第1自然段

One of the first great intellectual feats of a young child is learning how to talk, closely followed by learning how to count. From earliest childhood we are so bound up with our system of numeration that it is a feat of imagination to consider the problems faced by early humans who had not yet developed this facility. Careful consideration of our system of numeration leads to the conviction that, rather than being a facility that comes naturally to a person, it is one of the great and remarkable achievements of the human race.

 第1自然段：

一个年幼儿童的首个巨大智力成就之一是学会说话，紧随其后的是学会数数。从最早的童年开始，我们与我们的计数系统密不可分，要想考虑到早期人类面临的尚未发展起这种能力的问题，需要有相当的想象力。仔细考虑我们的计数系统会让人确信，它不是一个人天生具备的便利设施，而是人类的伟大而卓越的成就之一。

第2自然段

It is impossible to learn the sequence of events that led to our developing the concept of number. Even the earliest of tribes had a system of numeration that, if not advanced, was sufficient for the tasks that they had to perform. Our ancestors had little use for actual numbers; instead their considerations would have been more of the kind Is this enough? rather than How many? when they were engaged in food gathering, for example. However, when early humans first began to reflect on the nature of things around them, they discovered that they needed an idea of number simply to keep their thoughts in order. As they began to settle, grow plants and herd animals, the need for a sophisticated number system became paramount. It will never be known how and when this numeration ability developed, but it is certain that numeration was well developed by the time humans had formed even semi-permanent settlements.

第2自然段：

我们无法了解导致我们出现数字概念的事件序列。即使是最早的部落也有一套计数系统，尽管不够先进，但对他们所需完成的任务已经足够。我们的祖先几乎没有真正使用数字；相反，他们的考虑更多是“这够吗？”而不是“有多少？”例如，在进行食物采集时，他们的考虑可能更多是“这够用吗？”。然而，当早期人类开始思考周围事物的本质时，他们发现自己需要一个数字概念，只是为了保持他们的思维井然有序。随着他们开始定居、种植作物和放牧动物，对一个复杂的计数系统的需求变得至关重要。我们将永远不会知道这个计数能力是如何发展起来的，但可以肯定的是，到人类形成了甚至是半永久性定居点的时候，计数已经发展得相当完善。

第3自然段

Evidence of early stages of arithmetic and numeration can be readily found. The indigenous peoples of Tasmania were only able to count one, two, many; those of South Africa counted one, two, two and one, two twos, two twos and one, and so on. But in real situations the number and words are often accompanied by gestures to help resolve any confusion. For example, when using the one, two, many type of system, the word many would mean, Look at my hands and see how many fingers I am showing you. This basic approach is limited in the range of numbers that it can express, but this range will generally suffice when dealing with the simpler aspects of human existence.

第3自然段：

早期算术和计数能力的初级阶段的证据是很容易找到的。塔斯马尼亚的土著人只能数到一、两、多；南非的土著人数到一、两、两和一、两个两个、两个两个和一个，等等。但在实际情况中，数字和单词通常伴随着手势，以帮助解决任何混淆。例如，当使用“一、两、多”类型的系统时，“多”的意思是“看看我的手，看我展示给你多少个手指”。这种基本方法在可以表达的数字范围上有限，但在处理人类存在的较简单方面时，这种范围通常足够。

第4自然段

The lack of ability of some cultures to deal with large numbers is not really surprising. European languages, when traced back to their earlier version, are very poor in number words and expressions. The ancient Gothic word for ten, tachund, is used to express the number 100 as tachund tachund. By the seventh century, the word teon had become interchangeable with the tachund or hund of the Anglo-Saxon language, and so 100 was denoted as hund teontig, or ten times ten. The average person in the seventh century in Europe was not as familiar with numbers as we are today. In fact, to qualify as a witness in a court of law a man had to be able to count to nine!

