chapter 5 The symbolism of mathematics
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.
我们现在回到纯数学，仔细考虑构建这一科学的思想工具。我们首先关注的是该科学的符号系统，我们从最简单且普遍知晓的符号开始，即算术符号。 Let us assume for the present that we have sufficiently clear ideas about the integral number, 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 Hindu sources. The first known work in which it is systematically explained is a work by an Indian mathematician, Bhaskara (born A.D 1114). 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, For the detailed historical facts relating to pure mathematics, I am chiefly indebted to A Short History of Mathematics, By W.W.R.Ball. 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 seventeen century. Our modern power of easy reckoning with decimal fractions is the almost miraculous result of the gradual discovery of a perfect notation.
让我们暂时假设我们对整数的概念已经足够清晰，阿拉伯数字表示为0、1、2、...9、10、11、...100、101等。这种符号系统是通过阿拉伯人引入欧洲的，但他们显然是从印度来源获得的。第一个系统地解释这一符号系统的已知作品是印度数学家巴斯卡拉（公元1114年出生）所著。然而，实际的数字可以追溯到我们时代的第七世纪，或许最初是在西藏发明的。然而，就我们目前的目的而言，这种符号的历史是一个细节。值得注意的是，这种数字系统极好地说明了良好符号的巨大重要性。通过减轻大脑的所有不必要工作，一个好的符号系统使大脑能够集中精力解决更复杂的问题，从而实际上增强了人类的智力。在阿拉伯符号引入之前，乘法是困难的，而整数的除法甚至动用了最高级的数学能力。现代世界中，可能没有什么比让一位希腊数学家更感到惊讶的事情了，那就是在强制教育的影响下，西欧大量人口能够进行大数的除法运算。这个事实对他来说简直是不可能的。符号向小数的扩展直到十七世纪才完成。我们现代能够轻松计算小数的能力，几乎是完美符号逐渐发现的奇迹般结果。
Mathematics is often considered a difficult and mysterious science, because of the numerous symbols which it employs. Of curse , 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 idea of the subject and an almost pictorial representation of their relations to each other. If any one 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.
数学常常被认为是一门困难而神秘的科学，这主要是因为它使用了大量符号。当然，没有什么比我们不理解的符号更难以理解。此外，只有部分理解并且不习惯使用的符号也很难跟上。任何职业或行业的专业术语对那些没有接受过相关训练的人来说同样是难以理解的。但这并不是因为它们本身很难。相反，它们的引入通常是为了简化事情。因此，在数学中，假如我们认真关注数学思想，这些符号无疑是巨大的简化。它不仅在实际使用中有用，而且非常有趣。因为它代表了对主题概念的分析，以及它们彼此关系的几乎图像化表现。如果有人怀疑符号的实用性，那就让他完全不使用任何符号，写出以下方程所代表的代数基本定律的全部含义。
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 the first given number had been added to the second number.
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）是涉及加法和乘法的分配律。例如，如果不使用符号，（1）可以表达为：如果将第二个数字加到任何给定的数字上，结果与将第一个给定的数字加到第二个数字上是相同的。
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.
这个例子表明，通过符号的帮助，我们可以几乎通过视觉机械地进行推理的转变，而这在其他情况下则需要动用大脑的更高能力。
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 In reading these equations it must be noted that a bracket is used in mathematical symbolism to mean that the operations within it are to be performed first. Thus (1+3)+2 directs us first to add 3 to 1, and then to add 2 to the result; and 1+(3+2) directs us first to add 2 to 3, and then to add the result to 1. Again , a numerical example of equation (5) is
这是一种深刻错误的自明之理，所有的练习本和著名人物在演讲时都会重复这个观点：我们应该培养“在阅读这些方程时，必须注意在数学符号中使用括号表示括号内的运算要优先进行”的习惯。（而下文说，我们无需注意，而是将其变为不经思考的操作）因此，(1+3)+2指示我们首先将3加到1，然后再将2加到结果；而1+(3+2)则指示我们首先将2加到3，然后将结果加到1。同样，方程（5）的一个数值例子是
我们首先执行括号内的运算，得到
这显然是正确的
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.
