed attitudes in sociology. Thus, the Dialectics
of Nature of Friedrich Engels, and, much more shrilly,
the Materialism and Empirico-Criticism (1908) of V. I.
Lenin, were proclaiming the doctrine that a philosophy
which affirms the primacy of human freedom must be
based on a certain kind of metaphysical “materialism,”
and that this materialism must more or less be predi-
cated on a form of atomism.

Developments in twentieth-century science have
been undermining the possibilities of such firm corre-
lations. In the nineteenth century there were firm
distinctions and separations between materialism and
idealism, reality and imagination, phenomena and ob-
jects, experience and theory, experiments and explana-
tions. But in the twentieth century, the spreading prin-
ciples of duality for particle and field, for corpuscle
and wave, and the progressive and unrelenting mathe-
matization of all of theoretical physics, have been
dissolving the scientific foundations for such distinc-
tions and separations. Therefore, not only standard
“Marxist” tenets, but also, many other Victorian and
Edwardian correlations are losing their obvious justifi-
cations, and they will have to be re-thought from the
ground up.

IV. MATHEMATICS

In mathematics, continuity is an all-pervading concept.
Topology is a relatively recent major division of math-
ematics, and in the half-century 1890-1940 it was a
vast exercise in continuity from a novel comprehensive
approach. Also, this novel pursuit of continuity sup-
plemented but did not supersede the study of continu-
ity in “analysis,” in which, knowingly or not, it had
been a central conception since the fifth century B.C.

It will suit our purposes to distinguish, and keep
apart two aspects of continuity in mathematics.

Aspect (1). Continuity of linear ordering. This aspect
of continuity is suggested by, and is embodied in the
intrinsic continuity structure of the so-called linear
continuum of real numbers – ∞ < t < ∞.

Aspect (2). Continuity of a function y = f(x). The
simplest, and still very important case of a continuous
function f(x) arises if x and y are both real numbers,
and in this case a function is “equivalent” with an
ordinary graph or chart on ordinary graph paper. In
the general case, a function y = f(x) is a “mapping”
from any topological space X:(x) to any other topo-
logical space Y:(y).

Aspect (1) was envisioned by the Greeks, and they
worked long and hard at elucidating it. Aspect (2)
however eluded them. The Greeks had fleeting chance
encounters with it, but they were not inspired to focus
on it in any manner. This failure of the Greeks to
recognize aspect (2) far outweighed their ability to
identify aspect (1). By a purely scientific assessment,
this failure greatly contributed to the eventual decline
of Greek mathematics in its own phase.

Even in the recognition of aspect (1) the Greeks had
two blind spots. Firstly, Greek mathematics never

created the real numbers themselves. When the Greeks
formed the product of two quantities that were repre-
sented by lengths then, conceptually, the product had
to be represented by an area. Descartes may have been
the first to state expressly, as he did with some emphasis
at the very beginning of his La Géométrie (1637), that
the product may also be represented by a length. The
Greek substitute for our concept of real numbers was
their quasi-concept of magnitude (μέγεθοσ; megethos),
and the corresponding elementary “arithmetic” was
the Greek theory of proportions, as presented in
Euclid's Elements, Book 5.

The Greek magnitudes were a “substitute” for posi-
tive real numbers only; and we view it as a second
blind spot of the Greeks that they did not even intro-
duce a magnitude of value 0 (= zero), which, by con-
tinuity, would be the limiting case of magnitudes of
decreasing (positive) values. Thus, Greek mathematics
never had the thrill of conceiving that two coincident
lines form an angle of value 0; and Greek physics of
locomotion, as expounded in Aristotle's Physica, Books
5-8, always viewed “rest” (ἢρεμία) as a “contrary to
motion” (κίνησις) and never as a motion with velocity 0.

In fact, the first outright criticism of this Aristotelian
view is to be found only in Leibniz. It is strongly
implied in his pronouncement that “the law of bodies
at rest is, so to speak, only a special case of the general
rule for bodies in motion, the law of equality a special
case of inequality, the law for the rectilinear a sub-
species of the law for the curvilinear” (H. Weyl, p.
161).

