c MATHEMATICAL PSYCHOLOGY The Mathematical Psychology of Gratry and Boole TRANSLATED FROM THE LANGUAGE OF THE HIGHER CALCULUS INTO THAT OF ELEMENTARY GEOMETRY BY MARY EVEREST BOOLE Autlwr of" Logic Taught by Love," " Symbolical Methods of Study," etc LONDON SWAN SONNENSCHEIN & CO. LTD. NEW YORK: G. P. PUTNAM'S SONS 1897 BUTLER & TANNER, THE SELWOOD PRINTING WORKS, FROMB, AND LONDON. TO HENRY MAUDSLEY, M.D. DEAR DR. MAUDSLF.Y, You have often asked me to explain, for students unaqtiainted with the Infinitesimal Calculus, certain doctrines expressed in terms of that Calculus by P. Gratry and my late husband. That you permit me to dedicate my attempt to you will, at least, be a guarantee that the main ideas of mathematical psycho- logy are based, not on mystic dreams, but on scientific induction. I am glad of this opportunity of expressing to you my gratitude, both for your published works, and for much personal kindness to myself. I am, Dear Dr. Maudsley, Yours very truly, MARY EVEREST BOOLE. 2223198 PREFACE IT is usual, in a preface, to express the author's thanks to those who have been helpful in the evolution of the book. Were I to attempt to do this, I should have to mention the names of most of the persons with whom I ever conversed, and most of the authors whose works I have read. The existence of my little volume in its present form is due to Mr. Percy Furnival, F.R.C.S., of St. Bartholomew's ; who, on learn- ing the nature of the work which I have long had in hand, insisted that information which may possibly prove to be of some use to members of his profession, ought to be made accessible to them without further delay. I have to thank him for his kind help in the final arrangement of my material. M. E. B. CONTENTS CHAPTER PAGE I. INTRODUCTORY . . . . . . i II. GEOMETRIC CO-ORDINATES ... 28 III. THE DOCTRINE OF LIMITS . . .36 IV. NEWTON AND SOME OF His SUCCESSORS 42 V. THE LAW OF SACRIFICE .... 58 VI. INSPIRATION VERSUS HABIT ... 73 VII. EXAMPLES OF PRACTICAL APPLICATION OF THE MATHEMATICAL LAWS OF THOUGHT . 88 VIII. THE SANITY OF TRUE GENIUS . . 103 APPENDIX . . 114 CHAPTER I INTRODUCTORY T TOW is it possible that questions of general *" Psychology should be mathematically in- vestigated ? And if such a thing be possible, what class of readers does the matter concern ? In answer to the latter question, it is sufficient to say that I have been invited by members of the medical and educational professions to state, in the ordinary language common to all educated persons, certain conceptions about the nature of the process of original thought which have hitherto been expressed in a terminology intelli- gible only to mathematical specialists. Many to whom these conceptions have been explained viva voce seem to feel that it would be well if they were made accessible to all who are con- cerned either in fostering the development of original faculty, in distinguishing between genius and the mere inspirations of self-conceit, or in preventing the growth of hallucinations : that is to say, to young persons who either have B MATHEMATICAL PSYCHOLOGY or suppose themselves to have original ideas ; to parents and teachers ; and to those within the sphere of whose duties it may come either to minister to the special needs of genius, or to deal with that lurid shadow of genius which we call insanity. But how can there be such a science as Mathematical Psychology ? Mathematics is understood to be the science of number and space. It is true, indeed, that all large masses of data must be reduced statistically before the information which they are capable of affording can be made available as a basis for accurate reasoning ; and statistics, of course, belong to the domain of mathematics. In that sense the adjective " mathematical " might conceivably be used to qualify the name of any science ; but, in any other sense than that, how can mathe- matics be concerned in a treatise on human psychology ? This initial question can best be answered by a parable. Suppose that by some means the inhabitants of Mars became possessed of a ladder of human construction an ordinary ladder such as is used in our orchards ; and suppose they were informed that this instrument INTRODUCTORY is one in common use among the intelligent inhabitants of the Earth for the purpose of reaching certain articles of food which would otherwise be unattainable. The first impression of the Martians would probably be that the instrument had some direct relation to the articles desired by men ; either that its shape is adapted to catch the fruits, or that the ladder has some property of attracting the fruits and causing them to come of themselves towards the person possessed of it. On that basis they might possibly construct a pseudo-science ol earth-botany ; an elaborate scheme of hypothe- tical assumptions about the nature and growth of earth fruits. It would probably not occur to them at first that the ladder could reveal any- thing about the nature of man (except, of course, that they would infer a tendency in man to desire fruits growing out of his reach). But suppose it came to their knowledge that the ladder has no magnetic attraction for fruits, and that its shape has no relation to any property of fruit except the one property of growing out of human reach ; their hypothetical botany would then be seen to be chimerical, and they would be driven on to a different line of specu- MATHEMATICAL PSYCHOLOGY lation. If the ladder has no relation to the things to be reached, then it must be constructed in some very close relation to the normal action of the creatures who want to reach the things. Starting anew on this sounder basis, it might be possible for them to arrive at some know- ledge of the anatomy of the human climbing- limbs. Then suppose that beings on the other side of Mars acquired knowledge of our anatomy by some other means, suppose a book of anato- mical plates or the legs of a human body were conveyed to them, we can easily see how, if the two sets of Martians came together and compared notes, the set who had investigated the ladder might supplement the more direct knowledge acquired by the other set and add to it elements of considerable value. The science of mathematics is an intellectual ladder. The rather usual assumption that it is "a science of number, size, and form" is as erroneous as would be the assumption that a ladder is an instrument specially adapted to draw cherries from their natural level. Before we can truly grasp the conceptions of Mathe- matical Psychology, we must begin by realizing a little how far mathematical science is from INTRODUCTORY 5 being harmoniously adjusted to the laws of the subjects to which it is supposed to be necessarily attached. Mathematical processes may almost be said to have no relation to any property of the laws of number or of natural form, except, indeed, the one property of being too difficult for the intellect of man to grasp unaided. When we are clear on this point, when we have come to a true understanding of what mathe- matics is not, we shall be in a better position for entering on the question : What it is. Mankind have invented names for each of the numbers in order, and have agreed on certain written signs to represent those names : i, 2, 3, etc. Why, on arriving at ten, do we suddenly leave off writing fresh signs and begin to use combinations of the former ones ? No property of numbers is more absolutely cer- tain than that of following in an unbroken and homogeneous series without jolts or breaks. Why do we make a break in representing them ? Because man is a creature who finds it more convenient to use classified combinations of a few signs than an indefinite number of separate signs. But man easily uses twenty-six letters in his MATHEMATICAL PSYCHOLOGY alphabet; why, then, does he cut short his numerical alphabet at less than half that length ? Because, though our memory easily retains twenty-six or more separate signs of similar kind, few persons can visualize, or form a direct conception of, such a number as twenty-six. The largest number that an individual can clearly picture to himself or directly conceive varies, according to his degree of culture and his visualizing capacity, from three or four in the very young child to twelve or sixteen in the average adult. Only exceptional persons can conceive of rather larger numbers. Why, then, have we not chosen as the basis of our numeration the number twelve ? It is, by its own nature, an eminently J suitable number for a base, because it has several factors, and therefore is divisible in more than one way. Why did we not evolve a duodecimal instead of a decimal notation ? Because man has ten fingers to count on, five on each hand ; and, therefore, the only two notations which have ever come into ordinary use are that by fives (the so-called Roman, in which six is written "five and one," or the fingers of a hand plus one) and that by tens (the Arabic, in which INTRODUCTORY eleven is written "ten and one", or the fingers of two hands plus one). We see, then, that arithmetic corresponds from its very origin to properties which are distinctively human, not numerical. Its pro- cesses are, from the first, anthropomorphic, not, as is commonly assumed, purely abstract Our systems of numeration are permeated through and through with traces of an anatomical truth, the truth that man has ten fingers ; there is, therefore, nothing improbable a priori in the suggestion that mathematics may contain indications of psychological truths. Indeed, it has been shown that it does contain such indi- cations ; the very existence of any such inven- tion as a system of numeration points to the psychological doctrine that both man's capacity and his desire to register and combine numbers are greatly in excess of his power of directly conceiving them. This particular psychological doctrine happens to be one which we could have discovered without reference to mathe- matics ; therefore the specially mathematical evidence of it attracts little attention. We shall see in the sequel whether mathematics contains evidence of any other truths of psy- 8 MATHEMATICAL PSYCHOLOGY chology of any truths not easily discovered without its aid. As we trace arithmetical and algebraic reasoning through successive gradations of increasing complexity, we find at nearly every step the same artificiality. The processes are artificial in relation to the subject-matter, but they are well adapted to meet the limitations of our direct faculties, and to assist us in attaining knowledge unattainable by our un- aided powers. So far as we have now arrived, there is no great difference between arithmetic and the so- called natural sciences. Most of the implements and methods of science betray their distinc- tively human origin. Optical instruments, for instance, are implements whose primary func- tion is simply to extend the range of human vision ; special adaptations are in some cases made for the observing of special classes of objects ; but the function of the microscope is essentially to enable man to see more minute objects than he naturally could do ; and that of the telescope is to enable him to see more distant ones. The text-books of nearly all the natural sciences contain, mixed up with INTRODUCTORY information about the particular science, in- structions about devices for enabling man to supplement his limited faculties, physical or mental, and to extend the scope of their activities. Most scientific books, therefore, contain, im- plicitly, information about human anatomy or psychology, or both. When a teacher, in order to demonstrate to his class the structure of a leaf, shows a skeleton leaf, the cuticle under a microscope, and the cellular substance, and then appeals to his class to conceive these various portions combined, and acting on each other, in the growing leaf, his method follows, not the nature of vegetation, it is not the nature of a leaf to grow its cuticle, its ribs, and its parenchyma separately, and then combine them, but the nature of man, whose faculties are constructed to work under such conditions of alternate analysis and synthesis. We have arrived at nothing, as yet, to show that mathematics is psychologically more in- structive than botany ; but we have at least, it may be hoped, cleared out of our way the popular but erroneous conception of mathe- matics as a science differentiated from most other sciences by being purely abstract, non- human, out of line with vital processes. 