思考我们所做的事情。恰恰相反，文明的发展是通过扩展我们能够在不思考的情况下执行的重要操作的数量来实现的。思维的操作就像战斗中的骑兵冲锋——它们的数量严格有限，需要新鲜的马匹，并且必须在决定性时刻才能进行。
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 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 defining xy to stand for x*y . Thus the laws (3),(4), and (5) above are in general written,
一个非常重要的符号特性是，它应该简洁，以便能够在一瞥之间看清，并能够迅速书写。我们无法比将符号直接并列在一起更简洁地排列它们。因此，在一个好的符号系统中，重要符号的并列应该具有重要的意义。这就是阿拉伯数字表示法的优点之一；通过十个符号（0,
1, 2, 3, 4, 5, 6, 7, 8,
9）和简单的并列，它可以象征任何数字。同样，在代数中，当我们有两个变量x和y时，我们需要选择它们的并列xy所表示的含义。现在手头上最重要的两个概念是加法和乘法。数学家选择通过定义xy代表xy，使他们的符号系统更加简洁。因此，上述法则（3）、（4）和（5）通常写为：
从而在简洁性上获得了巨大的提升。相同的符号规则适用于一个确定数字和一个变量的并列：我们写3x代表3 * x，写30x代表30 * 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 2*3 for xy, and not 23 which means 20+3.
显然，在用确定的数字替代变量时，必须小心以恢复x，以避免与阿拉伯数字表示法发生冲突。因此，当我们在xy中将2替换为x，将3替换为y时，必须写成2*3，而不是23，因为23表示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 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 symbol 0, which stands for the number zero.
有趣的是，一个看似简单的符号对于科学发展的重要性。它可能代表了一个强调的思想呈现，往往是一个非常微妙的概念，通过它的存在，可以轻松展示该思想与所有复杂的思想链之间的关系。例如，考虑所有符号中最简单的符号，即0，它代表数字零。罗马数字没有零的符号，可能古代世界的大多数数学家对数字零的概念感到非常困惑。毕竟，零是一个非常微妙的概念，远非显而易见。在哲学著作中可以找到大量关于数量零含义的讨论。实际上，零的概念并不比其他基数更难或更微妙。我们对1、2或3的理解又是什么呢？但是我们对这些概念的使用是熟悉的，尽管我们大多数人可能会困惑于如何清晰地分析构成这些概念的更简单的思想。关于零的关键在于我们在日常生活中并不需要使用它。没有人会出去买零条鱼。某种程度上，它是所有基数中最文明的一个，其使用只是出于对复杂思维方式的需求。符号0代表数字零，提供了许多重要的服务。
The symbol developed in connexion with the Arabic notation for numbers of which it is an essential part. For in that notation the value of a digit depends on the position in which it occurs. Consider, for example, the digit 5 , as occurring in the numbers 25,51,3512,5213. In the first number 5 stands for five, in the second number 5 stands for fifty, in the third number for five hundred, and in the fourth number for five thousand. Now , when we write the number fifty-one in the symbolic form 51, the digit 1 pushes the digit 5 along to the second place (reckoning from right to left) and thus gives it the value fifty. But when we want to symbolize fifty by itself, we can have no digit 1 to perform this service; we want a digit in the units place to add nothing to the total and yet to push the five along to the second place. This service is performed by 0, the symbol for zero. It is extremely probable that the men who introduced 0 for this purpose had no definite conception in their minds of the number zero. They simply wanted a mark to symbolize the fact that nothing was contributed by the digit's place in which it occurs. The idea of zero probably took shape gradually from a desire to assimilate the meaning of this mark to that of the marks,1,2,...,9,which do represent cardinal numbers. This would not represent the only case in which a subtle idea has been introduces into mathematics by a symbolism which in its origin was dictated by practical convenience.
这个符号是在与阿拉伯数字记法相关的过程中发展起来的，它是其中的一个基本部分。因为在这种记法中，数字的值依赖于其出现的位置。例如，考虑数字 5 在 25、51、3512 和 5213 中的出现。在第一个数字中，5 代表五；在第二个数字中，5 代表五十；在第三个数字中，代表五百；在第四个数字中，代表五千。当我们用符号形式 51 写出五十一时，数字 1 将数字 5 推到第二位（从右到左计算），因此使其值变为五十。但当我们想单独用符号表示五十时，就不能有数字 1 来完成这个任务；我们需要一个在单位位置的数字，以便不增加总值，同时又将五推到第二位。这个功能是由 0 完成的，即零的符号。很可能，最初引入 0 用于此目的的人并没有清晰的零的概念。他们只想要一个符号，表示在该数字位置上没有贡献。零的概念可能逐渐形成，源于希望将这个符号的含义与代表基数的符号 1、2、…、9 进行结合。这并不是唯一一个通过一种起初因实用方便而产生的符号化方式，引入细致概念到数学中的例子。
Thus the first use of 0 was to make the arabic notation possible—no slight service. We can imagine that when it had been introduced for this purpose, practical men, of the sort who dislike fanciful ideas, deprecated the silly habit of identifying it with a number zero. But they were wrong , as such men always are when they desert their proper function of masticating food which others have prepared. For the next service performed by the symbol 0 essentially depends upon assigning to it the function of representing the number zero.
