Page 67 - Foundations Of Differential Calculus

50 3. On the Infinite and the Infinitely Small
infinitely divided. By this very fact, the existence of simple beings that
make up a body is completely refuted. He who denies that bodies are made
up of simple beings and he who claims that bodies are infinitely divisible
are both saying the same thing.
80. Nor is their position any better if they say that the ultimate particles
of a body are indeed extended, but because of the hardness they cannot
be broken apart. Since they admit extension in the ultimate particles, they
hold that the particles are composite. Whether or not they can be separated
from each other makes little difference, since they can assign no cause that
explains this hardness. For the most part, however, those who deny the
infinite divisibility of matter seem to have sufficiently felt the difficulties of
this latter position, since usually they cling to the former idea. But they
cannot escape these difficulties except with a few trivial metaphysical dis-
tinctions, which generally strive to keep us from trusting the consequences
that follow from mathematical principles. Nor should they admit that sim-
ple parts have dimensions. In the first place they should have demonstrated
that these ultimate parts, of which a determined number make up a body,
have no extension.
81. Since they can find no way out of this labyrinth, nor can they meet the
objections in a suitable way, they flee to distinctions, and to the objections
they reply with arguments supplied by the senses and the imagination. In
this situation one should rely solely on the intellect, since the senses and ar-
guments depending on them frequently are fallacious. Pure intellect admits
the possibility that one thousandth part of a cubic foot of matter might
lack all extension, while this seems absurd to the imagination. That which
frequently deceives the senses may be true, but it can be decided by no
one except mathematicians. Indeed, mathematics defends us in particular
against errors of the senses and teaches about objects that are perceived by
the senses, sometimes correctly, and sometimes only in appearance. This
is the safest science, whose teaching will save those who follow it from the
illusions of the senses. It is far removed from those responses by which
metaphysicians protect their doctrine and thus rather make it more sus-
pect.

82. But let us return to our proposition. Even if someone denies that
infinite numbers really exist in this world, still in mathematical speculations
there arise questions to which answers cannot be given unless we admit an
infinite number. Thus, if we want the sum of all the numbers that make
up the series 1+2+3+4+5+ ··· , since these numbers progress with
no end, and the sum increases, it certainly cannot be finite. By this fact it
becomes infinite. Hence, this quantity is so large that it is greater than any
finite quantity and cannot not be infinite. To designate a quantity of this
kind we use the symbol ∞, by which we mean a quantity greater than any
finite or assignable quantity. Thus, when a parabola needs to be defined in