DIVISIBILITY OF MOTION
INTRODUCTION
The subject or object of the science
of nature may be best designated as mobile or changeable being, with motion or
mobile taken in a sense wide enough to include any physical change. “In any
case, to say that things change is to say that simple bodies are constantly
being transformed into things through their internal principle of motion and by
the motion of other things.”[1] Therefore
mobile being is the most fundamental stuff or object of our science because it
is that with which our science is concerned primarily. Nature as we have seen
so far is the principle of motion and rest in that to which it belongs per se
(essentially and primarily) and not per accident. Therefore, we can say that
motion is involved in the very definition of nature; whoever ignores motion,
ignores nature itself.
However, motion according to
Aristotle is defined as the act of being (what exist) in potency insofar as it
exists in potency.
The
aim of this paper is to discuss the
divisibility of motion, i.e. how motion can be divided. But before we proceed,
we would like to clarify some terms which we shall discuss in this paper. The
terms are: Continuous, Contiguous, Extremity, and Intermediate. The knowledge
of these terms will help us to grasp the insight during the course of explanation.
CLARIFICATION
OF TERMS
Continuous:
it is that which is divided “ad infinitum”
(e.g. a straight line).
Contiguous:
it is that which has intermediate holding
them together because they can’t meet each other (e.g. a full water pipe which
has a connector. That connector has made it contiguous because it is not a
continuum).
Extremity: it
is the part of the contiguous. It stands on its own (e.g. Heaven and Earth).
Intermediate:
it is that which holds the contiguous together (e.g. the connector we see on
water pipes).
THE
DIVISIBILITY OF THINGS IN A CONTINUUM:
Things are said to be continuous when their extremities
are one (e.g. a line), if this is so, it is impossible for it to be composed of
indivisibles (e.g. points in a line). Therefore, if something is composed of
parts, the extremities must either be one i.e. continuous, or they must be
together i.e. contiguous. But the extremities of points cannot be together or
be one, because an extremity is that which is spoken of in relation to a part,
whereas an indivisible is not related to a part. Therefore, things are said to
be in contact when their extremities are together. Besides, if a continuum is
composed of points, they cannot touch, since everything that touches another
does so by a part touching the other. By these, Aristotle tries to say that a
continuum is a thing in motion which begins from a point to a point. For anything
in motion, as it moves, there are points. For example: if I move from the
library to the class, it is continuum. Now, as I move my legs from one point to
the other, these points are the divisibility in continuum.
Thus
between two points there must be a line, and between two “nows” there must be
time. This is because, if two points exist, they must differ in position;
otherwise they would not be two but one. No other intermediate is possible except
a line between two points and time between two “nows”, because if that which is
divisible becomes indivisible, then it will bring us back to the question-how a
divisible can be composed solely of indivisibles. But intermediate is always
divisible into further divisible, therefore it is a continuum.
THE
DIVISIBILITY IN MOTION:
Magnitude and Motion are correlative,
i.e. they work together. So that if magnitude is divisible then that will be
true for motion but if not then it also follows for motion.
To
illustrate this, take a magnitude (or road) that consists of points A, B and C.
when the mobile or thing in motion reaches point B from A it must be in motion
or must have completed its motion. If it is till in motion, then B must be
divisible; otherwise the motion would be complete at that point and the sum of
the motion from A to C would not be divisible motion but discrete moments.
Therefore discrete moments are the points of rest of a thing in motion.
Then
if a thing in motion consists of discrete moments, it would follow that
something has completed a motion without having been in motion. It also follows
that at each point on the route the mobile or thing in motion would be at rest,
while it was supposed to be in motion. Also the segment of motion corresponding
to each of the points on the route would also be at rest, and thus the whole
motion would be composed of non-motions.
THE
DIVISIBILITY IN TIME:
Time is the numbering of motion. Time
is divisible just as the magnitude is traversed in time. Thus a mobile going at
an equal speed covers half the distance in half the time of its journey.
Therefore the two are correlatively divisible into smaller and smaller segment.
