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Dictionary of the History of Ideas

Studies of Selected Pivotal Ideas

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Introduction. Space is a conception of many aspects,
and it has arisen—under various names, appellations,
and descriptions—in different areas of cognition and
knowledge: in cosmology, physics, mathematics, phi-
losophy, psychology, and theology.

There are stirring philosophemes about space in the
Timaeus of Plato and the Physica of Aristotle, and they
foreshadow our present-day space, or spaces, of labo-
ratory and cosmos, of mechanics and physics, of worlds
that are everlasting and stationary, and of universes
that are born and grow, and perhaps even age and
collapse. But, as a rule, Greek thoughts about space
were only about space in cosmology and physics, and
perhaps also theology; and seldom, if ever, did Greeks
compose a statement, or even an aphorism, about space
in any other area of insight.

Thus, the Greeks did not create a space of logical,
ontological, or psychological perception. There is
almost nothing about space in Plato's Meno,
Theaetetus, Sophistes,
or Parmenides, or in Aristotle's
De anima, or Metaphysica. In Aristotle's collection of
several treatises in logic, the so-called Organon, there
is mention of space only once, in Chapter 6 of Categor-
It is not a “research” conception of space, but an
indifferent schoolbook description of it, and Aristotle
had no occasion ever to recall it.

In modern philosophy, that is since 1600, any doc-
trine of perception since John Locke has dealt with
space as a matter of course; and, within this general
approach, a monumental construction, which kept the
nineteenth century enthralled, was the famed a priori
space of Immanuel Kant. In contrast, Greek philosophy
knew absolutely nothing about an a priori space (and
time) of pure intuition as expounded by Kant in the
“Transcendental Aesthetics” of his Critique of Pure
(1781). It is true (see sec. 13, below) that in
psychology of the twentieth century the role of space,
as a primary datum, has greatly shrunk. But this
shrinkage affected space in experimental psychology
rather than in metaphysical perception, and a marked
difference between ancient and modern attitudes re-

More conspicuous, and almost fate-sealing, was the
absence from Greek thought of a general conception
of space for geometry and geometrically oriented
analysis. Greek mathematics did not conceive an over-
all space to serve as a “background space” for geo-
metrical figures and loci. There is no such background
space for the configurations and constructs in the
mathematical works of Euclid, Archimedes, or Apol-
lonius, or even in the astronomical work Almagest of
Ptolemy. When Ptolemy designs a path of a celestial
body, it lies in the astronomical universe of Ptolemy;
but as a geometrical object of mathematical design and
purpose, it does not lie anywhere. In Archimedes, who,
in some respects was second only to Isaac Newton, the
mathematical constructs were placed in some kind of
metaphysical “Nowhere” from which there was “No
Exit” into a mathematical “Future.”

Such a background space was rather slow in coming.
Thus, Nicholas Copernicus did not have it yet. He was
an innovator in astronomical interpretation, and not
in mathematical operation. His mathematics was still
largely Ptolemaic, and only a bare outline of a mathe-
background space is discernible in his De revo-
lutionibus orbium coelestium.
Nevertheless, already a
century before Copernicus, Nicholas of Cusa, church-
man, theologian, mystic, and gifted mathematician, in
Book II of his leading work Of Learned Ignorance (De
docta ignorantia
), adumbrated an overall space of
mathematics by way of an overall mathematical
framework for the space of the universe. But the lead-
ing statements of the metaphysics of Cusa were
enveloped in theology and mysticism and not very
comprehensible to his contemporaries and to others
after him.

On the other hand, soon after the death of Coperni-
cus, in the second half of the sixteenth century, some
mathematicians began to grope for projective and
descriptive geometry, and this was bound to lead to
a background space. It did, but only after two hundred
years. In the meantime, in the first half of the seven-
teenth century a background space for geometry was
created, for all to see, in La Géométrie (1637) of René
Descartes. Half a century later, Isaac Newton in his
Principia (1687), created an ambitiously conceived
absolute space which was intended to be a background
space for mathematics, for terrestial and celestial me-
chanics, and for any space-seeking metaphysics.
Newton even made it into a “Sensorium of God,”
whatever that might be, and this aroused philosophical
passions which are still smoldering. A resulting corre-
spondence between Gottfried Wilhelm Leibniz and Dr.
Samuel Clarke (for a spirited account see Koyré,
[1957], Ch. XI) is much prized for what it reveals about
the philosophy of Leibniz. But it is less significant for
what it reveals about the role of space in science.

Beginning in the late eighteenth century, and
through the length of the nineteenth century, mathe-
matics developed a duality or polarity between space
and (geometrical) structure, by which, at long last—
two and a half millennia after Thales of Miletus—
mathematics became an artificer of space and spaces.
In the twentieth century this dualism of space and
structure greatly affected all of theoretical physics. For
instance, there would be no General Theory of Rela-


tivity without it. In this theory, space is gravitational
space; it is “curved,” and thus endowed with “form.”
This form is affected by the presence of gravitational
matter, which is the only kind of matter known to this
theory. In this way, the theory establishes a novel
intimacy between matter and space and between mat-
ter and form. An enthusiast might even declare that,
in this theory, matter is space (or form) and space is

In quantum theory, the de Broglie dualism of parti-
cles and waves offers a different version of the dualism
of matter and space, since matter is built of elementary
particles and waves are space-filling. Also, in quantum
“field” theory, the field is highly mathematical through
its conceptual provenance; and if mathematics is
equated with form, then a new version of the dualism
of matter and form emerges.

In the sections to follow we will have many other
assertions and details. Each section will be headed
either by a name for space, or by a formulaic descrip-
tion of space or of something resembling space. These
appellations will be introduced roughly in the chrono-
logical order in which they have arisen, and each
appellation will be then followed up in its develop-
ment. Our first appellation will be a Hebrew term from
the Old Testament, which seems to be the earliest on
record; and there is hardly another case from natural
philosophy in which the Old Testament fashioned a
“technical” term ahead of the Greeks.

1. Makom. Our term “space” derives from the Latin,
and is thus relatively late. The nearest to it among
earlier terms in the West are the Hebrew makom and
the Greek topos (τόπος). The literal meaning of these
two terms is the same, namely “place,” and even the
scope of connotations is virtually the same (Theol.
..., 1966). Either term denotes: area,
region, province; the room occupied by a person or
an object, or by a community of persons or arrange-
ments of objects. But by first occurrences in extant
sources, makom seems to be the earlier term and con-
cept. Apparently, topos is attested for the first time
in the early fifth century B.C., in plays of Aeschylus
and fragments of Parmenides, and its meaning there
is a rather literal one, even in Parmenides. Now, the
Hebrew book Job is more or less contemporary with
these Greek sources, but in chapter 16:18 makom
occurs in a rather figurative sense:

O earth, cover not thou my blood, and let my cry have
no place


Late antiquity was already debating whether this
makom is meant to be a “hiding place” or a “resting
place” (Dhorme, p. 217), and there have even been
suggestions that it might have the logical meaning of
“occasion,” “opportunity.”

