can be described in different ways. So it is with the mover and the
moved.
This view has a dialectical difficulty. Perhaps it is necessary that
the actuality of the agent and that of the patient should not be the
same. The one is 'agency' and the other 'patiency'; and the outcome
and completion of the one is an 'action', that of the other a
'passion'. Since then they are both motions, we may ask: in what are
they, if they are different? Either (a) both are in what is acted on
and moved, or (b) the agency is in the agent and the patiency in the
patient. (If we ought to call the latter also 'agency', the word would
be used in two senses.)
Now, in alternative (b), the motion will be in the mover, for the
same statement will hold of 'mover' and 'moved'. Hence either every
mover will be moved, or, though having motion, it will not be moved.
If on the other hand (a) both are in what is moved and acted on-both
the agency and the patiency (e.g. both teaching and learning, though
they are two, in the learner), then, first, the actuality of each will
not be present in each, and, a second absurdity, a thing will have two
motions at the same time. How will there be two alterations of quality
in one subject towards one definite quality? The thing is
impossible: the actualization will be one.
But (some one will say) it is contrary to reason to suppose that
there should be one identical actualization of two things which are
different in kind. Yet there will be, if teaching and learning are the
same, and agency and patiency. To teach will be the same as to
learn, and to act the same as to be acted on-the teacher will
necessarily be learning everything that he teaches, and the agent will
be acted on. One may reply:
(1) It is not absurd that the actualization of one thing should be
in another. Teaching is the activity of a person who can teach, yet
the operation is performed on some patient-it is not cut adrift from a
subject, but is of A on B.
(2) There is nothing to prevent two things having one and the same
actualization, provided the actualizations are not described in the
same way, but are related as what can act to what is acting.
(3) Nor is it necessary that the teacher should learn, even if to
act and to be acted on are one and the same, provided they are not the
same in definition (as 'raiment' and 'dress'), but are the same merely
in the sense in which the road from Thebes to Athens and the road from
Athens to Thebes are the same, as has been explained above. For it
is not things which are in a way the same that have all their
attributes the same, but only such as have the same definition. But
indeed it by no means follows from the fact that teaching is the
same as learning, that to learn is the same as to teach, any more than
it follows from the fact that there is one distance between two things
which are at a distance from each other, that the two vectors AB and
BA, are one and the same. To generalize, teaching is not the same as
learning, or agency as patiency, in the full sense, though they belong
to the same subject, the motion; for the 'actualization of X in Y' and
the 'actualization of Y through the action of X' differ in definition.
What then Motion is, has been stated both generally and
particularly. It is not difficult to see how each of its types will be
defined-alteration is the fulfillment of the alterable qua alterable
(or, more scientifically, the fulfilment of what can act and what
can be acted on, as such)-generally and again in each particular case,
building, healing, c. A similar definition will apply to each of
the other kinds of motion.
4
The science of nature is concerned with spatial magnitudes and
motion and time, and each of these at least is necessarily infinite or
finite, even if some things dealt with by the science are not, e.g.
a quality or a point-it is not necessary perhaps that such things
should be put under either head. Hence it is incumbent on the person
who specializes in physics to discuss the infinite and to inquire
The appropriateness to the science of this problem is clearly
indicated. All who have touched on this kind of science in a way worth
considering have formulated views about the infinite, and indeed, to a
man, make it a principle of things.
(1) Some, as the Pythagoreans and Plato, make the infinite a
principle in the sense of a self-subsistent substance, and not as a
mere attribute of some other thing. Only the Pythagoreans place the
infinite among the objects of sense (they do not regard number as
separable from these), and assert that what is outside the heaven is
infinite. Plato, on the other hand, holds that there is no body
outside (the Forms are not outside because they are nowhere),yet
that the infinite is present not only in the objects of sense but in
the Forms also.
Further, the Pythagoreans identify the infinite with the even. For
this, they say, when it is cut off and shut in by the odd, provides
things with the element of infinity. An indication of this is what
happens with numbers. If the gnomons are placed round the one, and
without the one, in the one construction the figure that results is
always different, in the other it is always the same. But Plato has
two infinites, the Great and the Small.