The
same can be seen in mobiles of different velocity (for example car race) when
the race begins the cars takeoff at the same time. But as they move on you
discover that the cars are in different points and this points are called
intermediate points. The fastest of them all gets to the finish line in less
time before other; at this time and magnitude can be divided. As we increase
velocity, we divide time, since the race is finished in shorter and shorter
times. But if we decrease the velocity and stop the mobile or the car after the
same length of time, we divide the magnitude, since less and less distance is
traversed.
Although
there are physical limits to division of a magnitude, just as there are
physical limits to the size of a natural thing, so there are natural limits to
velocity, but mathematically both magnitude and time are infinitely divisible,
and the same magnitude is traversable in ever faster speeds.
THE
DIVISIBILTY IN A CONTINUUM
Time corresponds to magnitude because
magnitude is a distance, and if magnitude could be infinite distance has no
beginning and end point such as a line would be. Some Philosophers said time
cannot be static and time at a finite velocity will only transverse a finite
magnitude (distance). If magnitude is infinitely divisible, the same is
applicable to time. The argument can be moved backward or reversed by saying,
if time is infinite in length, so must distance (magnitude) be. Because any
motion, however slow will eventually pass across a finite magnitude and if time
is infinitely divisible, so must magnitude be. There is also an argument raised
that why no continuous magnitude is indivisible which can be shown by a kind of
reasoning that constructs by positing two mobiles of different velocities-the
speed of something in a given direction, when the faster one crosses the first
supposedly indivisible part, the slower one will have crossed only part or half
way which is proving that the supposedly indivisible segment is divisible.
CONCEPTUAL
ANALYSIS OF THE DIVISIBILITY OF MOTION AND THE
INDIVISIBILITY
OF “NOW”.
The
term “now” is often used for a period, like “today”, but “now” is present in
every time without vagueness. “Now” is a
limit between the past and the future, and such must be one. We can also say that “now” stands in between
or in the middle which holds the past and the future. The “now” stopping the past and beginning of
future cannot be two “now” coming together, otherwise time would be of an
aggregate of indivisible “now”, which cannot be possible since we say time is
continuous and there cannot be a stop between two “nows” because there would be
a period of time or at least another “now”, in other case the “now” would be
divisible.[2] If the “now” is to be divisible, it will have
to involve some of the past or future in which some of past will be in the
future and some future in the past, therefore the “now” which stops the past
and begins the future must be one “now”.
There is no motion in a “now” since motion is continuous while the “now”
is not and this can be seen when positing two mobiles going at different
velocities whereby one moves faster than the other and reaches the same
distance in less than a “now” which makes it divisible but which is not the
case. There can be no rest in a “now”
because rest is a privation of motion and there is no privation where there is
no aptitude to have the object of privation.
Therefore, if there can be no motion in the “now” there can be no rest
either. Rest is sometimes understood as
something continuous like motion. Motion
occurs in time and must be intermediate between two extremes with part of the
mobiles towards one extremes and part of it towards the other. This is easily discovered or seen between the
three (3) species of motion. For example, the alteration of electricity in a
wire which is instantaneous. Here we can’t really say at what point the motion
began or ended.
THE
DIVISIONS OF MOTION
According to Aristotle, “motion can be divided
into two ways: according to its various
species or kinds, such as qualitative change and local motion and according to
its quantitative parts, such as halves, quarters, eighths, and so on.”[3]
E.g the motion of the part of the mobile since both time and the mobile is
continuous and divisible. Therefore the
entire motion belongs to the entire mobile just as parts of it belong to the
parts of the mobile. Time is the count or measure of motion since there is less
motion in less time. Time is a
consequent of beginning, it is limited and created time can also be said as a
dimension in which events can be ordered from the past through the present into
the future and also the measure of durations of events and the intervals
between them. An operational definition of
time wherein one says that observing a certain number of repetitions of one or
another standard cyclical event (such as the passage of free-swinging pendulum) constitutes one
standard unit such as the second, is highly useful in the conduct of both advanced
experiments and everyday affairs of life.
Furthermore, there are five things related to motion which are similarly
divided such as (1) time (2) motion which involves the notions of potency and
act (3) the very act of being moved (4) the mobile which is being moved and (5)
the species of motion which is the place, quantity and quality. Place which is local motion, quantity which
is growth and quality which is alteration.
The divisibility of the mobile is the basis of the divisibility of all
others in motion.
