applications. But granted that the mind can create and that its creations are, perforce, not self-destructive, but evolutionary, then we have a guaranty of that truth which really inheres in mathematics. Whether such truth can be made of use in our daily life is another question which we must discuss in another chapter, in the field of validity of mathematics. We may quote, as to the source of mathematical truth, Brunschvicg: The truth of the science does not imply the existence of a transcendental reality; it is bound to the processes of verification which are immanent in the development of mathe-. matics. It is this verification that we have believed we could uncover at the root of the constitutive notions of knowing; it is that which we have encountered at the decisive moments when the human mind saw wider horizons, as well in the book of the scribe Ahmes, who gave the proof of his calculations with fractions, as in the primary investigations of Newton and Leibniz in finding by arithmetic and algebra the results they had already obtained by the use of infinite series. Mathematical philosophy has ended its task by setting itself to follow the natural order of history, by becoming conscious of the two characters whose union is the specific mark of intelligence: indefinite capacity of progress, perpetual disquietude as to verification. REFERENCES Brunschvicg, Les étapes de la philosophie mathématique. Le rationnel. 1 Les étapes de la philosophie mathématique, p. 561. CHAPTER XV THE METHODS OF MATHEMATICS We may distinguish four distinct methods by which mathematical investigation proceeds. These are not exclusive of one another, of course, but may all appear in the same piece of research, and usually would appear. The names for the four have been chosen as roughly characterizing the methods. These are: the scientific method, the intuitive method, the deductive method, and the creative method. We will consider these in sufficient detail to make clear what we mean. 1. The scientific method.-It is commonly supposed that mathematics has nothing to do with observation, experimentation, analysis, and generalization, the chief features of the strictly scientific method. In answer to this we may quote Sylvester:1 Most, if not all, of the great ideas of modern mathematics have had their origin in observation. Take, for instance, the arithmetical theory of forms, of which the foundation was laid in the diophantine theorems of Fermat, left without proof by their author, which resisted all efforts of the myriad-minded Euler to reduce to demonstration, and only yielded up their cause of being when turned over in the blowpipe flame of Gauss' transcendent genius; or the doctrine of double periodicity, which resulted from the observation of Jacobi of a purely analytical fact of transformation; of Legendre's law of reciprocity; or Sturm's theorem about the roots of equations, which, as he informed me with his own lips, stared him in the I Nature, I (1869), p. 238. face in the midst of some mechanical investigations connected (if my memory serves me right) with the motion of compound pendulums; or Huyghen's method of continued fractions characterized by Lagrange as one of the principal discoveries of that great mathematician, and to which he appears to have been led by the construction of his Planetary Automaton; or the new algebra [invariants], speaking of which one of my predecessors (Mr. Spottiswoode) has said, not without just reason and authority, from this chair, "that it reaches out and indissolubly connects each year with fresh branches of mathematics, that the theory of equations has become almost new through it, algebraic geometry transfigured in its light, that the calculus of variations, molecular physics, and mechanics [he might, if speaking at the present moment, go on to add the theory of elasticity and the development of integral calculus] have all felt its influence." And more recently we have the remarks of Hobson' on the same subject: The actual evolution of mathematical theories proceeds by a process of induction strictly analogous to the method of induction employed in building up the physical sciences; observation, comparison, classification, trial, and generalization are essential in both cases. Not only are special results, obtained independently of one another, frequently seen to be really included in some generalization, but branches of the subject which have been developed quite independently of one another are sometimes found to have connections which enable them to be synthetized in one single body of doctrine. The essential nature of mathematical thought manifests itself in the discernment of fundamental identity in the mathematical aspects of what are superficially very different domains. One of the best examples we can find of this, as well as the other methods of mathematics, is Poincaré, whose 1 Nature, 84 (1910), p. 290. immense wideness of generalization, said Darwin, and abundance of possible applications are sometimes almost bewildering. He invented Fuchsian functions, then he found that they could be used to solve differential equations, to express the co-ordinates of algebraic curves, and to solve algebraic equations of any order. The very simple substitutions of sines and cosines or hyperbolic functions which enable us to solve quadratics and cubics were in this way generalized so that a single methodthe uniformization of the variables enables us to solve any algebraic equation, and to integrate any algebraic expression. The theory of continuous groups he applied to hypercomplex numbers and then applied hypercomplex numbers to the theory of Abelian integrals, and was able in this way to generalize the properties of the periods. He generalized the notion of Green's function, discovering the wide branch of fundamental functions and their uses. He generalized the notion of invariant to integrals over lines, surfaces, volumes, etc., and was able to reach a new point of attack on problems of dynamics. He generalized the figures of equilibrium for the heavenly bodies, discovering an infinity of new forms, and pointing out the transitions from one form to another. To state all his generalizations would take too much space and would only emphasize the great importance of the method. There is no essential difference between generalizations of this mathematical type and those of science. It is generalization to say that projective geometry merely states the invariants of the projective group, and that elementary geometry is a collection of statements about the invariants of the group of motions. Expansions in sines and cosines are particular cases of expansions of fundamental functions in general. It is generalization to . reduce the phenomena of light first to a wave-theory, then those of light, electricity, and magnetism, to the properties of the ether. It is generalization to reduce all the laws of mechanics to the geometry of four-dimensional Lobatchevskian space. When we say natural law, we mean generalization of some kind. Usually the process of generalization takes place by means of the various analogies present. The observation of these is necessary to generalization. But there is another mode also which leads to generalization, and that is the removal of premises in arguments, or at least of parts of premises. Much mathematical work of the present day consists in determining whether a conclusion can persist if the premises are made a little less restricted. Some element is removed from the postulates or from the defining character of the expressions, and it is then found that the conclusions still hold. For example, many theorems announced for functions of rather restricted type that are to be integrated are much more widely true if the integration is defined in Lebesgue's manner. The analogies found to exist between widely different theories enable us to see from the one theory and its developments how unsuspected developments may be made in another analogous theory. This is one of the reasons why mathematicians value even the most isolated investigations. As Whewell said: "If the Greeks had not cultivated conic sections, Kepler could not have superseded Ptolemy; if the Greeks had cultivated dynamics, Kepler might have anticipated Newton." And we may add that, if the Greeks had perceived the analogies between many theorems on conic sections, they would have invented projective geometry. If the world had seen the purely postulational character of much of geometry, long investi |