Alfred North Whitehead
"The Importance of Good Notation"
from Whitehead's An Introduction to Mathematics
Henry Holt & Co, 1911
from Chapter V “The Symbolism of Mathematics”
WE now return to pure mathematics, and consider more closely the apparatus of ideas out of which the science is built. Our first concern is with the symbolism of the science, and we start with the simplest and universally known symbols, namely those of arithmetic.
Let us assume for the present that we have sufficiently clear ideas about the integral numbers, represented in the Arabic notation by 0,1,2, . . ., 9, 10, 11, ... 100, 101, . . . and so on. This notation was introduced into Europe through the Arabs, but they apparently obtained it from Hindoo sources. The first known work in which it is systematically explained is a work by an Indian mathematician, Bhaskara (born 1114 A.D.). But the actual numerals can be traced back to the seventh century of our era, and perhaps were originally invented in Tibet. For our present purposes, however, the history of the notation is a detail. The interesting point to notice is the admirable illustration which this numeral system affords of the enormous importance of a good 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, under the influence of compulsory education, 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. The consequential extension of the notation to decimal fractions was not accomplished till the seventeenth century. Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation.
Mathematics is often considered a difficult and mysterious science, because of the numerous symbols which it employs. Of course, nothing is more incomprehensible than a symbolism which we do not understand. Also a symbolism, which we only partially understand and are unaccustomed to use, is difficult to follow. In exactly the same way the technical terms of any profession or trade are incomprehensible to those who have never been trained to use them. But this is not because they are difficult in themselves. On the contrary they have invariably been introduced to make things easy. So in mathematics, granted that we are giving any serious attention to mathematical ideas, the symbolism is invariably an immense simplification. It is not only of practical use, but is of great interest. For it represents an analysis of the ideas of the subject and an almost pictorial representation of their relations to each other. If anyone doubts the utility of symbols, let him write out in full, without any symbol whatever, the whole meaning of the following equations which represent some of the fundamental laws of algebra:
(3) x x y=y x x [read: “x times y equals y times x]
(4) (x x y) x z=x x (y x z)
(5) x x (y+z)=(x x y)+(x x z)
Here (1) and (2) are called the commutative and associative laws for addition, (3) and (4) 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 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 important symbols should have an important meaning. This is one of the merits of the Arabic notation for numbers ; by means of ten symbols, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and by simple juxtaposition it symbolizes any number whatever. Again in algebra, when we have two variable numbers x and y, we have to make a choice as to what shall be denoted by their juxtaposition xy. Now the two most important ideas on hand are those of addition and multiplication. Mathematicians have chosen to make their symbolism more concise by denning xy to stand for x x y. Thus the laws (3), (4), and (5) above are in general written,
xy=yx, (xy}z=x(yz) y x(y+z)=xy+xz
thus securing a great gain in conciseness. The same rule of symbolism is applied to the juxtaposition of a definite number and a variable : we write 3x for 3 x x, and 30x for 30 x x.
It is evident that in substituting definite numbers for the variables some care must be taken to restore the x, so as not to conflict with the Arabic notation. Thus when we substitute 2 for x and 3 for y in xy, we must write 2x3 for xy, and not 23 which means 20+3.
It is interesting to note how important for the development of science a modest-looking symbol may be. It may stand for the emphatic presentation of an idea, often a very subtle idea, and by its existence make it easy to exhibit the relation of this idea to all the complex trains of ideas in which it occurs. For example, take the most modest of all symbols, namely, 0, which stands for the number zero. The Roman notation for numbers had no symbol for zero, and probably most mathematicians of the ancient world would have been horribly puzzled by the idea of the number zero. For, after all, it is a very subtle idea, not at all obvious. A great deal of discussion on the meaning of the zero of quantity will be found in philosophic works. Zero is not, in real truth, more difficult or subtle in idea than the other cardinal numbers. What do we mean by 1 or by 2, or by 3? But we are familiar with the use of these ideas, though we should most of us be puzzled to give a clear analysis of the simpler ideas which go to form them. The point about zero is that we do not need to use it in the operations of daily life. No one goes out to buy zero fish. It is in a way the most civilized of all the cardinals, and its use is only forced on us by the needs of cultivated modes of thought. Many important services are rendered by the symbol 0, which stands for the number zero.
For anyone interested, the full text of Whitehead's An Introduction to Mathematics is available here.
Study Questions for Whitehead's "The Importance of Good Notation"