THE INDIVISIBILITY OF THE
BEGINNING AND END OF MOTION
It is evident that substantial
changes is said to occur at the instant moment when something or a thing ceases
to be what it was and takes another form (e.g. the burning of paper into ashes).
But the end of any motion is that similarly and indivisible moment, when a
thing can first be said to “have changed”. It is the period a thing changes its
state instantly where the time between the old and new state cannot be
measured. Thus the beginning of any motion is likewise an indivisible moment
since we cannot assign a first time when a thing began moving. In other words,
we can only talk about the end of a particular motion but cannot determine the
beginning of a motion because the starting point is undetermined.
EVERY “BEING MOVED” IS PRECEDED BY A
“HAVING BEING MOVED” AND EVERY “HAVING BEING MOVED” BY A “BEING MOVED”.
Since
motion is continuous infinitely divisible, while a thing is being moved, a part
of the motion can be designated where the thing can be said to “have been
moved” and in which motion ceases e.g. when an object is moving from point A to
point B it could be determined that the object began moving at point A because
we have already marked that point but is not the actual starting point of that
motion since it had already being moved. The same thing applies when we
conclude potentially that the motion ends at point B because a particular
target has been reached but does not imply that the motion actually ends there
since motion is infinite. For if the particular points (A-B) is not marked it
could be the “end” and “beginning” of another motion. This point which can be
designated marks a point of the motion which has been concluded. Such a point
is only potentially a part of motion, since the motion does not end there. It
becomes actual when the motion comes to a halt at a particular point. Any of
such point which is indicated as marking” the end point completed thus far” is
preceded by another segment of the motion, or a “being moved”; if ignored would
be the starting point of the motion. These two statements can be proved by
displaying how two mobiles of different speeds mark their various segments of
motion. Take for instance, a bicycle and a motorcycle which is of different
speeds. One would discover that at a particular point where the bicycle is
(“end point”), the motorcycle had already passed (“starting point”) and where
the motorcycle “ends” another motion will begin from. But this principle does
not apply to generation and corruption, since they are instantaneous and not
continuous changes. However, generation and corruption are the term of a
process of alteration and this alteration named after its term is divisible. As
such, “dying” is a motion that can be divided into stages.
THE SIMILARITY IN THE INFINITUDE AND
FINITUDE OF MAGNITUDE, MOTION, TIME AND THE MOBILE.
If we affirm that magnitude is finite, then
time cannot be infinite, and if time is finite magnitude cannot be infinite. It
means that time and magnitude is one. Magnitude is encompassed in time. In the
same vein, if time can be measured or calculated, the magnitude can be
measured, and if time cannot be
measured, magnitude too cannot be measured and vice versa. Similarly, it can be
proven that a mobile cannot be infinite if either the magnitude or the time is
finite. Time and magnitude determines the finite and infinity of a mobile. This
can also be applied to motion.
THE
DIVISION OF REST
At this point it is clear that
“coming to rest” is part of motion and it also takes place in time and
magnitude since the time and magnitude of rest can be determined or measured.
Just as we know that no part of motion can be said to be first, so also no part
of rest can be said to be first. This is as a result of the fact that the
actual rest point cannot be known. For when an object is at rest, it is so in
potency and not in actuality. In the same way, rest being continuous in time
can have no first part because each part is divisible by time and magnitude. We
say that something is at rest if throughout a definite period of time i.e. from
one “now” to another it is one and same state, for example to be in a place. As
such, nothing can be said to be at rest since even at “rest” it is still moving
in time and its state is never the same.
REFUTATIONS OF ZENO’S PARADOXES WHICH PURPORT TO SHOW
THE ABSURDITY OF MOTION'S
EXISTENCE.
Zeno's paradoxes have puzzled, challenged,
influenced, inspired, infuriated, and amused philosophers, mathematicians, and
physicists for over two millennia. The most famous are the so-called arguments
against motion described by Aristotle in his Physics.
Zeno the
Greek philosopher was a student of Parmenides and member of the Eleatic
philosophical tradition or school in support of parmenides’s doctrine that “all
is one” and that contrary to the evidence of one’s senses, the belief in
plurality and change is mistaken, and in particular that motion is nothing but
an illusion. Three of the strongest and most famous paradoxes are that of Achilles and the tortoise, the Dichotomy argument, and that of an arrow in flight.
