Marginal Notes From the Library of a Mathematician

THE books from which the following marginal notes are taken came into my possession at an auction sale in 18—. I made no formal bid for them, — indeed, no bid at all. They were a dejected-looking lot. Some one had evidently handled them frequently, but considerately, for, while worn, they had not been abused. Tied together in sets, as if unworthy of individual sale, they reminded me of slaves brought to the block after the master’s death, and as I raised my eye mechanically to the auctioneer they seemed to cower under his hammer. Possibly he detected some trace of sympathy in my face. At all events, any incautious lapse from stolidity of expression on my part received prompt interpretation, and I was declared the owner.

My unintentionally acquired possessions proved to be chiefly mathematical, and the penciled notes interpolated here and there in the text, extensions of theorems and hints at consequences which had escaped the author, were proof that their owner had been an intelligent reader. These, however, were not the notes which most interested me. Along the margins and above the captions, in fact wherever a pencil could wander, were others, whose contents, while evidently suggested by the text, contrasted as strangely with its prosaic subject as did their flowing characters with its rigid symbols. It is from these notes that the following extracts are made. Possibly some one may recognize in them the hand which ran so fluently over the margin of those dry pages, but which never wrote its master’s name on the fly-leaf. Should such prove to be the case, the undersigned would gladly receive any information of one whose random pencil alone sufficed for friendship. It remains for me only to add that, while these annotations are rarely connected with any particular passage in the text by visible signs, I have prefixed and inclosed in brackets the sentences which seem most probably to have suggested them.

A. S. Hardy.

[“ Force is the unknown cause of changes in the rate or direction of motion.” Here follows a reference to current speculations on the nature of force, in fine print.]

These fine-print notes have come into fashion since my day. May they never go out! One may theorize in brevier, but not in text-books, where theories on ultimates ought to appear only in diamond. I know a man who, after a course in chemical philosophy, believed that the atoms of a pyrophosphoric acid molecule were actually arranged as pictured on the page of his text. Why is it that people who know the evidence on certain questions is lacking do not say — just that? No one thinks to establish dynamics on “ pushes ” because we cannot understand “ pulls,” — that is, action at a distance, — simply because, in making the attempt, pushes would be found more embarrassing than pulls, the plenum more difficult to dogmatize in than the vacuum. One can dodge a question in the latter ! No, we fall back resignedly on the " unknown cause of changes in momentum,” and consign the speculations to a fine-print note. But this resignation, so commonly admired in physical science, is stamped elsewhere with the iron of heresy. Eminently desirable as it is to look at the pull-andpush question from more than one standpoint, in order that we may be brought to a modest admission of ignorance, free circulation about other questions presenting more than one aspect is treason. For such questions there are definite points of view, and all the adjacent territory is placarded “No trespassing ! ” The temple doctors may wage honorable warfare on those who occupy other standpoints, but for the unlucky wight who circulates about the object of interest, to obtain a view from all sides, they have a supreme contempt.

Moreover, the multitude follow the doctors. Most people believe the solid truth has only one side ; that, paradoxical as it may seem, it cannot be solid if it has more than one. It is the positive thinking, even of the most dogmatic kind, which charms. In the history of thought it is the positive thinker, though he be a fanatic and his truths half-truths, who leads the multitude and founds a school. The kings in philosophy and life, in the world of thought and the world of action, look steadily at one facet of the crystal, and it is from this steadfast vision, this preponderance of one aspect of truth at the expense of others, that the great empires, systems, sects, have sprung. Devotion to a single principle implies, it is true, contempt or neglect of other principles ; but the conservatism which lacks definite statements has ever been scattered and leaderless, for only about such definite centres will men gather, ready to die. The man of single purpose is the successful man, because he is the man of one dominant, controlling, positive idea, or set of ideas. They give the edge to his sword and the weight to its blows, whether he be a Descartes or a Mohammed.

Neutrals may be loved and respected, but have not the ear of the world. For intellectual leadership, we must have answers for questions, solvents for the mysteries which these questions concern, something definite to meet hopes and still fears. To explain, to modify, to condition, is to weaken. To see difficulties and acknowledge mysteries is to lessen authority. But an unlettered Arab, crying, “ This is the will of Allah ! ” founds a creed and air empire. Power lies in extremes, at the far end of the lever. If there is any in materialism, it is in Moleschott’s proposition that everything in the intellect has entered by the gate of the senses ; idealism is shorn of its strength when it admits the existence of matter ; the skeptic is at his best when he knows that he knows knowing is hopeless. So all the great currents of philosophic thought set strongly in one direction, and, like the magnetic needle, oscillate without ever resting on the true meridian. The authority and prestige of systems run parallel with boldness of assumption and definiteness of statement, even long after they have left a truly rational basis. What is not apparent in the founder appears in the disciple, who denies what the master neglected, pushes the premise to its ultimate conclusion, and develops the single limit towards which, like a series, it tends. It is not that a system ought to be thus autocratically exclusive in order to be influential, but that this association of exclusiveness and influence is a fact of history, and one we might predict from a study of individual development without any knowledge of history whatever. For while it is true that our ideal human development is a symmetrical one, in which body, mind, and heart move abreast; while we know that a healthy body is conclusive to sound mental action, that a man with a diseased liver is apt to take diseased views of life, that great intellectual powers would be softened and deepened by a strong emotional nature, as the latter, when in excess, would be toned and invigorated by a strong intellect; while our ideal, in short, is one of balance, we do not see this type markedly successful in life. The rounded spheres may be loved and admired, but fail of deep impressions. Whatever our ideals, we expect to find marked success in any one direction bought at the expense of development in other directions. It is the same with thought-systems. Our ideal one has not occupied the seat of power nor controlled the suffrages of men.