This pronouncement of Leibniz was part of a uni-
versal lex continui (“Law of Continuity”) which runs
through his entire metaphysics and science. Leibniz
did not present the law in a systematic study of its
own, but he frequently reverted to it, presenting some
of its aspects each time. Leibniz recognized, reflec-
tively, the importance of functions for mathematics.
He coined the name “function” in 1694, and, what is
decisive, he was well aware of our aspect (2) of con-
tinuity (Bochner, pp. 216-23). But he did not “create”
functions in mathematics. As rightly emphasized by
Oswald Spengler, the concept of function began to stir
in the late fourteenth century, and its emergence con-
stituted a remarkable difference between ancient and
post-medieval mathematics. Also, as early as 1604, that
is 90 years before Leibniz coined the name, Luca
Valerio had de facto introduced a rather general class
of (continuous) functions f(x) to a purpose, and had
operated with them competently in the spirit of the
mathematics then evolving. However, it was Leibniz
who was the first to assert, more or less, that functions
and functional dependencies in nature are usually con-
tinuous. Thus he states the maxim of cognition that
“when the essential determinations of one being ap-
proximate those of another, as a consequence, all the
properties of the former should also gradually approxi-
mate those of the latter” (Wiener, p. 187).

It is not easy to state the direct effect of Leibniz'
Law of Continuity on the growth of mathematics and
physics. In working mathematics, the meaning and role
of continuity unfolded excruciatingly slowly in the
course of the eighteenth and nineteenth centuries,
through cumulative work of Lagrange, Laplace,
Cauchy, Dirichlet, Riemann, Hankel, P. du Bois Rey-
mond, Georg Cantor, and others, without any manifest
reference to the metaphysically conceived lex continui
or Leibniz. Of course, the “Law” of Leibniz may have
been burrowing deep inside the texture of our intellec-
tual history, thus affecting the course of mathematics.
But to establish this in specific detail would be very
difficult.

Mathematics of the nineteenth century elucidated
basic facts about both aspects of continuity, for real
numbers, and for real and complex numbers. These
facts about continuity were intimately connected with
facts about infinity, especially about the infinitely small.
The efforts to elucidate these two sets of facts, severally
and connectedly, had begun with early Pythagoreans
and Zeno of Elea, and it took twenty-four centuries
to bring them to fruition.

The twentieth century greatly widened the scene of
continuity, especially of its aspect (2), by extending the
conception of continuity from functions from and to
real (and complex) numbers to functions from and to
general point-sets, that is general aggregates of mathe-
matical elements. In fact, a numerical function y = f(x)
is continuous if it transforms “nearby” numbers x into
“nearby” numbers y. Therefore, in order to apply the
notion of continuity to a function y = f(x) from a
general point-set X:(x) to a general point-set Y:(y) it
suffices to know what is meant by the statement that
points of X or points of Y are “sufficiently near” each
other. Now, in the twentieth century this has been
achieved by the introduction of a so-called topological
structure on a general point-set. For any given topo-
logical structure it is meaningful to say when two
points of the set are “near” each other, and when the
two point-sets X and Y are each endowed with a topo-
logical structure of its own it thus becomes meaningful
to say when a function y = f(x) is continuous. The
conception of a topological structure opened new
vistas, and it has become involved in most of the math-
ematics of today.

V. GEOLOGY AND BIOLOGY

Continuity plays a major role in descriptive science,
mainly in geology and biology, but also in psychology.

Aristotle's Historia animalium has already a renowned
aphorism to this effect:

(588b 4-7).

Also, in Aristotle's system of psychology there may
have been a hierarchy of souls corresponding to the
hierarchy of living things (Tricot, pp. 492-93, note 2).
Altogether Aristotle already envisioned the so-called
Great Chain of Being, which reached a dominant posi-
tion in the thinking of Leibniz and of the Age of
Enlightenment (Lovejoy).

Leibniz took pains to expound that the Great Chain
is indeed “great” in the sense that

(B. Glass, p. 37).

Also, in other contexts, Leibniz intensified, or diver-
sified, the adjective “great” by equating it variously
with “maximal,” “optimal,” “perfect,” “complete,”
“continuous,” etc.

Dr. Samuel Johnson, the redoubtable man of letters,
termed the Great Chain of Being the “Arabian Scale
of Existence,” and he made a very pertinent observa-
tion about its “greatness.” He compared this scale of
existence to the mathematical linear continuum, which
he probably knew from Aristotle's Physica, and he
pointed out that, notwithstanding a superficial similar-
ity, the two are very different from each other
(Lovejoy, pp. 253-54). In fact, as stated by Aristotle
in his Physica over and over again, the mathematical
linear continuum is “everywhere dense,” meaning that
between any two elements of it there are always some
other ones; in particular, no element of it is isolated.
However, in nature's Chain of Being, biological and
mineral, however great and complete it be, only a finite
number of Links can be discerned. Thus, even if the
Chain of Being has been made optimally great by
filling in all possible gaps in it, there still is only a
finite number of Links, all told. Because of that, each
individual member of the Chain is isolated; meaning
that there is a first neighbor that is hierarchically above
it, and another one that is hierarchically below it.