Long before it appears in Job, makom occurs in the
very first chapter of Genesis, in:

And God said, Let the waters under the heaven be gathered
together unto one place (makom) and the dry land appear,
and it was so

(Genesis 1:9).

This biblical account is more or less contemporary with
Hesiod's Theogony, but the makom of the biblical
account has a cosmological nuance as no corresponding
term in Hesiod.

Elsewhere in Genesis (for instance, 22:3; 28:11;
28:19), makom usually refers to a place of cultic sig-
nificance, where God might be worshipped, eventually
if not immediately. Similarly, in the Arabic language,
which however has been a written one only since the
seventh century A.D., the term makām designates the
place of a saint or of a holy tomb (Jammer, p. 27).

In post-biblical Hebrew and Aramaic, in the first
centuries A.D., makom became a theological synonym
for God, as expressed in the Talmudic sayings: “He
is the place of His world,” and “His world is His place”
(Jammer, p. 26). Pagan Hellenism of the same era did
not identify God with place, not noticeably so; except
that the One (τὸ ἕν) of Plotinus (third century A.D.)
was conceived as something very comprehensive (see
for instance J. M. Rist, pp. 21-27) and thus may have
been intended to subsume God and place, among other
concepts. In the much older One of Parmenides
(early fifth century B.C.), from which the Plotinian
One ultimately descended, the theological aspect was
only faintly discernible. But the spatial aspect was
clearly visible, even emphasized (Diels, frag. 8, lines

2. Chaos. In a connected essay on space (Physica,
Book 4, Chs. 1-5) Aristotle suggests (208b 29), rather
lightly, that Hesiod's “Chaos” (χάος) was one of the
earliest (Greek) designations for space, or perhaps uni-
verse, and he also quotes line 116 of Hesiod's Theogony:
“First of all things was Chaos, and next broad-bosomed
Earth.” It is true that by etymology of the word Chaos,
and in Hesiod's own vision, Chaos does not actually
represent space as we know it today (Kirk and Raven,
pp. 27ff.), that is, space in a cosmologically articulated
universe. But chaos was destined, by future develop-
ments, to have a certain relation to space, and it is
this that Aristotle's suggestion is hinting at. In fact,
in many “creation myths,” beginning with Plato's
Timaeus, there is an initial phase of a “primordial
chaos,” in which there is no ready-made space as yet,
but only a space in the making, and the structure of
this space unfolds not by itself but conjointly with the
structure of matter, energy, and other physical attri-
butes. Greek natural philosophy in general knew about
this initial phase, and, when in a mood of historical
retrospection, viewed Hesiod's Chaos as an aspect of


it. Aristotle's suggestion expresses such a view, in such
a mood.

In present-day cosmology there is an obvious need
for such an initial phase whenever a model of the
universe, be it expanding or pulsating, has a so-called
“point origin,” that is, a time point at which the radius
R of the universe has the value O or nearly so (H.
Bondi, pp. 82ff.). Or the point origin may be the time
point of a “big squeeze” for all matter and energy,
in consequence of a “collapse” of a universe just pre-
ceding (Gamow, p. 29). In either case, the resulting
situation has been described by Arthur Eddington
(1882-1944) thus:

If the world began with a single quantum, the notions of
space and time would altogether fail to have a sensible
meaning before the original quantum had been divided into
a sufficient number of quanta. If this suggestion is correct,
the beginning of the world happened a little before the
beginning of space and time

(Lemaître, p. 17).

It must be stated though that there is a contemporary
version of evolutionary theory in which there is no
“point” origin, and a space is preexistent. It assumes
that evolution began with a primordial plasma, or
rather “ambiplasma” (H. Alfvén, pp. 66ff.), that is with
a huge mass of gas composed of various particles of
energy, matter, and antimatter, and filling a spherical
volume of cosmic dimensions. Such a plasma is un-
stable. At some stage in the past a breakup set in which
led to the formation of galaxies, and this was the true
beginning of creation (ibid.).

In the Timaeus Plato imagined that Space, or rather,
Place, was preexistent, together with Being and Be-
coming (52D), but that Time began when creation
began (38B). With this fancy Plato outdid himself.

Whatever the mode of creation, cosmologists agree
that there was an initial phase of “disorder,” that is,
mathematically, of so-called turbulence. Greek natural
philosophers knew this, in thought patterns of theirs,
fairly early, certainly since Anaxagoras of Clazomenae
(500-428 B.C.), and possibly since Anaximander of
Miletus (610-545 B.C.). As scientists sometimes do even
today, the Greek philosophers projected back this
primordial disorder as far as they could. This led them
to attribute it to Hesiod's Chaos, and hence the famed

chaos: rudis indigestaque moles
(chaos: a “rude” and “undigested” heap),
in Metamorphoses 1, 7 of Ovid.

It has been noted long ago that Hesiod's Chaos, in
the light of later interpretations, brings to mind the
tohu wa bohu (“without form, and void”) of Genesis
1:2 and related biblical terms (see “Chaos” in Der
Kleine Pauly,
Vol. I, column 1129). This is of impor
tance, because, more than any other general concep-
tion from general philosophy, our conception of space
is just as much a biblical heritage as it is a Greek one
(Jammer, Ch. 2; here the emphasis is on space in

3. Cosmos. The term cosmos (κόσμος) is Homeric,
and classicists are studying it increasingly (even the
numerous bibliographical notices in Miss Jula
Kerschensteiner are not exhaustive). The basic meaning
in Homer is “order,” and throughout the length of
antiquity this original meaning remained active amidst
many figurative ones.

This “order” began to be “universe,” by way of
“world-order,” in the following saying of Heraclitus
of Ephesus (Diels, frag. 30):

This cosmos [κόσμον τόνδξ] did none of the gods or men
make, but it always was and is and shall be; an everliving
fire, kindling in measures and going out in measures

and Raven, p. 199).

The association of this cosmos with “everliving fire,”
whatever that be, need not disqualify it from repre-
senting cosmological space. In Albert Einstein's Gen-
eral Theory of Relativity cosmological space is most
intimately associated with gravitation (Whittaker, Vol.
2, Ch. 5). Yet the nuclear structure of gravitation is
so little known that a Heraclitus of today could not
be silenced, or even gainsaid, if he chose to declare
that gravitation is “everliving” and that “gravitational
waves” are alternately kindling and going out.

In this saying of Heraclitus, order is a principle of
the universe as a whole, but long afterwards, in the
logico-metaphysical outlook of Leibniz it is a schema
of the space around us. We quote.

Space is the order of coexisting things, or the order of exist-
ence for all things which are contemporaneous. In each of
both orders—in that of time as that of space—we can speak
of a propinquity or remoteness of the elements according
to whether fewer or more connecting links are required to
discern their mutual order

(Leibniz Selections, p. 202).

