terms in common; (2) their relation-numbers are the two relation-numbers in question, and then by defining by reference to R and S two other suitable relations whose relation-numbers are defined to be respectively the sum and product of the relation-numbers in question. We need not consider the details of this process. Now if n be any finite cardinal number, it can be proved that the class of those serial relations, which have a field whose cardinal number is n, is a relation-number. This relation-number is the ordinal number corresponding to n; let it be symbolized by ṅ. Thus, corresponding to the cardinal numbers 2, 3, 4 . . . there are the ordinal numbers 2., 3., 4. . . . The definition of the ordinal number 1 requires some little ingenuity owing to the fact that no serial relation can have a field whose cardinal number is 1; but we must omit here the explanation of the process. The ordinal number ȯ is the class whose sole member is the null relation—that is, the relation which never holds between any pair of entities. The definitions of the finite ordinals can be expressed without use of the corresponding cardinals, so there is no essential priority of cardinals to ordinals. Here also it can be seen that the science of the finite ordinals is a particular subdivision of the general theory of classes and relations. Thus the illusory nature of the traditional definition of mathematics is again illustrated.
Cantor’s Infinite Numbers.—Owing to the correspondence between the finite cardinals and the finite ordinals, the propositions of cardinal arithmetic and ordinal arithmetic correspond point by point. But the definition of the cardinal number of a class applies when the class is not finite, and it can be proved that there are different infinite cardinal numbers, and that there is a least infinite cardinal, now usually denoted by ℵ0, where ℵ is the Hebrew letter aleph. Similarly, a class of serial relations, called well-ordered serial relations, can be defined, such that their corresponding relation-numbers include the ordinary finite ordinals, but also include relation-numbers which have many properties like those of the finite ordinals, though the fields of the relations belonging to them are not finite. These relation-numbers are the infinite ordinal numbers. The arithmetic of the infinite cardinals does not correspond to that of the infinite ordinals. The theory of these extensions of the ideas of number is dealt with in the article Number. It will suffice to mention here that Peano’s fourth premiss of arithmetic does not hold for infinite cardinals or for infinite ordinals. Contrasting the above definitions of number, cardinal and ordinals, with the alternative theory that number is an ultimate idea incapable of definition, we notice that our procedure exacts a greater attention, combined with a smaller credulity; for every idea, assumed as ultimate, demands a separate act of faith.
The Data of Analysis.—Rational numbers and real numbers in general can now be defined according to the same general method. If m and n are finite cardinal numbers, the rational number m/n is the relation which any finite cardinal number x bears to any finite cardinal number y when n×x = m×y. Thus the rational number one, which we will denote by 1r, is not the cardinal number 1; for 1r is the relation 1/1 as defined above, and is thus a relation holding between certain pairs of cardinals. Similarly, the other rational integers must be distinguished from the corresponding cardinals. The arithmetic of rational numbers is now established by means of appropriate definitions, which indicate the entities meant by the operations of addition and multiplication. But the desire to obtain general enunciations of theorems without exceptional cases has led mathematicians to employ entities of ever-ascending types of elaboration. These entities are not created by mathematicians, they are employed by them, and their definitions should point out the construction of the new entities in terms of those already on hand. The real numbers, which include irrational numbers, have now to be defined. Consider the serial arrangement of the rationals in their order of magnitude. A real number is a class (α, say) of rational numbers which satisfies the condition that it is the same as the class of those rationals each of which precedes at least one member of α. Thus, consider the class of rationals less than 2r; any member of this class precedes some other members of the class—thus 1/2 precedes 4/3, 3/2 and so on; also the class of predecessors of predecessors of 2r is itself the class of predecessors of 2r. Accordingly this class is a real number; it will be called the real number 2R. Note that the class of rationals less than or equal to 2r is not a real number. For 2r is not a predecessor of some member of the class. In the above example 2R is an integral real number, which is distinct from a rational integer, and from a cardinal number. Similarly, any rational real number is distinct from the corresponding rational number. But now the irrational real numbers have all made their appearance. For example, the class of rationals whose squares are less than 2r satisfies the definition of a real number; it is the real number √2. The arithmetic of real numbers follows from appropriate definitions of the operations of addition and multiplication. Except for the immediate purposes of an explanation, such as the above, it is