10 MATHEMATICAL PSYCHOLOGY If we turn now to another department of elementary mathematics, that which relates, not to number, but to size, form, and motion, we are again confronted with an anomaly similar in kind to that which we encountered in arith- metic. Before we can enter on such a study as the line of a planet-path we have to become familiar with the properties of the plane ellipse. Now, no such thing occurs in Nature, so far as we know, as a plane ellipse. The path of a planet in space is an elliptical spiral. Could anything be more artificial, as regards the planet-path itself, than to pretend that we have cut it by a plane, and then pretend that our paper is that plane, and then pretend that the spiral path has cast a shadow on the paper ? Yet that is what we are virtually doing when we draw an ellipse on paper and use it to study or teach the action of natural forces in producing planetary motion. Even supposing there were a real, living ellipse, supposing that any thing in Nature did run in an elliptic line on one plane, what is the mean- ing of all the straight lines, the axes, tangents, and radii vectores pictured in our text-books ? What are the rectilinear co-ordinates of alge- INTRODUCTORY 1 1 braic geometry ? All these devices have been invented, not to suit the nature of the motion under investigation, most of them have no relation to that motion, but to meet the needs of the human mind in its efforts to understand what is beyond its comprehension. And they answer their purpose perfectly. Let us turn back now to elementary arith- metic. Suppose a student desires some knowledge which is not at present in his mind, say about the mode of fertilization of a particular plant, or the chemical constituents of a particular article of food. He must go outside of himself for the information which he lacks ; he must either investigate a specimen of the plant or substance, or he must refer to some book, or be instructed by some one better informed than himself. It may happen, of course, that he is able to recall to memory what he desires to know, but in that case he is only ransacking a library of impressions stored up within him which impressions were produced originally by information from the outside. If he never yet possessed the knowledge he now seeks, he cannot now re-collect it from within ; he must 12 seek it outside himself. The mythical philoso- pher who " evolved a lion out of his moral con- sciousness," instead of looking at the real lion in the Zoological Gardens, is proverbially ridiculous. But suppose that you, my reader, have occa- sion to know how many days there are in two million four hundred and eighty-one thousand seven hundred and forty-nine years, that is to say, how often the earth will turn on its own axis while it is going round in its orbit a number of times far, far too great for you to conceive or imagine. It is obvious that you cannot acquire the information you need by direct inspection of the facts ; you will not live to count the days in seven million odd years. But you will not, in such default ot direct source of knowledge, consult any written book or appeal to any human teacher; nor need you distress yourselt by laborious ransacking of your memory ; the chances are greatly against your ever having known the exact number of days in the precise number of years about which you are now inquiring. You will take a blank sheet of paper and a pencil, instruments whose sole function is to INTRODUCTORY 13 register successively your own mental processes as they take place, and so to assist you in keep- ing order among them, and within five minutes you will be able to state with absolute certainty the fact of which, five minutes before, you were ignorant. How did you acquire the new knowledge ? Not by any process of mere syllogistic logic ; there is no similarity between the process of arithmetical multiplication and such ordinary syllogism as "All men are mortal ; John is a man; therefore John will die." In any valid syllogism, the conclusion is really contained in the premisses ; the syllogistic process only puts into more convenient form knowledge already possessed. If any fanatic were to propound such an argument as this : "All men are mortal, there- fore the archangel Gabriel will die," how he would be laughed at! What know we of archangels ? Nobody ever saw one. Yet it is just as true that nobody ever saw a million years ; nobody ever saw truly saw and cognized a million of anything. Yet we are permitted to infer something about three hundred times two million from the mere datum that three times two are six. This pro- 14 MATHEMATICAL PSYCHOLOGY ceeding must certainly be non- syllogistic, extra- logical. We shall see this all the more plainly if we remember that, as was shown above, the names two million and three hundred are purely artificial. Numbers themselves form a homogeneous and unclassified series ; there is no more natural connection between the number which we have chosen to call three and the number which we have chosen to call three hundred than there is between three and nine ; the whole nomenclature and classification of numbers is an affair of purely human psycho- logy and human anatomy. Yet every school child makes statements about three hundred times two million for which he has no data, except that he has counted what three times two make. It will be noticed that the knowledge gained by the process of multiplication is, in one sense, relative to ourselves. The answer we get, if large, is not conceivable by us ; but it is ex- pressed in terms of the base of our system of numeration ; i.e. t of some number chosen by us as base because we can conceive it (e.g., ten, the number of our fingers). The process of multiplication is capable of INTRODUCTORY 15 indefinite expansion. It does not become more intricate as it is pushed further ; it is as easy to manipulate billions or trillions as hundreds or thousands ; the only extra cost is the labour of writing extra zeros. It is as if, when we reached the top of a cherry-ladder, we found it perfectly simple to add on another length and climb that ; and then another, and another, till we reached the stars. In physical sciences, progress is hampered by the constant need for some fresh facts to lean the ladder on ; the arithmetical ladder starts, indeed, from a physical basis, our ten fingers, but it never needs anything to lean on any more. And there is no limit to its expansibility. Scientific men of all " Fachs " agree in assert- ing that the rough guesses of mere average common sense and a priori probability need to be corrected by the dicta of science. And the delightful complacency of their consensus on this point has caused them to overlook the fact that the expression "correcting an unscientific guess by scientific investigation " may refer to either one of two processes ^as unlike each other as possible. A man widely known for fidelity and 1 6 MATHEMATICAL PSYCHOLOGY trustworthiness suddenly robs his employer. Three months later he becomes delirious, and soon after dies. Average common sense sup- poses that he yielded to the pressure of some exceptional temptation, and then went mad of remorse, and that the strain of that remorse caused his death. Medical science declares that common sense has mistaken cause for effect ; that the man died of softening of the brain, which must have been coming on before the date of the robbery, and that in that dis- ease decay of the moral instincts frequently precedes intellectual and physical failure. Or a girl appears depressed in spirits, and talks mysteriously about some one having ill- used her. She dies suddenly. General proba- bility suggests that she made away with herself; the probability seems all the greater when it is reported that she lately bought beetle-poison. Chemical investigation proves, however, that the beetle-poison contained no ingredient fatal to human life ; and at the autopsy no trace of poison is discovered, but a clot of blood or a tumour is found on her brain ; and medical science declares that this cause accounts for de- pression, for imaginations about being ill-used, and for sudden death. INTRODUCTORY I/ In these cases scientific experts can correct unscientific guesswork by reason of having ac- cess to facts not generally known. But now let us take the case of the Eastern potentate who, according to tradition, bade the inventor of the game of chess to name his own reward. The learned man asked that a grain of corn should be given to him for the first square, two for the second, four for the next, and eight for the next ; and so on till the end of the series of sixty-four squares. The monarch considered this request a modest one, till the wise man showed him that the year's harvest would not furnish the amount of corn necessary. In this case the scientific expert had access to no facts hidden from his unscientific hearer ; there were no facts to be taken into account, except the obvious one that the board was divided into sixty-four squares ; the two men differed because the one had gone through a certain set of mental processes which the other had not gone through. One remarkable difference between mathe- matical certainty and what is called scientific conviction must here be pointed out. " The general laws of Nature are not, for the most part, immediate objects of perception. c 1 8 MATHEMATICAL PSYCHOLOGY