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.