Some people maintain that position in space or time must be relative because, if we try to determine the position of a body A, if bodies B, C, D with respect to which the position of A could be determined were not present, we should be trying to determine something about A without having our senses affected by other things. These people seem to me to be like the cautious guest who refused to say anything about his host’s port-wine until he had tasted red ink.

“Wherein, then,” says Mr. Russell,[118] “lies the plausibility of the notion that all points are exactly alike? This notion is, I believe, a psychological illusion, due to the fact that we cannot remember a point so as to know it when we meet again. Among simultaneously presented points it is easy to distinguish; but though we are perpetually moving, and thus being brought among new points, we are quite unable to detect this fact by our senses, and we recognize places only by the objects they contain. But this seems to be a mere blindness on our part—there is no difficulty, so far as I can see, in supposing an immediate difference between points, as between colours, but a difference which our senses are not constructed to be aware of. Let us take an analogy: Suppose a man with a very bad memory for faces; he would be able to know, at any moment, whether he saw one face or many, but he would not be aware whether he had seen any of the faces before. Thus he might be led to define people by the rooms in which he saw them, and to suppose it self-contradictory that new people should come to his lectures, or that old people should cease to do so. In the latter point at least it will be admitted by lecturers that he would be mistaken. And as with faces, so with points—inability to recognize them must be attributed, not to the absence of individuality, but merely to our incapacity.”

Another form of this tendency is shown by Kronecker, Borel, PoincarÉ, and many other mathematicians, who refuse mere logical determination of a conception and require that it be actually described in a finite number of terms. These eminent mathematicians were anticipated by the empirical philosopher who would not pronounce that the “law of thought” that A is either in the place B or not is true until he had looked to make sure. This philosopher was of the same school as J. S. Mill and Buckle, who seem to have maintained implicitly not only that, in view of the fact that the breadth of a geometrical line depends upon the material out of which it is constructed, or upon which it is drawn, that there ought to be a paste-board geometry, a stone geometry, and so on;[119] but also that the foundations of logic are inductive in their nature.[120] “We cannot,” says Mill,[121] “conceive a round square, not merely because no such object has ever presented itself in our experience, for that would not be enough. Neither, for anything we know, are the two ideas in themselves incompatible. To conceive a body all black and yet white would only be to conceive two different sensations as produced in us simultaneously by the same object—a conception familiar to our experience—and we should probably be as well able to conceive a round square as a hard square, or a heavy square, if it were not that in our uniform experience, at the instant when a thing begins to be round, it ceases to be square, so that the beginning of the one impression is inseparably associated with the departure or cessation of the other. Thus our inability to form a conception always arises from our being compelled to form another contradictory to it.”

CHAPTER XLII