Leibniz must have been aware of this unbridgeable
difference between the Great Chain and the linear
continuum. He must have even been aware of the fact
that the linear continuum is not only “everywhere
dense,” as already known to Zeno of Elea and to
Aristotle, but also “complete,” in the sense that to any
bounded sequence of real numbers which is mono
tonely increasing or decreasing there corresponds a real
number which is a limit of the sequence. The com-
pleteness of the linear continuum was properly estab-
lished only in the nineteenth century by Dedekind and
Cantor; but Eudoxus and Archimedes had, more or less,
known it for their magnitudes, and Leibniz must have
half-known it for real numbers too.

But Leibniz pretended to be undeterred by such
differences. He desired to coalesce heterogeneous phe-
nomena from exact science, descriptive science, and
metaphysically oriented “moral” science into one
comprehensive law of continuity. The latter was ap-
parently also a law of optimality, and in this guise it
was closely allied to a principle of contradiction and
of sufficient reason. Yet, at other times Leibniz also
acknowledged that heterogeneity cannot be forcibly
overcome. Such an acknowledgment seems to be im-
plied in the following statement in which Leibniz
avows that his so-called labyrinth has two separate
aspects.

(Leibniz, Theodicy, p. 53).

In the nineteenth century, a quest for continuity was
particularly pronounced in geology and biology. As
already mentioned in section I, the hypothesis of con-
tinuity peculiar to geology is called uniformitarianism;
and its contrary was called catastrophism. Uniformi-
tarianism was introduced in 1795 in a treatise by James
Hutton (Gillispie, pp. 122-48; Albritton, chapter by
G. G. Simpson), and it became generally known
through a large-scale treatise of Charles Lyell, Princi-
ples of Geology, whose first edition appeared in three
volumes, 1830-33.

In biology of today, the hypothesis of continuity is
specifically the hypothesis of “transformism,” that is
the hypothesis that there is in operation an organic
evolution of life which proceeds by a transformation
of one species into another; the direct contrary to it
would be the doctrine of “fixed species” which the
French call “fixism” (P. Ostoya). Transformism as a
biological hypothesis fully began with Lamarck, and
evolution was assumed by him to come about by
adaptation. Charles Darwin presented an impressive
plea that evolution comes about by natural selection;
and, “popularly,” transformism is associated with this
kind of evolution only. Yet in present-day biology,
adaptation is not entirely ruled out, even if Natural
Selection remains the prime cause.

Geologists nowadays greatly favor uniformitarianism
over catastrophism, but it is easier to say what catas-
trophism asserts than what uniformitarianism actually
is. Catastrophism maintains that manifest discon-
tinuities in geological stratifications of mineral deposits
and imbedded fossils are due to discontinuities in the
physical processes which brought about the stratifica-
tions and perhaps even abrupt changes in the physi-
cal laws which produce the processes (Toulmin, Ch.
7). Uniformitarianism however wants to be a true con-
trary to catastrophism and not only a negation of it.
A mere negation would only demand that important
basic data and phenomena be continuous in time, and
nothing more; there would be no need for anything
to be a constant in time, say. Thus, the gravitational
force need not have at all times the same Newtonian
value 1/r2, but it might be a positive function of dis-
tance and time, provided that the dependence on all
its variables is a continuous one. This however is not
what uniformitarianism really wants to be. Its real aim
is to avow that there is “uniformity” in nature; and
this seems to imply that certain basic causes and laws
are not only continuous in time, but also constant in
time, and perhaps constant in some other parameters
too. By the prevalent interpretations of uniformitari-
anism, certain leading attributes of nature are recog-
nizably always the same, so that the “present deter-
mines the past” and, of course, the future.

It appears that our perception of “uniformity” and
of “continuity”—in whatever form these concepts
appear—is inseparable from our rational awareness of
the flow of time. The awareness of time, in its turn,
has come about by the presence of cyclical and recur-
rent phenomena in nature, although by cognitive
structure time is rectilinear and, in fact, a mathematical
linear continuum. It also appears that within our
Western civilization man's capacity for specific rigor-
ous knowledge has been awakened and shaped under
the impact of lunar, sidereal, and planetary events in
the external world which are recurring periodically (O.
Neugebauer, Ch. 1).

