In your schooldays most of you who read this book made acquaintance with the noble building of Euclid's geometry, and you remember — perhaps with more respect than love — the magnificent structure, on the lofty staircase of which you were chased about for uncounted hours by conscientious teachers. By reason of our past experience, you would certainly regard everyone with disdain who should pronounce even the most out-of-the-way proposition of this science to be untrue. But perhaps this feeling of proud certainty would leave you immediately if some one were to ask you: "What, then, do you mean by the assertion that these propositions are true?" Let us proceed to give this question a little consideration.
Geometry sets out form certain conceptions such as "plane," "point," and "straight line," with which we are able to associate more or less definite ideas, and from certain simple propositions (axioms) which, in virtue of these ideas, we are inclined to accept as "true." Then, on the basis of a logical process, the justification of which we feel ourselves compelled to admit, all remaining propositions are shown to follow from those axioms, i.e. they are proven. A proposition is then correct ("true") when it has been derived in the recognised manner from the axioms. The question of "truth" of the individual geometrical propositions is thus reduced to one of the "truth" of the axioms. Now it has long been known that the last question is not only unanswerable by the methods of geometry, but that it is in itself entirely without meaning. We cannot ask whether it is true that only one straight line goes through two points. We can only say that Euclidean geometry deals with things called "straight lines," to each of which is ascribed the property of being uniquely determined by two points situated on it. The concept "true" does not tally with the assertions of pure geometry, because by the word "true" we are eventually in the habit of designating always the correspondence with a "real" object; geometry, however, is not concerned with the relation of the ideas involved in it to objects of experience, but only with the logical connection of these ideas among themselves.
It is not difficult to understand why, in spite of this, we feel constrained to call the propositions of geometry "true." Geometrical ideas correspond to more or less exact objects in nature, and these last are undoubtedly the exclusive cause of the genesis of those ideas. Geometry ought to refrain from such a course, in order to give to its structure the largest possible logical unity. The practice, for example, of seeing in a "distance" two marked positions on a practically rigid body is something which is lodged deeply in our habit of thought. We are accustomed further to regard three points as being situated on a straight line, if their apparent positions can be made to coincide for observation with one eye, under suitable choice of our place of observation.
If, in pursuance of our habit of thought, we now supplement the propositions of Euclidean geometry by the single proposition that two points on a practically rigid body always correspond to the same distance (line-interval), independently of any changes in position to which we may subject the body, the propositions of Euclidean geometry then resolve themselves into propositions on the possible relative position of practically rigid bodies.1) Geometry which has been supplemented in this way is then to be treated as a branch of physics. We can now legitimately ask as to the "truth" of geometrical propositions interpreted in this way, since we are justified in asking whether these propositions are satisfied for those real things we have associated with the geometrical ideas. In less exact terms we can express this by saying that by the "truth" of a geometrical proposition in this sense we understand its validity for a construction with rule and compasses.
Of course the conviction of the "truth" of geometrical propositions in this sense is founded exclusively on rather incomplete experience. For the present we shall assume the "truth" of the geometrical propositions, then at a later stage (in the general theory of relativity) we shall see that this "truth" is limited, and we shall consider the extent of its limitation.
Notes
1) It follows that a natural object is associated also with a straight line. Three points A, B and C on a rigid body thus lie in a straight line when the points A and C being given, B is chosen such that the sum of the distances AB and BC is as short as possible. This incomplete suggestion will suffice for the present purpose.
译:
在你们的学生时代，大多数读过这本书的人都领略了欧几里得几何学的宏伟，你们还记得——也许是出于尊重而不是热爱——那座宏伟的建筑，在这座高大的楼梯上，你们被尽责的老师追了好几个小时。根据我们过去的经验，凡是对这门科学提出哪怕是质疑的人(哪怕是最偏僻的主张)，你一定会瞧不起的。
但是，如果有人问你:“那么，你断言这些命题一定是真的吗?” 也许之前的自信就会离开你。让我们开始考虑一下这个问题。
几何学从“平面”、“点”和“直线”等概念出发，我们可以将这些概念联系起来或多或少确定的概念，并从某些简单的命题（公理）中得出，凭借这些概念，我们倾向于接受为“真”。然后，在一个逻辑过程的基础上，我们不得不承认的理由是，所有剩余的命题都是从这些公理出发的，也就是说，它们被证明了。当一个命题以公认的方式从公理中推导出来时，它就是正确的（“真”）。因此，单个几何命题的“真”问题就被归结为公理的“真”问题之一。现在人们早就知道，最后一个问题不仅用几何方法无法回答，而且它本身完全没有意义。我们不能问是否只有一条直线穿过两点。我们只能说，欧几里德几何学处理的是所谓的“直线”，每一条直线都被归因于由位于直线上的两点唯一决定的性质。“真”的概念与纯几何的断言不符，因为用“真”这个词，我们最终习惯于总是指定与“真实”对象的对应关系；然而，几何并不关心它所涉及的思想与经验对象的关系，但只有这些想法之间的逻辑联系。
我们不难理解为什么尽管如此，我们还是觉得必须把几何学的命题称为“真”。几何思想与自然界中或多或少精确的物体相对应，而最后这些无疑是这些观念产生的唯一原因。几何学应该避免这样一个过程，以使其结构具有最大可能的逻辑统一性。例如，在“远处”看到一个实际上是刚性物体上的两个标记位置，这种做法深深地植根于我们的思维习惯中。我们还习惯于把三个点视为一条直线，只要在适当选择观察地点的情况下，使它们的明显位置与一只眼睛的观察相吻合。
如果按照我们的思维习惯，我们现在补充欧几里德几何的命题，用一个命题来补充欧几里德几何学命题，即一个实际刚体上的两点始终对应于相同的距离（线间距），而不受我们可能使物体受其支配的位置发生任何变化，然后，欧几里德几何的命题分解为关于实际刚体可能的相对位置的命题。1）以这种方式补充的几何，则被视为物理学的一个分支。我们现在可以合法地问这样解释的几何命题的“真理”，因为我们有理由问这些命题是否满足于我们与几何思想相关联的那些真实事物。更确切地说，我们可以通过这样一个几何命题的“真”来表达这一点，在这个意义上，我们理解它对于一个有规则和圆规的结构的有效性。
当然，在这个意义上，几何命题的“真理”的信念完全建立在相当不完整的经验基础上。现在我们假设几何命题的“真”，然后在以后的阶段（在广义相对论中），我们将看到这个“真理”是有限的，我们将考虑它的局限性。