APPLIED MATHEMATICS Selection of Topics.—The selection of the topics of mathe matical 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 annihilates some members of the class counted during the time and only during the time of its continuance. A legend of the Council of 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 ap proached the last of the series, he immediately turned into the likeness of his next neighbour." Such a story cannot be disproved by deductive reasoning from the premises of abstract logic. We can only assert that a universe in which such things are liable to happen on a large scale is unfitted for the practical applica tion of the theory of cardinal numbers. The application of the theory of real numbers to physical quantities involves analo gous 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 num bers 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 meas
urement, 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, the continuity of space apparently rests upon sheer assumption unsupported by any 'Due to Bertrand Russell, cf. "Mathematical Logic as based on the Theory of Types," Amer. Journ. of Math. vol. xxx. (i908) . It is more fully explained by him, with later simplifications, in Principia mathematica (Cambridge).
Existence of Applied Mathematics.—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 and viewed as mathe matics is independent of its supposed application, and hydrody namics remains a coherent and respected science though it is ex tremely 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 dif ference 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 hy potheses from which the "deductions" can be deduced are sought. Accordingly, every treatise on applied mathematics, prop erly 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.