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[5] 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 of classes and relations from the ultimate logical premisses. In the course of this process, undertaken for the first time with the rigour of mathematicians, some contradictions have become apparent. That first discovered is known as Burali-Forti’s contradiction,[6] and consists in the proof that there both is and is not a greatest infinite ordinal number. But these contradictions do not depend upon any theory of number, for Russell’s contradiction[7] does not involve number in any form. This contradiction arises from considering the class possessing as members all classes which are not members of themselves. Call this class w; then to say that x is a w is equivalent to saying that x is not an x. Accordingly, to say that w is a w is equivalent to saying that w is not a w. An analogous contradiction can be found for relations. It follows that a careful scrutiny of the very idea of classes and relations is required. Note that classes are here required in extension, so that the class of human beings and the class of rational featherless bipeds are identical; similarly for relations, which are to be determined by the entities related. Now a class in respect to its components is many. In what sense then can it be one? This problem of “the one and the many” has been discussed continuously by the philosophers.[8] All the contradictions can be avoided, and yet the use of classes and relations can be preserved as required by mathematics, and indeed by common sense, by a theory which denies to a class—or relation—existence or being in any sense in which the entities composing it—or related by it—exist. Thus, to say that a pen is an entity and the class of pens is an entity is merely a play upon the word “entity”; the second sense of “entity” (if any) is indeed derived from the first, but has a more complex signification. Consider an incomplete proposition, incomplete in the sense that some entity which ought to be involved in it is represented by an undetermined x, which may stand for any entity. Call it a propositional function; and, if φx be a propositional function, the undetermined variable x is the argument. Two propositional functions φx and ψx are “extensionally identical” if any determination of x in φx which converts φx into a true proposition also converts ψx into a true proposition, and conversely for ψ and φ. Now consider a propositional function Fχ in which the variable argument χ is itself a propositional function. If Fχ is true when, and only when, χ is determined to be either φ or some other propositional function extensionally equivalent to φ, then the proposition Fφ is of the form which is ordinarily recognized as being about the class determined by φx taken in extension—that is, the class of entities for which φx is a true proposition when x is determined to be any one of them. A similar theory holds for relations which arise from the consideration of propositional functions with two or more variable arguments. It is then possible to define by a parallel elaboration what is meant by classes of classes, classes of relations, relations between classes, and so on. Accordingly, the number of a class of relations can be defined, or of a class of classes, and so on. This theory[9] is in effect a theory of the use of classes and relations, and does not decide the philosophic question as to the sense (if any) in which a class in extension is one entity. It does indeed deny that it is an entity in the sense in which one of its members is an entity. Accordingly, it is a fallacy for any determination of x to consider “x is an x” or “x is not an x” as having the meaning of propositions. Note that for any determination of x, “x is an x” and “x is not an x,” are neither of them fallacies but are both meaningless, according to this theory. Thus Russell’s contradiction vanishes, and an examination of the other contradictions shows that they vanish also.
Applied Mathematics.—The selection of the topics of mathematical inquiry among the infinite variety open to it has been guided by the useful applications, and indeed the abstract theory has only recently been disentangled from the empirical elements connected with these applications. For example, the application of the theory of cardinal numbers to classes of physical entities involves in practice some process of counting. It is only recently that the succession of processes which is involved in any act of counting has been seen to be irrelevant to the idea of number. Indeed, it is only by experience that we can know that any definite process of counting will give the true cardinal number of some class of entities. It is perfectly possible to imagine a universe in which any act of counting by a being in it annihilated some members of the class counted during the time and only during the time of its continuance. A legend of the Council of Nicea[10] illustrates this point: “When the Bishops took their places on their thrones, they were 318; when they rose up to be called over, it appeared that they were 319; so that they never could make the number come right, and whenever they approached the last of the series, he immediately turned into the likeness of his next neighbour.” Whatever be the historical worth of this story, it may safely be said that it cannot be disproved by deductive reasoning from the premisses of abstract logic. The most we can do is to assert that a universe in which such things are liable to happen on a large scale is unfitted for the practical application of the theory of cardinal numbers. The application of the theory of real numbers to physical quantities involves analogous considerations. In the first place, some physical process of addition is presupposed, involving some inductively inferred law of permanence during that process. Thus in the theory of masses we must know that two pounds of lead when put together will counterbalance in the scales two pounds of sugar, or a pound of lead and a pound of sugar. Furthermore, the sort of continuity of the series (in order of magnitude) of rational numbers is known to be different from that of the series of real numbers. Indeed, mathematicians now reserve “continuity” as the term for the latter kind of continuity; the mere property of having an infinite number of terms between any two terms is called “compactness.” The compactness of the series of rational numbers is consistent with quasi-gaps in it—that is, with the possible absence of limits to classes in it. Thus the class of rational numbers whose squares are less than 2 has no upper limit among the rational numbers. But among the real numbers all classes have limits. Now, owing to the necessary inexactness of measurement, it is impossible to discriminate directly whether any kind of continuous physical quantity possesses the compactness of the series of rationals or the continuity of the series of real numbers. In calculations the latter hypothesis is made because of its mathematical simplicity. But, the assumption has certainly no a priori grounds in its favour, and it is not very easy to see how to base it upon experience. For example, if it should turn out that the mass of a body is to be estimated by counting the number of corpuscles (whatever they may be) which go to form it, then a body with an irrational measure of mass is intrinsically impossible. Similarly, the continuity of space apparently rests upon sheer assumption unsupported by any a priori or experimental grounds. Thus the current applications of mathematics to the analysis of phenomena can be justified by no a priori necessity.
