那么，什么是希尔伯特空间呢？在典型的高校数学课程中，它被定 义为“完备的内积空间”。修读这样一门课程的学生，理应从先修课程中 了解到，所谓“内积空间”是指配备了内积的向量空间，而所谓“完备”是 指空间中任意柯西列都收敛。当然，要想理解这样的定义，学生还必须 知道“向量空间”、“内积”、“柯西列”和“收敛”的定义。就拿其中一个举 例来说（这还并不是最长的一个）：序列x1 ，x2 ，x3 ，…若满足对于 任意正数ε，总存在整数N，使得对于任意大于N的整数p和q，xp 与xq 间 的距离不大于ε，则称这个序列为柯西列。
What, then, is a Hilbert space? In a typical university mathematics course it is defined as acomplete inner-product space. Students attending such a course are expected to know, fromprevious courses, that an inner-product space is a vector space equipped with an inner product, and that a space is complete if every Cauchy sequence in it converges.
模型
利用牛顿物理学和微积分的一些初步知识，可以计算得到最佳的折 中方案——石头离手时应与地面呈45度夹角。就这个问题而言，这基本 上是最简洁优美的答案了。同样的计算还可以告诉我们石头在空中的飞 行轨迹是个抛物线，甚至还能得出脱手后在空中任意时刻的速度有多 大。
看起来，科学与数学相结合能够使我们预测石块飞出去直至落地之 前的一切行为。然而，只有在我们作了许多的简化假设之后才能够如 此。其中最主要的假设是，作用在石头上的只有一种力，即地球的引 力，而且这种力的大小及方向在各处总是一样的。但实际上并非如此， 因为它忽略了空气阻力、地球自转，也没有计入月球的微弱引力，而且 越到高处地球引力越小，在地球表面上“垂直向下”的方向也随着具体位 置的不同而逐渐变化。即使你能够接受上述计算，45度角的结果也基于 另一个隐含假设：石头离手的初始速度与夹角无关。这也是不正确的： 实际上夹角越小，人能使上的力气越大。
上述这些缺陷的重要性各有不同，我们在计算和预测中应该采取怎 样的态度来对待这些偏差呢？把所有因素全部考虑在内进行计算固然是 一种办法，但还有一种远为明智的办法：首先决定你需要达到什么样的 精确度，然后用尽可能简单的办法达到它。如果经验表明一项简化的假 设只会对结果产生微不足道的影响，那就应当采取这样的假设。
The best compromise, which can be worked out using a combination of Newtonian physics andsome elementary calculus, turns out to be as neat as one could hope for under thecircumstances: the direction of the stone as it leaves your hand should be upwards at an angleof 45 degrees to the horizontal. The same calculations show that the stone will trace out aparabolic curve as it flies through the air, and they tell you how fast it will be travelling atany given moment after it leaves your hand.
It seems, therefore, that a combination of science and mathematics enables one to predict theentire behaviour of the stone from the moment it is launched until the moment it lands.However, it does so only if one is prepared to make a number of simplifying assumptions, themain one being that the only force acting on the stone is the earth’s gravity and that thisforce has the same magnitude and direction everywhere. That is not true, though, because itfails to take into account air resistance, the rotation of the earth, a small gravitationalinfluence from the moon, the fact that the earth’s gravitational field is weaker the higher youare, and the gradually changing direction of ‘vertically downwards’ as you move from one partof the earth’s surface to another. Even if you accept the calculations, the recommendation of45 degrees is based on another implicit assumption, namely that the speed of the stone as itleaves your hand does not depend on its direction. Again, this is untrue: one can throw astone harder when the angle is flatter.
In the light of these objections, some of which are clearly more serious than others, what attitude should one take to the calculations and the predictions that follow from them? One approach would be to take as many of the objections into account as possible. However, a much more sensible policy is the exact opposite: decide what level of accuracy you need, and then try to achieve it as simply as possible. If you know from experience that a simplifying assumption will have only a small effect on the answer, then you should make that assumption.