This the artist, the man of ideals, tacitly recognizes when he translates them into color and form. Every ideal figure must still represent a particular type. Cosmopolitanism, says Tourguéneff, is a zero, and outside of nationality there is no art.

Yet the neutral charms me. The world is like a huge nursery, with a few grave figures among the throngs of children. What a riot of laughter and crying ! What prayers and Te Deums! Our neutral friend had best hide himself in such company, lest his very modesty appear a garment of pride and his silence be mistaken for scorn. We are naturally believers. We are so persuaded that a thread of cause runs through the universe that any one who thinks he has discovered it is sure of followers. Childhood is one impulse to believe running riot. We grow more cautious with years, but never escape this spirit of inquiry, with its thirst to know. Out of it comes all progress. Theories run to the end of the tether, and the noose strangles them. Every eye that singles out one fact, or aspect of truth, challenges another eye. But when we see in this tumult the neutral who halts because he will not be towed, silent because he cannot affirm, we feel the power of his poise. If there were no danger in his conservatism, no shadow of fanaticism lurking in this very appeal to the judgment; if there were no skepticism, no despair, lying in wait for him, we should envy without fear his catholic heart, his calm, judicial eye.

[“ While the mechanism of what (for want of knowledge) we call gravitation has not been explained, we are none the less able to formulate its laws of action, and apply them to the practical problems of life.”]

Every teacher encounters those who would not challenge explanations of mysteries inscrutable as the mechanism of gravitation, but for whom the statement of our want of knowledge of this mechanism would prove a source of uneasiness. For such, ignorance of the action is a reason for restlessness under the action-law. If told that the sum of energy in the universe is a constant, they ask, “Who fixed the constant?” The desire for the absolute is so strong they would prefer to assert that π is exactly two sevenths rather than to admit it is the interminable decimal 3.141592+. Doubtless there would be something gratifying in the ability to write its exact value, but, were this possible, the mariner would not thereby be enabled to determine his latitude and longitude any more accurately, nor the astronomer to predict more closely the time of an eclipse, simply because, in each case, he can do so now as accurately as he pleases, and with far greater precision than he needs to. Yet this inability to express certain incommensurables in finite terms, though their exact evaluation would not appreciably affect the conduct of life, is as grave a source of concern to many as is this obstinate π to the circle-squarers. Like one who throws a coin in the air, they cannot know on which side it is to fall, yet must decide before it reaches the ground. Fate or free - will, mortality or immortality, — these are the stupendous head and tail of the coin, the life, and they must answer before it falls, before it is over. No matter how scanty or how conflicting the evidence, they must bring in a verdict.

My good friend the sailor is no whit safer in setting sail with an approximate value of π than are you, and if his bark sinks π will have nothing to do with the catastrophe. “ Be not curious in unnecessary matters, for more things are showed unto thee than men understand.” Meanwhile, “ add not more trouble to an heart that is vexed, and defer not to give to him that is in need; and let it not grieve thee to bow down thine ear to the poor and give him a friendly answer with meekness.”

[“ The Cartesian coördinate system is a device which enables us to discover and enunciate the properties of curves.”]

For proof of the fact that ultimate solutions are unnecessary for the right ruling of conduct, observe that all philosophical and theological systems, however divergent at first, and though they lead us through lands east of the sun and west of the moon, all meet at last and voice one golden rule. They are like the coördinate systems of geometry, — mere bridges and scaffoldings, — all proving the same propositions and enunciating the same truths. Why then look askance at our brother? Fancy a mathematician who had found the path of a projectile was a parabola by Cartesian coordinates assailing a colleague who had reached the same conclusion by the polar system! Imagine him eying the radius vector with suspicion, making fun of the pole, and holding up his hands with holy horror at the polar axis !

[“Any one who will try to imagine the state of a mind conscious of knowing the absolute position of a point will ever after be content with our relative knowledge.”]