In one sense there is no science of applied mathematics. When once the fixed conditions which any hypothetical group of entities are to satisfy have been precisely formulated, the deduction of the further propositions, which also will hold respecting them, can proceed in complete independence of the question as to whether or no any such group of entities can be found in the world of phenomena. Thus rational mechanics, based on the Newtonian Laws, viewed as mathematics is independent of its supposed application, and hydrodynamics remains a coherent and respected science though it is extremely improbable that any perfect fluid exists in the physical world. But this unbendingly logical point of view cannot be the last word upon the matter. For no one can doubt the essential difference between characteristic treatises upon “pure” and “applied” mathematics. The difference is a difference in method. In pure mathematics the hypotheses which a set of entities are to satisfy are given, and a group of interesting deductions are sought. In “applied mathematics” the “deductions” are given in the shape of the experimental evidence of natural science, and the hypotheses from which the “deductions” can be deduced are sought. Accordingly, every treatise on applied mathematics, properly so-called, is directed to the criticism of the “laws” from which the reasoning starts, or to a suggestion of results which experiment may hope to find. Thus if it calculates the result of some experiment, it is not the experimentalist’s well-attested results which are on their trial, but the basis of the calculation. Newton’s Hypotheses non fingo was a proud boast, but it rests upon an entire misconception of the capacities of the mind of man in dealing with external nature.
Synopsis of Existing Developments of Pure Mathematics.—A complete classification of mathematical sciences, as they at present exist, is to be found in the International Catalogue of Scientific Literature promoted by the Royal Society. The classification in question was drawn up by an international committee of eminent mathematicians, and thus has the highest authority. It would be unfair to criticize it from an exacting philosophical point of view. The practical object of the enterprise required that the proportionate quantity of yearly output in the various branches, and that the liability of various topics as a matter of fact to occur in connexion with each other, should modify the classification.
Section A deals with pure mathematics. Under the general heading “Fundamental Notions” occur the subheadings “Foundations of Arithmetic,” with the topics rational, irrational and transcendental numbers, and aggregates; “Universal Algebra,” with the topics complex numbers, quaternions, ausdehnungslehre, vector analysis, matrices, and algebra of logic; and “Theory of Groups,” with the topics finite and continuous groups. For the subjects of this general heading see the articles [Algebra, Universal]; [Groups, Theory of]; [Infinitesimal Calculus]; [Number]; [Quaternions]; [Vector Analysis]. Under the general heading “Algebra and Theory of Numbers” occur the subheadings “Elements of Algebra,” with the topics rational polynomials, permutations, &c., partitions, probabilities; “Linear Substitutions,” with the topics determinants, &c., linear substitutions, general theory of quantics; “Theory of Algebraic Equations,” with the topics existence of roots, separation of and approximation to, theory of Galois, &c.; “Theory of Numbers,” with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers. For the subjects of this general heading see the articles [Algebra]; [Algebraic Forms]; [Arithmetic]; [Combinatorial Analysis]; [Determinants]; [Equation]; [Fraction, Continued]; [Interpolation]; [Logarithms]; [Magic Square]; [Probability]. Under the general heading “Analysis” occur the subheadings “Foundations of Analysis,” with the topics theory of functions of real variables, series and other infinite processes, principles and elements of the differential and of the integral calculus, definite integrals, and calculus of variations; “Theory of Functions of Complex Variables,” with the topics functions of one variable and of several variables; “Algebraic Functions and their Integrals,” with the topics algebraic functions of one and of several variables, elliptic functions and single theta functions, Abelian integrals; “Other Special Functions,” with the topics Euler’s, Legendre’s, Bessel’s and automorphic functions; “Differential Equations,” with the topics existence theorems, methods of solution, general theory; “Differential Forms and Differential Invariants,” with