*Thc Handmaiden of the Sciences.* By Eric Temple Bell. New York: Reynal and Hitchcock. $2.00. *Mathematics for the Million.* By Lancelot Hogben, F. R. S. New York: W. W. Norton and Company. $3.75. *An Invitation to Mathematics.* By Arnold Dresden. New York: Henry Holt and Company. $2.80. *Men of Mathematics.* By E. T. Bell. New York: Simon and Schuster. $5.00.

To discuss any subject properly one must always have two adjectives, one for each side of the particular fence that separates the contending parties. Thus, in politics, we have liberals and reactionaries, or conservatives and radicals. In education, the brunt of the battle is borne by “vocational” and “cultural.” The terms are vague enough to adapt themselves to the demands of the most woolly-minded, and are particularly in demand at this time of year, when commencement speakers who have nothing to say are faced with the necessity of saying it. The meaning of “vocational” can be made fairly precise. The teacher would define it as “training the student for a particular activity in the general economy”; for this elaborate phrasing the student substitutes a briefer pragmatic equivalent: “having cash value.” But the more definitely we conceive the first of these terms, the vaguer become the outlines of the second. The tendency to make “cultural” synonymous with “non-vocational” is inevitable. Subjects without demonstrable cash value are cultural *per defintionem.* As these multiply and proliferate within the body pedagogic, any positive meaning that once attached to the word evaporates, leaving only a pious faith in the “somehow good” of all forms of instruction.

In this condition of affairs the status of mathematics in our schools and colleges is easily understood. Mathematics faces both ways. In its elementary aspects it is an indispensable tool in certain of the practical arts; it has a secure place in every engineering curriculum, and even in commercial courses. But in its higher aspects it is remote as few things are remote from a possibility of translation into dollars and cents, the possibility of a teaching career being always excepted. These aspects are therefore cultural, and on this plane mathematics must compete for the student’s time with the study of abnormal psychology or the history of art. In this respect the fate of mathematics is similar to, only worse than, that of other traditional disciplines, such as Greek and Latin. These latter can delay their final extinction by an appeal to snobbery. There are certain distinguishing marks of an “educated” man; certain information, certain accomplishments, certain social shibboleths, that label their possessor “gentleman.” Lacking these, the prospective lawyer, doctor, or man of business is severely handicapped. Here is cash value. The learned counsel’s happy use of a Vergilian tag wins the equally learned judge’s approval; the salesman’s intimate knowledge of Debussy enables him to put over the deal with the French banker. As the late Frank Moore Colby remarked, the average man “doesn’t expect culture to do anything to him, but he means to do a lot with it to you.” Not a lamp for our feet, but a jack-light with which to dazzle and bring down big game.

The word “culture” once expressed a definite concept, a concept of something positive and challenging, not a mere negation of utility. Whatever doubts we may feel about the wisdom or practicality of President Hutchins’ programme of educational reform, we should be grateful for his valiant attempt to restore some balance in educational ideals by his reassertion of the ancient values. Let me try to state these values in my own way.

Modern life conceals beneath a smooth surface an inconceivable complexity of structure. Each of us incurs daily a thousand debts—to those whom we never see but whose co-operation we take for granted. We discharge these debts by doing adequately our own share of the world’s work; failing in this, we become parasitic. But we have also a debt to the dead, to those whose creative genius has made possible all that we value. This debt we cannot pay, unless we too are of the rare company of creators. But we can acknowledge it. We all begin life in a state of spiritual parasitism, sucking from the tissues of the social structure in infantile insensibility all those ideas, habits of thought and action, and the like, that go to our upbuilding. Only in so far as we can become aware of what we owe do we become free individuals. It is this self-realization, attained by an understanding of our spiritual heritage, that is the essence of culture.

Of this heritage a great part is mathematical. The insights that man acquires into the nature of things are expressible only in symbols; where adequate symbolisms have been lacking, the mind of man has halted in its march. Mathematics, in its creative aspects, is the art of creation of symbolic form; art in its highest sense. In these aspects and these only are cultural values to be found. On the other hand, with this character of a creative art mathematics combines the progressive character of a science building new truth on truth already won. In this aspect it becomes a tool for other sciences. The contrast is between mathematics the queen of the sciences (say rather of the arts) and mathematics the handmaiden.

Of the four books before us, two accent one aspect and two the other. Eric T. Bell’s “The Handmaiden of the Sciences” is an account of the way mathematics has served in the development of our knowledge of nature. It is intended for readers with a minimum of mathematical preparation. The exposition is simple, and there is no attempt at profound analysis. On the other hand, the mathematically sophisticated will see behind the simplicity of statement the background of thorough knowledge; nowhere does the author falsify by making things too easy. I am too familiar perhaps with the ground covered by the book to estimate the skill of exposition. It seems to me workmanlike, but not distinguished.