I. Zeno’s Dichotomy (literally, “cutting in two”)
Paradox.
A.
Suppose (as we usually do) that a continuously moving runner in the stadium
traverses a finite distance S in a finite time T.
B. In
order to traverse finite distance S, a runner must first traverse S/2 (the
first half of S), then S/4 (the first half of what remains of S), then S/8 (the
first of half of what then remains of S), etc. ad infinitum.
C. But,
then, it follows that the runner “actually will have traversed an infinite
number of halves in a finite time” [CONTRADICTION]
Zeno infers from what he takes to be
this contradiction (by reductio or indirect argument) that the runner never
traverses the distance. This conclusion generalizes to the proposition that
local motion, that is, motion with respect to place, never occurs. Most of us
believe that our everyday experience of the world (through our senses) assures
us that motion does occur. It seems likely that Zeno–trusting what he takes to
be the conclusion of reason rather than the senses–simply responds as follows:
“so much the worse for the senses as sources of knowledge of the world around
us.”
Aristotle’s Resolution of Zeno’s Dichotomy
Paradox:
1. Each
‘sub-distance’ of total distance S traversed will be traversed in a
corresponding ‘sub-time’ of the finite time T which it takes to traverse S. If
the runner is moving at a constant speed, these sub-times will become
proportionally shorter as the sub-distances become shorter.
2. The
runner does not actually traverse a ‘complete’, infinitely large collection of
sub-distances.
3. So
step C in the argument above does not follow from steps A and B. The runner
will only have ‘potentially’ traversed an infinite number of sub-distances of
S, and there is nothing contradictory about that.
II. Zeno’s Arrow Paradox.
P1: A continuously
moving arrow traverses a certain finite distance S in some finite interval
(‘lapse’ or ‘duration’) T of time.
P2:
Whatever is occupying a space equal to its own dimensions is–while it is
occupying such a space–at rest.
P3: At
each temporal instant t during the temporal interval T of (supposed) motion, a
moving arrow is occupying a space equal to its own dimensions.
Step 4.
Therefore, at each temporal instant t during an interval T of (supposed)
motion, the arrow is at rest.
P4: If
anything is at rest at each instant t during an interval T of time, then that
thing is at rest throughout that interval T of time.
Step 6.
Therefore, the arrow moving throughout an interval T of time is at rest
throughout that interval T. [CONTRADICTION]
From this contradiction, one may
by reductio or in indirect argument deny at least one of the premises. Zeno, in
effect, chooses to deny premise P1–that is, he denies the existence of motion.
Other Resolutions of Zeno’s Arrow Paradox
1.
Denial of P2
A. Aristotle:
Nothing is properly said to be either moving or at rest with respect to
instants of time; both motion and rest presuppose a lapse or duration of time.
B. Some
contemporary accounts: Something that is said to be moving throughout an
interval of time is correctly said to be moving, in one sense of ‘moving’, with
respect to each instant contained in that interval.
But the
thing traverses no distance with respect to any such instant. Cf. the concept
of ‘instantaneous velocity’–roughly, the limit, with respect to a given instant
of time t, of a sequence of velocities (directed spatial distances traversed by
a moving body divided by the times it takes the body to traverse those
distances) as a sequence of ever-shorter temporal periods, converging to ‘null
time-lapse at t’, are considered.
2.
Denial of P4
Bertrand
Russell. The ‘At-At Conception of Motion: Motion is nothing more than the fact
that a body is in one position at a given temporal instant and at another
position at a later instant–and at intermediate positions at intermediate
temporal instants.
(Another
general constraint is usually assumed: the function from temporal instants to
positions is continuous at each argument t).
III. Zeno’s Achilles and the Tortoise paradox.
The [second] argument
was called “Achilles,” accordingly, from the fact that Achilles was taken [as a
character] in it, and the argument says that it is impossible for him to
overtake the tortoise when pursuing it. For in fact it is necessary that what
is to overtake [something], before overtaking [it], first reach the limit from
which what is fleeing set forth. In [the time in] which what is pursuing
arrives at this, what is fleeing will advance a certain interval, even if it is
less than that which what is pursuing advanced and in the time again in which
what is pursuing will traverse this [interval] which what is fleeing advanced,
in this time again what is fleeing will traverse some amount and thus in every
time in which what is pursuing will traverse the [interval] which what is
fleeing, being slower, has already advanced, what is fleeing will also advance
some amount.