Apparently in keeping with these basic ingredients
of our rationality, the demand of uniformitarianism is
a compound of constancy, continuity, and cyclicity;
and the relative magnitude of these three components
varies with the approach to the conception.

In the first half of the nineteenth century, in the
thinking of Charles Lyell at any rate, the component
of constancy was predominant; so much so that when
Lyell was extending uniformitarianism from geological
to organic matter, he had to give preference to fixed
species over evolving ones. But in the second half of
the nineteenth century, continuity proper was ever
more outweighing constancy; and there was a rising
consensus that, contrary to the view of Lyell, uniform-
itarianism and transformism mutually condition and
justify each other (Glass, pp. 367ff.). To Lyell, the
transition from one species to a next following one,
however short the distance, was a “catastrophe,” and
thus not admissible (de Beer, p. 104). To affirmers of
general continuity however, a sufficiently close transi-
tion from species to species ceased to be a “catas-
trophe,” that is a disquieting discontinuity, and became
“progress,” which bespoke the kind of continuity that
arises in the optical merger of rapidly succeeding visual
tableaus. (In the motion picture industry, “continuity”
refers to the coherence of the scenario, and not to the
flow of the optical illusion).

In the twentieth century, the continuity aspect of
organic evolution has been somewhat beclouded by the
fact that, genetically, evolutionary transition comes
about by a so-called mutation of the chromosomic
apparatus and that “the basis of spontaneous mutation
remains one of the great unsolved problems of genet-
ics” (McGraw-Hill Encyclopedia of Science and Tech-
nology, “Mutation”). Nevertheless, the fact that every-
thing proceeds by mutations is no more damaging to
the overall continuity in evolution than atomic and
quantum spontaneities in basic physics are prejudicial
to overall continuities in the foundations of modern
physics and related science.

Strictly speaking, every event in nature is probably
discrete, or a union of discrete subevents; that is, most
likely, no one event actually proceeds as mapped on
the mathematical continuum in its conceptual purity.
But the “fiction” that most events are best described
by continuous functions seems to be an operational
necessity, and there is nothing to suggest that it will
ever be possible to abandon it entirely. For instance,
the principles of our engineering mechanics, as taught
in engineering schools all over the world, were laid
down in Victorian and Edwardian treatises, and it
would be most cumbersome and inappropriate to make
this entire mechanics, in all its parts, nuclearly discon-
tinuous in accordance with some quantum field, or solid
state theory of our day.

VI. PHILOSOPHY

Continuity enters into all parts of philosophy. A
measure of continuity is involved in any conception
or philosopheme of Plato's, be it about man or God,
body or soul, memory or ideas, mathematics or morals,
poetry or artisanship, state or citizen. Without an
awareness of continuity there would have been no City
of God or Confessions of Saint Augustine; no medieval
problems about particulars and universals, creation and
eternity, fate and free will, Faith and Reason. And yet,
only in the philosophy of nature and of mathematics

is the presence of continuity immediate and tangible;
in other areas of philosophy the degree of its presence
is not easy to verify and the importance of its role
is not easy to determine.

Thus, Aristotle's Physica is full of concern for conti-
nuity, directly and emphatically, and it is easy to iden-
tify its presence in related treatises like De caelo and
De generatione et corruptione. But there is, directly,
very little about continuity in the Metaphysica, and
almost nothing indirectly to stir one's imagination.
Thus, at the beginning of Book 10 of the Metaphysica,
continuity is mentioned, directly, as one of several
meanings of Unity, but the context is philosophically
indifferent and little known. And even the continuity
in Physica deals, for the most part, only with the linear
continuum of mathematics, which, from our retrospect,
is a rather circumscribed topic, in a sense.

The first Western philosopher on record who tried
to visualize the problem of continuity in its entirety,
that is for philosophy in general, was G. W. Leibniz.
He set out to spread his lex continui over the vastnesses
of the theories of cognition, metaphysics, and sciences.
He even asserted that the mission of the Law of Conti-
nuity is to affirm that “the present is always pregnant
with the future,” and he also implied that to deny the
law would amount to denying the Principle of Suffi-
cient Reason, whatever that be (Leibniz Selections, p.
185). Yet, whenever Leibniz attempts to be specific and
to adduce some particular application of his general
Law of Continuity, the application usually becomes
a specific assertion within mathematics, or within nat-
ural philosophy, or within philosophy of mathematics.
It is true that in his mathematical allusions Leibniz
sometimes reaches out far into the future, but it is a
future of professional mathematics and not of extra-
mathematical philosophy.

