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.

学习讲话是孩子最早的重要智力技艺中的一方面，而学习计数紧随其后。从孩童初期起我们就和计数体系存在着密不可分的联系。在还没有发明这个计数体系之前，早期人类所面临的问题实在难以想象。仔细考虑一下计数体系的发明就会深信，它并不是人类能自然获得的工具，而是人类的一个伟大而又非凡的成就。

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.

不可能知道数字概念的发展形成过程。即使是最早的部落也有他们自己的计数体系。虽然并不是很先进，但对于他们所从事的活动而言，那已经足够了。我们的祖先很少使用具体数字，而他们考虑更多的是这类问题：这个够了吗?而不是多少才够。在他们分食物的时候就是个很好的例子。然而当他们开始仔细思考周围事物时，他们发现需要一个数字概念以便于有序的记忆。当他们开始定居、种植以及放牧时，他们更加意识到拥有一个复杂的计数体系的重要性。永远没有人知道这个计数体系是如何以及什么时候发展起来的，但是有一点可以确定的是，随着人类暂时定居的发展，计数体系得到良好发展。

早期的算术和计数的证据可以得到。塔斯马尼亚土著民族仅仅能够数一、二和多，而南非土著民族能够数一、二，二和一，两个二，两个二和一等。但是实际情况下数字和单词常常伴随着手势以避免误解。例如，当使用一、二和多这个体系时，“多”这个单词就可理解为：看我的手，并看我向你伸几根手指。基本的方式是限制在所能够表达的数字范围之内，而处理人们生活的各个较简单的方面时，这个范围就已经足够了。

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.

一些文化缺乏处理大数的能力并不令人惊奇。当追溯到欧洲语言早期的形式时，其在数字单词以及表达方面也很贫乏。在古哥特语中，十是用单词tachund来表达的，而一百就表达为 eachundtachund。到了7世纪，teon这个单词可以和tachund或盎格鲁-撒克逊语中的hund互换使用，因此一百被表示为hund teontig，或者十乘以十。公元7世纪欧洲的普通人对于数字的熟悉程度达不到我们今天的程度。实际上，有能力从一数到九的人才有资格成为法庭上的证人。

也许形成对数字的感觉的最基本的一步并不是数数的能力，而是把数字看成是一个抽象的概念而不是一组特殊事物的附属概念的能力。最早的人类就一定已经意识到四只鸟有别于二只鸟，但是把四只鸟的“四”和四块石头的“四”联系到一起并不是基础的一步。把数字联系到具体事物的性质对于数字感觉的发展是一个很大的障碍。当数字四被作为一个具体单词记录在人的思想里，而与所指的具体事物独立开来时，人们就迈出了发展数字符号体系的第一步，接着就是算术。

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!

计数早期阶段的发展痕迹可以在几种现今仍在使用的语言中找到。根据被数事物的种类，英属哥伦比亚的塔斯马尼亚语言的计数体系包含了七个不同的数字单词组：平面物体和动物、圆形物体和时间、人、长形物体和树、独木舟、尺度、当没有具体的物体用来计数时的数数。看起来最后一组是后来发展形成的，前六组显示了对更早体系的继承。数字名称的变化在一些广泛使用的语言如日语中仍然可以见到。

和数字感觉的发展相关联的是数数能力的发展。数散并不是和数字概念的形成直接相关的，因为通过拿要数的事物与一堆鹅卵石、玉米或者数数者的手指相比较也是可能数数的。对于早期人类而言，这些辅助物是必不可少的，没有这些机械性辅助物，数数就很难进行。出于方便考虑，这些辅助物，尽管不同，在今天的社会中仍然被一些受过最高等教育的人所采用。所有的数数除了与被数的事物相关外，最终都涉及指代物。最初可能是谷物或鹅卵石，而现在是恰好为数字名称的记忆单词序列。

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.

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.

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.