This paradox turns on much the same considerations as the last. Imagine Achilles
chasing a tortoise, and suppose that Achilles is running at 1 m/s,
that the tortoise is crawling at 0.1 m/sand that the tortoise
starts out 0.9 m ahead of Achilles. On the face of it Achilles
should catch the tortoise after 1s, at a distance of 1m from
where he starts (and so 0.1m from where the Tortoise starts). We
could break Achilles' motion up as we did Atalanta's, into halves, or we could
do it as follows: before Achilles can catch the tortoise he must reach the
point where the tortoise started. But in the time he takes to do this the
tortoise crawls a little further forward. So next Achilles must reach this new
point. But in the time it takes Achilles to achieve this, the tortoise crawls
forward a tiny bit further. And so on to infinity: every time that Achilles
reaches the place where the tortoise was, the tortoise has had enough time to
get a little bit further, and so Achilles has another run to make, and so
Achilles has in infinite number of finite catch-ups to do before he can catch
the tortoise, and so, Zeno concludes, he never catches the tortoise.
One aspect of the paradox is thus that Achilles must traverse the
following infinite series of distances before he catches the tortoise: first
0.9m, then an additional 0.09m, then 0.009m, These are the
series of distances ahead that the tortoise reaches at the start of each of
Achilles' catch-ups. Looked at this way the puzzle is identical to the
Dichotomy, for it is just to say that ‘that which is in locomotion must arrive
[nine tenths of the way] before it arrives at the goal’. And so everything we
said above applies here too. But what the paradox in this form brings out most
vividly is the problem of completing a series of actions that has no final
member—in this case the infinite series of catch-ups before Achilles reaches
the tortoise.
For Aristotle these infinite series of catch-ups are not actual, they
are only potential.
THE INCAPABILITY OF
INDIVISIBLES BEING PER SE MOTION.
For Democritus, indivisible atoms are per se
mobile. But Aristotle shows that there is neither motion nor rest in an
indivisible point of time and for him, that which is indivisible cannot be
moved. He clearly shows that the indivisible point of time is the “now”. In the
“now”, nothing is moved or is at rest. It must be first realized that “now” is
sometimes used, not in its proper meaning, but in an extended meaning. For
example, we say that something which is done in the whole of the present day is
done now. Now, the whole present day is not called present in the proper sense,
but in an extended sense. For it is clear that part of the present day has
passed and another part is yet to come. That which is past or future is not
now. Thus it is clear that the whole present day is not a “now” primarily and
per se, but only in regard to part of itself and the same is true of an hour
and of any other time. Hence, he says that, that which is called a “now”
primarily and per se, and not in extended sense is necessarily indivisible.
Furthermore, this “now” is necessarily in every time.
It is clear that for every finite
continuum there is some extremity outside of which there is nothing of that of
which it is the extremity. For example, there is no line outside of the point
which terminates the line. And past time is a continuum which is terminated at
the present. Also the changes that occur in between generation and corruption
are instantaneous, so we can’t say the particular time motion occurred because
it is not divisible.
CHANGE IS FINITE:
No
change is Infinite. Every mutation is from something to something. But when a
thing is in the terminus to which it is being changed, it is not being changed
anymore, but has been changed. When a
thing in respect to its whole and all of its parts is still in the terminus
from which it might change, it is not then being changed. For that which is the
same in itself and in all of its parts is not being moved, but rather is at
rest. Aristotle adds “all of its parts” because when a thing begins to be
changed, it leaves the place which it previously occupied, not totally, but
part by part.
