24. Arrangement of the Members.
Since we have started from the proposition that all members of a group are different from one another, we have perfect liberty to arrange them. The most obvious arrangement according to which some one definite member is followed by a single other member and so forth (as, for example, the arrangement of the letters of the alphabet) is by no means the only mode of arrangement, though it is the simplest. Besides this linear arrangement, there is also, for instance, the one in which two new members follow simultaneously upon each previous one, or the members may be disposed like a number of balls heaped up in a pyramid. However, we shall not have much occasion to occupy ourselves with these complex types of arrangement, and can therefore limit our considerations at first to the simplest, that is, to the linear arrangement.
This simplest of all possible forms expresses itself in the fact that the immediately experienced things of our consciousness are arranged in this way. In point of fact, the contents of our consciousness proceed in linear order, one single new member always attaching itself to an existing member. This law, however, is not strictly and invariably adhered to. It sometimes happens that our consciousness continues for a while to pursue the direction of thought it has once taken, although a branching off had already taken place at a former point, at which a new chain of thought had begun. Nevertheless, one of these chains usually breaks off very soon, and the linear character of the inner experience is immediately restored. Of certain specially powerful intellects it is recorded that they could keep up several lines of thought for a considerable length of time—Julius Cæsar, for instance.
The biologic peculiarity here mentioned of the linear juxtaposition of the contents of our consciousness has led to the concept of time, which has been appropriately called a form of inner life. That all our experiences succeed each other in time is equivalent to saying that our thought processes represent a group in linear arrangement. As appears from the above observations, this is by no means an absolute form, unalterable for all times. On the contrary, a few highly developed individuals have already begun to emancipate themselves from it. But the existing form is so firmly fixed through heredity and habit that it still seems impracticable for most men to imagine the succession of the inner experiences in a different way than by a line or by one dimension. Since, on the other hand, we have all learned to feel space as tri-dimensional, although optically it appears to possess only two dimensions (we see length and breadth, and only infer thickness from secondary characteristics), we come to recognize that the linear form by which we represent the succession of our experiences is a matter of adaptation, and that because the change has been extremely slight in the course of centuries it produces the impression of being unalterable.[D]
These discussions lead to a further difference that can exist in groups of linear arrangement. While in the first example we chose, the alphabet, the sequence was quite arbitrary, since any other sequence is just as possible, the same cannot be said of experiences into which the element of time enters. These are not arbitrary, but are arranged by special circumstances depending upon the aggregate of things which co-operate in the given experiences.
While, therefore, a group with free members, that is, members not determined in their arrangement by special circumstances, can be brought into linear order in very different ways, there are groups in which only one of those orders actually occurs. We see at once that in free groups the number of different orders possible is the greater, the greater the group itself. The theory of combinations teaches how to calculate these numbers which play a very important rôle in the various provinces of mathematics. The naturally ordered groups always represent a single instance out of these possibilities, the source of which always lies outside the group concept, that is, it proceeds from the things themselves which are united into a group.
25. Numbers.
An especially important group in the linear order is that of the integral numbers. Its origin is as follows:
First we abstract the difference of the things found in the group, that is, we determine, although they are different, to disregard their differences. Then we begin with some member of the group and form it into a group by itself. It does not matter which member is chosen, since all are regarded as equivalent. Then another member is added, and the group thus obtained is again characterized as a special type. Then one more member is added, and the corresponding type formed, and so on. Experience teaches that never has a hindrance arisen to the formation of new types of this kind by the addition of a single member at a time, so that the operation of this peculiar group formation may be regarded as unlimited or infinite.
The groups or types thus obtained are called the integral numbers. From the description of the process it follows that every number has two neighbors, the one the number from which it arose by the addition of a member, and the other the number which arose from it by the addition of a member. In the case of the number one with which the series begins, this characteristic is present in a peculiar form, the preceding group being group zero, that is, a group without content. This number in consequence reveals certain peculiarities into which we cannot enter here.