They are either inductive inferences from a large body of facts, the common truth in which they express, or, in their origin at least, physical hypotheses of a causal nature, serving to explain phenomena and to predict new combinations of them. They are, in all cases, and in the strict- est sense of the term, probable conclusions ; approaching, indeed, ever and ever nearer to certainty, as they receive more and more of the confirmation of experience ; but of the character of probability, in the strictest sense of that term, they are never wholly divested. On the other hand, the knowledge of the laws of mind does not require as its basis any extensive collection of observations. The general truth is seen in the particular instance, and it is not confirmed by the repetition of instances; . . . the per- ception of such general truths is not derived from an induction from many instances, but is involved in the clear apprehension of a single instance. In connection with this truth is seen the not less important one that our knowledge of the laws upon which the science of the intellect- ual powers rests, whatever may be its extent or its deficiency, is not probable knowledge. For we not only see in the particular example the INTRODUCTORY 19 general truth, but we see it as a certain truth, a truth our confidence in which will not con- tinue to increase with increasing experience of its verifications." l When a professor illustrates some principle he has been stating by one or two instances only, he uses those instances, not to prove the prin- ciple, but simply to make the pupils understand exactly what it is that he asserts. Now if he is teaching any physical science, the pupils are expected, if not to believe the prin- ciple on the word of the teacher, at least to accept it provisionally as a working hypothesis till they gain, by repeated and long-continued experience, the right to hold an opinion one way or other on its truth. But if the principle be a mathematical one, such, e.g n as that ax& = &xa, the pupils are not expected to suspend judgment till they gain certainty by experience ; when once they have grasped what the principle is that is being stated, they see and know for them- selves that it is absolutely, certainly, and always true. One singular characteristic of mathematics is 1 Boole : Laws of Thought^ Chapter I. 2O MATHEMATICAL PSYCHOLOGY the automatic power of self-correction which the mind possesses in relation to it. No such thing is possible as the existence of either a persistent or a widely spread error, or a serious difference of opinion, as to the result of a calculation. And this is not due to any special immunity from error which we have in connection with the sub- jects called mathematical. Every schoolboy knows that it is as possible to make a mistake in a sum as in any other exercise ; and the greatest mathematicians occasionally make mistakes in calculations. But there seems to be some mysterious court of arbitration within man which detects mathematical error. Any two persons who have come to different conclusions as to the result of a calculation can find out which of them was mistaken, without appealing to any external authority. Nor need they examine any outer facts ; they need look at nothing out- side of themselves except the slates or papers 'which contain the register of tJieir oivn successive mental acts. They judge between themselves, and judge infallibly. When one child only has done a sum, he may need to look at the " answer in the book " ; if his result does not correspond with that answer, he thinks that his processes INTRODUCTORY 21 must have been wrong somewhere. But when two persons have made the same calculation with differing results, there is no need to com- pare the results with the answer in a book ; they revise and compare their own mental processes as registered by themselves. This power of absolute and unerring correc- tion and mutual correction, which man pos- sesses in regard to arithmetic, is popularly supposed to be due to something in the nature of the subjects commonly treated mathemati- cally (/>., number and quantity). "Man can know number and quantity more accurately than he can know anything else," is the ordinary explanation. But when we examine it, we fail to find any warranty for the assumption that man has more natural power of being accurate about number and quantity than about any other kind of truth. The chess-board legend already alluded to, and such well-known puzzles as that about the nails in the shoes of a horse, prove that people make just as wild guesses about number as about any other department of the Unknown when they dis- pense with systematic investigation and trust to their unaided judgment. And, as was pre- 22 MATHEMATICAL PSYCHOLOGY viously shown, we use with ease an alphabet of twenty-six letters ; if our arithmetical alpha- bet were as large, we should become hopelessly fogged in our calculations. Most of us could probably recognise at sight any one of some hundreds of persons, of words, of flowers, of any sort of objects with which we are familiar ; but when we come to numbers our powers of direct cognition seem much feebler and more limited. If dots are arranged in patterns, each number having its own pattern, as is done in the print- ing of playing cards, we recognise any one of the first ten numbers easily, as we know our friends apart by their faces ; most of us would know at a glance the look of twelve dots, or per- haps even of any number up to twenty. After twenty we should probably have to begin to spell out the numbers i.e., to count or calculate. That is as if we could not read at a glance any words except some sixteen or twenty of the shortest and simplest in the language ; or as if we could not recognise by sight more than twenty of our most intimate acquaintance. This weakness of our power of direct cog- nition with regard to numbers, as compared to our faculty of cognition in some other direc- INTRODUCTORY 23 tions, may be partly due to lack of exercise. It might be possible to strengthen it by prac- tice. But at least it leaves us no excuse for as- serting that numbers, as such, are easier for us to cognise with certainty than facts of other kinds. What, then, is the true import of our easy and familiar access to the Great Unknown on the side of number ? What is the source of the unerring certainty which we can all attain with regard to it ? " Necessity is the mother of invention." Is it not possible that the very limitation of our direct numerical faculties has forced mathe- maticians to create, unconsciously, the ladder on which man can safely climb to The Infinite ? l Is it not possible that mathematics is, after all, a science, not of number and quantity, but of the conditions under which man can make his pro- gress towards unknown Truth uniform and safe, and can preserve himself from being seriously misled by the mistakes which he is sure to make on his way? This question unavoidably presents itself, process is a process of iprayer." Gratry : Logique. Chapter on L Induction Gfometrique, 24 MATHEMATICAL PSYCHOLOGY sooner or later, not, of course, to all mathema- ticians that is to say, all users of mathematical formulae, but to all serious students of mathe- matical philosophy. It is in this century that it has taken concrete form and come into open expression in print; but it seems to me that it must always have been haunting the imagi- nations of the more subtle thinkers, ever since the first Abacus was invented as an aid in reckoning ; ever since circles were first drawn by some inspired savage, by means, perhaps, of two thorns linked together by a strip of bark. But more than I have yet indicated has dawned on the imagination of a few mathema- ticians (in this century certainly, and I have reason to think earlier). Is not mathematics emphatically a miniature edition, so to speak, of the science of sane in- spiration ? Does it not contain the clue to an accurate distinction between the sane inspirations of genius and its aberrations ? (Or, to use old Asiatic phraseology, between inspir- ation from the true God and sorcery or magical possession.) It is easy for flippant critics to say that the whole distinction between inspir- INTRODUCTORY 25 ation and sorcery was a squabble between the priests of rival deities ; but no serious student of the old religious writings can doubt that more was involved than a mere contention as to whether the Source of Inspiration should be called Jehovah, Jove, or Lord. To Pan, the Unknown X, the Unity, the cosmic Force, the Great I Am, gave inspiration freely ; the question was what internal disposition, what mental attitude, what sequence of mental atti- tudes, create normal receptivity, putting the human machinery into such a condition that light from the Beyond enables it to see new truth without causing it to mix that truth with delusion. The belief of the mathematicians whose point of view I am trying to give is that mathematical science contains the answer to this great question. What they affirm is, not that studying mathematics as applied to number and quantity tends to promote sanity ; that has often been said, and may to some extent be true, but is not the special dictum of Mathe- matical Psychology. Nor do they say that all right and normal action of the human thinking- machinery must be mathematical in its se- quence ; that is very far, indeed, from being the 26 MATHEMATICAL PSYCHOLOGY case, as I hope in the sequel to show. But they affirm that when a human being desires to induce into himself light from an internal and unseen source, he should then set his mental machinery to work on the subject, whatever that may be, as to which he desires such light, in the same order of sequence as that followed the mathematician about number and quantity. When we desire to know how a particular flower is fertilized, how a certain bird builds its nest, how an operation is performed by those skilful in it, that knowledge must come to us from the outside : we must see the facts, or hear lectures, or read books. Or, if we wish to know what the learned suppose to be the interpretation of certain phenomena, we must get the learned to tell us, either by speech or by writing. But if we wish to "think out for ourselves" the meaning of phenomena i.e., to receive, without human instruction, new light about facts we already know then we must keep mathematical order in our sequence of mental operations, or a delusion may come upon us, and we shall be likely to believe a lie, and perhaps to fix it on the texture of our thinking- machinery. INTRODUCTORY 2/ Moreover, some mathematicians have be- lieved (though I have not met this statement in print) that such individuals as are, by natural organization, specially liable to receive sudden and involuntary illumination from above or within, should safeguard themselves from hallu- cinations and fixed delusions by frequently practising