The evolution of scientific thought from Newton to Einstein

by A. D'Abro

Whether or not, in the case of the retina, our appreciation of the coin’s roundness comes from some unconscious counting process is a problem for the specialist to decide; but at all events, when we consider that the retina and the eye are Euclidean bodies just like our rulers, the concordance between our computations of shape and size, as determined by rods, and our direct visual appreciation of shape does not appear to present much of a mystery.

But there is still a further point to be considered. Were we to have a direct intuition of congruence and of absolute shape or length, our measurements with rods would have to be adjusted so as to conform to this intuition. The introduction of rods would thus be contemplated merely as an adjunct, in order to obtain greater definiteness. But it so happens that such is not the case.

Our intuitive visual appreciations yield results which differ from results obtained with rods, not merely accidentally as a result of the imperfection of human observation, but systematically.

A coin that measures out as round will appear flattened to the eye.[13] This phenomenon is illustrated by the well-known optical illusion wherein two rigid rulers which coincide when placed side by side, appear of unequal magnitude when placed horizontally and vertically, respectively. It is a well-known fact that the vertical appears to be longer than the horizontal. For this reason, vertical stripes on a cloth cause the wearer to appear thinner and taller, whereas horizontal stripes produce the reverse effect. Now the mere fact that we have agreed to accept such discrepancies as due to optical illusions rather than to the untrustworthiness of our rods proves that we deliberately reject our intuitive judgment of shape and size in favour of more sophisticated rules of measurement. In other words, we have abandoned direct intuition for physical determinations, hence for convenient but conventional standards.

In short, we may state that we possess no direct intuition of shape and length, and that what little we appear to have is traceable in the final analysis to the properties of material bodies and light rays.

Let us consider a last example relating to distance. We realise, without resorting to measurement, that the distance across the street is less than the entire length of the street. A number of different reasons conspire to account for this conviction. First, the muscular sensations which accompany the convergence of the eyes and the focussing of images on the retina would in themselves suggest the origin of our appreciation of distance. In addition, we know from experience that it requires a greater effort and a longer time to travel along the entire length of the street than to step across from one sidewalk to the other. And if we enquired why it took a shorter time to cross the street, we should find that it was due to the fact that we advanced with definite steps.

Although in mathematical space the two distances would be equal or unequal according to our measuring conventions, in practical life we have unwittingly posited our measuring convention by walking. Our successive steps henceforth define in our estimation congruent distances; and under the circumstances distances become measurable in terms of these steps. But these steps are themselves controlled by our human frame; hence all we have done has been to measure space in terms of our legs. That measurement originated in this way is indicated clearly by such words as foot and cubit, or again by the definition of a yard as expressed by the distance between the tip of a certain king’s nose and the extremities of his fingers. And it is because our human limbs manifest the same type of congruence as the material bodies around us that this type of measurement once again imposes itself so strongly upon us. Indeed, measurements as computed with human limbs appear to be instinctive and are exemplified in children who, having grown, are surprised to find the rooms and buildings they have not seen for some years, appear considerably smaller. Instinctively, the child is measuring size in terms of his own height.

Thus, in whatever way we examine the matter, there appears to be nothing mysterious in our natural belief in the absoluteness of shape and size or in a definite geometry pertaining to space. None of the examples mentioned thus far entitle us to maintain that physical space manifests a definite metrics and that congruence is other than conventional.

If we consider the problem in its present state, we see that it is the physical behaviour of material bodies and light rays which is in the final analysis responsible for our natural belief in absolute shape. But this realisation brings with it the assurance that space itself has eluded us entirely in our discussions. Such was indeed Poincaré’s stand. He maintained that though for purposes of convenience it was only natural for us to measure space as we do, yet if needs be we could disregard the behaviour of material bodies entirely, adopt non-Euclidean standards of measurement, and proceed as before.

In spite of this change we could construct exactly the same engineering works, in fact rewrite the whole of physics. Needless to say, everything would be extremely complex, all our known laws would be disfigured, and hypotheses ad hoc would have to be introduced. But when all is said and done, the task would be theoretically possible; so that if we disregard the criteria of convenience and simplicity, there is nothing to choose between the various types of measurements, any more than between the metric system and the British units.