International Congress of Arts and Science, Volume 1 / Philosophy and Metaphysics

by Unknown

On the basis of the definitions just given, we can establish further propositions. If group A equals B, and B equals C, then A also equals C. The proof of this is that we can relate every member of A to a corresponding member of B and by hypothesis no member will be left. Then C is arranged with reference to B, and here also no member is left. By this process every member of A, through the connecting link of a member of B, is associated with a member of C, and this association is preserved even if we cut out the group B. Therefore A and C are equal. The same process of reasoning can be carried out for any number of groups.

Likewise it can be demonstrated that if A is poorer than B and B poorer than C, then A is also poorer than C. For in the association of B with A some members of B are left over by hypothesis, and likewise some members of C are left over if one associates C with B. Therefore in the association of C with A, not only those members are left over which could not be associated with B, but also those members of C which extend beyond B. This proposition can be extended to any number of groups, and permits the arrangement of a number of different groups in a simple series by beginning with the poorest and choosing each following so that it is richer than the preceding but poorer than the following. From the proposition just established, it follows that every group is so arranged with reference to all other groups that it is richer than all the preceding and poorer than all the following.[^[3]]

In this derivation of scientific proposition or laws of the simplest kinds, the process of derivation and the nature of the result becomes particularly clear. We arrive at such a proposition by performing an operation and expressing the result of it. This expression enables us to avoid the repetition of the operation in the future, since in accordance with the law we can indicate the result immediately. Thus an abbreviation and therefore, a facilitation of the problem is attained which is the more considerable the larger the number of operations saved.

If we have a number of equal groups, we know by the process of association that all of the operations with reference to arrangement which we can perform with one of them can be performed with all the others. It is sufficient, therefore, to determine the properties of arrangement of one of these groups in order to know forthwith the properties of all the others. This is an extremely important proposition, which is continually employed for the most various purposes. All speaking, writing, and reading rests upon the association of thoughts with sounds and symbols, and by arranging the signs in accordance with our thoughts we bring it to pass that our hearers or readers think like thoughts in like order. In a similar fashion we make use of various systems of formulæ in the different sciences, especially in the simpler sciences; and these formulæ we correlate with phenomena and use in place of the phenomena themselves, and can therefore derive from them certain characteristics of phenomena without being compelled to use the latter. The force of this process appears very strikingly in astronomy where, by the use of definite formulæ associated with the different heavenly bodies, we can foretell the future positions of these bodies with a high degree of approximation.

From the theory of order we come to the theory of number or arithmetic by the systematic arrangement or development of an operation just indicated (page 343). We can arrange any number of groups in such a way that a richer always follows a poorer. But the complex obtained in this manner is always accidental with reference to the number and the richness of its members. A regular and complete structure of all possible groups is evidently obtained only if we start from a group of one member or from a simple thing, and by the addition of one member at a time make further groups out of those that we have. Thus we obtain different groups arranged according to an increasing richness, and since we have advanced one member at a time, that is, made the smallest step which is possible, we are certain that we have left out no possible group which is poorer than the richest to which the operation has been carried.

This whole process is familiar; it gives the series of the positive whole numbers, that is, the cardinal numbers. It is to be noted that the concept of quantity has not yet been considered; what we have gained is the concept of number. The single things or members in this number are quite arbitrary, and especially they do not need to be alike in any manner. Every number forms a group-type, and arithmetic or the science of numbers has the task of investigating the properties of these different types with reference to their division and combination. If this is done in general form, without attention to the special amount of the number, the corresponding science is called algebra. On the other hand, by the application of formal rules of formation, the number system has had one extension after another beyond the territory of its original validity. Thus counting backward led to zero and to the negative numbers; the inversion of involution to the imaginary numbers. For the group-type of the positive whole numbers is the simplest but by no means the only possible one, and for the purpose of representing other manifolds than those which are met with in experience, these new types have proved themselves very useful.

At the same time the number series gives us an extremely useful type of arrangement. In the process of arising it is already ordered, and we make use of it for the purpose of arranging other groups. Thus, we are accustomed to furnish the pages in a book, the seats in a theatre, and countless other groups which we wish to make use of in any kind of order with the signs of the number series, and thereby we make the tacit assumption that the use of that corresponding group shall take place in the same order as the natural numbers follow each other. The ordinal numbers arising therefrom do not represent quantities, nor do they represent the only possible type of arrangement, but they are again the simplest of all. We come to the concept of magnitude only in the theory of time and space. The theory of time has not been developed as a special science; on the contrary, what we have to say about time first appears in mechanics. Meantime we can present the fundamental concepts, which arise in this connection, with reference to such well-known characteristics of time that the lack of a special science of time is no disadvantage.

The first and most important characteristic of time (and of space, too) is that it is a continuous manifold; that is, every portion of time chosen can be divided at any place whatever. In the number series this is not the case; it can be divided only between the single numbers. The series one to ten has only nine places of division and no more. A minute, or a second, on the other hand, has an unlimited number of places of division. In other words, there is nothing in the lapse of any time which hinders us from separating or distinguishing in thought at any given instant the time which has elapsed till then from the following time. It is just the same with space, except that time is a simple manifold and space a threefold, continuous manifold.

Nevertheless, when we measure them, we are accustomed to indicate times and spaces with numbers. If we first examine, for example, the process of measuring a length, it consists in our applying to the distance to be measured a length conceived as unchangeable, the unit of measure, until we have passed over the distance. The number of these applications gives us the measure or magnitude of the distance. The result is that by the indication of arbitrarily chosen points upon the continuous distance, we place upon it an artificial discontinuity which enables us to associate it with the discontinuous number series.

A still further assumption, however, belongs to the concept of measuring, namely, that the parts of the distance cut off by the unit used as a measure be equal, and it is taken for granted that this requirement will be fulfilled to whatever place the unit of measure is shifted. As may be seen, this is a definition of equality carried further than the former, for one cannot actually replace a part of the distance by another in order to convince one's self that it has not changed. Just as little can one assert or prove that the unit of measure in changing its place in space remains of the same length; we can only say that such distances as are determined by the unit of measure in different places are declared or defined as equal. Actually, for our eye, the unit of measure becomes smaller in perspective the farther away from it we find ourselves.