The physicists, on the other hand, all of them, always regard the
infinite as an attribute of a substance which is different from it and
belongs to the class of the so-called elements-water or air or what is
intermediate between them. Those who make them limited in number never
make them infinite in amount. But those who make the elements infinite
in number, as Anaxagoras and Democritus do, say that the infinite is
continuous by contact-compounded of the homogeneous parts according to
the one, of the seed-mass of the atomic shapes according to the other.
Further, Anaxagoras held that any part is a mixture in the same
way as the All, on the ground of the observed fact that anything comes
out of anything. For it is probably for this reason that he
maintains that once upon a time all things were together. (This
flesh and this bone were together, and so of any thing: therefore
all things: and at the same time too.) For there is a beginning of
separation, not only for each thing, but for all. Each thing that
comes to be comes from a similar body, and there is a coming to be
of all things, though not, it is true, at the same time. Hence there
must also be an origin of coming to be. One such source there is which
he calls Mind, and Mind begins its work of thinking from some
starting-point. So necessarily all things must have been together at a
certain time, and must have begun to be moved at a certain time.
Democritus, for his part, asserts the contrary, namely that no
element arises from another element. Nevertheless for him the common
body is a source of all things, differing from part to part in size
and in shape.
It is clear then from these considerations that the inquiry concerns
the physicist. Nor is it without reason that they all make it a
principle or source. We cannot say that the infinite has no effect,
and the only effectiveness which we can ascribe to it is that of a
principle. Everything is either a source or derived from a source. But
there cannot be a source of the infinite or limitless, for that
would be a limit of it. Further, as it is a beginning, it is both
uncreatable and indestructible. For there must be a point at which
what has come to be reaches completion, and also a termination of
all passing away. That is why, as we say, there is no principle of
this, but it is this which is held to be the principle of other
things, and to encompass all and to steer all, as those assert who
do not recognize, alongside the infinite, other causes, such as Mind
or Friendship. Further they identify it with the Divine, for it is
'deathless and imperishable' as Anaximander says, with the majority of
the physicists.
Belief in the existence of the infinite comes mainly from five
(1) From the nature of time-for it is infinite.
(2) From the division of magnitudes-for the mathematicians also
use the notion of the infinite.
(3) If coming to be and passing away do not give out, it is only
because that from which things come to be is infinite.
(4) Because the limited always finds its limit in something, so that
there must be no limit, if everything is always limited by something
different from itself.
(5) Most of all, a reason which is peculiarly appropriate and
presents the difficulty that is felt by everybody-not only number
but also mathematical magnitudes and what is outside the heaven are
supposed to be infinite because they never give out in our thought.
The last fact (that what is outside is infinite) leads people to
suppose that body also is infinite, and that there is an infinite
number of worlds. Why should there be body in one part of the void
rather than in another? Grant only that mass is anywhere and it
follows that it must be everywhere. Also, if void and place are
infinite, there must be infinite body too, for in the case of
eternal things what may be must be. But the problem of the infinite is
difficult: many contradictions result whether we suppose it to exist
or not to exist. If it exists, we have still to ask how it exists;
as a substance or as the essential attribute of some entity? Or in
neither way, yet none the less is there something which is infinite or
some things which are infinitely many?
The problem, however, which specially belongs to the physicist is to
investigate whether there is a sensible magnitude which is infinite.
We must begin by distinguishing the various senses in which the term
'infinite' is used.
(1) What is incapable of being gone through, because it is not in
its nature to be gone through (the sense in which the voice is
'invisible').
(2) What admits of being gone through, the process however having no
termination, or what scarcely admits of being gone through.
(3) What naturally admits of being gone through, but is not actually
gone through or does not actually reach an end.
Further, everything that is infinite may be so in respect of
addition or division or both.
