II
Unlike the Mayas, the ancient Greeks were not
obsessed by the temporal aspect of things. At the dawn
of Greek literature two contrasting points of view are
found in Homer and Hesiod. In the Iliad Olympian
theology and morality are dominated by spatial con-
cepts, the cardinal sin being hubris, that is going be-
yond one's assigned province. Homer was not inter-
ested in the origin of things and had no cosmogony.
On the other hand, Hesiod in his Works and Days gave
an account of the origin of the world, and his poem
can be regarded as a moralistic study based on the time
concept.
Two centuries or more later (sixth century B.C.) the
Ionian pioneers of natural philosophy visualized the
world as a geometrical organism or a live space-filling
substance. Heraclitus, on the other hand, believed the
world to be a soul involved in an endless cycle of death
and rebirth, the very essence of the universe being
transmutation. A similar emphasis on time and soul
characterized the Orphic religion which appears to
have provided the mythical background of Pytha-
goreanism. According to Plutarch, when asked what
Time was, Pythagoras replied that it was the soul, or
procreative element, of the universe. Pythagoras is
a shadowy figure but to him was attributed the
celebrated discovery, following experiments with a
monochord, that the concordant intervals of the musi-
cal scale can be expressed by simple ratios of whole
numbers. This was perhaps the most striking illustrative
example of Pythagoras' doctrine that the nature, or
ultimate principle, of things is not some kind of sub-
stance, as the Ionians thought, but is to be found in
number.
For the early Pythagoreans the concept of number
itself had both spatial and temporal significance. Num-
bers were represented by patterns of the type still
found on dominoes and dice. This led to an elementary
theory of numbers based on geometry. Number, how-
ever, was also regarded from a temporal point of view.
This is evident in the Pythagorean use of the
gnomon.
Originally, this was a time-measuring instrument—a
simple, upright sundial. The term then came to mean
the figure that remains when a square is cut out of
the corner of a larger square with its sides parallel to
the sides of the latter. Eventually it denoted any num-
ber which when added to a figurate number, for exam-
ple a square number, generates the next number of
the same shape. The early Pythagoreans regarded the
generation of numbers as an actual physical operation
in space and time, beginning with the initial unit or
monad. In general, they failed to make any clear dis-
tinction between the abstract and the concrete and
between logical and chronological priority.
These distinctions were clearly drawn by Parmen-
ides, the founding father of deductive argument and
logical analysis. In his Way of Truth he criticized
current cosmogonies for their common assumption that
the universe began at some moment of time. “And what
need,” he asked, “could have stirred it up, starting from
nothing, to be born later rather than sooner?” This
question was answered by Plato who claimed that time
is coexistent with the universe. But he was deeply
impressed by Parmenides' acute criticism of the ideas
of becoming and perishing and by his conclusion that
time does not pertain to anything that is truly real.
The difficulties involved in producing a logically
satisfactory theory of time and its measurement were
emphasized by Parmenides' pupil Zeno of Elea in his
famous paradoxes. For, although these paradoxes were
primarily concerned with the problem of motion, they
raised difficulties both for the idea of time as continu-
ous or infinitely divisible and for the idea of temporal
atomicity. Unlike the Pythagoreans, who tended to
identify the chronological with the logical, Parmenides
and Zeno argued that they are incompatible.
The influence of Parmenides and Zeno on Plato is
evident in the different treatment of space and time
in Plato's cosmological dialogue the Timaeus. Space
exists in its own right as a given frame for the visible
order of things, whereas time is merely a feature of
that order based on an ideal timeless archetype or realm
of static geometrical shapes (Eternity) of which it is
the “moving image,” being governed by a regular
numerical sequence made manifest by the motions of
the heavenly bodies. Plato's intimate association of
time with the universe led him to regard time as being
actually produced by the revolutions of the celestial
sphere.
This conclusion was not accepted by Aristotle who
rejected the idea that time can be identified with any
form of motion. For, he argued, motion can be uniform
or nonuniform and these terms are themselves defined
by time, whereas time cannot be defined by itself.
Nevertheless, although time is not identical with mo-
tion, it seemed to him to be dependent on motion.
Possibly influenced by the Pythagoreans, he argued that
time is a kind of number, being the numerable aspect
of motion. Time is therefore a numbering process
founded on our perception of “before” and “after” in
motion: “Time is the number of motion with respect
to earlier and later” (Physica, ed. W. D. Ross, Vol II,
Book IV, 219a). Aristotle regarded time and motion
as reciprocal. “Not only do we measure the movement
by the time, but also the time by the movement, be-
cause they define each other. The time marks the
movement, since it is number; and the movement the
time” (ibid.). Aristotle recognized that motion can
cease whereas time cannot, but there is one motion
that continues unceasingly, namely that of the heavens.
Clearly, although he did not agree with Plato, he too
was profoundly influenced by the cosmological view
of time. Moreover, although he began by rejecting any
association between time and a particular motion in
favor of one between time and motion in general, he
came to the conclusion that time is closely associated
with the circular motion of the heavens, which he
regarded as the perfect example of uniform motion.
For Aristotle the primary form of motion was uni-
form motion in a circle because it could continue
indefinitely, whereas uniform rectilinear motion could
not. Any straight line necessarily had finite end points,
since he did not have the modern mathematical con-
cept of the infinitely extended straight line. For Aris-
totle, therefore, time was intimately connected with
uniform circular motion.
Belief in the cyclic nature of time was widespread
in antiquity, since most ancient peoples tended to
regard time as essentially periodic. Long before Aris-
totle, this idea led the Greeks to formulate the concept
of the Great Year, and this is presumably what the
Pythagorean Archytas of Tarentum had in mind when
he said that time is the number of a certain movement
and is the interval appropriate to the nature of the
universe—a definition that may well have influenced
Aristotle. There were, however, two distinct inter-
pretations of the Great Year. On the one hand it was
simply the period required for the Sun, Moon, and
planets to attain the same positions in relation to each
other as they had at a given time. This appears to have
been the sense in which Plato used the idea in the
Timaeus. On the other hand, for Heraclitus it signified
the period of duration of the world from its formation
to its destruction and rebirth. Whereas Plato seems to
have refrained from giving any estimate of the length
of the Great Year, Heraclitus, with no particular astro-
nomical interpretation in mind, gave 10,800 years as
its duration. He may have arrived at this figure by
taking a generation of 30 years as a day and multi-
plying by 360, the (approximate) number of days in
the year.
The two interpretations of the Great Year were
combined by the Stoics who believed that, when the
heavenly bodies return at fixed intervals of time to the
same relative positions as they had at the beginning
of the world, everything would be destroyed by fire.
Then all would be restored anew just as it was before
and the entire cycle would be renewed in every detail.