可能有人会反对这种模型，他们会说，骰子在滚动时是遵循牛顿定 律的，至少在很高的精度上遵循，因此骰子落地的情况根本不是随机 的：原则上是完全能够被计算出来的。但是，“原则上”这个短语在这里 被过度使用了，因为这样的计算将会是
极端复杂的，并且需要知道骰子的形状、材料、初始速度、旋转速度等 更为精确的信息，而这般精确的信息在实际中是根本无法测出来的。因 为这一点，使用某种更为复杂的决定论模型是无论如何也不会有任何优 势的。
One might object to this model on the grounds that the dice, when rolled, are obeying Newton’s laws, at least to a very high degree of precision, so the way they land is anything but random: indeed, it could in principle be calculated. However, the phrase ‘in principle’ is being overworked here, since the calculations would be extraordinarily complicated, and would need to be based on more precise information about the shape, composition, initial velocities, and rotations of the dice than could ever be measured in practice. Because of this, there is no advantage whatsoever in using some more complicated deterministic model.
假设我们已知2002年初的总人口是p。根据上述模型，2002年的出 生人数和死亡人数将分别为bp和dp，因此2003年初的总人口将为p+bpdp=（1+b-d）p。其他年份亦然，因此我们就能够写出公式p（n+1） =（1+b-d）p（n），意即n+1年年初的人口是n年年初人口乘以（1+bd）。换句话说，每一年人口数量都会乘上（1+b-d）。那么20年后的人 口就是乘以（1+b-d）2 0 ，于是就得出了初始问题的答案。
Suppose we know that the population at the beginning of the year 2002 is P. According to the model just defined, the number of births and deaths during the year will be bP and dP respectively, so the population at the beginning of 2003 will be P + bP − dP = (1 + b − d ) P.This argument works for any year, so we have the formula P ( n + 1) = (1 + b − d ) P ( n ),meaning that the population at the beginning of year n + 1 is (1 + b − d ) times the population at the beginning of year n. In other words, each year the population multiplies by (1 + b − d ). It follows that in 20 years it multiplies by (1 + b − d)^20, which gives an answer to our original question.
因此，通过引入其他因素来使模型复杂化，这个想法相当诱人。我 们记出生率和死亡率分别为b（t）和d（t），使其可以随时间变化。我 们并不想单用一个数字p（t）来表示总人口，我们可能想要知道不同年 龄层各有多少人。如果同时还能尽可能多地知道各个年龄层的社会态度 和行为倾向，也会对预测未来的出生率和死亡率有所帮助。获取这样的 统计信息是十分昂贵且困难的，但这些信息确实能够大幅提高预测的精 度。因此，没有一种模型能够脱颖而出，声称比其他模型都好。关于社 会和政治的变迁，谁也不可能确切地说出情况会是什么样子。关于某种 模型，我们所能提出的合理问题，最多只能是问某种有条件的预测，也 就是说模型只能告诉我们，这样的社会或政治变迁如果发生的话会产生 怎样的影响。
It is therefore tempting to complicate the model by introducing other factors. One could have birth and death rates b ( t ) and d ( t ) that varied over time. Instead of a single number P (t ) representing the size of the population, one might also like to know how many people there are in various age groups. It would also be helpful to know as much as possible about social attitudes and behaviour in these age groups in order to predict what future birth anddeath rates are likely to be. Obtaining this sort of statistical information is expensive and difficult, but the information obtained can greatly improve the accuracy of one’s predictions.For this reason, no single model stands out as better than all others. As for social and political changes, it is impossible to say with any certainty what they will be. Therefore the most that one can reasonably ask of any model is predictions of a conditional kind: that is, ones that tell us what the effects of social and political changes will be if they happen.