... When it happens that one of these coexistent things
changes its relation to a multitude of others, which do not
change their relation among themselves; and that another
thing, newly come, acquire the same relation to the others,
as the former had; we then say it is come into the place
of the former; and this change we call a motion in that
body, wherein is the immediate cause of the change


In these reflections of Leibniz there is even a conflu-
ence of two properties of space, of ordering and of
relation; and the nearest to all this from classical
antiquity is in the following passage from Aristotle:

This is made plain also by the objects studied in mathe-
matics. Though they have no real place they nevertheless,
in respect of their position relative to us, have a right and


left, as ascribed to them only in consequence of their posi-
tion relative to us, not having by nature these various

(Physica 208b 23-24; Oxford translation).

Among forerunners of Leibniz' ideas after Aristotle,
if any, one might perhaps name the late Hellenistic
(or early medieval) Aristotle commentator Joannes
Philoponus (ca. 575). “For Philoponus conceives space
as pure dimensionality, lacking all qualitative differen-
tiation” (M. Jammer, p. 55), and to him “space and
void are identical,” with “void being a logical neces-
sity” (ibid., p. 54); and this creates a foretaste of
Leibniz, perhaps.

4. Apeiron. The generic meaning of apeiron
(ἄπειρον) is Infinity without a direct suggestion of space.
But the term has many connotations, and late tradition
makes it likely that Anaximander, the younger com-
patriot of Thales, denoted by it a generative substance
of the universe (Kirk and Raven, Ch. 3). If this was
so, then, in Anaximander's imagery, apeiron may have
also been a part-synonym for space, since matter and
space were probably proximate notions to him.

A token of this proximity is woven into the fabric
of Aristotle's Physica. Book 4 of this treatise is made
up of three essays, on place (Chs. 1-5), on void (Chs.
6-9), and on time (Chs. 10-14). Now, immediately
preceding these essays (Book 3, Chs. 4-8) is an essay
on apeiron, as if to indicate that there is a close link
between infinity and space (and void).

In the seventeenth century, Baruch Spinoza went
philosophically much farther, when, in his Ethics he
imparted infinity to Extension, that is, to space and
to other attributes of his God (Wolfson, 1, 154).

Before that, in the sixteenth century, Giordano Bruno
ecstatically fused space and infinity in an unbridled
vision of infinitely many worlds regularly distributed
over a wide-open all-infinite Euclidean space (D. W.
Singer, pp. 50-61). The cosmological facts were not
entirely new (ibid.), nor were they presented in ade-
quate detail to become meaningful as such, nor were
Bruno's insights greatly welcomed by his contem-
poraries. But somehow Bruno's outpourings made an
impression, and they created and fashioned, or only
activated, a philosopher's yearning for the infinitude
of space, which played a leading part on the stage of
philosophy until well into the twentieth century. In
a broad sense, the English philosophers Henry More
(1614-1687) and Richard Bentley (1662-1742) were
followers of Bruno (Koyré, Chs. 6-10), and so were
virtually all representatives of German idealism begin-
ning with Immanuel Kant, or even earlier.

Oswald Spengler advanced the thesis (Decline of the
Vol. 1, Ch. 5 and elsewhere), which probably
was not quite new either, that this hankering after the
infinitude of space, especially in its extra-rational
aspects, was a characteristic trait and a propellant of
Western European civilization since the early Middle
Ages, and he somehow also interpreted the emergence
of Gothic art and architecture as a response to this
hankering. This thesis, whatever its overall validity,
does not properly apply to leading scientists (Bochner,
Eclosion and Synthesis, Ch. 14). Most scientists, even
when adopting some features of Bruno's cosmology,
were circumspect and restrained. In scientific cos-
mology today, the Kinematic Relativity (J. D. North,
Ch. 8) of Edward Arthur Milne (1896-1950) seems
more compatible with Bruno's suggestions than other
viable theories; but even in Milne the physical presence
of infinity is considerably more restrained than in
Bruno's paradigm.

5. Kenon (“Void”). Early Pythagorean philosophy,
when still ingenuous, apparently identified space with
kenon (κενόν), which literally means “void.” In fact,
within his essay on the void in the Physica, Aristotle
has a passage about Pythagoreans which links kenon
with apeiron (“infinity”), pneuma (“breath”), ouranos
(“heaven”), and arithmos (“number”):

The Pythagoreans, too, held that the void exists and that
breath and void enter from the infinite into the heaven itself,
which as it were inhales; the void distinguishes the nature
of things, being a kind of separating and distinguishing
factor between terms in series. This happens primarily in
the case of numbers; for the void distinguishes their nature

(Physica 4, 6; 213b 22-28; Kirk and Raven, p. 252).

This fascinating report must be allowed to speak for
itself. Unlike some modern commentators, Aristotle,
very prudently, does not attempt to interpret it.

Otherwise, Aristotle's essay on the void in the
Physica suffers from an incurable weakness. As always
and everywhere, Aristotle maintains in this essay that
a void cannot exist, and in the present context Aristotle
would really like to give a general demonstration for
this thesis. But this he cannot do. Such a demonstration
would require that Aristotle first define his void
logically, and then argue metaphysically that it cannot
exist. However Aristotle finds it impossible to give a
logical definition of void that would not turn it into
a kind of space, or pseudo-space, or nonspace; and the
intended demonstration of his thesis dissolves into an
accumulation of remarks not easy to remember.

It is true that present-day physics is also unable to
define a void other than as a space devoid of matter
and energy, say. But this is of no harmful consequence
as long as nobody asserts, and wants to demonstrate,
that “a void cannot exist”; and nobody does.

6. Non-Being. The concept of Non-Being (τὸ μή ὄν)
occurs freely in Parmenides, and is probably due to


him. By an unimpeachable report in Aristotle's Meta-
(985 b4)—which is reinforced by a historical
analysis in De generatione et corruptione (325 a2;
Guthrie, 2, 392-94)—the Atomists Leucippus and
Democritus (fifth century B.C.), from their approach,
made Non-Being into an appellation for space, to al-
ternate with, or be a replacement for kenon. They
viewed the relation between material atoms and their
spatial setting as a contrast between the full and the
void (τὸ πλη̃ρες καὶ τὸ κενόν), and expressed it as a
duality between Being (τὸ ὄν) and Non-Being.

This duality, in a different outlook, had been created
by Parmenides. His Being, in fusion with Oneness and
Thought, constituted a universe of ontology. This uni-
verse, however ontological, was somehow also en-
dowed with physical attributes of a uniform finite
sphere (σψαι̃ρα), and as such it was continuous, indi-
visible, unchangeable, and ungenerated and imper-
ishable; whereas the Non-Being of Parmenides was only
an obverse of Being, vacuous of determination, a sham
polarity as it were. Now, the atomists heavily
emphasized the physical aspects of this Being. In their
atomistic conception, the evenly distributed Being was
shrunk from continuity to discreteness and had become
concentrated in discretely distributed atoms; and, by
the same token, Non-Being was metaphysically ele-
vated to the all-important role of a spatial setting for
the atoms, without which the activities of the atoms
cannot be imagined.