第4自然段：

某些文化缺乏处理大数字的能力并不令人意外。欧洲语言在追溯到早期版本时，在数字词汇和表达方面非常贫乏。古哥特语中，表示十的词是“tachund”，表示一百则是“tachund tachund”。到了七世纪，“teon”一词开始与盎格鲁撒克逊语中的“tachund”或“hund”互换使用，因此一百表示为“hund teontig”，即十乘以十。在七世纪的欧洲，普通人对数字的熟悉程度不如今天的人。事实上，要成为法庭证人，一个人必须能够数到九！

第5自然段

Perhaps the most fundamental step in developing a sense of number is not the ability to count, but rather to see that a number is really an abstract idea instead of a simple attachment to a group of particular objects. It must have been within the grasp of the earliest humans to conceive that four birds are distinct from two birds; however, it is not an elementary step to associate the number 4, as connected with four birds, to the number 4, as connected with four rocks. Associating a number as one of the qualities of a specific object is a great hindrance to the development of a true number sense. When the number 4 can be registered in the mind as a specific word, independent of the object being referenced, the individual is ready to take the first step toward the development of a notational system for numbers and, from there, to arithmetic.

第5自然段：

发展数字意识最基本的步骤可能并不是能够进行计数，而是要意识到数字实际上是一个抽象概念，而不仅仅是对一组特定物体的简单附属。对于最早期的人类来说，理解四只鸟与两只鸟是不同的，这应该是可以领会到的；然而，将与四只鸟有关的数字4与与四块石头有关的数字4联系起来并不是一个初级步骤。将数字作为特定对象的一种特性之一，对于真正的数字意识的发展是一个巨大的障碍。当数字4能够作为一个与所引用的对象无关的具体词语注册在大脑中时，个体已经准备好迈出向数字符号系统的发展的第一步，并从那里进入算术。

第6自然段

Traces of the very first stages in the development of numeration can be seen in several living languages today. The numeration system of the Tsimshian language in British Columbia contains seven distinct sets of words for numbers according to the class of the item being counted: for counting flat objects and animals, for round objects and time, for people, for long objects and trees, for canoes, for measures, and for counting when no particular object is being numerated. It seems that the last is a later development while the first six groups show the relics of an older system. This diversity of number names can also be found in some widely used languages such as Japanese.

第6自然段：

在今天的一些活语言中仍然可以看到数值发展的最初阶段的痕迹。不列颠哥伦比亚省的Tsimshian族语言的计数系统根据被计数的物品的类别包含了七种不同的数字词组：用于计算扁平物体和动物，用于圆形物体和时间，用于人，用于长形物体和树木，用于独木舟，用于度量单位，以及用于在没有特定物品被计数时计数。最后一种可能是后来的发展，而前六个群组显示了一种较早系统的遗迹。这种数字名称的多样性也可以在一些广泛使用的语言如日语中找到。

第7自然段

Intermixed with the development of a number sense is the development of an ability to count. Counting is not directly related to the formation of a number concept because it is possible to count by matching the items being counted against a group of pebbles, grains of corn, or the counter’s fingers. These aids would have been indispensable to very early people who would have found the process impossible without some form of mechanical aid. Such aids, while different, are still used even by the most educated in today’s society due to their convenience. All counting ultimately involves reference to something other than the things being counted. At first it may have been grains or pebbles but now it is a memorised sequence of words that happen to be the names of the numbers.

第7自然段：

与数字意识的发展交织在一起的是能够进行计数的能力的发展。计数与形成数字概念并不直接相关，因为可以通过将被计数的物品与一组卵石、玉米粒或计数者的手指进行匹配来进行计数。对于非常早期的人类来说，这些辅助工具在没有某种机械辅助的情况下，这一过程将是不可能完成的。即使在今天的社会中，这些辅助工具仍然在最受教育的人中使用，因为它们很方便。所有的计数最终都涉及与被计数的物品不同的参考对象。起初可能是谷粒或鹅卵石，但现在是一串记忆的单词序列，恰好是数字的名称。