N个分子完全相互独立运动的假设毫无疑问是过度简化的。比方 说，利用这个模型，我们就不可能理解，为什么当温度足够低时气体会液化：当你把模型中的各点运动速率降低，得到的还是相同的模型，无 非跑得慢一些而已。不过这个模型还是能够解释真实气体的许多行为。 例如，想象盒子被慢慢压缩的情形。分子仍然会继续以相同速率运动， 但由于盒子变小，分子撞击壁面更加频繁，可供撞击的壁面面积也变小 了。由于这两个缘故，单位面积的壁面每秒钟被撞击次数就增多了。这 些撞击正是气体压强的来源，于是我们可以总结出，气体体积减小时， 气体压强很可能增大——正如实际观测所证实的那样。类似的论证还可 以解释，为什么气体温度升高而体积不变时，压强会增大。要推算出压 强、温度与体积之间的数值关系也并不困难。
The assumption that our N molecules move entirely independently of one another is quite definitely an oversimplification. For example, it means that there is no hope of using this model to understand why a gas becomes a liquid at sufficiently low temperatures: if you slow down the points in the model you get the same model, but running more slowly. Nevertheless, it does explain much of the behaviour of real gases. For example, imagine what would happen if we were gradually to shrink the box. The molecules would continue to move at the same speed, but now, because the box was smaller, they would hit the walls more often and there would be less wall to hit. For these two reasons, the number of collisions per second in any given area of wall would be greater. These collisions account for the pressure that a gas exerts, so we can conclude that if you squeeze a gas into a smaller volume, then its pressure is likely to increase – as is confirmed by observation. A similar argument explains why, if you increase the temperature of a gas without increasing its volume, its pressure also increases.
And it is not too hard to work out what the numerical relationships between pressure,temperature, and volume should be.
然而，当p和q非常大时——比方说都是200位的素数，那么这一试 错过程会耗时极长，即使借助于强力计算机也是如此。（如果你想要体 会一下这种困难，不妨尝试找出6901的两个素因子，以及280 123 的。）可另一方面，似乎也不难感觉到，说不定这个问题存在着更聪明 的解决办法，基于它就可以编制出一种快速运转的计算机程序。如果能 找到这种好办法，我们就能破解作为大部分现代安全系统之基石的密 码，包括在互联网上以及其他各处——破解这些密码的难点就在大整数 的因子分解。反之，如果能够表明由pq计算出p和q的这种快速有效的方 法不存在的话，我们则能够安心。不幸的是，虽然计算机总在不断地让 我们惊叹它的各种能力，对于它们做不了的，我们却几乎毫无了解。
If, however, p and q are very large – with 200 digits each, say – then this process of trial and error takes an unimaginably long time, even with the help of a powerful computer. (If you want to get a feel for the difficulty, try finding two prime numbers that multiply to give 6901and another two that give 280123. ) On the other hand, it is not inconceivable that there is a much cleverer way to approach the problem, one that might be used as the basis for acomputer program that does not take too long to run. If such a method could be found, it would allow one to break the codes on which most modern security systems are based, on the Internet and elsewhere, since the difficulty of deciphering these codes depends on the difficulty of factorizing large numbers. It would therefore be reassuring if there were some way of showing that a quick, efficient procedure for calculating p and q from their product pq does not exist. Unfortunately, while computers continually surprise us with what they can be used for, almost nothing is known about what they cannot do.
对电路模型进行一点小小的修改，我们就能得到大脑的一种有用的
模型。这种模型不再使用01序列，而是使用0和1之间的任意值来表示强 度各异的信号。所有的门，即对应于神经元或者脑细胞，也有所不同， 但其行为还是很简单的。每个门从其他的门接收到一些信号，如果这些 信号的总强度——对应数字的总和——足够大，门就在某个特定的强度 水平上输出它自己的信号，否则不输出。这对应于神经元所作的是 否“激发”的决策。
A small modification to the circuit model leads to a useful model of the brain. Now, instead of 0s and 1s, one has signals of varying strengths that can be represented as numbers between 0 and 1. The gates, which correspond to neurons, or brain cells, are also different, but they still behave in a very simple way. Each one receives some signals from other gates. If the total strength of these signals – that is, the sum of all the corresponding numbers – is large enough, then the gate sends out its own signals of certain strengths. Otherwise, it does not.This corresponds to the decision of a neuron whether or not to ‘fire’.
