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.