This empirical origin of the spatial concept is stressed by Einstein in the following lines:
“We now come to our concepts and judgments of space. It is essential here, also, to pay strict attention to the relation of experience to our concepts. It seems to me that Poincaré clearly recognised the truth in the account he gave in his book, ‘La Science et l’Hypothèse.’ Among all the changes which we can perceive in a rigid body, those are marked by their simplicity which can be made reversibly by an arbitrary motion of the body; Poincaré calls these, changes in position. By means of simple changes in position we can bring two bodies into contact. The theorems of congruence, fundamental in geometry, have to do with the laws that govern such changes in position.”
When we realise that it is precisely these laws governing changes in position which govern our choice of a space among all those which the mathematician has to offer, we see how utterly dependent we are on experience when the problem of space is considered.
We may present these problems in a more vivid form. Suppose all we had ever seen of the world were given by its image in a reflecting spherical surface, such as a large door knob. The world of our visual perception would be very different from the one in which we normally live; the shapes of objects would squirm in a variety of ways as we displaced them before the curved mirror. And yet, however different our world might appear from the one of common observation, we should eventually succeed in co-ordinating our perceptions. We should still conceive of an outside space, but this space would no longer be Euclidean. If, then, all of a sudden the mirror were to be removed, and we to behold the world as other men perceive it, we should be completely at sea, accustomed as we were to the laws of our non-Euclidean world. In fact, the situation would be very similar to that of the man who tries to ride a bicycle through a crowded thoroughfare while crossing his arms over the handle-bars. He probably would come to grief; and yet had he always ridden his bicycle in this peculiar way, he would find it just as hard to alter his habits and ride it in the normal way.
Summarising, we may say that a belief in an outside universe of space, matter and change is arrived at as a result of a synthesis of sense impressions. These conclusions, which apply to commonplace knowledge, will be substantiated further when we consider illustrations taken from the more advanced fields of knowledge of the scientist. As mentioned previously, it is impossible to draw a line and say: “Here scientific knowledge begins and commonplace knowledge ends.” And since the methods of the scientist are easier to dissect, a study of the scientist’s procedure cannot help but shed light on the more obscure problem of the genesis of commonplace knowledge.
In scientific syntheses we do not restrict ourselves to co-ordinating mere sense impressions; we must also co-ordinate scientific facts. But scientific facts are themselves the results of previous co-ordinations of other scientific facts, and these in turn are traceable to a co-ordination of sense impressions. A few examples taken at random from science will make these points clearer. Why, for instance, does the astronomer maintain that the sun is spherical?
It is, as we know, in order to account for the continued circular aspect of the solar disk, for the passage of sunspots, for their flattened appearance when nearing the sun’s edges, suggesting that they are seen in perspective, for the protuberance of its equator, for the Doppler effect exhibited on its equatorial rim, for the brilliancy of the planets when illuminated edgewise. It is also in order to render compatible the sun’s shape with its fluidic nature imposed by its high temperature. In other words, the aim of the scientist is to frame one single hypothesis which will permit him to co-ordinate this wide variety of facts. We may note that all the facts that the astronomer is seeking to co-ordinate presuppose a knowledge of space and of material objects situated in space. But of course we are assuming that by the time men began to worry about the shape of the sun, they had advanced beyond the primitive stage of recognising the existence of objects in space. Now, when we decide that the sun is spherical, our first argument is based on its circular aspect. In order to account for this, we appeal to probability, arguing that it is very improbable that the sun should always turn the same face towards us. Of course the argument in itself does not carry much weight, since it is refuted in the case of the moon. Still, it serves as a suggestion, if nothing more.
Next consider the case of the planets. Were the sun a flat disk, it would appear strange that their brilliancy should remain appreciably the same regardless of their positions relatively to the sun. But this argument, be it noted, is highly sophisticated, for the natural view would be to assume that all bright points in the heavens shine of their own accord; and there would be no reason to differentiate between planets which reflected the solar light, and the stars which were in no wise dependent on the sun’s presence. It was only at a later stage that a differentiation between stars and planets became necessary. In short, the facts the astronomer is seeking to co-ordinate are of a highly sophisticated nature; it is only when we dissect them further and further, analysing the previous syntheses of science, that we are finally thrown back on our immediate awareness of sense impressions. We see, then, that science appears as an unending series of syntheses of other syntheses, but that in every case the synthetic method is the same.
Let us now pass to a less simple example, namely: “We know that molecules exist.” In the case of the existence of the table or the chair, all we needed was to co-ordinate certain immediate sense impressions. In the case of the shape of the sun, the procedure was more complicated, since we could not explore its surface with our hands and it was only by inference that we were led to believe we were viewing a spherical object from various positions in the course of a day or a year. But with molecules it is far worse, for no one has even seen or felt them. The inferences which we are led to make are based on others, these others again on others. The possibility of our co-ordination being proved incompatible with future discovery is therefore increased, and for this reason again our knowledge of molecules loses much of its certainty. Apart from questions of degree, however, this knowledge was arrived at in precisely the same way, by conceiving the simplest rational synthesis capable of co-ordinating a wide variety of facts of observation and experience.
It is probable that at a very remote stage in human history men noticed the difference in texture which existed between sand, which was grainy, and water, which appeared continuous and smooth. It would have been natural for them to wonder whether water would not turn out to be grainy if viewed microscopically. Some guessed one way, others another. There existed, however, a number of elementary facts of observation which had to be taken into consideration. For instance, a phenomenon on which the Greek thinkers laid due stress was the ability of wine and water to intermingle. The simplest manner of accounting for this was to assume that water and wine were formed of discrete particles which would exchange positions, much as two powders, one black and one white, would yield to a uniform grey mixture, when shaken together.