After considering the divine perfection we must consider the divine
infinity, and God's existence in things: for God is everywhere,
and in all things, inasmuch as He is boundless and infinite.
> Concerning the first, there are four points of inquiry:
> (1) Whether God is infinite?
> (2) Whether anything besides Him is infinite in essence?
> (3) Whether anything can be infinitude in magnitude?
> (4) Whether an infinite multitude can exist?
1: It seems that God is not infinite. For everything infinite
is imperfect, as the Philosopher says; because it has parts and
matter, as is said in Phys. iii. But God is most perfect; therefore
He is not infinite.
> Objection 2: Further,
according to the Philosopher (Phys. i), finite and infinite belong
to quantity. But there is no quantity in God, for He is not a body,
as was shown above (Question , Article ). Therefore
it does not belong to Him to be infinite.
3: Further, what is here in such a way as not to be elsewhere,
is finite according to place. Therefore that which is a thing in
such a way as not to be another thing, is finite according to substance.
But God is this, and not another; for He is not a stone or wood.
Therefore God is not infinite in substance.
the contrary, Damascene says (De Fide Orth. i, 4) that "God
is infinite and eternal, and boundless."
> I answer
that, All the ancient philosophers attribute infinitude
to the first principle, as is said (Phys. iii), and with reason;
for they considered that things flow forth infinitely from the
first principle. But because some erred concerning the nature of
the first principle, as a consequence they erred also concerning
its infinity; forasmuch as they asserted that matter was the first
principle; consequently they attributed to the first principle
a material infinity to the effect that some infinite body was the
first principle of things.
> We must consider therefore
that a thing is called infinite because it is not finite. Now matter
is in a way made finite by form, and the form by matter. Matter
indeed is made finite by form, inasmuch as matter, before it receives
its form, is in potentiality to many forms; but on receiving a
form, it is terminated by that one. Again, form is made finite by
matter, inasmuch as form, considered in itself, is common to many;
but when received in matter, the form is determined to this one
particular thing. Now matter is perfected by the form by which
it is made finite; therefore infinite as attributed to matter,
has the nature of something imperfect; for it is as it were formless
matter. On the other hand, form is not made perfect by matter,
but rather is contracted by matter; and hence the infinite, regarded
on the part of the form not determined by matter, has the nature
of something perfect. Now being is the most formal of all things,
as appears from what is shown above (Question
, Article , Objection ).
Since therefore the divine being is not a being received in anything,
but He is His own subsistent being as was shown above (Question , Article ), it is clear
that God Himself is infinite and perfect.
> From this appears the Reply to the First Objection.
> Reply to
Objection 2: Quantity is terminated by its form, which can
be seen in the fact that a figure which consists in quantity terminated,
is a kind of quantitative form. Hence the infinite of quantity
is the infinite of matter; such a kind of infinite cannot be attributed
to God; as was said above, in this article.
to Objection 3: The fact that the being of God is self-subsisting,
not received in any other, and is thus called infinite, shows
Him to be distinguished from all other beings, and all others to
be apart from Him. Even so, were there such a thing as a self-subsisting
whiteness, the very fact that it did not exist in anything else,
would make it distinct from every other whiteness existing in a
Objection 1: It seems that something else besides
God can be essentially infinite. For the power of anything is proportioned
to its essence. Now if the essence of God is infinite, His power
must also be infinite. Therefore He can produce an infinite effect,
since the extent of a power is known by its effect.
Objection 2: Further, whatever has infinite power,
has an infinite essence. Now the created intellect has an infinite
power; for it apprehends the universal, which can extend itself
to an infinitude of singular things. Therefore every created intellectual
substance is infinite.
On the contrary, The infinite cannot have a beginning,
as said in Phys. iii. But everything outside God is from God as
from its first principle. Therefore besides God nothing can be
> I answer that, Things other
than God can be relatively infinite, but not absolutely infinite.
For with regard to infinite as applied to matter, it is manifest
that everything actually existing possesses a form; and thus its
matter is determined by form. But because matter, considered as
existing under some substantial form, remains in potentiality to
many accidental forms, which is absolutely finite can be relatively
infinite; as, for example, wood is finite according to its own
form, but still it is relatively infinite, inasmuch as it is in
potentiality to an infinite number of shapes. But if we speak of
the infinite in reference to form, it is manifest that those things,
the forms of which are in matter, are absolutely finite, and in
no way infinite. If, however, any created forms are not received
into matter, but are self-subsisting, as some think is the case
with angels, these will be relatively infinite, inasmuch as such
kinds of forms are not terminated, nor contracted by any matter.
But because a created form thus subsisting has being, and yet is
not its own being, it follows that its being is received and contracted
to a determinate nature. Hence it cannot be absolutely infinite.
Reply to Objection 1: It is against the nature of
a made thing for its essence to be its existence; because subsisting
being is not a created being; hence it is against the nature of
a made thing to be absolutely infinite. Therefore, as God, although
He has infinite power, cannot make a thing to be not made (for
this would imply that two contradictories are true at the same
time), so likewise He cannot make anything to be absolutely infinite.
