Various Works Page 9

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

  whether there is such a thing or not, and, if there is, what it is.

  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

  considerations:

  (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