On the Hypotheses which lie at the Bases of Geometry: the Plan and the Application to Space

Plan of the Investigation研究计划#

It is known that geometry assumes, as things given, both the notion of space and the first principles of constructions in space. She gives definitions of them which are merely nominal, while the true determinations appear in the form of axioms. The relation of these assumptions remains consequently in darkness; we neither perceive whether and how far their connection is necessary, nor, a priori, whether it is possible.

众所周知，几何学把空间的概念和在空间中作图的基本原理，都当作已知的东西来假定。它给它们下的定义只是名义上的，而真正的规定则以公理的形式出现。因此，这些假定之间的关系始终处在黑暗之中：我们既看不出它们之间的联系是否必然、在多大程度上必然，也看不出先验地说，这种联系是否可能。

From Euclid to Legendre (to name the most famous of modern reforming geometers) this darkness was cleared up neither by mathematicians nor by such philosophers as concerned themselves with it. The reason of this is doubtless that the general notion of multiply extended magnitudes (in which space-magnitudes are included) remained entirely unworked. I have in the first place, therefore, set myself the task of constructing the notion of a multiply extended magnitude out of general notions of magnitude. It will follow from this that a multiply extended magnitude is capable of different measure-relations, and consequently that space is only a particular case of a triply extended magnitude. But hence flows as a necessary consequence that the propositions of geometry cannot be derived from general notions of magnitude, but that the properties which distinguish space from other conceivable triply extended magnitudes are only to be deduced from experience. Thus arises the problem, to discover the simplest matters of fact from which the measure-relations of space may be determined; a problem which from the nature of the case is not completely determinate, since there may be several systems of matters of fact which suffice to determine the measure-relations of space—the most important system for our present purpose being that which Euclid has laid down as a foundation. These matters of fact are—like all matters of fact—not necessary, but only of empirical certainty; they are hypotheses. We may therefore investigate their probability, which within the limits of observation is of course very great, and inquire about the justice of their extension beyond the limits of observation, on the side both of the infinitely great and of the infinitely small.

从欧几里得到勒让德（举出近代改革几何学的人中最著名的一位），无论是数学家，还是关心这个问题的哲学家，都没有把这片黑暗驱散。原因无疑在于：多重广延量（空间量包括在其中）的一般概念，始终完全没有得到研究。因此，我首先给自己定下的任务，是从量的一般概念出发，构造出多重广延量的概念。由此将会得出：一个多重广延量可以有不同的度量关系，因而空间只是三重广延量的一个特例。而由此必然推出：几何学的命题不能从量的一般概念中推导出来；那些使空间区别于其他可以设想的三重广延量的性质，只能从经验中得出。于是产生了这样一个问题：要找出能够确定空间度量关系的最简单的事实。就问题的本性而言，它不是完全确定的，因为可能有好几个事实体系都足以确定空间的度量关系——对我们眼下的目的来说，最重要的体系是欧几里得作为基础所奠定的那一个。这些事实，和一切事实一样，并不是必然的，而只具有经验的确定性；它们是假设。因此，我们可以考察它们的或然程度——在观测的范围之内，这种或然程度当然非常高——并且探究把它们推广到观测范围之外，无论朝着无穷大的一边还是朝着无穷小的一边，是否正当。

III. Application to Space三、对空间的应用#

§ 1.—By means of these inquiries into the determination of the measure relations of an n-fold extent the conditions may be declared which are necessary and sufficient to determine the metric properties of space, if we assume the independence of line-length from position and expressibility of the line-element as the square root of a quadric differential, that is to say, flatness in the smallest parts.

§ 1. 借助以上对 n 重广延的度量关系之确定所作的这些研究，如果我们假定线的长度与位置无关，并且线元可以表示为一个二次微分式的平方根，也就是说，在最小的部分中是平坦的，那么就可以说出确定空间度量性质所必要而又充分的条件。

First, they may be expressed thus: that the curvature at each point is zero in three surface-directions; and thence the metric properties of space are determined if the sum of the angles of a triangle is always equal to two right angles.

第一，这些条件可以这样表述：在每一点处，三个曲面方向上的曲率都为零；于是，只要三角形的内角和总是等于两直角，空间的度量性质就确定了。

Secondly, if we assume with Euclid not merely an existence of lines independent of position, but of bodies also, it follows that the curvature is everywhere constant; and then the sum of the angles is determined in all triangles when it is known in one.