Nor can
it be said that, while a thing is being moved, it is in both of the termini in
respect to both its whole and its parts. For then it will be in two places
simultaneously. Nor can it be said that it is in neither of the termini. What
is being referred to is the proximate terminus to which it is changed, and not
of the ultimate terminus. These changes occur between contradictory terminals,
which are affirmation or negation of something as in the case of generation and
corruption which are instantaneous and do not go on for anytime. There is also
limit to every alteration, for changes that occur between contrary terminals
have a maximum and minimum term according to the nature of the subject and the
species of change. This is similar to growth and decrease, because each nature
has a size that befits it. For example, the nature of a man and another for a
horse. Therefore, these changes cannot be infinite, they are rather finite.
QUANTUM THEORY
Quantum theory evolved as a new branch of theoretical
physics during the first few decades of the 20th century in an endeavour to
understand the fundamental properties of matter. It began with the study of the
interactions of matter and radiation. Certain radiation effects could neither
be explained by classical mechanics, nor by the theory of electromagnetism. In
particular, physicists were puzzled by the nature of light. Peculiar lines in
the spectrum of sunlight had been discovered earlier by Joseph von Fraunhofer
(1787-1826). These spectral lines were then systematically catalogued for
various substances, yet nobody could explain why the spectral lines are there
and why they would differ for each substance. It took about one hundred years,
until a plausible explanation was supplied by quantum theory.
Quantum
Theory is about the Nature of Matter.
In contrast to Einstein's
Relativity, which is about the largest things in the universe, quantum theory
deals with the tiniest things we know, the particles that atoms are made of,
which we call "subatomic" particles. In contrast to Relativity,
quantum theory was not the work of one individual, but the collaborative effort
of some of the most brilliant physicists of the 20th century, among them Niels
Bohr, Erwin Schrödinger, Wolfgang Pauli, and Max Born. Two names clearly stand
out: Max Planck (1858-1947) and Werner Heisenberg (1901-1976). Planck is
recognised as the originator of the quantum theory, while Heisenberg formulated
one of the most eminent laws of quantum theory, the Uncertainty Principle,
which is occasionally also referred to as the principle of indeterminacy.
Planck's
Constant: Energy is not Continuous.
Around 1900, Max Planck from the
University of Kiel concerned himself with observations of the radiation of
heated materials. He attempted to draw conclusions from the radiation to the
radiating atom. On basis of empirical data, he developed a new formula which
later showed remarkable agreement with accurate measurements of the spectrum of
heat radiation. The result of this formula was so that energy is always emitted
or absorbed in discrete units, which he called quanta. Planck developed his
quantum theory further and derived a universal constant, which came to be known
as Planck's constant. The resulting law states that the energy of each quantum
is equal to the frequency of the radiation multiplied by the universal
constant. The discovery of quanta revolutionised physics, because it contradicted
conventional ideas about the nature of radiation and energy. According to
Vincent smith, “this would seem to contradict our previous conclusion that
motion is gradual and successive and that magnitude, no matter how small, is
always continuous and divisible but when there is a conflict between some well-
established principle based on general experience and hypothetical conclusions
that are more or less dialectical, the more certain knowledge should always be
accepted in preference to the less certain.”[4]
CONCLUSION
In
an attempt to explain the material definition of the continuum, it was shown
that continuous quantity, e.g., a line, is always divisible into further
divisible parts. In explaining the formal definition of the continuum, it was
shown that the parts of a line, for example, are finite in number at any one
time and that there are indivisible divisors between the parts, e.g., the
points on a line. Motion is continuous because magnitude is continuous, and
time is continuous because motion is continuous. There is no smallest part of motion
and no smallest part of time; for any part, a smaller part can always be found.
There is no first moment or stage of any motion and no first period of time
during which the motion occurs. In any motion, no matter when we consider the
motion, a prior completion of motion can be identified; and in any completion
of motion a prior stage of motion can be found.
Zeno
erred by thinking that the parts of a continuum are infinite in act whereas
they are infinite only in potency. The view of quantum theory that motion is
discontinuous is hypothetical; it is dialectical. It cannot contradict the
conclusions based on more general experience that magnitude is continuous, that
matter cannot be resolved into indivisible components, that motion cannot be
resolved into a series of moments, and that time cannot be resolved into a
series of instants. Finally, in this paper, the divisibility of motion has been
considered and the quantitative parts of motion discussed.
[3] Vincent E. Smith, The General Science of Nature (Milwaukee:
The Bruce Publishing company, 1958), p. 327.
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