现在我们回到纯数学，更仔细地考虑一下构成科学的思想体系。我们首先关注的是科学的符号体系，我们从最简单和最普遍的符号开始，即算术符号。

现在我们假设我们对整数有足够清晰的认识，在阿拉伯符号中表示为 0、1、2、……、9、10、11、……100、101、……等等。这种符号是通过阿拉伯人引入欧洲的，但他们显然是从印度那里获得的。第一部系统解释它的已知著作是印度数学家 Bhaskara（生于公元 1114 年）的著作。但实际的数字可以追溯到我们时代的七世纪，也许最初是在西藏发明的。然而，就我们目前的目的而言，符号的历史只是一个细节。值得注意的有趣之处在于，这个数字系统很好地说明了良好符号的巨大重要性。良好的符号可以减轻大脑所有不必要的工作，使大脑能够集中精力解决更高级的问题，从而有效地增强人类的智力。在引入阿拉伯符号之前，乘法很难，甚至整数的除法也需要最高的数学能力。在现代世界中，可能没有什么比得知在义务教育的影响下，西欧大部分人能够对最大数字进行除法运算更让希腊数学家感到惊讶的了。在他看来，这一事实完全是不可能的。直到 17 世纪，符号才扩展到小数。我们现代人轻松计算小数的能力几乎是逐渐发现完美符号的奇迹。

数学通常被认为是一门困难而神秘的科学，因为它使用了众多符号。当然，没有什么比我们不理解的符号更令人费解的了。同样，我们只部分理解并且不习惯使用的符号也很难理解。同样，任何职业或行业的技术术语对于那些从未接受过使用训练的人来说也是难以理解的。但这并不是因为它们本身很难。相反，它们总是被引入来使事情变得简单。所以在数学中，假设我们认真关注数学思想，符号总是一种巨大的简化。它不仅具有实际用途，而且非常有趣。因为它代表了对主题思想的分析以及它们之间关系的几乎图形化表示。如果有人怀疑符号的效用，让他完整地写出以下方程的全部含义，而不用任何符号，这些方程代表了代数的一些基本定律：

(l)            x+y=y+x

(2)          (x+y)+z=x+(y+z)

(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)

这里 (1) 和 (2) 被称为加法的交换律和结合律，(3) 和 (4) 是乘法的交换律和结合律，(5) 是加法和乘法的分配律。例如，不使用符号，(1) 变成：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。

这个例子表明，借助符号，我们可以几乎机械地通过眼睛进行推理转换，否则将调用大脑的高级功能。

我们应该养成思考我们正在做的事情的习惯，这是一条极其错误的真理，所有的抄本和杰出人物在演讲时都重复了这一真理。事实恰恰相反。文明通过扩展我们可以不加思考地执行的重要操作的数量而进步。思维活动就像战斗中的骑兵冲锋，它们的数量受到严格限制，需要新鲜的马匹，并且只能在决定性时刻发起。

符号系统的一个非常重要的特性是它应该简洁，以便一目了然并能快速书写。现在，没有什么比将符号直接并列更简洁的了。因此，在好的符号系统中，重要符号的并列应该具有重要意义。这是阿拉伯数字符号的优点之一；通过十个符号，0、1、2、3、4、5、6、7、8、9，通过简单的并列，它可以表示任何数字。在代数中，当我们有两个变量数 x 和 y 时，我们必须选择它们的并列 xy 应该表示什么。现在，手头上最重要的两个想法是加法和乘法。数学家选择用 xy 表示 x x y，从而使其符号更简洁。因此，上述定律 (3)、(4) 和 (5) 一般写为：

xy=yx, (xy}z=x(yz) y x(y+z)=xy+xz

从而确保了简洁性的巨大提升。同样的符号规则也适用于确定数字和变量的并置：我们将 3 x x 写为 3x，将 30 x x 写为 30x。

显然，在用确定数字代替变量时，必须小心恢复 x，以免与阿拉伯符号冲突。因此，当我们在 xy 中用 2 代替 x 并用 3 代替 y 时，我们必须用 2x3 表示 xy，而不是 23，这意味着 20+3。

有趣的是，一个看似不起眼的符号对于科学的发展有多么重要。它可能代表一个想法的强调表达，通常是一个非常微妙的想法，并且通过它的存在，很容易展示这个想法与它出现的所有复杂想法之间的关系。例如，取所有符号中最不起眼的 0，它代表数字零。罗马数字符号中没有零的符号，古代世界上的大多数数学家可能对数字零的概念感到十分困惑。因为，毕竟，这是一个非常微妙的概念，一点也不明显。哲学著作中对数量零的含义进行了大量讨论。实际上，零的概念并不比其他基数更难理解或更微妙。我们用 1、2 或 3 来表示什么？但我们熟悉这些概念的用法，尽管我们大多数人可能都无法清楚地分析构成它们的更简单的想法。关于零的要点是，我们不需要在日常生活中使用它。没有人出去买零鱼。它是所有基数中最文明的，它的使用只是出于有教养的思维方式的需要。符号 0 代表数字零，它提供了许多重要的服务。

原文：

The Symbolism of Mathematics

http://introtologic.info/AboutLogicsite/whitehead%20Good%20Notation.html

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 Xizang. 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:

(l)            x+y=y+x

(2)          (x+y)+z=x+(y+z)

(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.

参考文献：

The Symbolism of Mathematics

http://introtologic.info/AboutLogicsite/whitehead%20Good%20Notation.html

https://archive.org/details/introductiontoma00whitiala/page/70/mode/2up