the specially mathematical sequence of mental actions, as a hygienic gymnastic for the brain. It is, therefore, surely desirable that all psy- chologists should learn to distinguish what is the essentially mathematical order of sequence, so as to be in a position to begin to investigate the two important questions : whether it is in itself hygienically valuable, and whether it affords the means of arriving at any test of the exact boundary line between the inspired and the insane conditions. The books in which the subject has hitherto been treated are written in the language of the Differential Calculus ; but the doctrines of Mathematical Psychology, so far, at least, as they mainly concern the ordinary psychologist, are expressible in terms of ordi- nary geometry ; and it is this task which I have been invited to undertake. CHAPTER II GEOMETRIC CO-ORDINATES \ T 7"E must now turn our attention again to * * that branch of mathematics which re- lates to the measuring of plane or solid figures. Its name, Geometry, indicates that its original function was to measure and compare areas on the earth's surface, both in the interests of land distribution and to assist men in finding their way about on the planet. One important prin- ciple of psychology comes out in the very words used to describe direction ; we say that a town lies to the " north-west " of us, or that the wind is blowing from " south - east and by east." Just as the signs by which we denote numbers are compounded from the primary signs by which we designate the first ten num- bers, so the names of all possible directions are compounded from the names of four directions, taken as primary north, south, east, and west. Why should four special directions be called by their own simple names, and all others be re- ferred to those ? Not for any reason which is GEOMETRIC CO-ORDINATES 29 re-al i.e., connected with the nature of the things to be studied ,but for a psychological reason. It happens, indeed, that the particular directions chosen as primary were not in themselves quite arbitrary as to the facts. The north-south direction is parallel to the axis of the earth's rotation, and the east-west direction perpen- dicular to it. The Equator affords a natural base line from which to measure latitude (i.e., north - south distance) ; for measuring longi- tude (i.e., the east-west distance) any meridian can be chosen at pleasure ; very often a writer chooses the meridian which passes through the chief observatory of his own country. When we leave navigation and turn to pure geometry, the system of construction lines (as they are called) i.e., the artificial scaffolding of measure- ment seems purely arbitrary ; each proposition of Euclid has its own set. But when we come to seriously complex form, the navigator's arti- fice comes into use again ; as a pure artifice this time, without any excuse derived from the earth's axis of rotation. It is a usual feature of mathematical artifice that it fastens itself at first to some natural fact, and by-and-by de- taches itself and becomes generalized. In 30 MATHEMATICAL PSYCHOLOGY books on algebraic geometry, we are pre- sented with two lines at right angles to each other ; these lines are called co-ordinates. One is usually called the axis of X, the other the axis of F; the point where they cross is called the origin. (The Equator is, for the navigator, an axis of X, and the particular meridian from which he reckons is his axis of F.) When the geometrician has to deal with solid forms, he uses a third axis, supposed perpen- dicular to the plane on which X and F lie ; it is called the axis of Z. When once the artifice of co-ordinates appears in pure mathematics it remains, so to speak in possession of the field. Algebraic geometry and the infinitesimal calculus rest upon it. As the science progresses, it shakes off its bondage to earth and to things ; it dissevers itself from all its attachments to outer fact one by one, but without losing its great psychological character- istic. First, as was said, it dislocates itself from the earth's special position in space, and refers itself no longer to a north-south and an east- west axis, but to a F and an X axis ; any two lines cutting each other at right angles will serve its purpose. Next, the lines need no GEOMETRIC CO-ORDINATES 31 longer be at right angles. Then it is found that straight lines are no longer necessary ; curvilinear co-ordinates are admissible ; or one co-ordinate may be the length of a revolving line and the other an angular measure. Or the tangent and radius may be co-ordinates, and in that case the actual curve-direction of one point is one of the co-ordinates for the next. This case is especially interesting psychologically. Later on, time is taken as one co-ordinate, and velocity or force may be the other. But throughout this long and varied chain of de- vices to facilitate the acquisition of knowledge there runs the same principle : Man when he withdraws from the observation of outer facts by means of his senses, to seek from beyond or within fresh light upon the meaning of facts already observed, conducts his mental opera- tions by referring them to two or a few selected ideals. These ideals may be partially or wholly fictitious ; the question of their objective reality in no way affects their value as psychological factors. The true thinker will use such ideals for his own education without being in bondage to any of them, will reserve to himself always the right to form a judgment about the reality of each in complete independence of its psycho- logical utility. To suppose, for instance, that, be- cause Buddhism has educated thinkers, therefore any such Being as Buddha must necessarily have existed is illogical. But no attitude of mind is more foolish or more superstitious than that of men who harden themselves against religious impressions on the ground that "we have no proof of the actual existence of either gods or devils," and who yet talk of mathematics as " pure truth," and of such ideals as co-ordinates, tangents, and normals as if they were realities. It is natural to man to climb to truth on a ladder of fiction ; it mathematical ideals have led to purer truth than theological ones, it is not because mathematicians have abstained from dealing in fictions. Probably the cause rather is that they have been from the first aware of the limits of their own knowledge, honest in avowing ignorance, and careful to distinguish between necessary psychological apparatus and objective truth. It may be worth noticing that geometric investigation is made easier by choosing for co-ordinates two standard lines or ideas which are essentially independent of each other GEOMETRIC CO-ORDINATES 33 The rectangular co-ordinates of algebraic geometry afford a good type of such independ- ence. The two co-ordinates are not opposite to each other, not contradictory : they are not one north and the other south ; but one may lie north-south, and the other east-west ; so that progress along the one would not necessarily involve either progress or retrogression along the other. The law of any particular curve is then ascertained by finding out, in that curve, according to what law progress in one of the polar directions is accompanied by progress or retrogression in the other e.g., what distance from axis X coincides in that curve with a certain distance from the axis Y. The co-ordinates of latitude and longitude, for instance, are so arranged that we can, on a map or globe, follow any meridian from the North to the South Pole without altering the longitude ; or a parallel of latitude through all the degrees of longitude without deviating to the north or south. By means of this arrangement, it is naade possible to mark exactly the track of any ship by saying that at such a parallel of latitude it was on such a degree of longitude. D 34 MATHEMATICAL PSYCHOLOGY When we have properly understood the doctrine of co-ordinates, we begin to see how psychology comes into contact with the law of the parallelogram of forces. If a body of men has to act as a whole, while the individuals composing it are swayed by a variety of motives, empirical common sense and good feeling declare that " each side must yield a little," that " a compromise must necessarily fall short of carrying out to the full the ideal of any one"; but when common sense and good feeling put themselves under the guidance of mathematics, they are introduced to a higher conception of what compromise may mean. In geometry it means something perfectly definite and accurate. It would, of course, be false reasoning to assume that because a ball pushed by two forces follows the diagonal, therefore a diagonal can always be found in human affairs ; but we have a right to point out that, so far as regards geometry, the human mind has long ago left off guessing empirically at the best compromise to come to between two motions, having found itself perfectly capable of know- ing exactly what is the true compromise ; and, moreover, that much inspiration of fresh know- GEOMETRIC CO-ORDINATES 35 ledge has been gained by insisting on arriving at such exact knowledge. We may observe, too, that the problems presented in life resemble those of the geo- metric parallelogram in this respect, at least : Two motives may partially neutralize each other's action, or they may partially assist each other, or they may be so adjusted to each other as to act quite independently. In mathematics, we have, in the first case, the short diagonal of an obtuse-angled parallelogram (with part of the force converted into heat within the body on which the conflicting forces act). In the second case, we have the long diagonal of the parallelogram. In the third case, we have the diagonal of a rectangular parallelogram ; that is to say, a complete resultant, which gives full effect to the whole action which each force would exert if acting alone. CHAPTER III THE DOCTRINE OF LIMITS /~\NE of the mathematical artifices which is Vr most important psychologically is what is called the doctrine of " limits." It may be illustrated by trying to sum up the series i + 1- + i + s + etc - It will be- noticed, first, that the series itself has no natural termination ; however long we write, we shall never write its last term. Next, that, however many terms we write, we shall never make up a sum quite equal to 2 ; the sum is always 2 minus a fraction equal to the last term we have written. Thirdly, that the more terms we write, the nearer their sum will approach to 2 ; so that by writing more terms we can make the sum as near to 2 as we please. The sum of the series approaches the limit 2, as the last term approaches the limit zero. These facts are expressed by the equations : Sum of i + | + | . . . adinfn. = 2. Last term of series 1 + 2 + i a d inf n > = > 36 THE DOCTRINE OF LIMITS 37 or by the statement that 2 is the limit of the series i + | + \ as the last term