因此，0 的首次使用是为了使阿拉伯记数法成为可能——这并不是一项微不足道的服务。我们可以想象，当它被引入用于这个目的时，那些厌恶幻想想法的实用主义者贬低了将其与数字零等同的愚蠢习惯。但他们错了，因为这样的男人总是在抛弃他们应有的咀嚼食物的功能时犯错，而这些食物是别人准备的。因为符号 0 所执行的下一个功能本质上依赖于将其赋予表示数字零的作用。
The second symbolic use is at first sight so absurdly simple that it is difficult to make a beginner realize its importance. Let us start with a simple example. In Chapter 2 we mentioned the correlation between two variable numbers x and y represented by the equation x+y=1. This can be represented in an indefinite number of ways ; for example, x=1-y, y=1-x, 2x+3y-1=x+2y, and so on. But the important way of stating it is
第二种符号的性用法乍一看是如此荒谬简单，以至于很难让初学者意识到它的重要性。让我们从一个简单的例子开始。在第二章中，我们提到了由方程
x+y=1 表示的两个变量数 x 和 y
之间的相关性。这可以用无数种方式表示；例如，x=1-y, y=1-x, 2x+3y-1=x+2y
等等。但重要的表述方式是
同样，写出方程 x=1 的重要方式是 x-1=0，而表示方程
This is an idea to which we shall have continually to recur; it is
not going too far to say that no part of modern mathematics can be
properly understood without constant recurrence to it. The conception of
form is so general that it is difficult to characterize it in abstract
terms. At this stage we shall do better merely to consider examples.
Thus the equations
这是一个我们将不断要重新考虑的想法；可以毫不夸张地说，现代数学的任何部分都无法在没有对它的不断回顾的情况下得到正确理解。形式的概念是如此广泛，以至于很难用抽象的术语来表征。在这个阶段，我们最好只是考虑一些例子。因此，方程
Then further there are the forms of equation stating correlations between two variables; for example , x+y-1=0, 2x+3y-8=0, and so on. there are examples of what is called the linear form of equation. The reason for the name of 'linear' is that the graphic method of representation , which is explained at the end of Chapter 2, always represents such equations by a straight line. Then there are other forms for two variables — for example , the quadratic form, the cubic form, and so on. But the point which we here insist upon is that this study of form is facilitated, and , indeed , made possible, by the standard method of writing equations with the symbol 0 on the right-hand side.
然后，还有一些方程形式用于表示两个变量之间的关联关系；例如，x + y - 1 = 0，2x + 3y - 8 = 0 等等。这些都是所谓的线性方程形式。'线性' 这一名称的由来是因为，正如第二章末尾所解释的那样，这类方程在图解表示中总是通过一条直线来展示。除此之外，还有其他形式的二元方程，例如二次方程形式、三次方程形式等等。然而，我们在此强调的要点是，研究这些方程形式之所以能够顺利进行，甚至成为可能，正是因为使用了将方程右侧写为符号 0 的标准方法。
There is yet another function performed by 0 in relation to the study
of form . Whatever number x may be ,
在形式研究中，0 还执行着另一项功能。无论 x 是什么数，0*x=0，且
x+0=x。借助这些性质，可以使形式上的细微差异得到统一。例如，上述提到的二次方程
For these three reasons the symbol 0 , representing the number zero, is essential to modern mathematics. It has rendered possible types of investigation which would have been impossible without it.
