Whether an Infinite Multitude Can Exist?
Objection 1: It seems that an actually infinite multitude is possible.
For it is not impossible for a potentiality to be made actual. But
number can be multiplied to infinity. Therefore it is possible for an
infinite multitude actually to exist.
Obj. 2: Further, it is possible for any individual of any species
to be made actual. But the species of figures are infinite. Therefore
an infinite number of actual figures is possible.
Obj. 3: Further, things not opposed to each other do not obstruct
each other. But supposing a multitude of things to exist, there can
still be many others not opposed to them. Therefore it is not
impossible for others also to coexist with them, and so on to
infinitude; therefore an actual infinite number of things is possible.
_On the contrary,_ It is written, "Thou hast ordered all things in
measure, and number, and weight" (Wis. 11:21).
_I answer that,_ A twofold opinion exists on this subject. Some, as
Avicenna and Algazel, said that it was impossible for an actually
infinite multitude to exist absolutely; but that an accidentally
infinite multitude was not impossible. A multitude is said to be
infinite absolutely, when an infinite multitude is necessary that
something may exist. Now this is impossible; because it would entail
something dependent on an infinity for its existence; and hence its
generation could never come to be, because it is impossible to pass
through an infinite medium.
A multitude is said to be accidentally infinite when its existence as
such is not necessary, but accidental. This can be shown, for example,
in the work of a carpenter requiring a certain absolute multitude;
namely, art in the soul, the movement of the hand, and a hammer; and
supposing that such things were infinitely multiplied, the
carpentering work would never be finished, forasmuch as it would
depend on an infinite number of causes. But the multitude of hammers,
inasmuch as one may be broken and another used, is an accidental
multitude; for it happens by accident that many hammers are used, and
it matters little whether one or two, or many are used, or an infinite
number, if the work is carried on for an infinite time. In this way
they said that there can be an accidentally infinite multitude.
This, however, is impossible; since every kind of multitude must
belong to a species of multitude. Now the species of multitude are to
be reckoned by the species of numbers. But no species of number is
infinite; for every number is multitude measured by one. Hence it is
impossible for there to be an actually infinite multitude, either
absolute or accidental. Likewise multitude in nature is created; and
everything created is comprehended under some clear intention of the
Creator; for no agent acts aimlessly. Hence everything created must be
comprehended in a certain number. Therefore it is impossible for an
actually infinite multitude to exist, even accidentally. But a
potentially infinite multitude is possible; because the increase of
multitude follows upon the division of magnitude; since the more a
thing is divided, the greater number of things result. Hence, as the
infinite is to be found potentially in the division of the continuous,
because we thus approach matter, as was shown in the preceding
article, by the same rule, the infinite can be also found potentially
in the addition of multitude.
Reply Obj. 1: Every potentiality is made actual according to
its mode of being; for instance, a day is reduced to act successively,
and not all at once. Likewise the infinite in multitude is reduced to
act successively, and not all at once; because every multitude can be
succeeded by another multitude to infinity.
Reply Obj. 2: Species of figures are infinite by infinitude of
number. Now there are various species of figures, such as trilateral,
quadrilateral and so on; and as an infinitely numerable multitude is
not all at once reduced to act, so neither is the multitude of
figures.
Reply Obj. 3: Although the supposition of some things does not
preclude the supposition of others, still the supposition of an
infinite number is opposed to any single species of multitude. Hence
it is not possible for an actually infinite multitude to exist.
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