The splendidly unchangeable ontological universe of
Parmenides the philosopher had of course nothing
whatsoever to do with the very changeable common
universe of Parmenides the citizen, which constantly
exhibited changes of day-and-night, light-and-dark,
hot-and-cold, dry-and-moist, etc. Parmenides the
philosopher knew this. But what he did not know was
that philosophically he need not, and must not concern
himself with the vulgar universe of Parmenides the
citizen, but that he ought to leave it in care of more
practical (scientific) experts who knew something about
such “vulgarities.” Instead, Parmenides the philosopher
considered himself “duty bound” to construct a
“model” of the other universe too, calling it, quite
unrealistically, the universe of mere “appearance”
(doxa, δόξα); which of course was the opposite of what
it really was. As could be foretold, the construction
turned out to be quite banal (Kirk and Raven, pp.
284-85), and late tradition has, mercifully, transmitted
but few original fragments of its description.

Even Leucippus and Democritus, great scientists
though they were, could not quite resist the tempta-
tions of having opinions on matters which others un-
derstood better. After having described, magnificently,
the workings of the “laboratory space” of their “atomic
physics,” and also the creation of the corresponding
“galactic space” of cosmogony (Diogenes Laërtius,
Book IX, Chs. 30-33; Loeb edition, 2, 438ff.), they felt
“duty bound” to discourse also on the astronomy of
the planetary system. About this they had nothing to
say that was in the least interesting (Kirk and Raven,
p. 412); and modern commentators since around 1900
have not ceased to point this out, gratuitously.

7. Chora and Topos. In Plato's creation myth in the
Timaeus space-in-the-making, that is, space in its
cosmogonic nascency and formation, is called chora
(χω̃ρα), but after its creation has been completed it
is called topos. In general usage, chora and topos have
approximately the same range of meanings, but chora
is used more loosely and informally, and it is less
specific than topos. “A locus in mathematics, that is,
a figure which is determined by, or results from, specific
requirements, became topos, not chora. In the Meteor-
when Aristotle wishes to single out a geo-
graphic district in a country, chora usually stands for
country and topos for district” (Bochner [1966], p. 152).

In De caelo Aristotle adheres to Plato's distinction,
but since his account is less cosmogonic than Plato's
the occurrence of topos prevails. However, going be-
yond Plato, markedly so, Aristotle also uses the name
of topos for an entirely different space, namely for the
space of physics proper, that is, for the operational
space of “laboratory physics” of today. Nowadays it
is imperative that this space be kept distinct from the
space of cosmology, and Aristotle confused the two but
little (Bochner [1966], pp. 154-55).

Aristotle presents his “laboratory space” in the spe-
cial essay on topos in Physica 4, 1-5. His leading asser-
tion is that in a scientific study of a physical system,
space is not given as the spread across the system, as
the naive view has it, but is given by the total structural
behavior as determined by the boundary configuration
of the system; and Aristotle's first succinct definition
is: “topos is the inner boundary of what contains”
(ibid.). When attempting to elaborate this first defini-
tion into a detailed description, Aristotle encounters
complications which are intrinsic to the subject matter,
and he arrives at alternate descriptions which are
seemingly not quite consistent with each other. How-
ever, on closer analysis these inconsistencies can be
reconciled (Bochner [1966], pp. 172-75).

Furthermore, it is important to realize that in
present-day physics the conception of space is prag-
matically used in alternate versions which are not
identical and that no serious harm arises. Thus, (i) in
engineering mechanics as taught in engineering schools
all over the world, and in large parts of so-called
“classical” mechanics and physics, space continues to
be Newtonian, that is Euclidean, as in Newton's


Principia (i.e., Philosophiae naturalis principia mathe-
London, 1687). But, (ii) the theory of single
electrons or other elementary particles—that is, the
so-called quantum field theory—operates in the space-
time of the special theory of relativity which is differ-
ent from the space-time of Newton's mechanics. How-
ever, (iii) in the physics of our galaxy at large (the
so-called Milky Way) and beyond, space is subject to
the general theory of relativity; and most “models” of
the universe presently under examination are different
from the two preceding ones. Finally, (iv) the “statis-
tical” space of quantum mechanics may be viewed as
being different from, and thus inconsistent with any
“non-statistical” space, Newtonian or relativistic
(Bochner [1966], p. 155).

Of course, in physics of today there is, as has always
been, a great quest for consistency, unity, and harmony.
But, in any one science, the volume of knowledge is
growing so fast and in so many subdivisions of the
science, and explanation is so far behind experi-
mentation that a detailed internal harmonization is not
attainable. In particular, the concept of space is so
ubiquitous, and is reached by so many avenues and
channels, that it would be stifling and sterile to force
upon it metaphysically a single logical schema, which,
even if acceptable today, might become unsuitable

8. To pan (“The all”). The Homeric term to pan
(τό πα̃ν) occurs several times in Aristotle's De caelo,
sometimes reinforced by to holon (τό ὅλον; the Whole),
and its meaning is a near-synonym for the leading term
ouranos (“Heaven,” “World”). However, in Physica,
Book 4, Ch. 5, at the end of the essay on topos, to
has a somewhat special connotation. There, Aris-
totle raises the following question in a rumination of
his: if one views the whole universe, to pan, not as
a cosmic datum but as a physical system however vast,
does it then have a physical topos, and how? (Bochner
[1966], p. 178). This is an intriguing question, and
various aspects of the question have been raised more
than once since.

Thus, Nicholas of Cusa, who had a mathematical
turn of thought, equated the would-be topos of the
universe with a mathematical substratum of it, and he
asked, implicitly but recognizably, whether the uni-
verse, in a suitable substratum, might escape the
dichotomy of having to be finite or infinite. He divined
that there are mathematical universes to which the
dichotomy does not apply (Bochner [1968], p. 325),
and he even knew that the space of the universe may
be endowed with a mathematical homogeneity by
which every point can be viewed as a center of it
(Koyré, Ch. 1). Or, if we envisage not the underlying
space of the universe but the matter in it, then in the
words of a present-day cosmologist: “It is theoretically
possible... for an unbounded distribution of matter
to have its circumference nowhere, and center every-
where” (G. J. Whitrow, p. 43).

After Cusa, Copernicus and Newton entertained
thoughts that were consonant with his. Newton may
have even been perturbed by the question (even if he
would not admit to it) of how to extend the mathe-
matical substratum of our solar system beyond itself,
in case some of the comets should move on hyperbolic
orbits, which are mathematically possible, but mathe-
matically are not contained within the substratum of
the planetary system proper (Bochner [1969], Ch. 14).

A version of Aristotle's problem arises in present-day
cosmology. In the general theory of relativity space
is gravitational space and is thus largely determined
by a distribution of gravitational masses. Now, if this
distribution is known and if the shape of the resulting
space is to be determined, then, for operational pur-
poses, a background space, that is a kind of topos in
the sense of Aristotle, must be chosen a priori; and it
would be desirable to have a procedure for making
this a priori choice in any one given case.

