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

SYMBOLISM OF MATHEMATICS 61
are the commutative and associative laws for
multiplication, and (5) is the distributive law
relating addition and multiplication. For example, without symbols, (1) becomes: If a
second number be added to any given number
the result is the same as if the first given
number had been added to the second number.
This example shows that, by the aid of symbolism, we can make transitions in reasoning
almost mechanically by the eye, which otherwise would call into play the higher faculties
of the brain.
It is a profoundlyerroneous truism,repeated
by all copy-books and by eminent people when
they are making speeches, that we should
cultivate the habit of thinking of what we are
doing. The precise opposite is the case.
Civilization advances by extending the number of important operations which we can
perform without thinking about them. Operations of thought are like cavalry charges in
a battle-they are strictly limited in number, they require fresh horses, and must only
be made at decisive moments.
One very important property for symbolism
to possess is that it should be concise, so as to
be visible at one glance of the eye and to be
rapidly written. Now we cannot place symbols more concisely together than by placing
them in immediate juxtaposition. In a good
symbolism therefore, the juxtaposition of im