unnecessary for mathematicians to have separate symbols, such as 2, 2r and 2R, or 2/3 and (2/3)R. Real numbers with signs (+ or −) are now defined. If a is a real number, +a is defined to be the relation which any real number of the form x+a bears to the real number x, and −a is the relation which any real number x bears to the real number x+a. The addition and multiplication of these “signed” real numbers is suitably defined, and it is proved that the usual arithmetic of such numbers follows. Finally, we reach a complex number of the nth order. Such a number is a “one-many” relation which relates n signed real numbers (or n algebraic complex numbers when they are already defined by this procedure) to the n cardinal numbers 1, 2 . . . n respectively. If such a complex number is written (as usual) in the form x1e1+x2e2+. . .+xnen, then this particular complex number relates x1 to 1, x2 to 2, . . . xn to n. Also the “unit” e1 (or es) considered as a number of the system is merely a shortened form for the complex number (+1) e1+oe2+. . .+oen. This last number exemplifies the fact that one signed real number, such as o, may be correlated to many of the n cardinals, such as 2 . . . n in the example, but that each cardinal is only correlated with one signed number. Hence the relation has been called above “one-many.” The sum of two complex numbers x1e1+x2e2+. . .+xnen and y1e1+y2e2+. . .+ynen is always defined to be the complex number (x1+y1)e1+(x2+y2)e2+. . .+(xn+yn)en. But an indefinite number of definitions of the product of two complex numbers yield interesting results. Each definition gives rise to a corresponding algebra of higher complex numbers. We will confine ourselves here to algebraic complex numbers—that is, to complex numbers of the second order taken in connexion with that definition of multiplication which leads to ordinary algebra. The product of two complex numbers of the second order—namely, x1e1+x2e2 and y1e1+y2e2, is in this case defined to mean the complex (x1y1−x2y2)e1+(x1y2+x2y1)e2. Thus e1×e1 = e1, e2×e2 = −e1, e1×e2 = e2×e1 = e2. With this definition it is usual to omit the first symbol e1, and to write i or √−1 instead of e2. Accordingly, the typical form for such a complex number is x+yi, and then with this notation the above-mentioned definition of multiplication is invariably adopted. The importance of this algebra arises from the fact that in terms of such complex numbers with this definition of multiplication the utmost generality of expression, to the exclusion of exceptional cases, can be obtained for theorems which occur in analogous forms, but complicated with exceptional cases, in the algebras of real numbers and of signed real numbers. This is exactly the same reason as that which has led mathematicians to work with signed real numbers in preference to real numbers, and with real numbers in preference to rational numbers. The evolution of mathematical thought in the invention of the data of analysis has thus been completely traced in outline.
Definition of Mathematics.—It has now become apparent that the traditional field of mathematics in the province of discrete and continuous number can only be separated from the general abstract theory of classes and relations by a wavering and indeterminate line. Of course a discussion as to the mere application of a word easily degenerates into the most fruitless logomachy. It is open to any one to use any word in any sense. But on the assumption that “mathematics” is to denote a science well marked out by its subject matter and its methods from other topics of thought, and that at least it is to include all topics habitually assigned to it, there is now no option but to employ “mathematics” in the general sense[1] of the “science concerned with the logical deduction of consequences from the general premisses of all reasoning.”
Geometry.—The typical mathematical proposition is: “If x, y, z . . . satisfy such and such conditions, then such and such other conditions hold with respect to them.” By taking fixed conditions for the hypothesis of such a proposition a definite department of mathematics is marked out. For example, geometry is such a department. The “axioms” of geometry are the fixed conditions which occur in the hypotheses of the geometrical propositions. The special nature of the “axioms” which constitute geometry is considered in the article Geometry (Axioms). It is sufficient to observe here that they are concerned with special types of classes of classes and of classes of relations, and that the connexion of geometry with number and magnitude is in no way an essential part of the foundation of the science. In fact, the whole theory of measurement in geometry arises at a comparatively late stage as the result of a variety of complicated considerations.
Classes and Relations.—The foregoing account of the nature of mathematics necessitates a strict deduction of the general properties
- ↑ The first unqualified explicit statement of part of this definition seems to be by B. Peirce, “Mathematics is the science which draws necessary conclusions” (Linear Associative Algebra, § i. (1870), republished in the Amer. Journ. of Math., vol. iv. (1881) ). But it will be noticed that the second half of the definition in the text—“from the general premisses of all reasoning”—is left unexpressed. The full expression of the idea and its development into a philosophy of mathematics is due to Russell, loc. cit.
