By reversing this process, and by starting from the theory of the Infinite, we may gain some idea of the discovery of the Finite. So Dedekind and Russell define finite numbers not only in the usual way as those which can be reached by mathematical induction, starting from 0 and increasing by 1 at each step, but also as those of classes which are _not_ similar to the parts of themselves obtained by taking away single terms. That is the reversal of the process applied just now. Dedekind also has deduced the Finite from the Infinite by a novel process. He predicates a world of thought which we each and all possess, filled with thoughts and things, to each thing corresponding a thought. There are thus two “trans-finite” series in the minds of each and all of us; we cannot say when the series of thoughts and things will end; but they have number, though it is infinite number. (Number exists wherever there is a correspondence, one to one, between two aggregates.) But in this _Gedankenwelt_, says Dedekind, there is one thing to which there is no corresponding thought: that is the ego. Each man is part of his own world of thought, but there is no thought of himself in his mind corresponding exactly to himself, as a thought in his mind corresponds to another object.[8] Two important results follow from Dedekind’s theory: first, the existence of a finite number one, the number of the ego, as deduced from the _Gedankenwelt_ of two infinite systems; second, by putting together all the _Gedankenwelts_ there are or may be, we get the notion of series of series, which seems to transcend Infinity, and it gives us the conditions which are possibly gathered up in the Absolute. Now the argument from the Finite to the Infinite and the converse process may both be employed in mathematics (or both may be neglected, as in the elementary methods of calculation used in arithmetic). A discussion has taken place in the _Hibbert Journal_ on the relative value of the two methods. Keyser, in an article called the Axiom of Infinity, argued that one method, that of Dedekind, should be exclusively developed. Russell answered him, stating that it was not necessary to hold exclusively to either. If the Finite and the Infinite can in turn be deduced from one another, neither conception can be truly called an axiom. The real axiom is existence, which includes both, and which is defined by mathematicians as that which is _not self-contradictory_. Now the problem of Infinity includes also that of continuity; in other words, the problem of number includes that of _cardinal_ and _ordinal_ number. It is time to get to the mathematical definition of number, which we have found as a conception can be attached both to the Finite and to the Infinite. What is number in mathematics?