9. Ouranos. The word ouranos (ούρανός) means
heavens, and it is a keyword in De caelo. In the asser-
tion that the world is finite, or rather in the arguments
that the “body” (soma) of the world is finite, Aristotle
uses either ouranos or to pan. But he uses mainly
ouranos in the speculation, which in the Renaissance
brought down upon him much condemnatory criticism,
that the heavens rotate around the earth in concentric
spheres. He also uses ouranos in his beautifully
reasoned assertion that there is only one world (De
Book I, Chs. 8 and 9).

Ouranos is a word of uncertain etymology. It occurs
in Homer and other ancient poetry and has there
always one complex meaning of “the region which
contains the stars and in which the phenomena of
weather take place, a region which was personified and
considered to be divine or to be the dwelling place
of the gods” (L. Elders, pp. 140-41). It thus had a
well-established standing even before Aristotle put his
imprint on it. Yet, Aristotle made it the center of a
system “which, although Aristotle was a naturalist
rather than a physicist, held the stage of physics for
almost two thousand years, and which, by its flashes
of insight and uncanny anticipations, evokes fascination
even today” (Bochner [1966], p. 178).

10. Spatium. This is the main term for space in
classical Latin, and it has given rise to space (English),
espace (French), spazio (Italian), espacio (Spanish), etc.
The common Teutonic stem ruum, which gave rise to
English room and German Raum, had no lexical spread
of comparable compass.


Within Western civilization, with spatium, began a
widespread imposition of the vocabulary of space on
the parallel conception of time. Thus Cicero uses the
expression spatium praeteriti temporis, in the meaning
of: “the space (i.e., interval) of time gone by,” and his
usage has the ease of a colloquialism. Furthermore,
according to the Oxford English Dictionary, the term
space in English had from the first, that is since around
1300, two meanings, a temporal and a spatial, and the
Dictionary lists the temporal meaning first. A corre-
sponding French Dictionary (Paul Robert, 3, 1703) also
lists both meanings for espace, and it observes that from
the twelfth to the sixteenth centuries the temporal
meaning was the leading one. Spanish and Italian
dictionaries also have both meanings, and, according
to one of them, spazio occurs in a temporal meaning
in the Purgatory of Dante and in a story of Boccaccio.
(Niccolò Tommaseo, Dizionario della lingua italiana,
Turin [1915], 6, 135).

Yet, two thousand years after Cicero, the fin-de-siècle
philosopher Henri Bergson was able to build a career
and reputation on an intellectual opposition to the
quantitative subordination of time to space (J. A. Gunn,
Ch. 6). He was pressing his conceptions of durée, élan
vital, évolution créatrice,
etc. into a lifelong campaign
for reconstituting the data of human consciousness in
their original intuition that was free from the idea of
space and from the scientific notion of time; and he
was apparently greatly admired for this by many.

Bergson's finding, which so alarmed him, that time
is dominated by space is not even correct. The true
fact is that both space and time are dominated by one
common paradigm, namely the mathematical linear
continuum, which in the early part of Bergson's career
had just been perfected by Richard Dedekind and
Georg Cantor; and the seemingly spatial vocabulary
is in fact a joint mathematical one. Aristotle in his
Physica, in the context of Zeno's paradoxes, had stated
over and over again, in words of his own, that there
must be a common paradigm for space and time if
there is to be any conception of movement at all. Also,
for Aristotle, movement in a broad sense, which he
termed kinesis, separated the animate from the
inanimate, and without kinesis there would be no soul,
and thus no kind of consciousness or intuition. For
Aristotle, space (and time) were features of what he
viewed as “nature.” He did not have a space (or time)
of perception, but he also could not imagine any kind
of perception without a suitable kinesis, and for the
latter (his) space and time, in coordination, were
undoubtedly prerequisite. Whatever will endure of
Bergson's philosophy, his opposition to a coordination
of space and time will not.

11. Extensio. Our term extension comes from the
late Latin term extensio—itself derived from the clas-
sical verb extendere—and it became a philosophical
term in the Middle Ages. The exact philosophical status
in the Middle Ages is not easy to determine, and this
is due to a general difficulty which is tellingly presented
in the following passage from a book on Duns Scotus:

Thus the nature of Space is discussed with reference to
transubstantiation and the nature of angelic operation,
while that of Time, though treated more thoroughly and
at greater length in the De rerum principio, is once more
mooted in the commentaries on the Sentences in connection
with the angelic experience. Nor is this all. Our difficulties
are increased by the fact that the scholastic terminology
is almost impossible to translate exactly. For spatial relations
are expressed in terms of accidens, respectus and funda-
all logical rather than mathematical symbols. In
fact, the entire physics of the medieval world reflects this
logical view of things so strange to our modern scientific
modes of thinking

(C. R. S. Harris, 2, 173).

In philosophy after 1600, extension leapt into
prominence when Descartes used it, together with the
equivalent étendue, in his Philosophical Principles. Oc-
casionally, Descartes writes espace for it, but only
informally, because formally espace is something else
for him. In fact, in La Géométrie Descartes introduces
an espace (qui a trois dimensions) as an operational
background space for coordinate geometry in mathe-
matics, and this is the true role of espace in the thinking
of Descartes. Extension however is for him something
conceptually different, namely the space of physics and
of the universe. In this role, extension is coextensive
with matter, certainly with matière subtile, and it is
the carrier of Cartesian vortices (Hesse, pp. 102-08).

After Descartes, extension gradually diminished in
importance, or at least in prominence. In Spinoza's
Ethics it is “identified” with Spinoza's God (Wolfson,
Ch. VII), and it then occurs in Leibniz' reaction to
Spinoza (Leibniz Selections, pp. 485ff.). It still has a
standing in the theory of perception of George
Berkeley, Bishop of Cloyne (Jammer, p. 133) but after
that it began to be a philosophical term of second rank.

But the “subtle matter” of Descartes, which filled
his extension, maintained itself longer, although it had
already had a long career, starting out with the role
of Aristotle's body (soma) which filled his topos. Philos-
ophers of the eighteenth century showed signs of tiring
of this “subtle matter,” but, unperturbed by this, it
somehow managed to become the front-page aether
of James Clerk Maxwell in the nineteenth century
(Whittaker, Vol. 1, Ch. IX). Only the early twentieth
century finally sent it into retirement, but it took an
Albert Einstein to bring this about.

Instead of aether there are nowadays various “fields;”
gravitational field, electromagnetic field, fields of vari-


ous de Broglie waves. The fields are dual to particles
of matter or energy, and energy is equivalent with
mass, so that a return to a “subtle matter” has been
effected. Physics has but a limited budget of ideas of
cognition with which to operate, and the same ideas
are likely to return every so often for reassignment.

12. Space of Perspective. The sixteenth century
created the space of standard (rectilinear) perspective
for use in representational arts. This perspective was
intended to secure a two-dimensional mimetic illusion
of three dimensional actuality, and the central struc-
tural device for achieving this was the introduction of
a “vanishing point” at infinity. Also, this theory of
perspective advanced the presumption that it created
the one and only space of “true” optical vision.