After Leibniz, the eighteenth century contributed
nothing notable to the comprehension of continuity in
philosophy. This fully applies even to Immanuel Kant.
To judge by the entries “Kontinuität,” and “Stetigkeit”
in a recent Kant dictionary (R. Eisler, Kant-Lexikon),
Kant made no pronouncements on continuity that
contributed anything new to what had been said by
philosophers from Aristotle to Leibniz.

In the nineteenth century, in general philosophy,
most pronouncements on continuity were likewise
monotonous and uninspiring (R. Eisler, Wörterbuch).
But a few philosophers did try to break out of the
monotony; and by an odd coincidence, or perhaps
concurrence, they designated continuity in philosophy
not by names that are cognates to the Latin verb
continere, but by names which they coined from the
Greek verb synechein.

Thus, Johann Friedrich Herbart (1776-1841) has a
section on “Synechology” in his Metaphysics. Next,
Gustav Theodor Fechner (1801-87), co-founder of the
famed Weber-Fechner law of quantitative psychology
(intensity of sensation varies as the logarithm of the
stimulus), the first of its kind, has, in an impenetrably
obscure book of his, a section on the “synechological
outlook versus the monadological outlook” (Fechner,
p. 204). Finally, and most importantly, the American
philosopher Charles S. Peirce (1839-1914), the leading
architect of the algebra of relations in symbolic logic,
denotes by “synechism” what, from a certain retro-
spect, was a revival of Leibniz' Law of Continuity.
But Peirce made the lex continui genuinely universal,
and he updated it in its scope and intent, so as to make
it measure up to the exigencies of the late Victorian
and Edwardian eras.

Herbart's synechology is a peculiar philosophical
compound of realism and psychology. As a realist
Herbart finds that data from natural philosophy like
space, time, and matter exist outside ourselves. As a
psychologist however he finds that all attributes of such
data, continuity among them, are created by the psy-
chological process which operates on the intuition
through which such data reveal themselves to us. These
two findings seem to be divergent, but Herbart some-
how reconciles them.

In connection with this we wish to point out that
a passage in Aristotle's De anima apparently argues
against identifying the continuity of the process of
thought with the continuity of data conceived by
thought:

(407a 6-11).

Herbart has been lauded for the saying: “Continuity
is union in separation, and separation in union”
(Mauxion, p. 107). The saying is interesting enough,
but there are plenty of similar statements in Aristotle.
Also, after Herbart, in Fechner, there is a counterpart
to Herbart's saying which seems more original. We
translate it thus:

(Fechner, p. 247).

Following this, Fechner asserts in a very difficult
sentence of his that this “contraction” leads to a kind
of “synechological” equidistribution in the world,

which Fechner opposes to a “monadological” concen-
tration at points, and, Fechner continues, of this equi-
distribution we have a divinely inspired awareness.

C. S. Peirce, finally, being a master of mathematical
logic and also of philosophy of mathematics, knew
about the importance of continuity for mathematics
in considerable detail; he also knew how the concep-
tion of continuity, when fanning out from mathematics,
was reaching into large areas of cognition. Being thus
equipped, Peirce was elaborating aspects of continuity
which are recognizably mathematical, and he was also
endeavoring to establish a presence of continuity,
under the name of “synechism,” in most of philosophy.

It is however not clear from the statements in Peirce,
and it may have never become clear to himself,
whether synechism is indeed effectively present outside
of areas of philosophy of mathematics, or whether,
conversely, philosophy of mathematics extends into
every precinct of metaphysics in which the presence
of synechism is detectable. Peirce was one of the first
of a species of philosophers who by trend, intent, or
circumstances had been blurring the several demarca-
tions between mathematics, mathematical logic, phi-
losophy of mathematics, and general philosophy.

Being a logician by intellectual faculty, Peirce con-
ceived his synechism within a logical setting. Peirce
established a certain triad of metaphysical constructs
which he called categories, in which he placed “Syne-
chism” along with “Tychism” and “Agapism.”

Peirce called his three categories “cenopythago-
rean”: “Firstness, Secondness, Thirdness,” and they
recognizably corresponded to the triads in Kant's table
of twelve categories, but also resembled the stages of
Hegel's phenomenology of mind (Peirce, pp. 384ff.).

