Page:IntroductionToMathematicsWhitehead.pdf/65
are the commutative and associative laws for multiplication, and is the distributive law relating addition and multiplication. For example, without symbols, 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 profoundly erroneous 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-