The Value of Notation

In An Introduction to Mathematics, Alfred North Whitehead speaks to the value of notation:

By relieving the brain of all unnecessary work, a good notation sets it free to concentrate on more advanced problems, and, in effect, increases the mental power of the race.

Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world would have more astonished a Greek mathematician than to learn that … a large proportion of the population of Western Europe could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility … Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation. […]

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. Civilisation 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.

Lest we become too enamored of notation, however, Abraham Kaplan, in The Conduct of Inquiry: Methodology for Behavioral Science, offers the following words of caution:

[T]he complexity of a notation does not of itself endow the notation with scientific importance. Behavioral science has suffered often from the illusion that a commonplace formulated in an uncommon notation becomes profound—rich with scientific promise.


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