Now, “syn-thesis” means literally “putting-
together,” and in analogy to this, Peirce associated
various aspects of Thirdness with “synechism,” which
means literally “hanging-together.” And Peirce's aim
becomes clear if one contemplates the actual content
of his Thirdness, which a commentator of his has de-
scribed thus.

(Freeman, p. 19).

Thus Peirce's design for his Synechism was even
more ambitious than Leibniz' design for his lex con-
tinui, but Peirce was even less successful than Leibniz
in carrying out his plans. Even friendly critics of Peirce,
like Morris R. Cohen, were complaining that Pierce
had been promising a vast philosophical system, but
had never been able to erect it. And a recent critic
puts it thus:

(Murphey, p. 407).

This harsh verdict against Peirce is true as to fact;
and yet it can be mellowed by the fact that Peirce
was reaching out for the impossible and stumbled over
his own genius when attempting this. Peirce wanted
a conception of continuity that would be philo-
sophically as all-pervasive as Leibniz had envisioned
it, and, at the same time, logically as rigorous as math-
ematics of his own day was capable of making it. But
Peirce was striving after an impossibility. Mathematics
cannot be thus fused with philosophy in entirety, and
mathematics is in no justifiable sense sufficient to de-
termine philosophy in its general scope. If a conception
from general philosophy has been made mathe-
matically rigorous, then it can wear the vestments of
mathematical rigor to advantage only when moving
about in areas of mathematics proper, or, at best, in
border areas which mathematics is in the process of
penetrating, but certainly not when moving about in
areas which are well outside of mathematics' sphere
of influence. There are differences between mathe-
matics and philosophy which cannot be winked at with
impunity. A. N. Whitehead and Bertrand Russell were
frequently musing that it ought to be possible to tres-
pass on philosophy proper with conceptions from
mathematics. But they were prudent enough, especially
Whitehead, not to become entangled in difficulties into
which Peirce was stepping only too boldly.

VII. HISTORIOGRAPHY

Excepting simpleminded chronicles and listings, any
historical work has a theme of continuity inside of it.
The theme of the Old Testament, whatever the many
digressions, is the gradual erection of the Israelite
theocracy. Thucydides fused the two parts of the Pelo-
ponnesian war, which were separated by the peace of
Nicias, into one continuous event. Aristotle created our
academic field of the history of philosophy by conceiv-
ing a closely-knit continuity of development in natural
philosophy from Thales to Democritus. Within this
development he even created, rather forcedly, the
subdevelopments of “monism” and “pluralism.” Aris-
totle says himself that it may appear incongruous to
create a continuity of transition from the “materialists”
Thales, Anaximander, Anaximenes, to the “ontologists”

Parmenides and Melissus by calling them all “monists,”
but that he is going to do so anyhow (Physica, Book
1, Ch. 2). Aristotle also knew well that there had been
a great difference between the four “roots” (= ele-
ments) of Empedocles and the infinitely many atoms
of Leucippus and Democritus, but he subsumed them
under the rubric of “pluralists” nonetheless.

Continuity as a methodology in history does not at
all mean that all leading developments are presumed
to be continuous, that is composed of accumulations
of small-step events, let alone that developments are
presumed to be always “progressive,” that is positively
accented forward advances.

After the Renaissance, under the spell of a wide-
spread “idea of progress” (J. B. Bury), and lasting deep
into the nineteenth century (Bochner, pp. 73-74), such
presumptions sometimes did assert themselves. Thus,
in the history of science, the inductivism of Francis
Bacon presented such an idea of progress fairly closely.
It presumed that science advances gradually from ob-
servation to theoretization, univalently, forcibly, uner-
ringly. It also assumed that there are ways in science
of deciding between right and wrong and that an
experimenting and observing scientist can report on
facts “faithfully” without at all rendering an opinion
on them (Bochner, p. 62).

The twentieth century has become very critical of
inductivism in the history of science, but it has not
decided what to put in its stead. It is not properly
known what brings about significant changes in science,
and what the actual mechanism of change is. Some-
times a major change in science appears to be literally
a “revolution” which came about in a single step, but
at other times a major change, an equally significant
one, may appear to be the sum of many relatively small
changes in rapid succession. It is very difficult to find
a rationale for difference of these two types of change
or a schema common to both types.