We attack our problems in geometry and kinematics with “ fixed axes, but the fixity is only relative. Think of a geometer doubting his results because he had not referred his data to an absolute origin; or, what is worse, since he at least knows the absolute is unthinkable, declaring his results absolute when his origin is not! What endless vistas of hopeless search open before the ambition that is dissatisfied with a statement of relative velocity ! And yet our theological colleagues have absolute systems of reference! We should have to face all their dilemmas if we attempted to refer our relative data to the “ absolute centre,” and these dilemmas of a mind conscious of knowing the attributes and ways of the absolute ought long ago to have made us content with our relative knowledge.

Then again, not being tied up to an absolute (!) origin, the geometer often recognizes the desirability of a change of axes, and transforms his equations unhesitatingly when such a change facilitates his search. But what a stir a transformation to new axes occasions in theology! And the most remarkable accompanying phenomenon of such a change is the fact that men who once effected similar changes thereafter oppose new ones. It seems as if those who once pushed the triumphant car of intellectual progress ahead an inch always got into it afterwards, and objected to further advance.

Why should not a movable origin, such as is used in certain kinematical problems, be introduced also into morals ? X is naturally avaricious. It requires a tremendous effort on his part not to steal. Y does not even see the opportunity. Yet, practically, both are judged by the same standard. Such a method of evaluation is to oloose for mathematics. Absolute standards which are not absolute serve only to distort every relative.

[“ The quaternion analysis is only a new organ of expression.”]

We need a new language for all logical processes. English does well enough for poetry and fiction, but is it the proper vehicle for exact thought? Huxley and Gladstone both use x in the argument, but it turns out that Huxley meant x’ and Gladstone x.” A writer makes a like mistake twenty times in a chapter, and no one discovers it for two centuries ! Such blunders in the Mathematical Journal would be set down to the printer, — all the errors are clerical, that is, easily avoided; or if, as may be contended, easily made, then also easily detected and corrected. The truth is x means x. Mathematics has no dictionary of synonyms. How few books we should have on some subjects if we had a better “ organ of expression,” and how much more we should know ! How we should be obliged to recognize our premise as such in our conclusion, and to eliminate our infinites, infinitesimals, and imaginaries before stating our results.

True, it is impossible to write, even in the language of mathematics, without, like Horatio, putting more into a combination of symbols than we dreamed of. But its interpretation is not left to the imagination. A terrible necessity, inherent in the nature of these symbols, governs their meaning. The mask they wear may cover the unknown, never the uncertain. But who knows what lies under the mask of an English word ? No logical necessity strong enough to bind the fancy of the reader determines its signification. O la belle exégèse!

What sermons thou wringest from the text! Not Rumford boring his cannon could have been more surprised to find that the more he bored the more caloric he got.

In our desire for a new language for logical forms, let us not, however, dream to translate into it all our Bible, lest we fall into the difficulty of one who, in his zeal to make the decimal system universal, was obliged to suppress two of the twelve apostles. This language of algebra has no perspective effects, no sense of proportion. In it x is as good as y, though x be a mud-puddle and y a rainbow. If the realist is in earnest, let him write in algebraic characters, where, if x be the unknown quantity, it must be called so ; where the minus is as important as the plus, and the uninteresting as vital as the interesting; where there are no softening shades, no melting tones, no Claude Lorraine glasses, and the inventor of the cosmos described monopolized all the imagination he created.

[“ All the great advances of the century grow out of our increasing knowledge of the properties of matter.”]

We are signing away our liberties to these vanities of the century, — telegraphs, railroads, and the rest. If we could reverse the experience of Rip Van Winkle, and sleep backwards a hundred years, we should wake more helpless than he did. Moreover, everything that increases the delicacy and sensitiveness of organization (of nervous tissue or of society) turns a stronger light on the fact of our dependence upon matter. Pascal boasted that he was greater than the universe, because, though the universe crushed him, he knew it, whereas of this the universe knows nothing. Precisely, says Leopardi, savoir sans rien pouvoir. To realize our spirituality is simply to tug at our chains. What a poem might be written on compensation ! — the spiritual shadow, ever more distinct as we rise into the sunlight.

Are we then so ungrateful that, after the molecule has acquired a vortical motion and a complex, unstable equilibrium for our especial delectation, we turn and revile it ? Fancy an “ aquosity,” as Huxley would say, declaring water to be a nasty compound ! — kicking against its very raison d’être. Still, when we rise to the summits of being, we too kick at our chains. There in the rarer medium, where thought plays with space and time as if they were the fictions they may be, the mile-per-hour velocityunit which the body, dragged so painfully up the hill, imposes on the spirit seems a ridiculous one for a rational being, — a second infinitely too large, and a mile infinitely too small. No, we will be something better than matter. But let us waste no time in deciding what that something may be, lest the ever - increasing “ promise and potency ” of these wondrous atoms outstrip our dreams.