by the aid of symbolism, we can make transitions in reasoning almost mechanically by the eye which would otherwise call into play the higher faculties of the brain.
…Civilization advances by extending the number of important operations that 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 be rapidly written.
– Whitehead, A.N. (1911). An introduction to mathematics, Chapter 5, “The Symbolism of Mathematics” (page 61 in this version)
HT to Santiago Nuñez-Corrales (Illinois page for Santiago Nuñez-Corrales, LinkedIn for Santiago Núñez-Corrales) who used part of this quote in a Conceptual Foundations Group talk, Nov 29.
From my point of view, this is why memorizing multiplication tables is not now irrelevant; why new words for concepts are important; and underlies a lot of scientific advancement.