There exists a more popular interpretation of illusions. It results from a confusion between ignorance and illusion. Suppose, for instance, that we were to receive visual sensations which led us to assert that a table was present, but that on approaching the table we were to find that we could walk through it. If we, and we alone among men, could see the table, we should be justified in assuming, in view of past experience, that we were suffering from some hallucination, since any other alternative would deny us the possibility of conceiving of an objective world the same for all men. But if all who came into the room perceived the same table and failed, as we had done, to derive any tactual impressions from its presence, it would be taking too much for granted to assert that we had all of us been the victims of error. The mystery might be cleared up without difficulty were we to find it possible to account for this appearance in terms of phenomena already known; but if this procedure failed, the wiser course would be to agree that some new mysterious manifestation had been discovered. At all events, it would be wrong to maintain that the appearance must necessarily be the result of a collective illusion on the ground that its reality would conflict with the laws of matter. Our knowledge of natural laws is obtained only by generalising from experience; and where experience is incomplete, as it must always be, the laws can lay claim to no measure of certainty. For example, a man knowing nothing of atmospheric pressure might well assume that a balloon could never rise in the air without coming into conflict with the law of gravitation, or that a firefly could not emit light without getting burnt. The reason why the theoretical scientist is compelled to approach problems of a seemingly mysterious nature with great open-mindedness is because on so many previous occasions he has been confronted with discoveries which yield nothing in strangeness to the one we have mentioned; and strangeness is not necessarily the product of illusion. More often than not, it is the result of ignorance and prejudice. In this respect, we have only to conceive of the surprise we should experience, had we never heard of an electromagnet, on discovering that distant objects flew towards it without being pulled by strings.
And now let us return to our objective world. We will assume that illusions have been barred therefrom; and the world we have thus obtained can be conceived of as existing in precisely the same way for all observers. If we assume that there is a real world of things in themselves, constituting the underlying cause of our sense experience, the point we wish to ascertain is whether scientific procedure can throw any light on its nature.
In the first place, when discussing the outside world as an existent reality, we must differentiate between its substance and its structure, or form. If, with science, we consider that our knowledge of the external universe can be arrived at only by a rational synthesis of facts of experience, we must recognise that substance escapes us completely; all we can hope to approach is structure, or relationships. Of course, a view of this sort presupposes that we are justified in asserting that knowledge can be acquired only by means of rational co-ordinations. If we dispute the legitimacy of this point the entire argument collapses. Hence the scientific argument can constitute no refutation of the views of the mystic. A man who “knows” of things through intuition or faith, or a man who tells us that he knows that in his previous incarnation he was Julius Cæsar, is a type of opponent with whom the scientist cannot even argue; for whatever can be neither proved nor disproved by observation or reason constitutes a form of knowledge which is meaningless, or at least completely irrelevant to science. If we did not place some kind of limitation on what we were to regard as knowledge, there would be no reason to prefer the opinions of a Newton or an Einstein to the ravings of an ignoramus or a lunatic; and human knowledge would become so conflicting as to lose all significance. It is not denied that intuition may often lead to discovery. Even in mathematics examples are numerous; but the fact remains that intuition alone has so often led men astray that unless its disclosures can be submitted to some kind of test, no reliance can be placed in them.
Furthermore, it is not asserted that intelligence is everything. We know that with animals and, still more, with insects, instinct plays a major rôle. But it is to be noted, first of all, that instinct is scarcely apparent in human beings. And in the second place, with bees, for example, which have selected the hexagonal shape for their combs, a choice which mathematical calculation proves to have been the best they could have selected, instinct has yielded exactly the same results that intelligence would have done; so that aside from the lack of consciousness which accompanies instinct, no difference is detected in the results to which it leads. Hence, it can scarcely be deemed to yield a new form of knowledge. If these points be granted, we may return to our subject and enquire why it is that substance escapes us completely when by “knowledge” we refer to a rational co-ordination of facts and sense impressions.
Rational co-ordinations have their most perfect prototype in mathematical co-ordinations—in those of mathematical physics, for example. Not all co-ordinations can be constructed mathematically; chemistry, and of course, to a still greater degree, biology, afford us illustrations where mathematics is comparatively useless. Nevertheless, as the methods of co-ordination are essentially the same in all cases, the mathematical ordinations, in view of their greater clarity, can be studied to advantage as typical instances in this respect.
Now mathematical equations are nothing but relations, and from initial relations all we can deduce are other relations. In other words, our equations can never yield us more than we originally put into them. It follows that were all relationships in nature to be preserved and the substances changed, no observable difference could be detected; and we should never be able to differentiate between a whole class of worlds identical in structure but differing in substance. If, then, we discard the procedure of the mystic, or of the metaphysician who claims a knowledge that cannot be submitted to the control of experiment, we must recognise that substance escapes us completely and that our knowledge of the real world can at best reduce to a skeleton or structure.
An illustration given by Eddington presents the same problem in a more concrete manner. He assumes that in some future age the game of chess may be dead and forgotten; and he compares our position in respect to the real world with that of archaeologists who will have unearthed curious records of the game written in the usual obscure symbolism. Eventually they would succeed in reconstructing the moves of the different pieces, the two-dimensional ordering relation of the partitions, and the rules of the game. To this extent they could claim to have understood the game of chess. Yet certain aspects would escape them completely. The shape of the partitions, whether square, round or oblong, the aspect of the pieces, the nature of the board, whether of wood or of stone, would remain unknown. But it is to be noted that these elements of knowledge that escaped them would be irrelevant. The board might be of wood or stone, the partitions of diamonds or squares, and yet the game could be played as before. If, on the other hand, the ordering relation of the partitions were changed, if the permissible moves of the pieces were modified, the written records would make no sense. It is the same with the real world. The substance may change in nature, yet no difference will be perceived; but if the relationships are modified, a new world will divulge itself to our observation. To this extent, therefore, we are justified in saying that the most our observations can reveal reduces to the structure of the real world. As the French mathematician Bertrand once said, the age of the captain, the number of the crew, the height of the mast can yield us no information about the position of the ship.
Structure, as outlined above, resolves itself into those mathematical relationships or equations which appear to account for the sequence of phenomena. But structure in the more usual sense would apply to the hidden mechanisms themselves, to the deep-seated, underlying causes. When, however, we wish to attain this more concrete knowledge of structure, we encounter a number of difficulties. As Poincaré points out, every time conservation of energy and the principle of least action are found to be satisfied (as is the case in electromagnetics, for example), an indefinite number of different mechanical models are possible; so that our knowledge of the structure of the world is vitiated by the fact that the substructure remains indefinite. Clifford expresses the same idea when he says:
“Whatever can be explained by the motion of a fluid can be equally well explained either by the attraction of particles or by the strains of a solid substance; the very same mathematical calculations result from the three distinct hypotheses; and science, though completely independent of all three, may yet choose one of them as serving to link together different trains of physical enquiry.”
In a certain measure, difficulties of this sort may prove temporary. It may happen that further experiment will enable us to determine which of the various hypotheses corresponds to concrete reality. Such, indeed, was the hope of the scientists of the eighteenth and early nineteenth centuries. And in our days the practical isolation of electrons and molecules is a proof that these hopes were not always unfounded. Yet, even so, we should have advanced but a step, and should again be faced with the problem of determining the structure of these electrons, and so on ad infinitum. It would be of no avail to say that our aim had been realised when the ultimate constituent elements were isolated. For either these supposedly ultimate elements would possess no structure, in which case we could never know anything further about them, or else they would present a structure; but then they would not constitute ultimate elements, since this structure would imply relationships between their various parts, and we should not be at the end of our journey.