approaches the limit o. These are fictitious statements ; no series is ever written out ad infu. In the same way it is stated that the para- bola touches a certain imaginary straight line, the asymptote, " at infinity," or that the asymptote is the "limit" of breadth of the parabola, as it approaches GO in length. The parabola is a re-al line, the curve traced by a projectile. In any given case it comes to an abrupt termination, because the projectile is stopped by the earth ; but it does not naturally join ends like an ellipse ; it is, potentially, of indefinite length. Man, in investigating the parabola, finds it useful to invent the imaginary asymptote, and then to show that, supposing the asymptote existed, the parabola would touch it " at infinity " (i.e., never), but be always approaching nearer and nearer to it the longer it grew. That this device assists man is a psychological fact. We are all familiar with the device of the reductio ad absurduin, as employed by Euclid. He supposes, or guesses, or thinks he sees that a certain thing cannot be ; but, instead of 38 MATHEMATICAL PSYCHOLOGY arguing : " I should think it cannot be," or " Of course it is not," etc., he boldly faces the supposition : " Suppose it were," and then traces out the logical consequences of that supposi- tion till he lands himself in some impossible absurdity or sheer logical contradiction. This device has found its way to some extent into ordinary logic. But in mathematical reasoning it allies itself with another, to which I have already alluded : the fact that man is assisted in understanding a curve by the device of inventing the notion of tangents. Take any real curve, say the path of a planet ; the pious dogmatist asserts that this curve is traced, as a whole, by the Will of God. The mathemati- cian analyzes the action, at any point, of God- Will, or One-Force, into two imaginary forces : the centrifugal, which is supposed to act along the tangent, and the centripetal, which pulls towards the sun. These two forces are, of course, as imaginary as the conception of Baldur and Loki, or of Ormuzd and Ahriman, or any other pair of antithetical deities. No one ever saw a planet either fly off at a tangent or rush into the sun, nor is the student intended to calculate the probability of any planet doing THE DOCTRINE OF LIMITS 39 so. But it is a fact of human psychology that man understands the action of the One-Force better by first splitting it, in thought, into anti- thetic imaginations, and then combining the conception of them. We have now, I think, reached a point where we can gain valuable light from mathematics on the solution of certain discussions which disturb mankind. The ordinary good citizen, whether of the unlearned or of the learned world, is fond of saying that you must not push any logical argument to extremes, or it lands you in some absurdity. Wherever mathematics touches, it instructs us to push everything to the extreme (in thought, in words), in order to be landed in absurdities ; because only so can our premisses be thoroughly tested. It is only by pushing in thought our tangent to infinity and our radius to the focus that we can properly understand our curve. The object of so pushing to the ultimatum is that we may understand the curve. When we understand it, we will discard altogether those fictitious entities, tangent and radius ; that is, for the mathematician, a matter of course. But the world imagines that the tangent or the radius 40 MATHEMATICAL PSYCHOLOGY or both are meant for lines of direction as to conduct ; and is shocked, or argues to prove that such conduct would be unwise. When told that the suggestion in question was never meant to be acted on, the world asks what is the use of discussing unpractical theories that were not meant to be acted on. The reply to that question is to ask another : What is the use of a tangent ? The mathematical step next in order of sequence, after following the tangent into space in its centrifugal flight, and the radius vector to its origin at the focus, is to find out what would be the effect of two forces acting in combination on the same body, one of which forces has been imagined as carrying it along the tangent, and the other as carrying it down the radius vector ; in other words, we have to calculate to what kind of curve our two imaginary straight lines are radius and tangent. If any student should stop short of this, should insist that either the tangent or the radius is the planet path, we should remind him that he has not completed his investigation ; just as we should do if a child handed in one line only of a sum in long multiplication, mistaking it for the answer to the sum itself. Whenever in THE DOCTRINE OF LIMITS 4! the sequel I speak of uncompleted mathe- matical sequence in thought process, I would wish the reader to understand that I mean something more or less analogous to mistaking the tangent or the radius for the planet-path. I call it completed mathematical sequence when the mind has gone through a succession of processes analogous to those which, in mathe- matics, lead to a true conception of an orbit. It is interesting to note that biologists in not a few cases find their way, empirically or by instinct, to the exact mode of procedure dic- tated by Mathematical Psychology. E.g. : one such instance has been given in Chapter I. : that of the three fictitious elements of the leaf, by the union of which the student has to construct within his mind a conception of the leaf as it grows. In this case, and I think most others, the analysis, or separation into fictitious ele- ments, is made by projecting the mind out- wards, as it were ; by the observation of outer facts. But the synthesis which completes the sequence takes place within. CHAPTER IV NEWTON AND SOME OF HIS SUCCESSORS T TNTIL the time of Newton, the Science of *^ Mathematics was comparatively slow in its growth ; mathematicians became accustomed to the use of artificial processes so gradually as hardly to notice their unreality. Newton and his contemporary, Leibnitz, created an instrument called the Infinitesimal Calculus, or Differential and Integral Calculus, the unreality of which, its lack of conformity to the nature of things, was so glaring that it called forth a momentary protest I have no right to assume any knowledge of the Calculus on the part of my readers ; and, of course, it would be futile to attempt to analyse it here. It is sufficient to say that the method rests on a certain modification of the theory of the diagonal of forces combined with a peculiar extension of the doctrine of limits. In the case of rectilinear motion along a diagonal, two or more forces are supposed to act for a definite time, each in one straight line. In the case of NEWTON AND SOME OF HIS SUCCESSORS 43 a curve, no force acts for an instant, or for any fraction of an instant, in one straight line ; change of direction is as continuous as the motion itself; we have nowhere to deal with the diagonal of any actual parallelogram. By Newton's method, we, as it were, invent a non- existing parallelogram, E M P N, two opposite angles of which are on the curve. Then we suppose N to draw nearer to M along the curve. By reason of the curvature, the shape of the parallelogram alters as well as its size ; the ratio between the sides changes as well as their actual length ; the direction of the diagonal changes as the diagonal diminishes. The Newton method investigates the limiting condi- tion to which the ratio between the sides 44 MATHEMATICAL PSYCHOLOGY approaches as the length of the sides approaches zero. This limiting ratio expresses the limiting direction of the diagonal ; or, in other words, the actual direction of the tangent at M, MO. In speaking of this device, Gratry says : "We have analysed the finite in order to know the infinitesimal. From what we have learned about the finite we have effaced the character of finiteness ; what remains is found to be true for the infinitesimal element, that is to say for the analysis and the study of the indivisible and the infinite (curve). We have analysed the discontinuous, the divisible, the finite, and have thus found the law of the continuous, the indivisible." That is what Gratry writes in this century. In Newton's time, however, the mathematical world was startled at the roundabout and apparently senseless process invented by him. How can any truth be arrived at by first taking the ratio between two sides of a parallelogram imagined as existing, and then supposing them not to exist ? Surely when they vanish to nothing, all reasoning based upon their sup- posed existence ceases to have any validity. NEWTON AND SOME OF HIS SUCCESSORS 45 Reasoning about the relation of black men to white men may be still valid in an island where there remain only two black men and one white one ; but surely no such reasoning can prove anything about a country where all the human inhabitants have been killed off. The answer is that Newton's method enables men to solve satisfactorily problems hitherto unsolvable. Not only has astronomy made giant strides by means of it, but modern inves- tigations in electricity, mechanics, statistics and actuarial science depend on it. Mathematicians at, or soon after, Newton's time settled down to the use of the artifice invented by him ; accept- ing it for the most part, without understanding why it should be helpful. And less than half a century ago, it was still possible for a Cam- bridge graduate to introduce a pupil to the Calculus with the remark that "no one can understand why this method should answer ; it seems contrary to the nature of things ; but if we trust to it the results come out right ; it is like putting corn into a mill-hopper ; you cannot see what happens, but it comes out flour at the end." What happens is the same as happens when 46 MATHEMATICAL PSYCHOLOGY we do a multiplication sum ; we put our in- spiration-receiving machinery through a pro- cess consonant to its nature, though unrelated to the facts we wish to investigate; and we thereby raise it to a position where it can grasp truths previously out of reach. The lines whose evanishing to zero give us the knowledge which we seek, were construction lines, fictions, not portions of the curve, but made by the investigator as helps in the investigation. For those who are seeking from the Calculus information about human psychology, it is a fact of overwhelming importance that the special knowledge about the curve which is to be got by means of the varying ratio between two fictitious elements comes only at the moment when those elements vanish ; i.e., when the subject-matter which has been under investigation vanishes to No-Thing. When once this principle has been grasped, it becomes easy to see that it is latent in mathematics from the very beginning. The vivid sense of the sacredness of that psychological moment when the differential elements are reduced to