第二，如果我们像欧几里得那样，不仅假定线的存在与位置无关，而且假定物体的存在也与位置无关，那么就得出：曲率处处是常数；于是，只要知道了一个三角形的内角和，所有三角形的内角和也就确定了。

Thirdly, one might, instead of taking the length of lines to be independent of position and direction, assume also an independence of their length and direction from position. According to this conception changes or differences of position are complex magnitudes expressible in three independent units.

第三，人们还可以不只是假定线的长度与位置和方向无关，而且假定线的长度和方向都与位置无关。按照这种观念，位置的变化或差异是可以用三个独立单位来表示的复量。

§ 2.—In the course of our previous inquiries, we first distinguished between the relations of extension or partition and the relations of measure, and found that with the same extensive properties, different measure-relations were conceivable; we then investigated the system of simple size-fixings by which the measure-relations of space are completely determined, and of which all propositions about them are a necessary consequence; it remains to discuss the question how, in what degree, and to what extent these assumptions are borne out by experience. In this respect there is a real distinction between mere extensive relations, and measure relations; in so far as in the former, where the possible cases form a discrete manifoldness, the declarations of experience are indeed not quite certain, but still not inaccurate; while in the latter, where the possible cases form a continuous manifoldness, every determination from experience remains always inaccurate, be the probability ever so great that it is nearly exact. This consideration becomes important in the extensions of these empirical determinations beyond the limits of observation to the infinitely great and infinitely small; since the latter may clearly become more inaccurate beyond the limits of observation, but not the former.

§ 2. 在前面的研究过程中，我们首先区分了广延关系或区域关系与度量关系，发现在同样的广延性质下，可以设想不同的度量关系；接着我们研究了一个简单的定量体系，空间的度量关系由它完全确定，关于空间度量关系的一切命题都是它的必然结果；剩下要讨论的问题是：这些假定在多大程度上、在什么范围内为经验所证实。在这方面，单纯的广延关系与度量关系之间有一个真正的区别：在前者，可能的情形构成一个离散的流形，经验的判断虽然并不完全确定，却并非不精确；而在后者，可能的情形构成一个连续的流形，无论它近乎精确的或然性有多大，从经验得来的每一个规定始终都是不精确的。当我们把这些经验规定推广到观测范围之外，推向无穷大和无穷小时，这个考虑就变得重要了；因为后者在观测范围之外显然可能变得更不精确，前者却不会。

In the extension of space-construction to the infinitely great, we must distinguish between unboundedness and infinite extent, the former belongs to the extent relations, the latter to the measure-relations. That space is an unbounded three-fold manifoldness, is an assumption which is developed by every conception of the outer world; according to which every instant the region of real perception is completed and the possible positions of a sought object are constructed, and which by these applications is for ever confirming itself. The unboundedness of space possesses in this way a greater empirical certainty than any external experience. But its infinite extent by no means follows from this; on the other hand if we assume independence of bodies from position, and therefore ascribe to space constant curvature, it must necessarily be finite provided this curvature has ever so small a positive value. If we prolong all the geodesics starting in a given surface-element, we should obtain an unbounded surface of constant curvature, i.e., a surface which in a flat manifoldness of three dimensions would take the form of a sphere, and consequently be finite.

在把空间构造推广到无穷大时，我们必须区分无界与无限延伸：前者属于广延关系，后者属于度量关系。空间是一个无界的三重流形，这是我们对外部世界的每一种认识中都会用到的一个假定；按照这个假定，我们每时每刻都在补全实际感知的区域，构造出所寻求的对象可能所在的位置；而它在这些应用中永远在证实着自己。因此，空间的无界性比任何外部经验都具有更大的经验确定性。但它的无限延伸却决不能由此推出；相反，如果我们假定物体与位置无关，因而认为空间具有常曲率，那么只要这个曲率有一个哪怕极小的正值，空间就必然是有限的。如果我们把从某一曲面元出发的所有测地线延长，就会得到一个具有常曲率的无界曲面，即这样一个曲面：在一个平坦的三维流形中，它会呈现为球面的形状，因而是有限的。

§ 3. The questions about the infinitely great are for the interpretation of nature useless questions. But this is not the case with the questions about the infinitely small. It is upon the exactness with which we follow phenomena into the infinitely small that our knowledge of their causal relations essentially depends. The progress of recent centuries in the knowledge of mechanics depends almost entirely on the exactness of the construction which has become possible through the invention of the infinitesimal calculus, and through the simple principles discovered by Archimedes, Galileo, and Newton, and used by modern physic. But in the natural sciences which are still in want of simple principles for such constructions, we seek to discover the causal relations by following the phenomena into great minuteness, so far as the microscope permits. Questions about the measure-relations of space in the infinitely small are not therefore superfluous questions.

