The continuum
Spacial extension, motion, and time are often thought of as continua—as
wholes made up of a series of smaller parts. Aristotle develops a subtle
analysis of the nature of such continuous quantities. Two entities are
continuous, he says, when there is only a single common boundary between them.
On the basis of this definition, he seeks to show that a continuum cannot be
composed of indivisible atoms. A line, for example, cannot be composed of
points that lack magnitude. Since a point has no parts, it cannot have a
boundary distinct from itself; two points, therefore, cannot be either adjacent
or continuous. Between any two points on a continuous line there will always be
other points on the same line.
Similar reasoning, Aristotle says, applies to time and to motion. Time
cannot be composed of indivisible moments, because between any two moments
there is always a period of time. Likewise, an atom of motion would in fact
have to be an atom of rest. Moments or points that were indivisible would lack
magnitude, and zero magnitude, however often repeated, can never add up to any
magnitude.
Any magnitude, then, is infinitely divisible. But this means “unendingly
divisible,” not “divisible into infinitely many parts.” However often a
magnitude has been divided, it can always be divided further. It is infinitely
divisible in the sense that there is no end to its divisibility. The continuum
does not have an infinite number of parts; indeed, Aristotle regarded the idea
of an actually infinite number as incoherent. The infinite, he says, has only a
“potential” existence.
Motion
Motion (kinesis) was for Aristotle a broad term, encompassing changes in
several different categories. A paradigm of his theory of motion, which appeals
to the key notions of actuality and potentiality, is local motion, or movement
from place to place. If a body X is to move from point A to point B, it must be
able to do so: when it is at A it is only potentially at B. When this
potentiality has been realized, then X is at B. But it is then at rest and not
in motion. So motion from A to B is not simply the actualization of a potential
at A for being at B. Is it then a partial actualization of that potentiality?
That will not do either, because a body stationary at the midpoint between A
and B might be said to have partially actualized that potentiality. One must
say that motion is an actualization of a potentiality that is still being
actualized. In the Physics Aristotle accordingly defines motion as “the
actuality of what is in potentiality, insofar as it is in potentiality.”
Motion is a continuum: a mere series of positions between A and B is not
a motion from A to B. If X is to move from A to B, however, it must pass
through any intermediate point between A and B. But passing through a point is
not the same as being located at that point. Aristotle argues that whatever is
in motion has already been in motion. If X, traveling from A to B, passes
through the intermediate point K, it must have already passed through an
earlier point J, intermediate between A and K. But however short the distance
between A and J, that too is divisible, and so on ad infinitum. At any point at
which X is moving, therefore, there will be an earlier point at which it was
already moving. It follows that there is no such thing as a first instant of
motion.
Time
For Aristotle, extension, motion, and time are three fundamental
continua in an intimate and ordered relation to each other. Local motion
derives its continuity from the continuity of extension, and time derives its
continuity from the continuity of motion. Time, Aristotle says, is the number
of motion with respect to before and after. Where there is no motion, there is
no time. This does not imply that time is identical with motion: motions are
motions of particular things, and different kinds of changes are motions of
different kinds, but time is universal and uniform. Motions, again, may be
faster or slower; not so time. Indeed, it is by the time they take that the
speed of motions is determined. Nonetheless, Aristotle says, “we perceive
motion and time together.” One observes how much time has passed by observing
the process of some change. In particular, for Aristotle, the days, months, and
years are measured by observing the Sun, the Moon, and the stars upon their
celestial travels.
The part of a journey that is nearer its starting point comes before the
part that is nearer its end. The spatial relation of nearer and farther
underpins the relation of before and after in motion, and the relation of
before and after in motion underpins the relation of earlier and later in time.
Thus, on Aristotle’s view, temporal order is ultimately derived from the
spatial ordering of stretches of motion.
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