似乎很难相信这个模型能够捕捉大脑全部的复杂性，但这部分缘于 我并没有提到应当有多少个门以及如何安排这些门。一个典型的人类大 脑包含大约1000亿个神经元，它们以非常复杂的方式排列着；以我们当 前对大脑的认识，还不可能谈及太多——至少在精微的细节方面不可能 说清。不过，上述模型提供了一种有益的理论框架，供我们思考大脑可 能是如何工作的，也使我们能够模拟某些类似于大脑运行的行为。
It may seem hard to believe that this model could capture the full complexity of the brain.However, that is partly because I have said nothing about how many gates there should be or how they should be arranged. A typical human brain contains about 100 billion neurons arranged in a very complicated way, and in the present state of knowledge about the brain it is not possible to say all that much more, at least about the fine detail. Nevertheless, them odel provides a useful theoretical framework for thinking about how the brain might work, and it has allowed people to simulate certain sorts of brain-like behaviour.
这两个问题看似截然不同，但一种合理的模型能够说明，从数学的 观点来看它们其实是一样的。在这两个问题中，都需要给一些对象（国 家、课程）赋予一些属性（颜色、时间）。对象中有某些两两组合（相 邻的国家，不能冲突的课程）是不能相容的，也就是说它们不能被赋予 相同的属性。在这两个问题中，我们其实并不关心具体的对象是什么、 要赋予的属性是什么，所以我们也可以仅用点来表示它们。为了表示那 些不相容的成对的点，我们可以将它们用线段连结起来。这样一组边和 边连结起来的点的集合，就是“图”这种数学结构。图5给出了一个简单 的例子。通常称图中的点为顶点，称线段为边。
These two problems appear to be quite different, but an appropriate choice of model shows that from the mathematical point of view they are the same. In both cases there are some objects (countries, modules) to which something must be assigned (colours, times). Some pairs of objects are incompatible (neighbouring countries, modules that must not clash) in the sense that they are not allowed to receive the same assignment. In neither problem do we really care what the objects are or what is being assigned to them, so we may as well just represent them as points. To show which pairs of points are incompatible we can link them with lines. A collection of points, some of which are joined by lines, is a mathematical structure known as a graph. It is customary to call the points in a graph vertices, and the lines edges.
数与抽象
《泰晤士报文学增刊》的一篇评论在开篇写到：
已知0×0=0以及1×1=1，就可以得到：平方等于自身的数是存在的。但是再进 一步，我们就可以得到：数是存在的。经过这简简单单朴实无华的一步，我们似乎 就从一个基本的算术式得到了一个令人吃惊的、极具争议的哲学结论：数是存在 的。你可能还以为这有多么困难呢。
A few years ago, a review in the Times Literary Supplement opened with the following paragraph:
Given that 0 × 0 = 0 and 1 × 1 = 1, it follows that there are numbers that are their own squares. But then it follows in turn that there are numbers. In a single step of artless simplicity, we seem to have advanced from a piece of elementary arithmetic to a startling and highly controversial philosophical conclusion: that numbers exist. You would have thought that it should have been more difficult.
如果我们把同样的论证方法搬到国际象棋上来，这个“简简单单朴 实无华”的关于数的存在的论证就明显荒谬了。已知象棋中的黑色国王 有时可以斜向移动一格，由此得出，存在着可以斜向移动一格的棋子。 但接下来将进而得出：国际象棋的棋子是存在的。当然了，我想说的并 不是那句平平常常的断言：人们有时候会制造国际象棋的棋具。毕竟， 没有棋具同样也能下棋。我所指的是一个更加“惊人”的哲学结论：象棋 棋子的存在独立于它们的物理形态。
The absurdity of the ‘artlessly simple’ argument for the existence of numbers becomes very clear if one looks at a parallel argument about the game of chess. Given that the black king, in chess, is sometimes allowed to move diagonally by one square, it follows that there are chess pieces that are sometimes allowed to move diagonally by one square. But then it follows in turn that there are chess pieces. Of course, I do not mean by this the mundane statement that people sometimes build chess sets – after all, it is possible to play the game without them – but the far more ‘startling’ philosophical conclusion that chess pieces exist independently of their physical manifestations.