5
Now it is impossible that the infinite should be a thing which is
itself infinite, separable from sensible objects. If the infinite is
neither a magnitude nor an aggregate, but is itself a substance and
not an attribute, it will be indivisible; for the divisible must be
either a magnitude or an aggregate. But if indivisible, then not
infinite, except in the sense (1) in which the voice is 'invisible'.
But this is not the sense in which it is used by those who say that
the infinite exists, nor that in which we are investigating it, namely
as (2) 'that which cannot be gone through'. But if the infinite exists
as an attribute, it would not be, qua infinite an element in
substances, any more than the invisible would be an element of speech,
though the voice is invisible.
Further, how can the infinite be itself any thing, unless both
number and magnitude, of which it is an essential attribute, exist
in that way? If they are not substances, a fortiori the infinite is
not.
It is plain, too, that the infinite cannot be an actual thing and
a substance and principle. For any part of it that is taken will be
infinite, if it has parts: for 'to be infinite' and 'the infinite' are
the same, if it is a substance and not predicated of a subject.
Hence it will be either indivisible or divisible into infinites. But
the same thing cannot be many infinites. (Yet just as part of air is
air, so a part of the infinite would be infinite, if it is supposed to
be a substance and principle.) Therefore the infinite must be
without parts and indivisible. But this cannot be true of what is
infinite in full completion: for it must be a definite quantity.
Suppose then that infinity belongs to substance as an attribute.
But, if so, it cannot, as we have said, be described as a principle,
but rather that of which it is an attribute-the air or the even
number.
Thus the view of those who speak after the manner of the
Pythagoreans is absurd. With the same breath they treat the infinite
as substance, and divide it into parts.
This discussion, however, involves the more general question whether
the infinite can be present in mathematical objects and things which
are intelligible and do not have extension, as well as among
sensible objects. Our inquiry (as physicists) is limited to its
special subject-matter, the objects of sense, and we have to ask
whether there is or is not among them a body which is infinite in
the direction of increase.
We may begin with a dialectical argument and show as follows that
there is no such thing. If 'bounded by a surface' is the definition of
body there cannot be an infinite body either intelligible or sensible.
Nor can number taken in abstraction be infinite, for number or that
which has number is numerable. If then the numerable can be
numbered, it would also be possible to go through the infinite.
If, on the other hand, we investigate the question more in
accordance with principles appropriate to physics, we are led as
follows to the same result.
The infinite body must be either (1) compound, or (2) simple; yet
neither alternative is possible.
(1) Compound the infinite body will not be, if the elements are
finite in number. For they must be more than one, and the contraries
must always balance, and no one of them can be infinite. If one of the
bodies falls in any degree short of the other in potency-suppose
fire is finite in amount while air is infinite and a given quantity of
fire exceeds in power the same amount of air in any ratio provided
it is numerically definite-the
infinite body will obviously prevail
over and annihilate the finite body. On the other hand, it is
impossible that each should be infinite. 'Body' is what has
extension in all directions and the infinite is what is boundlessly
extended, so that the infinite body would be extended in all
directions ad infinitum.
Nor (2) can the infinite body be one and simple, whether it is, as
some hold, a thing over and above the elements (from which they
generate the elements) or is not thus qualified.
(a) We must consider the former alternative; for there are some
people who make this the infinite, and not air or water, in order that
the other elements may not be annihilated by the element which is
infinite. They have contrariety with each other-air is cold, water
moist, fire hot; if one were infinite, the others by now would have
ceased to be. As it is, they say, the infinite is different from
them and is their source.
It is impossible, however, that there should be such a body; not
because it is infinite on that point a general proof can be given
which applies equally to all, air, water, or anything else-but
simply because there is, as a matter of fact, no such sensible body,
alongside the so-called elements. Everything can be resolved into
the elements of which it is composed. Hence the body in question would
have been present in our world here, alongside air and fire and
earth and water: but nothing of the kind is observed.
(b) Nor can fire or any other of the elements be infinite. For
generally, and apart from the question of how any of them could be
infinite, the All, even if it were limited, cannot either be or become
one of them, as Heraclitus says that at some time all things become
fire. (The same argument applies also to the one which the