It belongs to the history of art to determine the
extent to which this presumption was or was not
heeded in the seventeenth and eighteenth centuries,
but it is a matter of public record that in the nineteenth
century a school of French painting openly revolted
against it. The leading revolutionary in the nineteenth
century was Paul Cézanne, and he replaced the space
of classical perspective by a space of illusion of his
own, which although not objectively fixed, was never-
theless subjectively controlled. The twentieth century
went much further. Beginning with cubism, the visual
arts began to take much greater liberties with space
than Cézanne had ever done or envisaged, but this
again is a topic for the history of art only.

13. Absolute Space. The sixteenth century also
initiated descriptive and projective geometry (J. L.
Coolidge, Chs. 5 and 6), and when, much later, in the
nineteenth century, projective geometry was fully
developing, its unfolding was part of the creation of
many novel structures, Euclidean and other (see sec.
16, below). In the seventeenth century there were
remarkable achievements by Gérard Desargues, Blaise
Pascal, and others. But after that there was a long
period of very slow advance, and non-Euclidean ge-
ometry, for instance, presented itself only in the nine-
teenth century, although, by content and method, the
eighteenth century was just as ready for it.

This retardation may have been caused in part by
Isaac Newton's insistence on the Euclidean character
of his absolute Space (for other such retardations
caused by Newton see Bochner [1966], pp. 346f.). In
Newton's Principia, the program was to erect a mathe-
matical theory of mechanics, based on the inverse
square law of gravitation, from which to deduce the
three planetary laws of Kepler and Galileo's parabolic
trajectory of a cannon ball, all in one. Newton suc-
ceeded in this endeavor, but virtually every step of
his reasoning required and presupposed that his under-
lying space be Euclidean. Newton was keenly aware
of this prerequisite, and following a general philo-
sophical trend of his age, he endowed his Euclidean
background space with extra-formal features of physi-
cal and metaphysical uniqueness and theological
excellence, by which it became “absolute.” These
extra-formal features are not needed for the deductions
of the main results, and Newton discourses on these
features in supplementary scholia only (Bochner [1969],
Ch. 12).

In support of his contention that there is an absolute
space, Newton adduces two arguments (experiment
with two globes, and, more importantly, with the rota-
ting bucket) which physicists find arresting even today,
although the arguments do not demonstrate that there
is space which is absolute in Newton's own sense. In
the Victorian era, the physicist-philosopher Ernst Mach
in his The Science of Mechanics... (Die Mechanik
in ihrer Entwicklung;
many editions and translations),
which was composed from a post-Comtean positivist
stance of his age, was quite critical of Newton's argu-
ments and conclusions (Jammer, Ch. 5, esp. pp.
140-42); but a recent reassessment by Max Born leads
to a balanced appraisal of importance (Born, pp.

The opposition to absolute space by philosophers
began immediately with the Leibniz-Clarke corre-
spondence (see Introduction, above), and has not quite
abated since. Yet the Encyclopédie of Diderot and
d'Alembert, under the heading Espace (1755), pro-
nounced the debate sterile: “cette question obscure est
inutile à la Géométrie & à la Physique” (Jammer, pp.

As a background for mechanics, Newton's Euclidean
space eventually evolved a variant of non-Euclidean
structure out of itself (Bochner [1966], pp. 192-201,
338). In fact, one hundred years after the Principia,
Louis de Lagrange in his Mécanique analytique (1786),
when analyzing a mechanical system of finitely many
mass points with so-called “constraints,” introduced de
facto a multidimensional space of so-called “gener-
alized coordinates” (or “free parameters”) as a sub-
space of a higher-dimensional space. Implicitly, though
not at all by express assertion or even awareness,
Lagrange endowed this space with the non-Euclidean
Riemannian metric which the imbedding in the higher
dimensional Euclidean space is bound to induce.

Analysts in the nineteenth century knew this part
of the Lagrangian mechanics extremely well. This may
help to explain why, for instance, Carl Jacobi showed
no reaction of surprise at the news of the Bolyai-
Lobachevsky non-Euclidean hyperbolic geometry
around 1830; nor, apparently, did William Rowan
Hamilton ever mention it, or even Bernhard Riemann,
who should have felt “urged” to speak about it in his


great memoir on general “Riemannian” geometry, in
which non-Euclidean spherical geometry is adduced
as a particular case.

14. Space of Perception. As already stated in the
Introduction, the space of John Locke led to the a priori
space of Immanuel Kant, which is a durable creation
indeed. But Kant unnecessarily (Spengler, I, 170-71)
and imprudently fused it with Euclidean space; and
partisans of Kant do not quite know how to disembar-
rass themselves of the fact that mathematics of the
nineteenth century constructed other spaces, and
physics of the twentieth century adopted some of these.

In the first half of the nineteenth century, psychology
became experimental psychology and broke away from
philosophy. This brought into being a space of psy-
chology and physiology to which the Victorian era was
very attentive (H. Weyl, secs. 14 and 18). Thus,
H. L. F. Helmholtz investigated the mathematical struc-
ture of the space of experience under certain assump-
tions of “free mobility,” in his dual capacity of physicist
and physiologist.

An active preoccupation with the space of psychol-
ogy continued into the beginnings of the twentieth
century. Thus, a two-volume treatise of the psycholo-
gist William James had a chapter (20) of 150 pages
on “the perception of space.” But, not long afterwards,
psychology began to lose interest in space as the
Victorian age had known it, and all that it still wanted
to know about space were such un-Kantian topics as:
Visual angle, Monocular Movement, Parallax, Stereo-
scopic vision, etc. A very voluminous Handbook from
around mid-twentieth century (S. S. Stevens) devotes
only 30 pages out of 1435 to the topic of space.

15. The Night-Sky. At night, only our own sun is
turned away from us, but all the other suns (that is
fixed stars) of the universe shine upon us as by daytime.
Yet, the night-sky is dark, meaning that only very little
radiation energy from all the other stars reaches us and
falls on our retinas. It is a leading problem about the
structure of the universe to explain why this is so, that
is, why and how so much radiative energy is “lost”
in space, as it appears to be.

The problem was posed in the first half of the eight-
eenth century; first, somewhat casually, in 1720 by
Edmund Halley, eminent British astronomer, translator
of difficult works from antiquity, and personal friend
to Isaac Newton; and then, quite formally, in 1743,
by the youthful Swiss gentlemen astronomer Jean
Philippe Loys de Chéseaux, owner of an observatory
on his estate (North, p. 18). After that, most unbeliev-
ably, for eighty years nothing happened. Then, in 1823,
the problem was stated anew, quite emphatically, by
the German astronomer H. W. M. Olbers. This stirred
up some notice, but not much, and, unbelievably, the
problem did not move into an area of active attention
for another hundred years. But after Hermann Bondi
had dubbed the problem the “Olbers Paradox” it be-
came generally known, among professionals at any rate
(North, p. 18; Bondi, Ch. 3).