zero and the sides 6f the parallelogram vanish into No-Thing, has led NEWTON AND SOME OF HIS SUCCESSORS 47 many mathematicians to feel an impassioned reverence for all forms of worship of The Eternal No-Thing, in Whom all contrasts vanish, Who " is at once the slayer and the slain " ; and to conceive an abhorrence of idolatry ; by which such men always mean, not the worship of images nor the worship of the wrong mental Eidolon, but the act of worship of any concrete entity, any mentally conceivable Eidolon at all. The ordinary pious clamour on this topic, founded on the perfections of Jesus, affect them much as medical men would be affected by a claim that some particular foreign body, being of pure gold, must be fit to introduce into the heart-valves and can do nothing but good. From the point of view of the higher mathematics, the adoration of the concrete and manifested is not a mistake in theology but a brain-vice. The mathematical standpoint is not : There cannot be an incar- nation of good in contrast with a personal devil, but : If there be an incarnation, the worst use we could put it to is to worship it or mistake it for God. Boulanger, a devout mathematician of the last century and one of the great forces of the intellectual revolution led by the Encyclo- pedists, wrote a book on the Origin of Eastern 48 MATHEMATICAL PSYCHOLOGY Despotism, with the motto Monstrum Hor- rendum Ingens Difforme^ in which he expounds the doctrine that no tyranny could keep itself in existence were not the minds of the peoples weakened by some form or other of the practice of putting a concrete ideal in the place of God. He has defined idolatry in language on which it would be difficult to improve. " Idolatry does not consist, necessarily, in taking a statue, an animal, or a man, as the representative of God ; to define it fully we must say that every form of worship or code of law is idolatrous which takes as divine that which is not divine. It is not only idolatrous to treat a stone, a beast, or a mortal as if it were God ; we are also guilty of idolatry if we imagine that the words of that man, or the oracles pronounced through that statue, are the very words and decrees of Deity. We are guilty of idolatry when we prefer speculations and mystical chimeras to reason ; when we treat any legislative code as if it were dictated by the Almighty ; when we endow with a divine charac- ter the servants of a theocracy ; when we try to regulate the conduct of men here below by laws suited only to celestial beings ; when we confuse Heaven with earth ; when we mistake our own position and pretend to be more than mortal ; and when we forsake our own place as citizens of this world and subjects of the civil govern- ment, either to tyrannize over other men in the name of God, or to live as recluses, despising or forgetting our fellow-men." He adds that he considers it essentially idolatrous to regulate our conduct in this world as' if we were already in a better. Boulanger believed in a Revelation from God to man ; but he believed that Inspiration comes through the Zeit-Geist of the age ; and will surely so come if men are not hindered from receiving that Zeit-Geist in its entirety. But if we are to take the Zeit-Geist as a whole, we must take into account, amongst other essen- tial psychological factors, the passionate protest of so many earnest men as to the necessity of some concrete human centre for the religious, moral, and emotional life. The desire to under- stand and do justice to the Christian conception gave, forty years ago, a great impetus to the study of what are called Singular Solutions of Differential Equations, by means of which a whole set, or, as mathematicians say, a "family," of curves is investigated from the point of view E 50 MATHEMATICAL PSYCHOLOGY of their common relation to some special point or curve. The nature of the method may be in- dicated by saying that the Singular Solution of a family of Differential Equations is subject to the law which is the characteristic law of the Family, but in a different way : elements which are constant in one of the others are variable in the Singular Solution ; and elements which are variable in one of them are constant in the Singular Solution. It would be possible, and might for some purposes be instructive, to con- ceive of an extra planet, moving among the planets and comets of our system, and cutting across their orbits, in a complex curve so arranged as to "touch" each of the existing orbits once. (When two lines coincide at a point without cutting across each other, they are said, in mathematical language, to " touch.") The Equation of this imaginary orbit would be a Singular Solution of the System of Equations of the actual planetary orbits. Now, if certain calculations were found easier to arrange, by referring them to such a fictitious Singular Solution of the general equation of the planetary family, no astronomer would therefore think that there must exist an actual planet running in the orbit he had formulated. NEWTON AND SOME OF HIS SUCCESSORS 51 The new science of Singular Solutions turned out to be not only useful in certain kinds of in- vestigation, but intensely fascinating in itself. Its wonderful charm exerted a softening influ- ence on certain mathematicians of whom De Morgan may be taken as a type. It corrected the tendency to conceive Humanity as a col- lection of individual intellectual powers, each holding isolated communion with The Eternal No-Thing ; and brought them, not indeed to actual worship of any concrete eidolon, but to the conception that it may be helpful to prepare for seeking inspiration from the Inconceivable No-Thing by mental contact with an Ideal of Manhood. An eminent mathematician once wrote to me enthusiastically about the inspired utterances of a Christian preacher ; and added : " I have made out what puts the whole subject of Singular Solutions into a state of Unity." Unscientific persons of pious tendency catch only too willingly at the suggestion that there are Singular Solutions in nature, and that this fact somehow proves the doctrine of Incar- nation ; on the other hand scientific people accustomed to rigid inference object that the 52 MATHEMATICAL PSYCHOLOGY existence of curves in space cannot prove the historic truth of Incarnation. Both parties forget that in the first place Singular Solutions are not objective facts, but psychological entities in the mind of Man ; and, in the next place, Mathematical Psychology cannot deal with historic facts ; the most it could be supposed to attempt to prove is that a tendency to refer a group of individuals to an Ideal Type, in contact with all the members, yet differing from all, leads in mathematics to a better knowledge of the laws which govern them ; and that there- fore, when the same tendency is discovered in writings on religion and ethics, it should not be assumed to be abnormal, idolatrous, or un- meaning. Those who wish to pursue further the sub- ject of the connection between Mathematical Psychology and the higher religious life should read the second volume of Gratry's Logique. It is full of most instructive suggestions. We have, however, to bear in mind in reading Gratry that, at the time when he wrote, com- paratively little was known about Singular Solutions ; and there is no evidence that he was acquainted with even that little. He was NEWTON AND SOME OF HIS SUCCESSORS 53 groping his way through that part of the subject by sheer instinct, the instinct, moreover, of a Catholic ; and the portions of his treatise which refer to " Le Verbe Incarne " are less clear than might be desired. It has now, I hope, been made sufficiently clear that the modern Science of Mathematical Psychology does not rest on any such false basis as that of assuming analogies between facts of different orders. The analogies on which Mathematical Psychology rest are anal- ogies of logical process, not of facts or things. It would indeed be false reasoning to assume that because a certain order prevails in the star- paths, it must therefore manifest itself among phenomena of other kinds ; but it is valid argument to say that a mode of sequence in reasoning which is considered admissible, and which leads to true results, in mathematics, ought not to be treated with contempt as baseless and fanciful when applied to human affairs. But though modern mathematicians do not offer to the world as proven, statements in- capable of proof in the strict scientific sense of the term, those of them who have approached their subject from the standpoint of psychology 54 MATHEMATICAL PSYCHOLOGY have received suggestive clues from certain ancient sages who were free from modern con- ventions and restrictions. The mighty founders of ancient religions did not feel themselves bound to prove their statements ; they found out, by instinct and experience, certain things about the conditions on which man can ap- proach Unknown Truth with safety and profit ; and they boldly stated what they knew, leav- ing any one to contradict who dared. They pointed, for instance, to the Rainbow, a tem- porary explosion of contrasted colours, which soon fades into the Unity of white light, leaving No-Thing behind ; and declared that it was the token of God's covenant with man, the ever-young and always fresh messenger from the gods ; or, as our sturdy Scandinavian fathers put it, a bridge by which the brave soul reaches the abode of the Divine on its own feet. They instructed disciples to pray that the Will of The Eternal may be done on earth as it is in the Heavens ; i.e., by a rhyth- mic alternation of apparently contrary motions ; by incessant, unopposed, and orderly Re-volu- tion. They cut a forked stick from a tree, and held it first in its natural position, and then NEWTON AND SOME OF HIS SUCCESSORS 55 reversed, putting Unity where there had been separation, and separation where Unity had been; and they called this symbol of analysis and synthesis a " rod of divination," that is to say, of inspired knowledge. They held it as the symbol and instrument of man's conquest over the difficulties which beset his path. What part mathematicians took in ancient days in the investigation of psychological problems, we have now no means of finding out. Mathematicians were often considered as wizards ; and their books were burned by the ignorant and superstitious. It seems to me certain that ancient sages had arrived, by what process we cannot now ascertain, at a knowledge of the value, for true inspiration, of that psychological element on which the Calculus is founded ; the impor- tance of the moment when something which had been elaborated with accurate care vanishes and is reduced to nothing. They tried to teach this psychologic truth to the unthinking masses by object-lessons ; declaring that the true Inspirer would descend at the moment when the suppliant had burned to ashes some product of his labour which he considered 56 