§ 3. 关于无穷大的问题，对于解释自然来说是无用的问题。但关于无穷小的问题却并非如此。我们对现象之间因果关系的认识，本质上取决于我们把现象追踪到无穷小时的精确程度。近几个世纪在力学知识上的进步，几乎完全取决于构造的精确性，而这种精确性之所以成为可能，是由于发明了无穷小演算，是由于阿基米德、伽利略和牛顿所发现、并为现代物理学所使用的那些简单原理。而在那些仍然缺少这类构造所需之简单原理的自然科学中，我们就在显微镜所允许的范围内，把现象追踪到极其细微之处，以求发现其中的因果关系。所以，关于空间在无穷小中的度量关系的问题，并不是多余的问题。

If we suppose that bodies exist independently of position, the curvature is everywhere constant, and it then results from astronomical measurements that it cannot be different from zero; or at any rate its reciprocal must be an area in comparison with which the range of our telescopes may be neglected. But if this independence of bodies from position does not exist, we cannot draw conclusions from metric relations of the great, to those of the infinitely small; in that case the curvature at each point may have an arbitrary value in three directions, provided that the total curvature of every measurable portion of space does not differ sensibly from zero. Still more complicated relations may exist if we no longer suppose the linear element expressible as the square root of a quadric differential. Now it seems that the empirical notions on which the metrical determinations of space are founded, the notion of a solid body and of a ray of light, cease to be valid for the infinitely small. We are therefore quite at liberty to suppose that the metric relations of space in the infinitely small do not conform to the hypotheses of geometry; and we ought in fact to suppose it, if we can thereby obtain a simpler explanation of phenomena.

如果我们假定物体的存在与位置无关，那么曲率处处是常数，而从天文测量就可以得出：这个曲率不可能不等于零；或者至少，它的倒数必定是这样一个面积，与之相比，我们望远镜所能达到的范围可以忽略不计。但如果物体与位置无关这一点并不成立，我们就不能从大尺度的度量关系推断无穷小尺度的度量关系；在那种情况下，每一点处三个方向上的曲率都可以取任意的值，只要空间每一个可测量部分的总曲率与零没有明显差别。如果我们不再假定线元可以表示为一个二次微分式的平方根，还可能存在更复杂的关系。而看来，空间度量规定所依据的那些经验概念——固体的概念和光线的概念——在无穷小中不再有效。因此，我们完全可以假定：空间在无穷小中的度量关系并不符合几何学的假设；而且，如果这样做能对现象作出更简单的解释，我们事实上就应当这样假定。

The question of the validity of the hypotheses of geometry in the infinitely small is bound up with the question of the ground of the metric relations of space. In this last question, which we may still regard as belonging to the doctrine of space, is found the application of the remark made above; that in a discrete manifoldness, the ground of its metric relations is given in the notion of it, while in a continuous manifoldness, this ground must come from outside. Either therefore the reality which underlies space must form a discrete manifoldness, or we must seek the ground of its metric relations outside it, in binding forces which act upon it.

几何学的假设在无穷小中是否成立的问题，与空间度量关系的根据问题联系在一起。在后一个问题中——我们仍然可以把它看作属于空间学说的问题——上面所作的那个说明就有了用处：在一个离散流形中，其度量关系的根据已经包含在它的概念之中；而在一个连续流形中，这种根据必须来自外部。因此，要么构成空间之基础的实在形成一个离散流形，要么我们必须到空间之外，到作用于它的结合力中，去寻找它的度量关系的根据。

The answer to these questions can only be got by starting from the conception of phenomena which has hitherto been justified by experience, and which Newton assumed as a foundation, and by making in this conception the successive changes required by facts which it cannot explain. Researches starting from general notions, like the investigation we have just made, can only be useful in preventing this work from being hampered by too narrow views, and progress in knowledge of the interdependence of things from being checked by traditional prejudices.

这些问题的答案，只能这样得到：从迄今为经验所证明、并为牛顿当作基础的那种现象观念出发，再根据它所不能解释的事实，逐步对这种观念作出必要的修改。像我们刚才所作的这样从一般概念出发的研究，只能起这样的作用：防止这项工作受到过于狭隘的观点的妨碍，防止对事物相互依存关系的认识的进步受到传统偏见的阻挠。

This leads us into the domain of another science, of physic, into which the object of this work does not allow us to go to-day.

这就把我们引入另一门科学，即物理学的领域；而本文的目的，不容许我们今天走进去。