the topics differential forms, including Pfaffians, transformation of differential forms, including tangential (or contact) transformations, differential invariants; “Analytical Methods connected with Physical Subjects,” with the topics harmonic analysis, Fourier’s series, the differential equations of applied mathematics, Dirichlet’s problem; “Difference Equations and Functional Equations,” with the topics recurring series, solution of equations of finite differences and functional equations. For the subjects of this heading see the articles [Differential Equations]; [Fourier’s Series]; [Continued Fractions]; [Function]; [Function of Real Variables]; [Function Complex]; [Groups, Theory of]; [Infinitesimal Calculus]; [Maxima and Minima]; [Series]; [Spherical Harmonics]; [Trigonometry]; [Variations, Calculus of]. Under the general heading “Geometry” occur the subheadings “Foundations,” with the topics principles of geometry, non-Euclidean geometries, hyperspace, methods of analytical geometry; “Elementary Geometry,” with the topics planimetry, stereometry, trigonometry, descriptive geometry; “Geometry of Conics and Quadrics,” with the implied topics; “Algebraic Curves and Surfaces of Degree higher than the Second,” with the implied topics; “Transformations and General Methods for Algebraic Configurations,” with the topics collineation, duality, transformations, correspondence, groups of points on algebraic curves and surfaces, genus of curves and surfaces, enumerative geometry, connexes, complexes, congruences, higher elements in space, algebraic configurations in hyperspace; “Infinitesimal Geometry: applications of Differential and Integral Calculus to Geometry,” with the topics kinematic geometry, curvature, rectification and quadrature, special transcendental curves and surfaces; “Differential Geometry: applications of Differential Equations to Geometry,” with the topics curves on surfaces, minimal surfaces, surfaces determined by differential properties, conformal and other representation of surfaces on others, deformation of surfaces, orthogonal and isothermic surfaces. For the subjects under this heading see the articles [Conic Sections]; [Circle]; [Curve]; [Geometrical Continuity]; [Geometry, Axioms of]; [Geometry, Euclidean]; [Geometry, Projective]; [Geometry, Analytical]; [Geometry, Line]; [Knots, Mathematical Theory of]; [Mensuration]; [Models]; [Projection]; [Surface]; [Trigonometry].
This survey of the existing developments of pure mathematics confirms the conclusions arrived at from the previous survey of the theoretical principles of the subject. Functions, operations, transformations, substitutions, correspondences, are but names for various types of relations. A group is a class of relations possessing a special property. Thus the modern ideas, which have so powerfully extended and unified the subject, have loosened its connexion with “number” and “quantity,” while bringing ideas of form and structure into increasing prominence. Number must indeed ever remain the great topic of mathematical interest, because it is in reality the great topic of applied mathematics. All the world, including savages who cannot count beyond five, daily “apply” theorems of number. But the complexity of the idea of number is practically illustrated by the fact that it is best studied as a department of a science wider than itself.
Synopsis of Existing Developments of Applied Mathematics.—Section B of the International Catalogue deals with mechanics. The heading “Measurement of Dynamical Quantities” includes the topics units, measurements, and the constant of gravitation. The topics of the other headings do not require express mention. These headings are: “Geometry and Kinematics of Particles and Solid Bodies”; “Principles of Rational Mechanics”; “Statics of Particles, Rigid Bodies, &c.”; “Kinetics of Particles, Rigid Bodies, &c.”; “General Analytical Mechanics”; “Statics and Dynamics of Fluids”; “Hydraulics and Fluid Resistances”; “Elasticity.” For the subjects of this general heading see the articles [Mechanics]; [Dynamics, Analytical]; [Gyroscope]; [Harmonic Analysis]; [Wave]; [Hydromechanics]; [Elasticity]; [Motion, Laws of]; [Energy]; [Energetics]; [Astronomy] (Celestial Mechanics); [Tide]. Mechanics (including dynamical astronomy) is that subject among those traditionally classed as “applied” which has been most completely transfused by mathematics—that is to say, which is studied with the deductive spirit of the pure mathematician, and not with the covert inductive intention overlaid with the superficial forms of deduction, characteristic of the applied mathematician.