MATHEMATICAL PSYCHOLOGY specially valuable : (a perfect animal, or the firstfruits of tillage). Their attempts were fol- lowed by results often ludicrous, sometimes disastrous. The psychological doctrine became confused with and lost in one pseudo-inspira- tion after another : in heathendom with the mere wild animal love of killing, with childish delight in big blazing fires, with triumph over human foes ; in corrupted civilizations, with the diseased love of inflicting torture ; in Palestine, with the desire of hygienists and moralists to insist on the public slaughter and inspection of animals intended for food ; in some cases with that abject lust for indulging the sensation of reckless devotion which is a mere perversion of the sex-instinct ; in other cases with the ascetic disease, which may be described as a form of suicidal mania, bearing the same kind of relation of inversion to man's normal desire for pleasant sensations that actual suicidal mania does to normal Will- to-live. The particular afflatus or pseudo-in- spiration which is just at present obscuring and confounding the true mathematical law of inspiration by extinction is what is called Altruism ; a doctrine which is partly true, but NEWTON AND SOME OF HIS SUCCESSORS 5/ which sometimes runs into the belief that how- ever foolish an action may be in itself, it is glorified if done at the sacrifice of something valuable to the doer for the sake of somebody else. Through all these ideas one can detect here and there, in the writings of the more inspired teachers of them, an ever-recurring conscious- ness of the great truth : that the true direction for progress is revealed to man at the moment when something which he has been construct- ing with elaborate care vanishes into No- Thing. CHAPTER V THE LAW OF SACRIFICE * I ''HE word sacrifice may seem to strike a *- keynote which does not belong to any scale in which mathematical conceptions can be expressed ; but this should not prevent us from inquiring whether the Laws of Sacrifice find expression in mathematical operations. When we study the question impartially, we are led to see that they do find such expres- sion. Sacrifice forms an integral portion of every true inspirational cycle. In the life of religious inspiration the necessary sacrifices are large enough to involve a serious and often painful moral effort ; whereas in mathe- matics they are so slight and so habitual that we are hardly aware of making any sacrifice at all. But this difference is one of size, not of kind ; the acts of sacrifice imposed by the Laws of Mathematics are proportioned to the minute scale on which the sequence is being carried on, and to the comparatively small value of the knowledge sought. 58 THE LAW OF SACRIFICE 59 Of course there is a sense in which all study involves sacrifice ; the sacrifice of inclination, of laziness, of amusement, and so on. But in mathematics, as in the higher spiritual life, something quite different from this is in- volved : the continual giving up, not only of impulses hostile to study, but also of the very results which we have been toiling to attain. When a child looks out a word in a dic- tionary, it is one of the words in the passage he will have to read ; it is also a real word of the language we wish him to learn. When we tell him a fact, in History or Natural History, we intend him to remember that fact. He may forget it ; but, if he does so, he will have to learn it again, for it forms part of the sub- ject he is learning. Every result of his labour, in these cases, is a something which we intend he shall retain as part of the furniture of his memory. The process of learning arithmetic differs in this respect from the acquirement of knowledge from any finite source. Take as an instance the problem, mentioned in a pre- vious chapter, of finding the number of days in 2,481,749 years. The child has to plod 60 MATHEMATICAL PSYCHOLOGY conscientiously through the labour (to him no light one) of multiplying the long row of figures by 5 ; then by 6 ; then by 3 ; but none of these laboriously attained results are shown in the final answer. They are, so to speak, immediately effaced, merged in a total which bears no trace of their existence. In a complex calculation the worker goes through a long series of whole sums, the answer to each of which is merged in the next. Even when a child has finished his sum, he has acquired no tangible knowledge which we wish him to retain in his memory ; the result of his toil is, not a stock of information, but skill in drawing information from the Unseen Stores whenever he may need it. This apparently useless labour, this seeming waste of painstaking effort, is not made neces- sary by the laws of number ; there is no reason in the essential nature of things why the num- ber of days in a year should be expressed by three separate figures, why we should not have a sign to express that number as a whole, and multiply by it directly ; the sole reason lies obviously in the nature of man's faculties ; the effaced and unrecognised steps in the cal- THE LAW OF SACRIFICE 6 1 culation are rungs of the ladder by means of which our finite intellect reaches up to re- ceive knowledge, direct from The Unseen, about truths beyond our own power to grasp directly. A touching letter from James Hinton to Ellice Hopkins may be quoted here: "A little girl of ten said to me, 'Do tell me about the fluxion.' (I'd been talking in her hearing about Newton's fluxions, which I don't at all understand, a great deal of it.) So what do you think I did ? Such a happy thought came to me. I told her plainly, so that she quite understood all I meant. " I said (but I can only give you the barest notion), 'Multiply 17 by 3. So you know we get 3 times 7 is 21, i and carry 2; 3 times I is 3 and 2 is 5 = 51. Now,' I said, 'do you see what you have done with that 2 ? You have put it down, and then rubbed it out; it was necessary to have it, but not to keep it. Now, a fluxion is this : it is a thing we need to have, but are not intended to hold ; a thing we rightly make, but in order to unmake.' And indeed that is the whole point. But this simple case shows also perfectly how it comes, how the law of it comes upon our life. For you 62 MATHEMATICAL PSYCHOLOGY see our making 21 comes only from our taking the 7 of 17 by itself, isolating it from the 10. In the 3 times 17 there is no 21 ; we make the 21 simply by separating the 17 into 2 parts and taking one only first. That is, the 21 is an isolation-right, a right or truth that comes by leaving out. " Now, all isolation-rights are fluxions in this sense ; we have to do them, but in order that we may undo them. " And this is the law of man's life, because Nature is so great and rich that he is compelled to take her piecemeal ; and, in all things, he makes for himself, by his inevitable leaving out at first, ' isolation-rights.' " And the difficulty of his life the difficulty above all others is that these isolation-rights have got to be not kept, but used ; not held as they come, but ' carried ' on into another mode of being, which seems like their being lost. " This is the pathos of man's life. He makes isolation-rights, and has not known the law of it and recognised its meaning. "Now, this applies to all things. It is so pretty I can't show you even a fraction of its THE LAW OF SACRIFICE 63 bearings. But you can think of many of them if you choose. Look at all our sciences. Are they not, clearly enough, every one of them, parts no more the whole than 7 is, not only of 17, but of a whole page full of figures? Each one of them, therefore, makes for itself an isolation-right or true, a truth which comes only by its being isolated, wants not holding, but to be lost, as the 2 of the 21 is lost in 5, when the one is taken in (i.e., the I of the 17). "And the very relation of the figures in numeration, whereby each term means not one, but many of those before it, comes to have a distinct significance all through our mental life. " In fact, the world is so beautiful I don't know what to do ; only, as you know, the condition of that joy is consenting to bear pain ; and one" scarcely dares to say one is happy, because it makes the pain confront one, and the words have lost their meaning ere they have passed one's lips. " Look at this very fluxion ; it is such joy to see it, such pain to have to live it. ... Oh, me ! I am happy and sorry ; and just 64 MATHEMATICAL PSYCHOLOGY now I cannot see a bit whether that gladness I think is coming on the earth is coming or not." * We adults have become so accustomed to sacrifices in arithmetic that we make them without reflection, as mechanically and calmly as the farmer throws down some of his pre- cious crop to rot in the ground ; the " sowing in tears" goes on perpetually, only the tears are omitted. When the process goes on on a larger scale, we are tempted to become pes- simistic and talk of the "futility of human effort," "the disappointing nature of things," "the unsatisfactoriness of life," and so on. But no one can study much of Mathematical Psychology without discovering that, for crea- tures constituted as we are, the unsatisfactori- ness and futility are conditions of any true inspirational life. In mathematics, however, much of the pain- ful sense of futility is avoided by our knowing exactly what sacrifice is necessary for obtain- ing the special knowledge sought Nearly nineteen centuries ago, a suggestion was made, 1 Life and Letters of James Hinton> Chapter XV. THE LAW OF SACRIFICE 65 which, when followed up, imports into the sub- ject of sacrifice for Inspiration much of the exactness which robs mathematical sacrifice of its sting; and changes the sowing in tears re- commended by piety from the condition of vague scattering of treasures at random, to that of well-calculated planting of good seed in appro- priate soil. The suggestion occurs in a passage in the New Testament, on which much mystical comment has been written by theologians. It was pointed out to me long ago, by Professor Boole, as containing a psychological clue specially well worth following up. It seems to me to sum up and make available for purposes of practical guidance that theoretical knowledge about the nature of our own inspir- ation-receiving powers which we gain from mathematics generally. " The wind bloweth where it listeth, and thou nearest the sound thereof, but canst not tell whence it cometh or whither it goeth : so is every one that is born of the Spirit." x This evidently cannot refer to wind regarded as travelling in a straight line ; we do know of 66 MATHEMATICAL PSYCHOLOGY a trade wind whence it comes and whither it goes. The path of the whirlwind was a matter of observation long before the Christian Era, and might have been suggested at any time to an intelligent observer by the motion of particles of dust. If the reader will look at DIAGRAM OF A DUST-WHIRL. the diagram of a dust-whirl, and suppose