看出这一点很有意思——尽管我的论述并不直接依赖于它：象棋， 或者任何类似的游戏，都可以以图为模型。（图已经在前一章末定义过 了。）图的顶点代表游戏的某种可能的局面。如果两个顶点P和Q有边 相连，那就意味着可以从局面P出发，经过合乎规则的一步之后达到局 面Q。因为有可能无法从Q返回到P，所以这样的边需要用箭头来指示方 向。某些顶点可以看作白棋获胜，还有某些顶点可以看作黑棋获胜。游 戏从一个特定顶点，即游戏的开始局面出发。两位棋手相继沿着边移 动。
It is amusing to see, though my argument does not depend on it, that chess, or any other similar game, can be modelled by a graph. (Graphs were defined at the end of the previous chapter. ) The vertices of the graph represent possible positions in the game. Two vertices P and Q are linked by an edge if the person whose turn it is to play in position P has a legal move that results in position Q. Since it may not be possible to get back from Q to P again, the edges need arrows on them to indicate their direction. Certain vertices are considered wins for white, and others wins for black. The game begins at one particular vertex, corresponding to the starting position of the game. Then the players take turns to move forwards along edges. The first player is trying to reach one of white’s winning vertices, and the second one of black’s. A far simpler game of this kind is illustrated in Figure 6. (It is not hard to see that for this game white has a winning strategy. )
不过在定义它的时候，我完全没有提及关于 棋子的任何事情。从这个角度来看，黑色国王是否存在的问题就是个离 奇的问题了：棋盘和棋子只不过是方便我们将这么大的图中一团乱麻似 的顶点和边组织起来的一种原则而已。如果我们说出“黑色国王被将军 了”这样的句子，那么这只是一种简化的说法，它所指的无非是两位棋 手到达了极多的顶点中的某一个。
And yet, when I defined it I made no mention of chess pieces at all. From this perspective, it seems quite extraordinary to ask whether the black king exists: the chessboard and pieces are nothing more than convenient organizing principles that help us think about the bewildering array of vertices and edges in a huge graph. If we say something like, ‘The black king is in check’, then this is just an abbreviation of a sentence that specifies an extremely long list of vertices and tells us that the players have reached one of them.
然而，当我们考虑更大的数时，其中的纯粹性就变少了。图8向我 们表示了7，12和47这几个数。可能有部分人能够立刻把握第一张图中 的“七性”，但大多数人可能会在一瞬间有这样的思考：“外围的点形成 了一个六边形，再加上中心的一个得到6+1=7。”类似地，12可能会被考 虑为3×4或2×6。至于47，和别的数字比起来，比方和46相比，一组这个 数量的物体就缺乏特别之处。如果这组物体以某种模式排列起来，例如 排成少两个点的7×7的阵列，那么我们的知识就可以通过7×7-2=49-2=47 迅速地得出总数一共有多少。如若不然，我们就只能去一个个地数，这 时我们则是将47视为46之后的那个数，而46又是45之后的那个数，依此 类推。
Perhaps some people instantly grasp the sevenness of the first picture, but in most people’s minds there will be a fleeting thought such as, ‘The outer dots form a hexagon, so together with the central one we get 6 + 1 = 7. ’ Likewise, 12 will probably be thought of as 3 × 4, or 2 × 6. As for 47, there is nothing particularly distinctive about a group of that number of objects, as opposed to, say, 46. If they are arranged in a pattern, such as a 7 × 7 grid with two points missing, then we can use our knowledge that 7 × 7 − 2 = 49 − 2 = 47 to tell quickly how many there are. If not, then we have little choice but to count them, this time thinking of 47 as the number that comes after 46, which itself is the number that comes after 45, and so on.