The nineteenth century brought to the fore an inter-
pretation of continuity in history which is much less
naive than the ordinary belief in “inevitable progress,”
although it is deceptively similar to it. If we adopt
the term “continuism” (Bochner, p. 61), its relation to
the idea of progress may be seen as follows. Continuism
also assumes that any event of today was directly
preceded by some event which must have taken place
yesterday. However, the event of today is not neces-
sarily an “advance” over the event of yesterday, but
it is only a “reaction” to it, and the reaction may be
a positive or negative one. That is, the event of today
may concur with yesterday's event and carry it for-
ward, or it may disagree with it, and oppose it with
something different.

An unmistakably continuist enterprise is evident in
the large-scale work of Pierre Duhem in the history
of science, which was achieved in the beginning of the
twentieth century. In this work the author, una-
bashedly,

(Bochner, p. 117).

Duhem's continuism is obvious and obtrusive, and
therefore somewhat tedious; but subtler forms of con-
tinuism have been fully operative in many areas of
academic activity since the early nineteenth century.
Continuism has greatly influenced the routines of aca-
demic research, and it has been involved in an un-
precedented growth of scholarship and of historically
oriented analyses in many compartments of knowledge.
Whether it be the study of the origins of the Iliad or
the Old Testament, of Herodotus or Diogenes Laërtius,
of a play of Shakespeare or the Opticks of Newton,
there is always a strain of continuism involved in the
investigation. Finally, like all methods in historiogra-
phy, continuism had its distant roots in antiquity. In
fact, Aristotle's conception of Pre-Socratic philosophy
was entirely continuist, and has remained so since.
After Aristotle, versions of continuism are identifiable
in scholarship of any period, but it was the nineteenth
century which made the most of it.

In the twentieth century a major challenge to
straightforward continuism has come from the problem
of the rise of Western civilization as a whole. It has
long been recognized that in Western civilization in
its total course there had been, at various stages of
its growth, component civilizations with distinctive
characteristics of their own. This finding by itself is
not in conflict with continuism. But a conflict might
arise if one posits that two component civilizations did
not affect each other in a major way although they
were temporally contiguous or even overlapping. And
that there had indeed been such component civili-
zations has been proposed, respectively, by Oswald
Spengler and by Arnold Toynbee. The novelty of such
proposals is wearing off, yet the echo of them lingers
on and is likely to persist.

Very intriguing is a certain “continuist” question
relating to the origin of Western civilization in its
Mediterranean littoral. The oldest components of this
total civilization were the Old Egyptian and the Old

Mesopotamian civilizations. From the distance of our
retrospect the two arose “almost simultaneously” in
the fifth and fourth millennia B.C. This poses the prob-
lem whether there were any links between them, and,
if so, what the links were. They both initiated the art
of writing in a major way; and the absorbing problem
is whether there was any “stimulus diffusion” (Toynbee,
12, 344ff.) from the one to the other, and also what
it really was that made the Mediterranean littoral
eligible for the rise of both.