由于这三个原因，表示数字零的符号 0 对现代数学至关重要。它使一些原本无法进行的研究成为可能。
The symbolism of mathematics is in truth the outcome of the general ideas which dominate the science. We have now two such general ideas before us, that of the variable and that of algebraic form. The junction of these concepts has imposed on mathematics another type of symbolism almost quaint in its character, but none the less effective. We have seen that an equation involving two variables, x and y, represents a particular correlation between the pair of variables. This x+y=1 represents one definite correlation, and 3x+2y-5=0 represents another definite correlation between the variables x and y; and both correlations have the form of what we have called linear correlations. But now, how can we represent any linear correlations between the variable numbers x and y? Here we want to symbolize any linear correlation; just as x symbolizes any number. This is done by turning the numbers which occur in the definite correlation 3x+2y-5=0 in to letters. We obtain ax+by+c=0. Here a, b, c, stand for variable numbers just as do x and y : but there is a difference in the use of the two sets of variables. We study the general properties of the relationship between x and y while a,b, and c have unchanged values. We do not determine what the values of a, b, and c are; but whatever they are, they remain fixed while we study the relation between the variables x and y for the whole group of possible values of x and y. But when we have obtained the properties of this correlation, we note that, because a, b, and c have not in fact been determined, we have proved properties which must belong to any such relation. Thus , by now varying a, b, and c, we arrive at the idea that ax+by+c=0 represents a variable linear correlation between x and y. In comparison with x and y , the three variables a, b, and c are called constants. Variables used in this wat are sometimes also called parameters.
数学的符号实际上是支配这门科学的普遍思想的结果。我们现在面前有两个这样的普遍思想：变量的概念和代数形式的概念。这些概念的交汇为数学强加了另一种几乎古雅的符号体系，但同样有效。我们已经看到，涉及两个变量 x 和 y 的方程表示这对变量之间的特定相关性。方程 x+y=1 代表一种明确的相关性，而 3x+2y−5=0 则代表 x 和 y 之间的另一种明确的相关性；这两种相关性都具有我们所称的线性相关性的形式。但现在，我们如何表示变量 x 和 y 之间的任何线性相关性呢？在这里，我们想要象征任何线性相关性，正如 x 代表任何数字一样。这是通过将出现在确定相关性 3x+2y−5=0 中的数字转换为字母来实现的。我们得到 ax+by+c=0。在这里，a、b、c 代表变量数，就像 x 和 y 一样；但两组变量的使用是有区别的。我们研究 x 和 y 之间关系的一般性质，而a、b 和 c 的值保持不变。我们并不确定 a、b 和 c 的具体值；但无论它们是什么，它们在我们研究变量 x 和 y 之间的关系时始终是固定的，涵盖了 x 和 y 的所有可能值。但是，当我们获得这种相关性的性质时，我们注意到，由于a、b 和 c 的值实际上并未被确定，因此我们证明了必须属于任何这种关系的性质。因此，通过改变 a、b 和 c，我们得出了 ax+by+c=0 代表 x 和y 之间的一个变量线性相关的概念。与 x 和 y 相比，三个变量 a、b 和 cc 被称为常数。在这种情况下使用的变量有时也称为参数。
Now, mathematicians habitually save the trouble of explaining which of their variables are to be treated as 'constants', and which as variables, considered as correlated in their equations, by using letters at the end of the alphabet for the 'variable' variables. and letters at the beginning of the alphabet for the 'constant' variables, or parameters. The two systems meet naturally about the middle of the alphabet. Sometimes a word or two of explanation id necessary; but as matter of fact custom and common sense are usually sufficient, and surprisingly little confusion is caused by a procedure which seems so lax.
现在，数学家们习惯于省去解释哪些变量要视为“常量”，哪些要作为方程中相互关联的“变量”的麻烦，而是通过采用字母表末尾的字母表示“变量”变量，用字母表开头的字母表示“常量”变量或参数。这两种系统自然地在字母表中间部分交汇。有时需要几句话来进行解释；但实际上，惯例和常识通常就足够了，这种看似松懈的做法竟然很少引起混乱。
The result of this continual elimination of definite numbers by successive layers of parameters is that the amount of arithmetic performed by mathematicians is extremely small. Many mathematicians dislike all numerical computation and are not particularly expert at it. The terroitory of arithmetic ends where the two ideas of 'variables' and of 'algebraic form' commence their sway.
通过连续地用一层层的参数消除特定数字的结果是，数学家们所进行的算术运算量极少。许多数学家不喜欢所有的数值计算，也并不特别擅长这方面的工作。算术的领域在“变量”和“代数形式”这两个概念开始发挥影响力的地方结束。