也就是说，当数字变得还不算太大时，我们就已经不再将其视作一 些独立的客体了，而开始通过它们的内在属性，它们与其他数字的关 联，以及它们在数系 中的作用来理解。这也就是我之前说数能“做”什么 所要表达的意思。
In other words, numbers do not have to be very large before we stop thinking of them as isolated objects and start to understand them through their properties, through how they relate to other numbers, through their role in a number system. This is what I mean by what a number ‘does’.
一个数系并不仅仅是一堆数字，而是由数字 及算术规则共同构成的。我们还可以这样来总结这种抽象方法：考虑规 则，而不是考虑数字本身。按这种观点，数字就可以被当作某种游戏中 的记号（或许应该被称为计数子）。
A number system is not just a collection of numbers but a collection of numbers together with rules for how to do arithmetic. Another way to summarize the abstract approach is: think about the rules rather than the numbers themselves. The numbers, from this point of view, are tokens in a sort of game (or perhaps one should call them counters).
从非抽象的角度出发，可能会这样去论证：0×2的意思是指，总共0
个2相加 ，没有2，就是0。但用这种思考方式，我们将不太容易回答诸 如我儿子约翰问我的这个问题（在他六岁时）：既然无和无相乘的意思 是没有 无，那为什么结果又会是无呢？
An alternative, non-abstract argument might be something like this: ‘0 × 2 means add up no twos, and if you do that you are left with nothing, that is, 0. ’ But this way of thinking makes it hard to answer questions such as the one asked by my son John (when six): how can nought times nought be nought, since nought times nought means that you have no noughts?
为什么在很多人看来，负数的实在性要低于正数呢？大概因为对数 量不多的物体的计数是人类的基本活动，在这其中并不会用到负数。但 这只不过意味着，作为模型的自然数系在某种特定场合下比较有用，而 扩充数系则不太用得上。但如果我们考虑温度、日期或者银行账户，那 负数就的确 能够发挥作用了。只要扩充数系是逻辑自洽的——实际上 也正是如此，用它作为模型就没有任何害处。
Why does it seem to many people that negative numbers are less real than positive ones?Probably it is because counting small groups of objects is a fundamental human activity, and when we do it we do not use negative numbers. But all this means is that the natural number system, thought of as a model, is useful in certain circumstances where an enlarged number system is not. If we want to think about temperatures, or dates, or bank accounts, then negative numbers do become useful. As long as the extended number system is logicallyconsistent, which it is, there is no harm in using it as a model.
从用来描述数系每次扩充所得到的新数的名字，我们能够发现在历 史上对抽象方法质疑的一些痕迹，比如“负的”和“无理的”。但更让人难 以下咽的还在后面，这就是“虚幻的”，或者“复杂的”数，即形如a+bi的 数，其中a和b均为实数，i是-1的平方根。
i和 之间最主要的区别就是我们被迫 抽象地去思考i，而对于 我们则还有备选方案，可以将它具体地表示为1.4142…，或者看作单位 正方形的对角线长度。要看出为什么i没有这样的表示方法，不妨问问 自己这个问题：-1的两个平方根中，哪个是i哪个是-i呢？这个问题是没 有意义的，因为我们对i所定义的唯一 的性质就是平方等于-1。既然-i也 有同样的性质，那么关于i成立的那些命题，如替换为关于-i的相应命 题，必定依然成立。一旦领会了这一点，就很难再赞同i指示一个独立 存在的实在的客体。
但对数学家来说，复数系已经必不可 少。对科学家和工程师同样也是。比如，量子力学的理论就高度依赖于 复数。复数作为最佳的例证之一，向我们表明了一条概括性原则：一种 抽象的数学构造若是充分自然的，则基本上必能作为模型找到它的用 途。
but the complex number system has become indispensable to mathematicians, and to scientists and engineers as well – the theory of quantum mechanics, for example, depends heavily on complex numbers. It provides one of the best illustrations of ageneral principle: if an abstract mathematical construction is sufficiently natural, then it will almost certainly find a use as a model.