Specifically the problem is as follows. If, à la
Giordano Bruno we make the assumption that the
universe is Euclidean and unchanging; that it houses
infinitely many stars which, on a suitable average, are
uniformly distributed; and that the universe does not
change in time, so that in particular it has “always”
existed in the past; then, by a simple calculation, the
total radiation energy reaching, at any time, a general
point of the universe is not only not small, but is in
fact “infinitely large.” Which means that under these
assumptions the sky would be just as bright by night
as by day. However it is not so, and we thus have the
Olbers paradox.

In the calculation which leads to the paradox most
of the energy comes from distant stars, and the paradox
will be overcome if a suitable change in the above
assumptions will imply that distant stars contribute
little or no radiation (Bondi, Ch. 3). For instance, it
suffices to assume that the universe has not existed
“always,” but, on the contrary, has been created “rela-
tively recently.”

Another change of assumptions, a highly favored
one, is the hypothesis that the universe is expanding.
The expansion produces the red-shift in the traveling
energy waves, that is a decrease of their energy. Very
informally it can be said that a fraction of the radiative
energy is absorbed by the space as “nutriment” for its
growth, and that the fraction is the larger the more
distant the source from which the radiation is emitted.

Finally, the paradox can be overcome by the as-
sumption that the stars, or rather the galaxies, are not
distributed homogeneously, but, on the contrary, are
concentrated in clusters, “hierarachically” so. Thus,
between 1908 and 1922, C. V. I. Charlier advanced
the hypothesis that there are clusters of galaxies
(clusters of the first order), clusters of clusters (clusters
of the second order), clusters of clusters of clusters,
etc. (North, pp. 20-22). This hypothesis is of interest
in our context because it revived a suggestion made
in the eighteenth century by the imaginative mathe-
matician and natural philosopher Johann Heinrich
Lambert in his Kosmologische Briefe... (1761).

This “hierarchic hypothesis” does not have many
adherents, probably because it cannot be easily recon-
ciled with the so-called “Cosmological Principle”
which is in great demand for applications. This princi-
ple was expressly enunciated, and so named, by
Edward Arthur Milne in the 1930's (North, pp.
156-58), and it maintains rather broadly, and not too


specifically, that the total cosmological picture of the
universe, in its meaningful features, is independent of
the vantage point of the observer composing the pic-
ture. The principle is flexible in its specific formula-
tions, and is in this way a great aid in speculative
deductions (Bondi, Ch. 3).

It also ought to be realized that if no radiation from
the stars were lost in space, “... no planet anywhere
in the universe would be cool enough to permit bio-
logical life of any kind” (Coleman, p. 67), as we know
it today.

16. Space of Geometry. The nineteenth century
finally and fully created Euclidean space; and the ven-
erable geometry of Euclid finally acquired a space in
which to house its figures and constructs. As a mathe-
matical object, Euclidean space had already been
clearly present in Descartes and very actively so in
Lagrange. But only the nineteenth century created a
duality between Euclidean space and Euclidean struc-
ture, as a particular case of a general duality of “space
and structure.” In the twentieth century this general
mathematical duality conquered and captured basic
physics from within.

Metaphysically this duality revealed itself with the
advent of the Bolyai-Lobachevsky non-Euclidean ge-
ometry in 1829-30; but mathematically it had mani-
fested itself before (Bochner [1969], Ch. 13), in the
differential and descriptive geometries of Gaspar
Monge and the projective geometry of Jean-Victor
Poncelet; and it had been foreshadowed in the work
of Lagrange.

A memorable event occurred in 1854 when Riemann
farsightedly set forth this duality in his renowned
“Habilitationsschrift” (1868); and as an immediate ap-
plication of it he outlined the so-called Riemannian
Geometry, defining it as a duality between a general
manifold and a so-called Riemannian metric. Leaping
into the twentieth century, Riemann stated, in expres-
sions of his, that a manifold is a space which is locally
Cartesian, so that in local neighborhoods it is deter-
mined by a system of ordinary real numbers as known
from ordinary mensuration.

Riemann's paper was published only in 1868, and
one of the first to plumb its depth was the philosopher
and mathematician William Kingdon Clifford. But the
philosophers J. B. Stallo and Bertrand Russell (see
Bibliography) did not appreciate Riemann's visionary
thrust, and Stallo was almost abusively critical. Con-
temporary mathematicians were telling these philoso-
phers that what distinguished nineteenth-century
mathematics was the creation of projective geometry
in which the numerical and metrical aspects are some-
how derived from the descriptive and qualitative ones.
Riemann, however, anticipating twentieth-century
valuations, did in no wise attempt to hide the numeri-
cal behind a facade of the descriptive, and some
philosophers were puzzled and even dismayed.

In the twentieth century Riemann became philo-
sophically unassailable; and his status became enhanced
when his geometry was elected to be the setting for
the General Theory of Relativity, which filled philoso-
phers with awe. As if to make it quite clear who in
the past had been right and who not, the physicist and
philosopher Percy Williams Bridgman, in an introduc-
tion to a 1960-reissue of the book by Stallo says that
“the discussion of transcendental geometry is definitely
the weakest part of the book” (p. xxiii).

In a sense, the most abstractly conceived general
space in the nineteenth century was the phase space
of statistical mechanics, especially in the general ver-
sion of Josiah Willard Gibbs. In the twentieth century
this space developed into the infinitely dimensional
space of quantum mechanics, as a setting for its physi-
cal states and statistical interpretations. This space is
an outright intentional mathematical construction,
pure and simple. Yet by physicists' constant preoccu-
pation with it, this space is gradually being transformed
from a tool in mathematics to a “reality” in nature,
if by “reality” we understand anything that evokes a
sense of being immediate, familiar, inevitable, and

17. Logical Space and World of Analytical Philoso-
Georg Cantor, the creator of set theory, fruitfully
applied his theory to what he termed “point sets,” that
is general aggregates of points in general spaces, and
he somehow began to view such an aggregate of points,
as a “subspace” of the general space and then as a
space in its own right. In consequence of this, in work-
ing mathematics of the twentieth century, the concepts
of “general set,” “general point set,” and “general
space” have gradually become nearly synonymous.

Thus, in the theory of probability and statistics, a
“probability space” is a general set in which, subject
to appropriate rules, certain subsets have been marked
off as “events.” With each event there is associated
a probability value which is a non-negative real num-
ber between 0 and 1, and the total set has probability
value 1. If, in Aristotelian terminology, a general set
is a probability space not actually but only potentially,
that is, if the set is not given as a probability space
but is only supposed and expected to be one and is
analyzed accordingly, then, relative to such an analysis,
the general set is called a “sample space.”