Reply to Objection 2: The fact that the power of
the intellect extends itself in a way to infinite things, is because
the intellect is a form not in matter, but either wholly separated
from matter, as is the angelic substance, or at least an intellectual
power, which is not the act of any organ, in the intellectual soul
joined to a body.
> Reply to Objection 3: Primary
matter does not exist by itself in nature, since it is not actually
being, but potentially only; hence it is something concreated rather
than created. Nevertheless, primary matter even as a potentiality
is not absolutely infinite, but relatively, because its potentiality
extends only to natural forms.
Objection 1: It seems that there can be something
actually infinite in magnitude. For in mathematics there is no
error, since "there is no lie in things abstract," as the Philosopher
says (Phys. ii). But mathematics uses the infinite in magnitude;
thus, the geometrician in his demonstrations says, "Let this line be
infinite." Therefore it is not impossible for a thing to be infinite
> Objection 2: Further, what
is not against the nature of anything, can agree with it. Now to
be infinite is not against the nature of magnitude; but rather
both the finite and the infinite seem to be properties of quantity.
Therefore it is not impossible for some magnitude to be infinite.
Objection 3: Further, magnitude is infinitely divisible,
for the continuous is defined that which is infinitely divisible,
as is clear from Phys. iii. But contraries are concerned about
one and the same thing. Since therefore addition is opposed to
division, and increase opposed to diminution, it appears that magnitude
can be increased to infinity. Therefore it is possible for magnitude
to be infinite.
> Objection 4: Further,
movement and time have quantity and continuity derived from the
magnitude over which movement passes, as is said in Phys. iv. But
it is not against the nature of time and movement to be infinite,
since every determinate indivisible in time and circular movement
is both a beginning and an end. Therefore neither is it against
the nature of magnitude to be infinite.
> On the
contrary, Every body has a surface. But every body which
has a surface is finite; because surface is the term of a finite
body. Therefore all bodies are finite. The same applies both to
surface and to a line. Therefore nothing is infinite in magnitude.
I answer that, It is one thing to be infinite in
essence, and another to be infinite in magnitude. For granted
that a body exists infinite in magnitude, as fire or air, yet this
could not be infinite in essence, because its essence would be
terminated in a species by its form, and confined to individuality
by matter. And so assuming from these premises that no creature
is infinite in essence, it still remains to inquire whether any
creature can be infinite in magnitude.
> We must therefore
observe that a body, which is a complete magnitude, can be considered
in two ways; mathematically, in respect to its quantity only; and
naturally, as regards its matter and form.
> Now it
is manifest that a natural body cannot be actually infinite. For
every natural body has some determined substantial form. Since
therefore the accidents follow upon the substantial form, it is
necessary that determinate accidents should follow upon a determinate
form; and among these accidents is quantity. So every natural body
has a greater or smaller determinate quantity. Hence it is impossible
for a natural body to be infinite. The same appears from movement;
because every natural body has some natural movement; whereas an
infinite body could not have any natural movement; neither direct,
because nothing moves naturally by a direct movement unless it
is out of its place; and this could not happen to an infinite body,
for it would occupy every place, and thus every place would be
indifferently its own place. Neither could it move circularly;
forasmuch as circular motion requires that one part of the body
is necessarily transferred to a place occupied by another part,
and this could not happen as regards an infinite circular body:
for if two lines be drawn from the centre, the farther they extend
from the centre, the farther they are from each other; therefore,
if a body were infinite, the lines would be infinitely distant
from each other; and thus one could never occupy the place belonging
to any other.
> The same applies to a mathematical body.
For if we imagine a mathematical body actually existing, we must
imagine it under some form, because nothing is actual except by
its form; hence, since the form of quantity as such is figure,
such a body must have some figure, and so would be finite; for figure
is confined by a term or boundary.
> Reply to Objection
1: A geometrician does not need to assume a line actually
infinite, but takes some actually finite line, from which he subtracts
whatever he finds necessary; which line he calls infinite.
> Reply to Objection 2: Although the infinite is not against the nature of magnitude in general, still it is against the nature of any species of it; thus, for instance, it is against the nature of a bicubical or tricubical magnitude, whether circular or triangular, and so on. Now what is not possible in any species cannot exist in the genus; hence there cannot be any infinite magnitude, since no species of magnitude is infinite.
to Objection 3: The infinite in quantity, as was shown above,
belongs to matter. Now by division of the whole we approach to
matter, forasmuch as parts have the aspect of matter; but by addition
we approach to the whole which has the aspect of a form. Therefore
the infinite is not in the addition of magnitude, but only in division.
Reply to Objection 4: Movement and time are whole,
not actually but successively; hence they have potentiality mixed
with actuality. But magnitude is an actual whole; therefore the
infinite in quantity refers to matter, and does not agree with
the totality of magnitude; yet it agrees with the totality of time
and movement: for it is proper to matter to be in potentiality.
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.
Objection 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.
Objection 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
> 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 to Objection
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 to Objection 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 to Objection
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.