Lancelot Hogben’s “Mathematics for the Million” emphasizes practical utility, giving an outline of elementary mathematics in easily digestible form. A syllabus of the ground covered would show a quite conventional content of algebra, geometry, trigonometry, with the elements of calculus and of statistics. The presentation is less conven-

tional. At times the author shows great skill in leading up to some central concept; at other times the exposition is halted while he slops over into social theory, history made

to order, and general irrelevance. Mr. Hogben is some sort of Marxist. I hesitate to characterize him more exactly but it were putting it mildly to say that he takes no pains to conceal his antipathy to priests, aristocrats, and capitalists. It is not improbable therefore that I judge his Marxian digressions too harshly; the sauce with which he douses algebra and geometry, and which nauseates my bourgeois stomach, may make his downtrodden millions lick their lips in anticipation. He prefaces the work with a quotation from Dantzig’s “Number”:

It is a remarkable fact that the mathematical inventions which have proved to be most accessible to the masses are also those which exercised the greatest influence on the development of pure mathematics.

It is indeed a fact, remarkable or not, that the great ideas of mathematics are simple ideas, and hence accessible to all. But Mr. Hogben interprets otherwise. He thinks that mathematics has stagnated when it has been the exclusive possession of a few, and that its concepts must become common property before progress is possible. The difficulty with this thesis is that it is unsupported by any facts of history known to us; the facts necessary to support it must be invented for the purpose, or unimportant improvements in notation must be exalted above the insights of men of genius and called “turning points” in the history of mathematics. Professor Bell’s biographical sketches, which will be discussed further on, furnish a good corrective to this Marxian mythologizing.

Mr. Hogben is no mathematician; there are passages in “Mathematics for the Million” that would make a genuine mathematician shudder. Also there is throughout a failure to distinguish between the abstract logical system which is mathematics proper and the application which interprets it. In taking leave of this book, as a specimen of irrelevance I quote the following gem:

The French language is especially suitable for the exercise of ironical wit. The English language is especially suitable to convey scientific truths concisely. The tortuous prolixity of German diction can be used to befuddle sensible and decent people till they believe that Hegel’s dialectic makes sense, and that Jew-baiting makes a nation prosperous.

My insufficient knowledge of French stands in the way of appropriate comment.

Professor Dresden has written consciously for college students. His object was to provide a course of mathematical training which should ignore utilitarian considerations and bring the student into contact with a wide field of genuine mathematical thought, instead of furnishing him with a language of science or a bag of tricks. In “Invitation to Mathematics” it is not mathematics the handmaiden invited to serve at the table of physics or economics; it is mathematics the queen of the sciences who invites us to her royal board. Here we must observe fully the ceremonial and decorum appropriate to majesty. Mr. Hogben and Professor Bell have been at pains to show us how really simple the apparently complex is. Professor Dresden reveals to us the profundities hidden in the apparently obvious; only those who can delight in the beauty of clean logic will be welcome to sit above the salt at this table. Being myself something of a propagandist and proselytizer, I expect to give away numerous copies of “Invitation to Mathematics” to friends not past hope of redemption.

j Eric T. Bell’s “Men of Mathematics” is a collection of short biographies of eminent mathematicians. It is a fascinating book. For the serious student of mathematics it unrolls in panoramic view the long procession of those who have furnished our modern world with its vast equipment

of patterns of abstract thought. Whitehead has said truly

that “the science of mathematics, in its modern developments, may claim to be the most original creation of the human spirit.” Those who accept Matthew Arnold’s definition of culture as knowledge of the best that has been said and thought, should take this saying of Whitehead’s to heart. Not all of them would or could acquire any conception of the world in which the mathematician moves; they might at least be less complacent in their ignorance, and might be moved to make the acquaintance of some of the

figures in Professor Bell’s roster of the greatest. Of the thirty-odd names in this list not more than half would

awaken even a flicker of responsive recognition by students of *literae humaniores.* In Merz’s four large volumes on the “History of European Thought in the Nineteenth Century” the name of Abel, probably the second greatest mathematician of that period, is not even mentioned, nor is that of Galois, another of mathematics, immortals, killed in a duel in his twenty-first year. Merz devotes much space to Sir William Hamilton, second-rate commentator on Kant, but of Sir William Rowan Hamilton, the Irish mathematician, a rarely original creative genius whose work has paved the way for our knowledge of the atom, there is not a whisper. This in a work that professes to follow the growth of scientific thought.

The thought that Professor Hogben’s book will be widely read fills me with pity for the million. It is too much to hope that a large body of readers will have the hardihood to accept Professor Dresden’s invitation. But “Men of Mathematics” can be and should be read by everyone. The mathematically initiated will find delight in the human figures that walk across Professor Bell’s stage. Those to whom mathematics is a book with seven seals will be made to realize something of the existence of a world of high art whose creative impulse has worked like a ferment in western thought and whose achievements are among the major glories of the human intellect.