Page 65 - Foundations Of Differential Calculus

48 3. On the Infinite and the Infinitely Small
74. Although these things are clear enough, so that anyone who would
deny them must contradict himself, still, this theory of the infinite has been
so obfuscated by so many difficulties and even involved in contradictions
by many who have tried to explain it, that no way is open by which they
may extricate themselves. From the fact that a quantity can be increased
to infinity, some have concluded that there is actually an infinite quantity,
and they have described it in such a way that it cannot be increased.
In this way they overturn the very idea of quantity, since they propose a
quantity of such a kind that it cannot be increased. Furthermore, those who
admit such an infinity contradict themselves; when they put an end to the
capacity a quantity has of being increased, they simultaneously deny that
the quantity can be increased without limit, since these two statements
come to the same thing. Thus while they admit an infinite quantity, they
also deny it. Indeed, if a quantity cannot be increased without limit, that
is to infinity, then certainly no infinite quantity can exist.

75. Hence, from the fact that every quantity can be increased to infinity, it
seems to follow that there is no infinite quantity. A quantity increased con-
tinuously by increments does not become infinite unless it shall have already
increased without limit. However, that which must increase without limit
cannot be conceived of as having already become infinite. Nevertheless, not
only is it possible to give a quantity of this kind, to which increments are
added without limit, a certain character, and with due care to introduce it
into calculus, as we shall soon see at length, but also there exist real cases,
at least they can be conceived, in which an infinite number actually exists.
Thus, if there are things that are infinitely divisible, as many philosophers
have held to be the case, the number of parts of which this thing is con-
stituted is really infinite. Indeed, if it be claimed that the number is finite,
then the thing is not really infinitely divisible. In a like manner, if the whole
world were infinite, as many have held, then the number of bodies making
up the world would certainly not be finite, and would hence be infinite.
76. Although there seems to be a contradiction here, if we consider it
carefully we can free ourselves from all difficulties. Whoever claims that
some material is infinitely divisible denies that in the continuous division
of the material one ever arrives at parts so small that they can no longer be
divided. Hence, this material does not have ultimate indivisible parts, since
the individual particles at which one arrives by continued division must be
able to be further subdivided. Therefore, whoever says, in this case, that
the number of parts is infinite, also understands that the ultimate parts
are indivisible; he tries to count those parts that are never reached, and
hence do not exist. If some material can always be further subdivided, it
lacks indivisible or absolutely simple parts. For this reason, whoever claims
that some material can be infinitely subdivided denies that the material is
made up of simple parts.