Before entering upon them we must repeat that the question of number and series in mathematics is independent of the assumptions of space and time. As a science, mathematics could exist outside them: order is not necessarily spatial or temporal. Our conclusions, therefore, cannot be attacked on the ground that they are based on Euclidean conceptions of space: they are based on the laws of logic.
I. The Indefinite Regress
Hume and, later, Kant argued that by the principle of association when we think of one quality of a thing the others are naturally brought before our minds, and thus that we get into the habit of attributing to the notion of the thing a certain group of qualities. And it is true that we do attend to a thing all at once, including in the notion of it all the qualities which we know belong to it.
Now experience, according to Leibniz, gives us an example of a unity which embraces a multiplicity of detail. Thus a thing is one substance as embodying an individual experience, and its qualities belong to it in the same sense as the constituents of experience belong to the single experience. These qualities are in relation. (The Pragmatist denies the existence of relations as part of a higher unity.[23]) But they are not only relation, since relation always implies something more than itself. Let us take the example of number. Numbers could never have been counted if there had not been things to count. Now suppose each quality could be analysed into a new relation, we should still not get rid of the quality. At each stage there remains a quality in relation, and this goes on to Infinity. Such a constant subdivision perhaps results from our finite experience seizing facts in a disjointed way. When we analyse a law in its working, we always do seem to come to this Indefinite Regress. Now it has been the reproach against metaphysics, as uttered by the Pragmatist, that there is no correspondence in scientific fact to this road into Infinity.
W. James asserts: “But in point of fact, nature doesn’t make eggs by making first half an egg, then a quarter, then an eighth, &c., and adding them together. She either makes a whole egg at once or none at all, and so of all her other units. It is only in the sphere of change, then, where one phase of a thing must needs come into being before another phase can come, that Zeno’s paradox gives trouble. And it gives trouble then only if the successive steps of change be infinitely divisible.”[24]
The sphere of change, however, one would answer, includes all nature, and science in its discoveries acts on the hypothesis that these steps of change may be infinitely divisible. Royce held to it firmly that any consistent attempt to make an orderly arrangement of the terms of an infinite whole must lead to the Indefinite Regress. And he further shows the connection with the fact that an infinite series can be adequately represented by a part of itself.
In the Boyle Lecture, delivered in Oxford in 1908, on the properties of radium, two facts emerged which show that the Indefinite Regress is now recognised in science.
First, that in the region of experiment we become aware of groups of elements allied to radium, which seem, in the number of individuals in their groups, to follow a simple arithmetical progression.
Secondly, that radio-active elements lose in activity at a certain rate, which always represents an exact proportion of the mass which remains. The tremendous disintegrating force slackens in exact relation to the time which passes, so that the smaller the morsel the less the relative loss of mass. Here, then, is the Indefinite Regress. In the world of fact as well as of ideas we are dealing with aspects of Infinity.[25]
II. Infinite Series