Once one has learned to think abstractly, it can be exhilarating, a bit like suddenly being ableto ride a bicycle without having to worry about keeping one’s balance. However, I do not wishto give the impression that the abstract method is like a licence to print money. It isinteresting to contrast the introduction of i into the number system with what happens whenone tries to introduce the number infinity. At first there seems to be nothing stopping us:infinity should mean something like 1 divided by 0, so why not let ∞ be an abstract symboland regard it as a solution to the equation 0 x = 1?
The trouble with this idea emerges as soon as one tries to do arithmetic. Here, for example, isa simple consequence of M2, the associative law for multiplication, and the fact that 0 × 2 = 0.
1 = ∞ × 0 = ∞ ×(0 × 2) = ( ∞ × 0) × 2 = 1 × 2 = 2
What this shows is that the existence of a solution to the equation 0 x = 1 leads to aninconsistency. Does that mean that infinity does not exist? No, it simply means that no naturalnotion of infinity is compatible with the laws of arithmetic. It is sometimes useful to extendthe number system to include the symbol ∞, accepting that in the enlarged system these lawsare not always valid.
But with this as adefinition, it is not easy to interpret an expression such as 2^ 3/2 , since you cannot take oneand a half twos and multiply them together. What is the abstract method for dealing with aproblem like this? Once again, it is not to look for intrinsic meaning – in this case ofexpressions like a^n– but to think about rules.
关于指数运算的两条基本规则是：
E1 对任意实数a，a1 =a。 E2 对任意实数a和任意一对自然数m、n，有am + n =am ×an 。
从上述两条规则出发，我们可以迅速重新得到已经知道的一些事 实。比如，根据E2 即知a2 =a1 + 1 等于a1 ×a1 ；再根据E1 ，此即为a×a， 正如我们所了解的。除此以外，我们现在还能够做更多的事情。让我们 用x来表示23 / 2 。那么x×x=23 / 2 ×23 / 2 ，由E2得知它就是23 / 2 + 3 / 2 =23 =8。也就是说x2 =8。这并没有完全确定下x，因为8有两个平方根。所 以我们通常会采取如下的准则。
E3 如果a＞0且b是实数，那么ab 为正数。
对数是另一个抽象地来看会变得更加容易的概念。关于对数，我在 本书中要说的不多。但如果它确实困扰你，那么你可以消除顾虑，只要 了解它们遵循如下三条规则就足以使你去应用对数了。（如果你希望对 数是以e为底而不是以10为底的，只需要在L1 中把10替换为e即可。）
L1 log（10）=1。 L2 log（xy）=log（x）+log（y）。 L3 若x＜y，则log（x）＜log（y）。
例如，要得到log（30）小于 ，可以应用L1 和L2 得出 log（1000）=log（10）+log（100）=log（10）+log（10）+log（10） =3。而由L2 得出2log（30）=log（30）+log（30）=log（900），又 由L3 得到log（900）＜log（1000）。因此2log（30）＜3，即得 log（30）＜32。
证明
虽然如此，争论在原则上 必然能够解决这一事实的确使数学独一无二。没有任何一个学科像数学一样具有这一特性：有些天文学家仍然固守着宇宙的稳态理论；关于自然选择究竟有多大的解释力，生物学家 各自都抱有极为不同的坚定信念；关于意识与物质世界的关系，哲学家 们具有根本性的分歧；经济学家也追随着观点截然相反的不同学派，如 货币主义和新凯恩斯主义。
Nevertheless, the fact that disputes can in principle be resolved does make mathematics unique. There is no mathematical equivalent of astronomers who still believe in the steady-state theory of the universe, or of biologists who hold, with great conviction, very different views about how much is explained by natural selection, or of philosophers who disagreefundamentally about the relationship between consciousness and the physical world, or of economists who follow opposing schools of thought such as monetarism and neo-Keynesianism.
