04

CHAPTER XIII

THEORY OF EQUATIONS

One of the chief sources of mathematical advance is the consideration of problems. We do not say the solution of problems, for frequently the problems are not solved, indeed, may not be solvable, yet in their consideration the mathematician has been led to invent new methods, new concepts, new branches of mathematics. One of the sources of problems, from which flows a growing stream as knowledge progresses, is natural science. We need only remember the famous problem of three bodies and the attendant series of memoirs on mathematics which has been its outcome. The equations of mathematical physics have suggested many widely diverse branches of mathematics. Problems in geometry suggest theorems in arithmetic, and problems in arithmetic have suggested geometrical advance. But by far the largest number of problems emanate from the mind itself in its study of mathematics. The inventions of the mathematician bring a constantly growing number of problems which in turn suggest a still wider field of investigation. Says Hilbert: "If we do not succeed in solving a mathematical problem, the reason frequently consists in our failure to recognize the more general standpoint from which the problem before us appears as a single link in a chain of related problems. After finding this standpoint, not only is this problem frequently more accessible to our investigation, but at the

I

1 Bull. Amer. Math. Soc. (2), 8 (1902), p. 443.

same time we come into possession of a method which is applicable to related problems."

The chief point of interest to us in the present investigation is, however, the creation of new ideas-new entities, we may well say in mathematics, from the attempts to solve problems. These we will look at in detail. One of the first attempts to solve problems was that of the solution of what we call diophantine equations, which have to be solved in integers. At present this kind of problem culminates in the famous last theorem of Fermat. In the attempt to prove this, Kummer was led to invent the ideal numbers which he used, and which in turn lead to the general theory of algebraic numbers and algebraic integers. We find created here the branch of mathematics called higher-number theory, as well as the introduction of the various domains of integrity in which we intend to work. Along the same line is the Galois theory of equations, which consists in finding the domain of integrity in which the solutions or roots of a given equation lie. On the one hand, we find from these problems the notion of rationality and, on the other, the notion of hypernumber, springing up spontaneously. The negative and the imaginary owed their origin to the necessity of finding solutions for certain equations. They were not known for centuries afterward to exist in nature in any way, and neither were they objects of intuition in any ordinary sense of the term. We might call them products of that faculty denominated by Winter, the transintuition, which is, so to speak, the intuition of the pure reason alone.

In the same region of solutions of problems arising from a single range, we have the list of functions invented to solve the ordinary differential equations of a single

independent variable. Hyperbolic functions, elliptic functions, hyperelliptic functions, Abelian functions-these were invented, as well as hosts of others, in order to complete the solution of the differential equations that arose in the course of the work of the mathematician. They in turn brought up the question of a functional domain of integrity, that is, a study of the conditions under which such differential equations could be solved in terms of given functions, as, for example, when a differential equation can be solved in terms of algebraic functions, circular functions, elliptic functions, etc. This is the Picard-Vessiot theory, similar to the Galois theory of equations and dependent upon the groups of the differential equations. The creation of new functions which are derived not from experience, but from their properties, is a sufficient phenomenon in itself to prove the autonomy of mathematics and its self-determination. Further, we find in the recent developments of difference-equations an opening of the new field which will lead to further inventions.

The theory of differential equations of several independent variables is responsible for the invention of spherical harmonics, ellipsoidal harmonics, harmonics in general, and a wide variety of unnamed functions. We also have the enormous list of solutions of equations with total differentials, which lead to functions that are not easily representable in the ordinary way. The greater part of mathematical physics lies in this region, since mathematically it is merely a consideration of the solutions of differential equations. The invention of the Green's functions alone and the expansion of this notion to cover a large class of functions of several variables, which are defined by differential equations with given boundary

conditions, is another branch of mathematics quite capable of demonstrating the fertility of the mind.

We need not stop to consider the solutions of problems of construction or problems of logic. They have their place, and what small synthetic character logistic has, lies in its few contributions in this direction. We find in the forms of atoms, molecules, and multimolecules ideals of mathematical chemistry.

In the theory of operators we have the invention of the automorphic functions as the functions which are solutions of certain equations of operators, particularly operators that form groups. The periodic functions, the doubly periodic functions, and others have extended mathematics very far. The study of integral equations, which is properly the study of functions that satisfy certain operational equations of a linear character, or to a small degree non-linear character, has introduced, not only new methods and new solutions, but a new point of view for the treatment of a wide range of mathematics. It enables us to define orthogonal functions in general and suggests other functions than the orthogonal, which remain for the future to study. Closely following it is the theory of functional equations in general, in which we undertake to find functions as the solution of certain functional equations. This includes the theory of operations and leads up to a theory we may call functional analysis. The calculus of variations belongs here, one of the oldest branches of mathematics of this type. Many problems in physics may be stated as problems in the calculus of variations, indeed, this method of statement seems to be the most unifying we have today. The determination of the solutions of variational questions is one of the important divisions of functional analysis.

The solutions of problems arising in a similar way from the functions of complex variables are intimately connected with the preceding forms, and usually little distinction would be made between them. However, the problems involving functions of several complex variables have peculiarities that must be taken into account. The problems arising in the consideration of functions of quaternions have yet to be investigated, and, when they have been studied in full, they will no doubt lead to many new ideas.

Problems in games such as the endings in chess are so far only amusements, and problems in the solutions of questions of deductions are purely in a tentative state. When mathematics has devised methods of producing the theories of scientific elementary ideals, the progress of science will be rapid. And these solutions will come in time, for all science is approximating a mathematical statement. The methods of science and those of mathematics are practically the same, and this identity will be revealed more plainly as the advances of mathematics enable us to handle problems of deduction.

This is the most important central principle of mathematics, namely, that of inversion, or of creating a class of objects that will satisfy certain defining statements. If the mathematician does not find these at hand in natural phenomena, he creates them and goes on in his uninterrupted progress. This might be considered to be the central principle of mathematics, for with the new creation we start a new line of mathematics, just as the imaginary started the division of hypernumbers, just as the creation of the algebraic fields started a new growth in the theory of numbers. Thus it is evident that mathematics is in no sense a closed book; that its chief concern