An introduction to mathematics, by A. N. Whitehead.

246 INTRODUCTION TO MATHEMATICS
applied, so as to secure lengths being contiguous and not overlapping, we say that the
lengths have been added to form one whole
length. But we cannot arbitrarily take any
test as the test of equality and any process
as the process of addition. The results of
operations of addition and of judgments of
equality must be in accordance with certain
preconceived conditions. For example, the
addition of two greater lengths must yield
a length greater than that yielded by the
addition of two smaller lengths. These preconceived conditions when accurately formulated may be called axioms of quantity. The
only question as to their truth or falsehood
which can arise is whether, when the axioms
are satisfied, we necessarily get what ordinary
people call quantities. If we do not, then
the name "axioms of quantity" is ill-judged
that is all.
These axioms of quantity are entirely abstract, just as are the mathematical properties of space. They are the same for all
quantities, and they presuppose no special
mode of perception. The ideas associated
with the notion of quantity are the means by
which a continuum like a line, an area, or a
volume can be split up into definite parts.
Then these parts are counted; so that numbers can be used to determine the exact properties of a continuous whole.