理解前面“在原则上”这个短语是很重要的。没有哪个数学家愿意费 心写出证明的完整细节——从基本公理开始，仅通过最明显、最易于检 查的步骤来推导出结果。即使可行，也并不必要：数学论文是写给经过 严格训练的读者的，无须事无巨细详细说明。而如果某人提出一个重要 的论断，其他数学家发觉难以理解其证明，他们就会要求作者详细解 释。这时，把证明步骤分解为较易理解的小步骤的过程就会开始。同样 因为听众都是经过严格训练的，这个过程通常不需要进行太久，只要给 出了必要的解释或者发现了其中的错误就可以了。因此，对某个结果的 一个证明如果的确是 正确的，那几乎总会被数学家当作是正确的。
It is important to understand the phrase ‘in principle’ above. No mathematician would ever bother to write out a proof in complete detail – that is, as a deduction from basic axioms using only the most utterly obvious and easily checked steps. Even if this were feasible it would be quite unnecessary: mathematical papers are written for highly trained readers who do not need everything spelled out. However, if somebody makes an important claim and other mathematicians find it hard to follow the proof, they will ask for clarification, and the process will then begin of dividing steps of the proof into smaller, more easily understood substeps. Usually, again because the audience is highly trained, this process does not need to go on for long until either the necessary clarification has been provided or a mistake comes to light. Thus, a purported proof of a result that other mathematicians care about is almost always accepted as correct only if it is correct.
这里有个著名的难题。画八横八纵正方形网格，去掉相对的两个 角。你能用多米诺骨牌形状的地砖——每一块正好覆盖两个相邻方格， 把剩余部分覆盖吗？我在图中（图14）表明，如果用四横四纵来代替八 横八纵，你是办不到的。假设你决定用一块地砖覆盖我图中标为A的区 域，那么容易看出，你必须还要把地砖放到B、C、D和E的位置上，剩 下一个小方格无法覆盖。既然右上角的格子无论如何总要覆盖住，而仅 剩的另一种覆盖的方式也会导致类似的问题（通过位置的对称关系）， 所以覆盖整个图形是不可能的。
极限与无穷
It is an accepted truth of mathematics that one point nine recurring equals two, but this truth was not discovered by some process of metaphysical reasoning. Rather, it is a convention. However, it is by no means an arbitrary convention, because not adopting it forces one either to invent strange new objects or to abandon some of the familiar rules of arithmetic.
Do you get ‘infinity minus one’ noughts followed by a one? What is the decimal expansion of 1/3? Now multiply that number by 3. Is the answer 1 or 0. 999999. . . ? If you follow the usual convention, then tricky questions of this kind do not arise. (Tricky but not impossible: a coherent notion of ‘infinitesimal’ numbers was discovered by Abraham Robinson in the 1960s, but non-standard analysis, as his theory is called, has not become part of the mathematical mainstream. )
Notice that what we have done is to ‘tame’ the infinite, by interpreting a statement that involves infinity as nothing more than a colourful shorthand for a more cumbersome statement that doesn’t. The neat infinite statement is ‘ x is an infinite decimal that squares to 2’. The translation is something like, ‘There is a rule that, for any n, unambiguously determines the n th digit of x. This allows us to form arbitrarily long finite decimals, and their squares can be made as close as we like to 2 simply by choosing them long enough. ’
It is hard work to define addition and multiplication of infinite decimals without mentioning infinity, and one must check that the resulting complicated definitions obey the rules set out in Chapter 2, such as the commutative and associative laws. However, once this has been done, we are free to think abstractly again. What matters about x is that it squares to two. What matters about the word ‘squares’ is that its meaning is based on some definition of multiplication that obeys the appropriate rules. It doesn’t really matter what the trillionth digit of x is and it doesn’t really matter that the definition of multiplication is somewhat complicated.
Our problems begin when we start to look at shapes with curved boundaries. It is not possible to cut a circle up into a few triangles. So what are we talking about when we say that its area is πr^ 2 ?
This is another example where the abstract approach is very helpful. Let us concentrate not on what area is, but on what it does. This suggestion needs to be clarified since area doesn’t seem to do all that much – surely it is just there. What I mean is that we should focus on the properties that any reasonable concept of area will have.