САМВАЛЫН BASS

SEWANEE REVIEW

VOL XIV.]

JANUARY, 1906.

[No. I.

SOME PHASES OF MODERN MATHEMATICS

The study of mathematics needs no defense. As some one has said, "every one must count and measure or perish." This sentiment may, however, seem a little irrelevant in speaking of higher mathematics, with which we are chiefly concerned in this article. I have not time or space here to refer to the recognized utility of mathematics as a mental drill or as the useful handmaiden of the arts and sciences. "I hate mathematics" is not an uncommon saying, but surely only the ignoramus would seriously venture to add "it is of no use." The empirical and observational sciences all come to mathematics for help and all bow before it as a science superior to themselves; and the pure mathematician is only too glad to have an opportunity of lending aid to engineering, astronomy, physics, chemistry and other sciences, and on the other hand he realizes that he gets much inspiration and much material aid from these sciences.

But while the larger and more valuable fruits of pure mathematics are to be found in applied mathematics, we can not overlook the fact that, though such abstract reasoning seems dry and unprofitable to some minds, the study of pure mathematics, even outside of its sphere as a mental exercise of the highest order, is uplifting in itself and opens up visions of the true and the beautiful in a way that no inexact science can do. While the writer has in mind, in this paper, these sweeter and better flavored fruits of pure mathematics, he does not mean in the least to disparage applied mathematics, for which he has the greatest respect and admiration. The former is really the mathematics of

precision, the latter, the mathematics of approximation, but the gulf between them is not a broad one.

I think I may safely say that very few people, even among the most cultivated and best educated, have the faintest idea what the great mathematicians of the day are doing, and in what channels mathematical thought has directed itself during the past hundred years. The very names of the world's best mathematical workers are scarcely known outside of a very limited circle of scientists. Now there are some facts concerning the development of mathematics in recent times that can perhaps, even in the limited time at my disposal, be amplified enough to be of some interest to the general reader. Although I propose to write of modern mathematics, let us go back for a moment (by so doing we shall really be sounding the key-note of modern thought) to first principles by asking the question: What is mathematics?

For curiosity, if for no better reason, I turn to the "Century Dictionary," and find its definition to be "mathematics is the science of quantity: the study of ideal constructions (often applicable to real problems) and the discovery thereby of relations between the parts of these constructions before unknown." Owing to necessary limitations of space in a work such as the "Century Dictionary," perhaps this definition could not be greatly improved upon; but it is nevertheless entirely inadequate when we call to mind such subjects as projective geometry, the theory of groups, and many other phases of recent mathematical development. Not to do injustice to this dictionary, I ought to add that further light is thrown upon the subject by quotations from Clifford, and there is also given the celebrated definition of Benjamin Peirce: "Mathematics is the science which draws necessary conclusions."

That keen thinker, Professor Simon Newcomb,' defines mathematics as "the science which reasons about the relations of magnitudes and numbers, considered simply as quantities admitting of increase, decrease and comparison." Professor Chrystal, the well-known English mathematician, suggests in part the

1 In Johnson's "Universal Cyclopædia." 2 Encyclopædia Britannica, Vol. XV.

following definition: "Any conception which is definitely and completely determined by means of a finite number of specifications, say by assigning a finite number of elements, is a mathematical conception." "A triangle," for example, being defined by three elements (a finite number) is a mathematical conception; "a man" is a non-mathematical conception, for no finite number of elements is sufficient for an adequate definition.

Now these scholars had no intention of laying down dogmatically a precise definition. No mathematician was ever satisfied with a definition of mathematics. Professor Bôcher, of Harvard, at the St. Louis Congress of Arts and Sciences, delivered an address, on "Fundamental Conceptions and Methods of Mathematics," in which he discusses at some length possible answers to the question: What is mathematics? Among other things, he says: "in order to reach a satisfactory conclusion as to what really characterizes mathematics, one of two methods is open to us. On the one hand we may seek some hidden resemblance in the various objects of mathematical investigation, and having found an aspect common to them all we may fix on this as the one true object of mathematical study. Or, on the other hand, we may abandon the attempt to characterize mathematics by means of its objects of study, and seek in its methods its distinguishing characteristics. Finally there is the possibility of our combining the two points of view. The first of these methods is that of Kempe, the second will lead us to the definition of Benjamin Peirce, while the third has recently been elaborated at great length by Russell. Other mathematicians have naturally followed out more or less consistently the same ideas, but I shall nevertheless take the liberty of using the names Kempe, Peirce and Russell as convenient designations for these three points of view."

For a rather lengthy and able discussion of these three methods the reader is referred to Bôcher's published address,' wherein he shows that the three methods of approach to the question lead in the end to results that stand in intimate relation to one another.

3 Bulletin of the American Mathematical Society, Vol. XI, No. 3.