The theory of isomorphism in arrangements can be placed properly here as well as problems of transitivity. The whole of this branch is too undeveloped to expect much knowledge of the invariant characters it may have.

The logic of classes, relatives, and propositional functions in general possesses few invariants that have been systematically developed. The rules of the calculus constitute about the only logical invariants known so far, although recent investigations are drifting this way.

When we come to the field of operators, we find a rich harvest of invariants. It is sufficient merely to mention projective geometry, with regard to which Steiner1 said: "By a proper appropriation of a few fundamental relations one becomes master of the whole subject; order takes the place of chaos, one beholds how all the parts fit naturally into each other and arrange themselves serially in the most beautiful order, and how related parts combine into well-defined groups. In this manner one arrives, as it were, at the elements, which nature herself employs in order to endow figures with numberless properties with the utmost economy and simplicity." We notice geometric transformations in general, of which Lie2 said: "In our century the conceptions of substitutions and substitution group, transformation and transformation group, operation and operation group, invariant, differential invariant, differential parameter, appear more and more clearly as the most important conceptions of mathematics." We must not leave out analysis situs, the study of continuous one-to-one transformations, such transformations as can happen to a rubber surface or to a battered tin can. This is the most fundamental of all the I Works, 1 (1881), p. 233.

2 Leip. Ber., 47 (1895), p. 261.

branches of geometry, its theorems remain true under the most trying conditions of deformation, they come the nearest to representing the necessities in an infinite evanescence that any theory can furnish. If we were to add to it a new analysis situs of an infinitely discontinuous character, we might hope that some day we could furnish certain laws of the natural world that would hold under the most chaotic transformations. If we increase this already tremendous list with the grand theories of differential and integral invariants, we can almost feel ourselves the masters of the flowing universe. We find ourselves able to see the changeless in that which is smaller than the ultra-microscopic and also to ride on the permanent and indestructible filaments of whirling smoke wreaths throughout their courses to infinity. Wars may come and go, man may dream and achieve, may aspire and struggle, the aeons of geology and of celestial systems may ponderously go their way, electrons and dizzy cycles of spinning magnetons, or the intricate web of ether filaments may "write in the twinkling of an eye differential equations that would belt the globe," yet under all, and in all, the invariants of the mathematician persist, from the beginning even unto the end.

In the branch of hypernumbers the list of invariants is not extensive as yet. The automorphisms of an algebra, however, are necessary for the investigations of its structure and of its applications. The invariant equations of an algebra define it and also show to what things its numbers belong naturally. This field will become as large in time as that of algebraic invariants is now.

processes

The invariant theory of the branch we called is not touched. Nor is the invariant theory of schemes

I

Herschel, Familiar Lectures on Scientific Subjects, p. 458.

of inference yet investigated. When it is developed, we may really talk about laws of thought.

Just as in the principle of form we are studying chiefly the synthetic character of mathematics, so in the principle of invariance we are studying the permanent character of mathematical constructions. Its results are everlasting, and we have in them a growing monument to the human intellect. But we cannot afford to confuse the determination of invariants in mathematical constructions with the whole of mathematics and with the permanent character of mathematical theorems. In other words, the theorems of mathematics are the invariants of the field of mathematical investigation. Among these are theorems regarding the invariants of some of the objects of investigation under transformation. Mathematics contains many theorems which are invariants of thought, but are not theorems about invariants of mathematical objects. The fact that every mathematician comes to the same conclusion with regard to the same subject of investigation shows the invariant character of the intellect. The subject of investigation itself, however, need not be a study as to invariancy, but anything in the realm of mathematics. For instance, problems as to the theory of functions may not deal with invariants at all. Of course this is the same as saying that the questions of the mathematician are not always questions as to the permanency of something, but may be questions as to synthetic construction, as to correspondence, or questions as to the solution of equations of various types.

The theory of invariants is evidently one of the central principles of mathematics, yet mathematics cannot be reduced to mere problems of invariance. The invariants of mathematical objects serve to characterize them, but

not to define them completely, nor do they give other properties of that which they define. These must be sought for along the lines of the other central principles.

We find in the invariants of mathematics a source of objective truth. So far as the creations of the mathematician fit the objects of nature, just so far must the inherent invariants point to objective reality. Indeed, much of the value of mathematics in its applications lies in the fact that its invariants have an objective meaning. When a geometric invariant vanishes, it points to a very definite character in the corresponding class of figures. When a physical invariant vanishes or has particular values, there must correspond to it physical facts. When a set of equations that represent physical phenomena have a set of invariants or covariants which they admit, then the physical phenomena have a corresponding character, and the physicist is forced to explain the law resulting. The unnoticed invariants of the electromagnetic equations have overturned physical theories, and have threatened philosophy. Consequently the importance of invariants cannot be too much magnified, from a practical point of view. But for the pure mathematician there are the other phases that must also be considered and which are important. The theory of invariants, like the theory of form, is not the most important theory in mathematics-that high place is reserved for the theory of solutions of equations of all kinds.

REFERENCES

Meyer (trans. by Fehr), Sur les progrès de la théorie des invariants projectifs, 1897.

CHAPTER XII

MATHEMATICS AS THE THEORY OF FUNCTIONS

In his lectures on the development of analysis, at Clark University, Picard says:

The whole science of mathematics rests upon the notion of function, that is to say, of dependence between two or more magnitudes, whose study constitutes the principal object of analysis. It was a long time before account was taken of the extraordinary extent of this notion, a circumstance which was very happy for the progress of the science. If Newton and Leibniz had thought that continuous functions need not have derivatives, which is in general the case, the differential calculus would not have been born, likewise, the inexact ideas of Lagrange upon the possibility of developments in Taylor's series rendered immense service.

When we look over the range of modern analysis or the differential and integral calculus, with all its applications, including the functions of complex and other variables, we may at first consider that it would be safe. to define mathematics as the whole theory of functions in general. We shall inspect the field, however, a little more closely, remembering that, in order to classify any branch of mathematics under the theory of functions, it must deal with the idea of dependence or correspondence.

We meet in the theory of general ranges, first of all, functions that are determined by the assignment of a finite number of numerical values. These are of little interest in the present discussion. Then come functions that run over a denumerably infinite range of values.