Such developments were not limited to mathematics
and science. Thus the Tractatus logico-philosophicus
(German edition 1918, first English translation 1922)
of the linguo-philosopher Ludwig Wittgenstein makes
statements about a “logical space” (logischer Raum),


and, by the text of the Tractatus, this space is some
kind of aggregate or congeries of logical entities like
“facts,” “atomic facts,” “states of affairs,” “proposi-
tions,” etc. Some commentators of the Tractatus ascribe
to this space some specific structural features, but even
these features are not very geometrical in a traditional
sense. The Tractatus also refers to a “world” or “uni-
verse” (Welt). This universe has some ontological traits,
and in a sense the logical space is but a background
space to it. Nevertheless, the logical space seems to
be more primary than the world, inasmuch as the
constituents of the world are only some kind of “picto-
rial” representation of the constituents of the logical

The air of indeterminacy and vagueness which ad-
heres to the notions of space and world in the Tractatus
is indicative of the fact that Wittgenstein was never
greatly interested in these notions as such; in his later
work, the linguo-analytical Philosophical Investigations
(1953), these notions hardly occur at all. Also, in the
Tractatus Wittgenstein asserted, quite unnecessarily,
that his logical space is “infinite”; this was simply a
standard philosophers' assertion since Giordano Bruno,
and nothing more. This reduced interest in space was
not an innovation of linguo-philosophers but was a
neo-Hegelian trend in which even “phenomenologists”
like Edmund Husserl shared.

The ontological traits of the world of the Tractatus
could be taken straight out of the universe of
Parmenides, which fused Being with Thought, and
added some dosage of Truth (Aletheia); except that the
Truth in Parmenides, although already “two-valued,”
was still ontological rather than logical. But with re-
gard to the question of the size of the universe,
Parmenides made the splendid assertion, which beauti-
fully conforms with twentieth-century cosmology, that
his “sphere” is both “finite” and “complete” (Bochner,
The Size of the Universe). This assertion of Parmenides
was so subtle that even his leading disciple Melissus
of Samos did not comprehend it at all, and—to
Aristotle's uncontrollable chagrin—made the universe
infinite instead. The great handicap of Parmenides was,
that, as a Greek, he did not have the concept, or even
percept, of a background space in his thinking. There-
fore he could not separate his universe into a “space”
and a “world,” and it is this which makes his fusion
of ontology with physics so puzzlingly “antiquarian.”

In developments since the Renaissance, the first
aspects of a “logical” world are discernible in pro-
nouncements of Leibniz, even in his pronouncements
about a world which purports to be a best possible
one. The innovation of Leibniz was not at all that he
fused physics with metaphysics—this was done by
everybody, even including Kant, his hottest pro
testations notwithstanding—but that he added logic as
well, and that this logic, as a partner of physics and
metaphysics, had an equal standing with them. It is
a fusion of logical ordering with metaphysical being,
and not some specific achievements in logical theory,
which makes Leibniz a precursor of analytical philoso-
phy of the twentieth century, and which makes his
universe of “monads,” however permeated with meta-
physics, congenial and even challenging to many an
analytic “skeptic” of today.

18. The Expanding Universe. This thrilling epithet,
originally l'expansion de l'univers, was created in
1927 by Abbé Georges Édouard Lemaître, and cos-
mology has not been the same since. There had been
models of an expanding universe earlier, namely since
1922 (North, pp. 113ff.), but they were described in
words which had no appeal. However, an “expanding
universe” caught everybody's attention, and Arthur
Stanley Eddington soon began to write a book to fit
this title.

Most cosmologists of today aver that the universe
is expanding, meaning that the nebulae (that is,
galaxies) of the universe are receding from our galaxy,
that is, are moving away from our telescopes in their
lines of sight. By the Cosmological Principle (sec. 15
above), if applicable, it then follows that any nebula
is receding from the others.

The evidence adduced is the so-called red-shift in
the (visible) spectrum of a nebula, that is the displace-
ment of the total set of spectral “lines of absorption”
towards the “red” end of the spectrum and thus away
from its violet end. Also, the redistribution of the
spectral lines is such that it is possible to associate a
well-defined positive real number with each nebula.
On the assumption that the red-shift is caused by the
so-called “Doppler effect”—which asserts that a wave
emanating from a receding source gains in wavelength
in transit—this real number is proportional to the
velocity of the recession.

Working entirely within our galaxy, and using the
Doppler effect in this way, Sir William Huggins had
asserted already in 1868 that the star Sirius is moving
away from the Sun, and he calculated a velocity. The
assertion was later confirmed and the velocity found
tolerably good (Coleman, p. 48). But only the twentieth
century was equipped to apply this spectroscopic pro-
cedure to nebulae as a whole.

The red-shift appears to be the greater the fainter
the nebula, and in 1929 this led the American astrono-
mer Edwin P. Hubble to suggest that the velocity of
recession of a nebula is proportional to the nebula's
distance from our galaxy
(North, p. 145). This is a
renowned law, called “Hubble's Law.” It also permits
a rough estimate of the limit of the observable universe,


and in 1963 the universe was thus estimated to be about
13 billion light years (Coleman, p. 65).

In 1935 the British astronomer E. A. Milne gave an
extremely simple derivation of Hubble's Law, within
a purely kinematic study of the motion of nebulae. He
treated the assemblage of nebulae of the universe
almost as if it were a large assemblage of particles
composing an expanding gas (North, p. 160).

It was this approach which suggested to Milne,
justifiably, to expressly introduce his Cosmological
Principle (sec. 15). Before Milne, the principle used
to be introduced, somewhat haphazardly, as the occa-
sion would arise; but in one form or another it had
appeared in virtually every cosmological theory since
the beginning of the century (North, p. 156).

Milne credited the Cosmological Principle to a re-
mark of Albert Einstein in 1931 that Alle Stellen des
Universums sind gleichwertig
(ibid; “all the stars of the
universe are equivalent”), but philosophically the
principle had a long past. A certain version of it can
be identified in utterances of Nicholas of Cusa
(Jammer, p. 54), and a rather explicit passage is the

The fabric of the world (machina mundi) will quasi have
its center everywhere and its circumference nowhere, be-
cause the circumference and the center are God who is
everywhere and nowhere

(Koyré, p. 17).

This statement refers both to the universe and to
God. But long before Cusa there was a statement about
God only, namely that he is “a sphere of which the
center is everywhere, and the circumference nowhere”
(sphaera infinita cuius centrum est ubique, circum-
ferencia nusquam
). The saying occurs in the Book of
XXIV Philosophers,
which appears to be a pseudo-
Hermetic compilation of the twelfth century; but the
Renaissance philosopher Marsilio Ficino attributes it
to Hermes Trismegistus, which would make the saying
even older, however shadowy the figure of this Hermes
may be (Yates, p. 247).

On the other hand, long after Cusa, Johannes Kepler
was opposed to a Cosmological Principle; or so it
would appear from the following utterance of his:

It will never be the case that the [starry heavens] would
appear to those whom we may imagine observing them from
these stars as they appear to us. From which it follows that
this place, in which we are, will always have a certain
peculiarity that cannot be attributed to any other place
in all this infinity

(Koyré, p. 67).

First impressions notwithstanding, it is possible that
this argument is only a special pleading for something
like the uniqueness of life on earth, which in itself is
not a contravention of the Cosmological Principle.
Even cosmologists today sometimes contemplate that
“our corner of the universe” may have some peculiari-
ties by which to explain certain occurrences that can-
not be explained otherwise. Altogether it appears that
the meaning, history, origin, and past interpretations
of the Cosmological Principle are still to be investi-


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(Chicago, 1964).


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