ALGEBRA, Definitions and Fundamental Concepts. The word is occasionally used with the same meaning as in the sense of a body of mathe matical entities which obey a given set of laws or formula. Accordingly we speak of the algebra of logic, the algebra of quaternions, the algebra of relations, the algebra of groups and so on indefinitely. The word is often used to cover those systems only whose formal struc ture is of a certain type. In this article we shall consider algebra in a still narrower sense —in fact in that sense in which the term is used in the high-school books of our child hood. That is, we shall consider the special algebra of ordinary complex numbers and the various simpler algebras of real numbers, of rational numbers and of integers, through which we pass in reaching the algebra of ordinary complex numbers.
Mathematics has been defined by Benjamin Pierce as that science which draws necessary conclusions. Logic, however, is not only a science which draws necessary conclusions, but is in fact the science of the method in which necessary conclusions may be drawn. The rela tion between logic and mathematics is thus peculiarly intimate and in fact the present view of the matter is that deductive logic is simply that part of mathematics which possesses the least limited hypotheses. We shall therefore search for the basis of algebra among such logical entities as classes, properties, relations, etc., and shall see how the entire complicated fabric of algebra can be woven from these simple threads.
Assemblages.— We wish to exhibit how the algebra which is familiar to the schoolboy has its roots in the much simpler algorithm which will be found treated in extenso under the heads Lome., SYMBOLIC and ASSEMBLAGES, GEN ERAL THEORY or. Here it is sufficient for us to call attention to one or two of the simpler as pects of this important theory. Though many difficult and fascinating paradoxes have been found underlying this simple assumption, we shall take it for granted that the things which enter into our daily life — (Shoes and ships and sealing-wax and cabbages and kinge— can be arranged in assemblages or. classes in such a
manner that a class is uniquely determined by the specification of all its members. These classes we shall represent by small Greek char acters, and their members by small Roman letters.
Simple Properties of Assemblages.— One of the most obvious of these is that of contain ing members. Strange to say, this is not com mon to all assemblages. It is conspicuously absent in the assemblage of round squares, the assemblage of purple cows, etc. Another para dox about assemblages is that an assemblage of one member is a different thing from that mem ber. The ink-bottle now in front of me may be the only one of its sort, but it is not the sort of ink-bottle of which it is the sole repre sentative. In the symbolism which we shall use, the assemblage of all purple cows shall be represented by the symbol A, and if x is my ink-bottle, and is the only one of its sort, the assemblage of all the ink-bottles of the sort shall be called ex. This is the symbolism of A. N. Whitehead and Bertrand Russell, the most recent writers on this subject as a whole.
Every two classes determine several new classes. If a stands for lawyers and (3 for congressmen, there are classes determined by a and R, one of which is made up of all the lawyers in Congress, the other of those who either sit in Congress or practice the law, or possibly carr7 on both pursuits. The first class we shall indicate by a ni3, the second by a u R. It is clear that this use of it and u can be ex tended to any two classes whatsoever.
Similarity.— In a properly equipped army, every private has at least one gun issued to him. Let us suppose that work at the arsenals has ceased, and that every gun has been issued to some private. It is clear that the class of guns and that of privates will stand in a peculiarly intimate mutual relation— that of one—one correspondence or similarity. We may express the same fact by the statement that there are as many guns as soldiers. Ob serve that this identity in number is defined prior to and independently of number itself.