him- self to be one of a cloud of gnats caught in the gust, or to be steering one of a flotilla of ships caught in a hurricane, he will see that the sentence contains both an admirable picture of a society on which has descended an in- spiration to think on some new topic, and a THE LAW OF SACRIFICE 67 canon of guidance for an individual who desires to find the main line of progress and to pre- serve his own sanity and safety. The Zeit-Geist sets men thinking on some common topic, but in all manner of ways; some feel impelled to go north, some south, some east, some west ; no one knows how to steer, for no one can judge, merely from the direction in which he feels impelled to go, what is the main direction of progress, or where- abouts safety can be found. If we could trust always to mere momentary inspiration, the matter would be simple enough; each indi- vidual would be carried round and round, and advance on the whole with the wind as a whole. But man, when once he has arrived at consciousness, cannot let himself be blown about by every wind ; he must organize, formu- late, translate momentary direction-impulse into definite tangential direction. It is useless to rail at this tendency ; it is a law of the human mind (else, why need mathematicians have in- vented tangents ?). But each man, or each group of men, mistakes tJie human tendency to invent a tangent for divine inspiration acting in one continuous straight line. This gives 68 MATHEMATICAL PSYCHOLOGY birth to all kinds of conflicting and erroneous opinion ; each sect or group believing that the tangential direction it has formulated for itself, by inference from some inspired utterance of its leader, is the true line of inspiration ; where- as, in fact, the still centre of calm and uniform progress lies between any two parties whose paths lie in opposite directions ; i.e., between any two whose original inspirations were dia- metrically opposite. Therefore, any individual who sees that another has been blown in the direction exactly opposite to the one in which he himself is drifting, has the clue he requires fof finding out where lies the still centre of calm and orderly progress. By steering to- wards one's absolute opponent, not in the direction towards which he is drifting, but to- wards himself, one would reach the calm centre of the storm. This gives a distinct rule for discovering where lies the still centre. Of course, as a general rule only the philo- sopher even desires actually to live in that cen- tral position. Most people are content to be swayed to some extent by personal afflatus. But as tangential direction leads at last to drifting away from the centre of force, safety, THE LAW OF SACRIFICE 69 for every one, depends on keeping not too far from the centre ; every one should treat the direction in which he feels impelled to go as one co-ordinate, and the path which leads to- wards those who seem to him in error as the other co-ordinate, and should frequently correct his own aberrations by acts of conversion to- wards the centre. Therefore the sacrifice appro- priate to any given situation of difficulty or doubt is that which converts the force which is taking the individual along his habitual path into energy of union with opponents. Unfor- tunately, religious conversion too often takes the form of joining the ranks of former oppo- nents, of adopting their opinions and imitating their customs. A mere glance at the figure is sufficient to show that this is not the kind of conversion which leads to the true centre of progress and of peace ; that centre is not on the line which the opponents are following, but on the line which runs from us to our opponents. The act of conversion from the path of personal afflatus to that along which union with opponents is effected should precede the attempt to learn what is the true path of pro- MATHEMATICAL PSYCHOLOGY gress along which the inspiring storm is to carry humanity as a whole. This interpretation of the geometric figure used by Jesus would seem to be in line with many of His most distinctive utterances, which have been supposed to be canons of some peculiar code of ethics taught by Him. It would seem to me probable that they were hardly intended to supersede, in ordinary cases, the ordinary ethics of the Jewish hearth and home ; but were rather meant as suggestions of methods for bringing about the special state of mind in which pseudo-inspiration or personal afflatus becomes " converted " into energy for reaching the centre of progress. It is interesting to note that ever since the time of Christ, the Christian world has been overrun and perplexed with various theories of " conversion." It seems agreed that true inspiration is to be attained, and the favour of the Inspiring Deity to be secured, when the individual is truly " con- verted"; but there have been many discussions as to what constitutes valid conversion. About these discussions nothing need here be said, except that the disputants, however they may disagree, are unanimous in neglecting to pay to THE LAW OF SACRIFICE "Jl Christ the ordinary courtesy expected by every geometrician from his brothers of the craft : that they will take the trouble to draw the figure which he describes, and study it care- fully, before publishing speculations about his probable meaning. It is to be hoped that students of Psychology have been trained in scientific rather than theologic conceptions of the respect due to a Teacher from those who aspire to learn from him. The Parable of the Wind-storm, when closely analysed, bears out Boulanger's definition of Idolatry quoted above : We need no higher inspiration (it seems to teach) than that of the Zeit-Geist of our own age ; but we must take that inspiration as a whole ; we must not let ourselves be tempted too far out of the general movement by mistaking for it any tangential direction ; we must try to get to the calm centre by unifying contrasted motions. Authorities of one sort or another, religious, political, or edu- cational, priests of this or that Eidolon, are constantly trying to arrest this vital process of conversion, by telling us that this or the other portion of the common afflatus is specially wrong or dangerous. As Boulanger says : MATHEMATICAL PSYCHOLOGY "Priests were appointed to lead men into truth ; but in all ages they have feared lest men should find the truth and walk in it." 1 From the point which we have now reached, we are able to distinguish pseudo-inspiration from genuine revealed knowledge of the des- tined path of progress for humanity. Pseudo- inspiration is personal afflatus, formulated into a tangent, and not yet corrected by conver- sion towards those who are impelled by the opposite afflatus. 1 Boulanger : Origine du Despotisme Oriental. CHAPTER VI INSPIRATION VERSUS HABIT T T will be well to give here a summary of -*- what has been found out mathematically about the contrast between the cultivation of the Inspirational powers, and the training which forms habit and what (in non-mathematical affairs) we call character. The apparent con- flict between the two modes of treatment is similar to that between the actions of a gardener who is making soil fertile for plants and that of his assistant who is making paths as unfit as possible to grow weeds ; almost everything that is right for the one to do, is, for the same reason, wrong for the other to do ; but the two workers come into no conflict because the territory is well marked out between them ; whereas, in the treatment of young minds, the psychologist who is endeavouring to foster the growth of original faculty often finds himself in antagonism with the experienced school- master. He whose mission is practical education has to equip his pupils for life. He has to secure 73 74 MATHEMATICAL PSYCHOLOGY for them, as best he can, such knowledge of facts as is necessary for conquering Nature ; such knowledge of prevailing human opinion and sentiment as is necessary for finding a place in society ; a set of habits convenient to the indi- vidual and not annoying to those with whom he will have to live ; and an upright and kindly character. Now the knowledge of facts is derived, not from inspiration or mathematical induction, but from observation and reading. The know- ledge of human opinion and feeling should be based on a solid habit of study. Owing to what is now called suggestion or tele- pathy, it is easy to form pseudo-inspirational guesses as to what others are thinking or feel- ing ; and this instinctive imbibing of other men's thoughts is among the most dangerous and misleading forms of pseudo-inspiration, un- less severely corrected by studying the actual utterances of others. Besides, even for the kind of knowledge which inspiration is an appro- priate means of receiving, inspiration, in its true sense of completed synthesis, is a slow and laborious process. He who trusts too many of his actions to the guidance of inspiration will INSPIRATION VERSUS HABIT 75 find that he has not time to complete either his analyses or his syntheses ; he must act in the meantime ; and he will find himself driven to act on pseudo-inspiration, i.e., incomplete se- quence ; on uncorrected, untested flashes of im- pulse. In the interests, therefore, of the in- spirational life itself, it is essential that a large part of the conduct of the practical life should be guided by habit. Now all formation of habit is destructive of that mental elasticity on the free play of which the possibility of pure inspiration depends. Habit means the arrest- ing of free play in certain directions ; it means setting up a preponderating tendency to do this rather than that, so that the organism, unless absolutely controlled to the contrary by the higher centres, will automatically do this and not that. Whereas the cultivation of genius, of original power, of intuitive and inspirational faculties, depends as a writer in the last cen- tury said on " keeping the mind in a live and sensible (i.e., sensitive) state," l ready to respond freely to the slightest stimulus of fresh sugges- tion. For we must never forget that true 1 T. Wedgwood : MS. Treatise on Genius. MATHEMATICAL PSYCHOLOGY genius, true inspiration, depends on correction of error \ and this implies the "right to go wrong." It would seem impossible to com- bine this with the formation of good habits, except by a careful demarcation, made by each adult for himself, and by parents for each child, of the field in which his genius shall be allowed to play. The peculiar antithesis between the forma- tion of character and the cultivation of genius (or the exercise of the inspirational faculty) seems to me to throw light on some utterances of certain great ethical teachers, which appear to conflict with family duty and social harmony. For the formation of habit and character, it is of primary importance to cultivate kindly sen- timents towards those with whom we come most frequently in contact ; ?'.