BIBLIOGRAPHY

Al-Ghazali, The Destruction of Philosophers (Tahafut al-
Falasifah), trans. A. Kamali (Lahore, 1958). This is his lead-
ing philosophical work. Aristotle, History of Animals; our
text is adapted from D'Arcy W. Thompson's version in the
Oxford translation of Aristotle's works under the general
editorship of W. D. Ross (Oxford, 1910), Vol. 4. Averroës
(Ibn Rushd), The Destruction of the Destruction (Tahafut
al-Tahafut), trans. S. van Bergh (London, 1954); this is a
refutation of al-Ghazali. Salomon Bochner, The Role of
Mathematics in the Rise of Science (Princeton, 1966). Pierre
Boyancé, Lucrèce et l'Épicurisme (Paris, 1963). J. B. Bury,
The Idea of Progress (London, 1920; New York, 1932; reprint
1955). Florian Cajori, ed., Sir Isaac Newton's Mathematical
Principles of Natural Philosophy and his System of the
World, trans. Andrew Motte (1729) revised by F. Cajori
(Berkeley, 1934; many reprints); the Latin title Philosophiae
naturalis principia mathematica (London, 1687) is briefly
identified as Principia. Gavin de Beer, Charles Darwin
(London, 1963; New York, 1964). René Descartes, La
Géométrie, original text, with an English translation by
David Eugene Smith and M. L. Latham (Chicago, 1925).
Diogenes Laërtius, Lives of Eminent Philosophers, 2 vols.
(London and Cambridge, Mass., 1935), Book IX, secs. 28-51
on Leucippus and Democritus; Book X on Epicurus; II,
439ff. Pierre Duhem, Études sur Léonard de Vinci, 3 vols.
(Paris, 1906-13); idem, Le Système du monde. Histoire des
doctrines cosmologiques de Platon à Copernic, 10 vols. (Paris,
1913-59). Rudolf Eisler, Kant-Lexikon (Hildesheim, 1961);
idem, Wörterbuch der philosophischen Begriffe, 4th ed.
(Berlin, 1930), article, “Stetigkeit.” Friedrich Engels, Dia-
lectics of Nature, ed. C. Duff (New York, 1940). Majid
Fakhry, Islamic Occasionalism and its Critique of Averroës
and Aquinas (London and New York, 1958). Gustav Theodor
Fechner, Die Tagesansicht gegenüber der Nachtansicht, 2nd
ed. (Leipzig, 1904). Eugene Freeman, The Categories of
Charles S. Peirce (Chicago, 1931). Charles Coulston Gillispie,
Genesis and Geology (New York, 1960). Bentley Glass, Fore-
runners of Darwin, 1745-1859 (Baltimore, 1958). Johann
Friedrich Herbart, Allgemeine Metaphysik (Königsberg,
1829). Isaac Husik, A History of Medieval Jewish Philosophy
(Philadelphia, 1916), Ch. XVI. Max Jammer, The Conceptual
Development of Quantum Mechanics (New York, 1966).
Robert Hugh Kargon, Atomism in England from Hariot to
Newton (Oxford and New York, 1966). G. S. Kirk and J. E.
Raven, The Presocratic Philosophers (Cambridge and New
York, 1962). Kurd Lasswitz, Geschichte der Atomistik...,
2 vols. (Hamburg, 1892); Vol. I contains a scathing criticism
of Bruno's atomism after a full account of its scope. Gott-
fried Wilhelm Leibniz, Theodicy, trans. E. M. Huggard
(New Haven, 1953). Leibniz Selections, ed. Philip P. Wiener,
revised ed. (New York, 1959). Arthur O. Lovejoy, The Great
Chain of Being: A Study of the History of an Idea
(Cambridge, Mass., 1936; New York, 1960). Charles Lyell,
Principles of Geology, being an attempt to explain the former
changes of the Earth's Surface, by Reference to Causes Now
in Operation (London, 1830-33). Moses Maimonides, Guide
of the Perplexed (Moreh Nebuchem, ca. 1170). Three leading
translations are: (1) S. Munk, in French, with copious notes,
3 vols. (Paris, 1856-66); (2) M. Friedländer, 3 vols. (London,
1881-85, reissue 1 vol. 1925; reprint, New York, 1940),
copiously annotated in English; (3) Shlomo Pines, in English
also (Chicago, 1963), closer to the original than Fried-
länder's, and has important long introductions. Marcel
Mauxion, La Métaphysique de Herbart (Paris, 1894). Murray
G. Murphey, The Development of Peirce's Philosophy (Cam-
bridge, Mass., 1964). Otto Neugebauer, The Exact Sciences
in Antiquity, 2nd ed. (Providence, 1957; New York, 1962).
Parmenides, Works, trans. Leonardo Tarán (Princeton,
1965). Charles S. Peirce, Selected Writings (New York, 1966).
Shlomo Pines, Beiträge zur Islamische Atomenlehre (Berlin,
1936). Otto Pretzl, “Die Frühislamische Atomenlehre,” Der
Islam, 19 (1931), 117-30. Robert H. Silliman, “Smoke Rings
and Nineteenth-Century Atomism,” Isis, 54 (1963), 461-74.
Solvay Institute, 13th Physics Conference. Structure and
Evolution of Galaxies (New York, 1965). Oswald Spengler,
The Decline of the West, trans. C. F. Atkinson, 2 vols. (New
York, 1926-28). Stephen Toulmin and June Goodfield, The
Discovery of Time (London, 1965). Arnold Toynbee, A Study
of History, 12 vols. (London and New York, 1935-61). J.
Tricot, Aristote: histoire des animaux (Paris, 1957). Andrew
G. van Melsen, From Atomos to Atom (New York, 1960).
Hermann Weyl, Philosophy of Mathematics and Natural
Science (Princeton, 1960).

SALOMON BOCHNER

[See also Baconianism; Cycles; Historiography; Infinity;
Platonism; Pragmatism; Pre-Platonic Conceptions; Progress;
Pythagorean...; Revolution; Uniformitarianism and Ca-
tastrophism.]