Natural Philosophy

by Wilhelm Ostwald

As for space, the presence of the three dimensions conditions a great manifoldness of possible relations, and hence the existence of a very extensive science of bodies in space, of geometry. Geometry is divided into various parts depending upon whether or not the concept of measurement enters. When dealing with purely spacial relations apart from the concept of measurement it is called geometry of position. In order to introduce the element of measurement a certain hypothesis is necessary which is undemonstrable, and therefore appears to be arbitrary and can be justified only because it is the simplest of all possible hypotheses. This hypothesis takes for granted that a rigid body can be moved in all directions in space without changing in measure. Or, to state the inverse of this hypothesis, in space those parts are called equal which a rigid body occupies, no matter how it is moved about.

We are not conscious of the extreme arbitrariness of this assumption simply because we have become accustomed to it in school. But if we reflect that in daily experience the space occupied by a rigid body, say a stick, seems to the eye to undergo radical changes as it shifts its position in space and that we can maintain that hypothesis only by declaring these changes to be "apparent," we recognize the arbitrariness which really resides in that assumption. We could represent all the relations just as well if we were to assume that those changes are real, and that they are successively undone when we restore the stick to its former relation to our eye. But though such a conception is fundamentally practicable in so far as it deals merely with the space picture of the stick, we nevertheless find that it would lead to such extreme complications with regard to other relations (for example, the fact that the weight of the stick is not affected by the change of the optic picture) that we do better if we adhere to the usual assumption that the optical changes are merely apparent.

In this connection we learn what an enormous influence the various parts of experience exert upon one another in the development of science. In every special generalization of experiences, that is, in every individual scientific theory, our aim is not only to generalize this special group of experiences in themselves, but at the same time to join such other experiences to them as expedience demands. If the effect of this necessity is on the one hand to render the elaboration of an appropriate theory more difficult, it has on the other hand the great advantage of affording a choice among several theories of apparently like value, and thus making possible a more precise notion of the reality. For example, for the understanding of the mutual movements of the sun and the earth it is the same whether we assume that the sun moves about the earth or the earth about the sun. It is not until we try to represent theoretically the position of the other planets that we see the economic advantage of the second conception, and facts like Foucault's experiment with a pendulum can be represented only according to this second conception in our present state of knowledge.

Likewise, the assumption on which scientific geometry goes, that space has the same properties in all directions, conflicts with immediate experience. In immediate experience we make a sharp distinction between below and above, although we are prepared to admit the "homogeneity" of space in the horizontal direction. This is due, as physics teaches, to the fact that we are placed in a field of gravitation which acts only from above downward and which permits free horizontal turnings, although it imparts a characteristic difference to the third direction. Since considerations of another kind enable us to place ourselves in a position in which we ignore this field of gravitation in the investigation of space, geometry abstracts this element and disregards the corresponding manifoldness. In the theory of the gravitation potential, on the other hand, this very manifoldness is made the subject of scientific investigation.

The common application of the concepts of space and time results in the concept of motion, the science of which is called phoronomics. In order to make this new variable subject to measurement we must arrive at an agreement or convention as to the way in which to measure time. For since past time can never be reproduced we actually experience only unextended moments, and have no means of recognizing or defining the equality of two periods of time by placing them side by side, as we can in the case of spacial magnitudes. We help ourselves by saying that in uninfluenced motions equal periods of time must correspond to the equal changes in space. We regard the rotation of the earth on its axis and its revolution about the sun as such uninfluenced motions. The two depend upon dissimilar conditions, and the empirical fact that the relation of the two motions, or the relation between the day and the year, remains practically the same, sustains that assumption, and at the same time shows the expediency of the given definition of time.

Analytic geometry, the application of algebra to geometric relations, occupies a noteworthy position, from the point of view of method, in the science of space. It yields geometric results by means of calculation, that is, by the application of the algebraic material of symbols we can obtain data concerning unknown spacial relations. An explanation is necessary of how by a method apparently so extraneous such results as these can be attained.

The answer lies again in the general principle of co-ordination, which in this very case receives a particularly cogent illustration. Three algebraic signs, x, y, and z, are co-ordinated with the three variable dimensions of space. First, the same independent and constant variability is ascribed to these signs, and, further, the same mutual relations are assumed to subsist between them as actually exist between the three-spacial dimensions. In other words, precisely the same kind of manifoldness is imparted to these algebraic signs as the spacial dimensions possess to which they are co-ordinated, and we may therefore expect that all the conclusions arising from these assumptions will find their corresponding parts in the spacial manifoldness. Accordingly, a co-ordinated spacial relation corresponds to every change of those algebraic formulas resulting from calculation, and if such changes lead to an algebraically simple form, then the spacial form corresponding to it must show an analogous simplicity. Here, therefore, we have a case such as was described under simpler conditions on [p. 86] of operations undertaken with one group and repeated correspondingly in the co-ordinated group. And it is only the great difference in the things of which in this case the two groups are composed—spacial relations on the one side and algebraic signs on the other—that creates the impression of astonishment which was felt very strongly at the invention of this method, and which is still felt by students with talent for mathematics when they first become acquainted with analytical geometry.

41. Recapitulation.

Before we proceed to consider the fundamentals of other sciences, it is well to make a general résumé of the field so far traversed. Since the later sciences, as we have already observed, make use of the entire apparatus of the earlier sciences, the mastery of them must be assured in order to render their special application possible.

This does not mean that one must have complete command of the entire range of those earlier sciences in order to pursue a later one. Mere human limitations would prevent the fulfilment of such a demand. As a matter of fact, successful work can be done in one of the later sciences even if only the most general features of the earlier ones have been clearly grasped. Nevertheless, the rapidity and certainty of the results are very considerably increased by a more thorough knowledge of the earlier sciences, and the investigator, accordingly, should seek a middle road between the danger of insufficient preparation for his special science and the danger of never getting to it from sheer preparation. In any circumstances he must be prepared always, even though it be in later age, to acquire those fundamental aids so soon as he feels the need of them for carrying out any special work. It is generally acceded that without logic the adequate pursuit of science is impossible. Nevertheless, the opinion is widely current, even among men of science, that everybody has command of the needful logic without having studied it. No more than a man can learn of himself to use the calculus, even if he may have discovered unaided some of its elementary principles, can he acquire certainty and readiness in the use of the logical rules generally necessary, unless he has made the necessary studies. It is true that the scientific works of the great pioneers and leaders in the special sciences furnish practical examples of such logical activity. But complete freedom and security are acquired only on the basis of conscious knowledge.