One of the first great intellectual feats of a young child is learning how to talk, closely followed by learning how to count. 小孩子最重要的智力表现之一，就是学习如何讲话，紧随其后的就是学习如何数数。 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. 从很小的年纪开始，我们就与计数体系存在着非常紧密的联系，以至于它已经变成了我们考虑早期人类所面临的问题时用到的重要依据之一，而这些问题产生之时，这一体系尚未存在。 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. 认真地分析一下我们的计数体系，不难得出这样的结论：计数体系不是某个人水到渠成的想象，而是整个人类最伟大、最杰出的成就之一。

It is impossible to learn the sequence of events that led to our developing the concept of number. 我们无法找到某些事件的先后顺序来证明数字概念的进化。 Even the earliest of tribes had a system of numeration that, if not advanced, was sufficient for the tasks that they had to perform. 即便最早期的部落也拥有计数体系，尽管它可能尚不先进，却也足以用来完成他们的各项工作。 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. 我们的祖先很少真正用到数字，比如，在收集食物的过程中，他们用到的大都是“够吗？”，而不是“多少？”这样的概念。 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. 然而，当早期人类开始思考周遭事物特性的时候，他们发现自己需要一种数字的思维以保持思维顺畅。 As they began to settle, grow plants and herd animals, the need for a sophisticated number system became paramount. 随着定居生活的开始，他们开始种植作物并养殖家畜，对复杂完整的数字体系的需求变得极为迫切。 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. 我们永远无法知晓计数体系是在何时以何种方式发展起来的，但是我们可以确定一点，即计数体系在人类形成半定居生活伊始，就已经非常完善了。

Evidence of early stages of arithmetic and numeration can be readily found. 早期算数和计数方式的证据倒是不难获取。 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. Tasmania 的土著人只能数出“一、二、很多”，南非的土著人可以数出“一、二，二和一，两个二，两个二和一”，诸如此类。 But in real situations the number and words are often accompanied by gestures to help resolve any confusion. 但是在真实环境中，手势经常伴随着数字和语言以解决很多歧义。 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. 例如，在使用“一、二、很多”计数体系时，“很多”可能意味着“瞧，我伸出了几根手指就有多少”。 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. 这种原始的方法局限于它能够表达的数字范围，但是这个范围基本上足以解决人类生活的简单方面。

The lack of ability of some cultures to deal with large numbers is not really surprising. 某些文化处理较大数字的能力并不强，这不足为奇。 European languages, when traced back to their earlier version, are very poor in number words and expressions. 欧洲语言，倘若追溯到早期阶段，在表达数字的文字方面也是非常弱的。 The ancient Gothic word for ten, tachund, is used to express the number 100 as tachund. 古代哥特语中的“十”是“tachund”，而“百”是“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. 到了7世 纪 ，teon 这个字可以与 tachund 或者古盎格鲁－撒克逊语的 hund 替换 ，因 而 100 就被描述 成“hund teontig”，意思是十乘十。 The average person in the seventh century in Europe was not as familiar with numbers as we are today. 7 世纪欧洲的普通人对于我们今天使用的数字并不熟悉。 In fact, to qualify as a witness in a court of law a man had to be able to count to nine! 在法庭上，一个人要证明自己有资格出庭作证，就必须要能够数到九。

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. 在数字认知的发展过程中，最重要的阶段也许并不是数数的能力，而是明白一个道理，即数字是一个抽象的概念，而不是只能用来描述某些物品的附属概念。 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. 早期人类的理解应该是这样的：四只鸟与两只鸟不是一回事。然而，把四只鸟的“四”与四块石头的“四”关联起来，就不再是基础阶段的问题了。 Associating a number as one of the qualities of a specific object is a great hindrance to the development of a true number sense. 把数字与具体物品联系起来是数字概念形成的一个巨大障碍。 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. 当“四”这个数字在思维中可以被当成一个独立于某些物品之外的字的时候，具有这种思维的人就可以使第一个阶段向符号化数字的阶段发展了，然后，再向算数阶段发展。

Traces of the very first stages in the development of numeration can be seen in several living languages today. 计数体系最初阶段的痕迹仍存在于当今的某些语言当中。 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. 英属 Columbia 地区的 Tsimshian 语中，计数体系包括 7 套不同的字，使用哪套要取决于所数物品的不同级别：形状平的物品和动物、圆形物品和时间、人、长条形物品和树、木船、量度、和没有具体物品时用到的计数等。 It seems that the last is a later development while the first six groups show the relics of an older system. 似乎是这样：最后这种是最后发展起来的，而前面六种只是古老计数体系的遗留而已。 This diversity of number names can also be found in some widely used languages such as Japanese. 这种数字叫法的多样化在某些广泛使用的语言（如日语）中也还存在。

Intermixed with the development of a number sense is the development of an ability to count. 与数字认知的发展混杂在一起的是数数能力的提高。 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. 数数与数字概念的形成并无直接联系，因为通过把所数物品与石头，玉米粒或者手指匹配在一起，也可以数数。 These aids would have been indispensable to very early people who would have found the process impossible without some form of mechanical aid. 这些物品的辅助对于早期人类来说是必不可缺的，因为他们没有这种机械形式的帮助，就无法完成数数的过程。 Such aids, while different, are still used even by the most educated in today's society due to their convenience. 出于便利性的考虑，这种协助在当今社会受教育程度最高的人群中仍有使用，虽然稍有不同。 All counting ultimately involves reference to something other than the things being counted. 所有数数的行为最终都要以某些非所数物品做参照。 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. 最初，这种参照可能是谷粒或者石块，而现在，它是一个文字序列，只是这个序列刚好是数字的名称而已。