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diff --git a/77011-0.txt b/77011-0.txt new file mode 100644 index 0000000..cab6c4c --- /dev/null +++ b/77011-0.txt @@ -0,0 +1,5606 @@ + +*** START OF THE PROJECT GUTENBERG EBOOK 77011 *** + + + + + + THE ORGANISATION + OF THOUGHT + + + + + THE ORGANISATION + OF THOUGHT + + EDUCATIONAL AND SCIENTIFIC + + + + + BY + A. N. WHITEHEAD, Sc.D., F.R.S. + + FELLOW OF TRINITY COLLEGE, CAMBRIDGE, AND PROFESSOR OF + APPLIED MATHEMATICS AT THE IMPERIAL COLLEGE OF + SCIENCE AND TECHNOLOGY + + + + + LONDON + WILLIAMS AND NORGATE + 14 HENRIETTA STREET, COVENT GARDEN, W.C.2 + 1917 + + + _The rights of translation are reserved_ + + + + + PREFACE + + +THE discourses included in this volume have been delivered as addresses +on various occasions which are duly noted; the only exception is +_The Anatomy of Some Scientific Ideas_, which is now published +for the first time. These discourses fall into two sections, the first +five chapters deal with education, and the remaining three embody +discussions on certain points arising in the philosophy of science. +But a common line of reflection extends through the whole, and the two +sections influence each other. + +I have left in each chapter the reference to the particular occasion of +its first production, and I have not sought for a verbal consistency +covering perplexity. But the various parts of the book were in fact +composed with express reference to each other, so as to form one whole. + +I have to thank the Syndics of the Cambridge University Press for +permission to republish the contents of Chapter V. + +_Imperial College of Science and Technology,_ +_April 1917._ + + + + + TABLE OF CONTENTS + + +CHAP. PAGE + + PREFACE v + + I. THE AIMS OF EDUCATION--A PLEA FOR REFORM 1 + (Presidential Address to the Mathematical Association, + January, 1916.) + + II. TECHNICAL EDUCATION AND ITS RELATION TO + SCIENCE AND LITERATURE 29 + (Presidential Address to the Mathematical Association, + January, 1917.) + + III. A POLYTECHNIC IN WAR-TIME 58 + (Address at the Prize Distribution, Borough Polytechnic + Institute, Southwark, February, 1917.) + + IV. THE MATHEMATICAL CURRICULUM 69 + (Presidential Address to the London Branch of the + Mathematical Association, 1912.) + + V. THE PRINCIPLES OF MATHEMATICS IN RELATION + TO ELEMENTARY TEACHING 92 + (Paper read in the Educational Section of the International + Congress of Mathematicians, Cambridge, + 1912.) + + VI. THE ORGANISATION OF THOUGHT 105 + (Presidential Address to Section A, British Association, + Newcastle, 1916; also read subsequently before the + Aristotelian Society.) + + VII. THE ANATOMY OF SOME SCIENTIFIC IDEAS 134 + + VIII. SPACE, TIME, AND RELATIVITY 191 + (Paper read to Section A at the Manchester Meeting of + Section A, 1915; also, with the appended Commentary, + read subsequently before the Aristotelian + Society.) + + + + + ORGANISATION OF THOUGHT + + + + + CHAPTER I + + THE AIMS OF EDUCATION--A PLEA FOR REFORM + + (_Presidential Address to the Mathematical Association, January, + 1916_) + + +WHEN I had the honour of being made President of the Mathematical +Association, I did not foresee the unusual responsibility which it +entailed. It was my intention to take as the theme of a presidential +address the consideration of some aspect of those special subjects to +which my own researches have principally been directed. Events have +forced me to abandon that intention. It is useless to discuss abstract +questions in the midst of dominant practical preoccupation. We cannot +disregard the present crisis in European civilisation. It affects every +function of life. In the harder struggle for existence which lies +before the nation, all departments of national effort will be reviewed +for judgment. The mere necessity for economy in resources will provoke +this reformation. + +We are concerned with education. This Association, so rich in its +membership of educationalists, with the conception of reform as the +very reason of its being, is among those bodies which must take the +lead in guiding that educational reconstruction which by a sociological +law follows every social revolution. We do not want impracticable +ideals, only to be realised beyond the clouds in + + "Some wild, weird clime, + Out of Space, and out of Time." + +We require to know what is possible now in England, a nation conscious +of its high achievements, and of great failures, shaken to its +foundations, distrustful of the old ways, and dreading fantastic +novelties. + +I will take my courage in both hands, and put before you an outline of +educational principles. What I am going to say is of course entirely +without your authority, and does not pledge or prejudge any action of +the Association. We are primarily concerned only with the intellectual +side of education, and, as mathematicians, are naturally concerned to +illustrate details more particularly by reference to mathematics. Thus +much to explain deliberate omissions in what follows. + +Consider now the general and special education of two types of boys, +namely those in secondary schools who in after life must form the +professional and directing classes in commerce, industry, and public +administration, and again those in junior technical schools and later +in advanced continuation classes, who are going to form the class of +skilled artisans and foremen of workshops. These two sets compose the +educated strength of the nation. We must form no ideals which include +less than these entire classes within their scope. What I shall say, +will in phraseology apply more directly to the secondary schools, but +with unessential changes it will apply equally to the other group. + +What is the first commandment to be obeyed in any educational scheme? +It is this: Do not teach too many subjects. The second command is this: +What you teach, teach thoroughly. The devil in the scholastic world +has assumed the form of a general education consisting of scraps of a +large number of disconnected subjects; and, with the artfulness of the +serpent, he has entrenched himself behind the matriculation examination +of the University of London, with a wire entanglement formed by the +Oxford and Cambridge schools' examination. + +Culture is activity of thought, and receptiveness to beauty, and humane +feeling. Scraps of information have nothing to do with it. A merely +well-informed man is the most useless bore on God's earth. What we +should aim at producing is men who possess both culture and expert +knowledge in some special direction. Their expert knowledge will give +them the ground to start from, and their culture will lead them as deep +as philosophy and as high as art. We have to remember that the valuable +intellectual development is self-development, and that it mostly takes +place between the ages of sixteen and thirty. As to training, the most +important part is given by mothers before the age of twelve. A saying +due to Archbishop Temple illustrates my meaning. Surprise was expressed +at the success in after-life of a man, who as a boy at Rugby had been +somewhat undistinguished. He answered, "It is not what they are at +eighteen, it is what they become afterwards that matters." + +In training a child to activity of thought, above all things we must +beware of what I will call "inert ideas"--that is to say, ideas that +are merely received into the mind without being utilised, or tested, or +thrown into fresh combinations. + +In the history of education, the most striking phenomenon is that +schools of learning, which at one epoch are alive with a ferment of +genius, in a succeeding generation exhibit merely pedantry and routine. +The reason is, that they are overladen with inert ideas. Education +with inert ideas is not only useless: it is, above all things, +harmful--_Corruptio optimi, pessima_. Except at rare intervals +of intellectual ferment, education in the past has been radically +infected with inert ideas. That is the reason why uneducated clever +women, who have seen much of the world, are in middle life so much +the most cultured part of the community. They have been saved from +this horrible burden of inert ideas. Every intellectual revolution +which has ever stirred humanity into greatness has been a passionate +protest against inert ideas. Then, alas, with pathetic ignorance of +human psychology, it has proceeded by some educational scheme to bind +humanity afresh with inert ideas of its own fashioning. + +Let us now ask how in our system of education we are to guard against +this mental dry rot. We recur to our two educational commandments, +"Do not teach too many subjects," and again, "What you teach, teach +thoroughly." + +The result of teaching small parts of a large number of subjects is the +passive reception of disconnected ideas, not illumined with any spark +of vitality. Let the main ideas which are introduced into a child's +education be few and important, and let them be thrown into every +combination possible. The child should make them his own, and should +understand their application here and now in the circumstances of his +actual life. From the very beginning of his education, the child should +experience the joy of discovery. The discovery which he has to make, +is that general ideas give an understanding of that stream of events +which pours through his life, which is his life. By understanding I +mean more than a mere logical analysis, though that is included. I mean +"understanding" in the sense in which it is used in the French proverb, +"To understand all, is to forgive all." Pedants sneer at an education +which is useful. But if education is not useful, what is it? Is it a +talent, to be hidden away in a napkin? Of course, education should be +useful, whatever your aim in life. It was useful to Saint Augustine +and it was useful to Napoleon. It is useful, because understanding is +useful. + +I pass lightly over that understanding which should be given by the +literary side of education. It is not peculiarly the function of this +Association to consider it. Nor do I wish to be supposed to pronounce +on the relative merits of a classical or a modern curriculum. I would +only remark that the understanding which we want is an understanding +of an insistent present. The only use of a knowledge of the past +is to equip us for the present. No more deadly harm can be done to +young minds than by depreciation of the present. The present contains +all that there is. It is holy ground; for it is the past, and it is +the future. At the same time it must be observed that an age is no +less past if it existed two hundred years ago than if it existed two +thousand years ago. Do not be deceived by the pedantry of dates. The +ages of Shakespeare and of Molière are no less past than are the ages +of Sophocles and of Virgil. The communion of saints is a great and +inspiring assemblage, but it has only one possible hall of meeting, +and that is, the present; and the mere lapse of time through which any +particular group of saints must travel to reach that meeting-place, +makes very little difference. + +Passing now to the scientific and logical side of education, we +remember that here also ideas which are not utilised are positively +harmful. By utilising an idea, I mean relating it to that stream, +compounded of sense perceptions, feelings, hopes, desires, and of +mental activities relating thought to thought, which forms our life. +I can imagine a set of beings which might fortify their souls by +passively reviewing disconnected ideas. Humanity is not built that +way--except perhaps some editors of newspapers. + +In scientific training, the first thing to do with an idea is to prove +it. But allow me for one moment to extend the meaning of "prove"; I +mean--to prove its worth. Now an idea is not worth much unless the +propositions in which it is embodied are true. Accordingly an essential +part of the proof of an idea is the proof, either by experiment or by +logic, of the truth of the propositions. But it is not essential that +this proof of the truth should constitute the first introduction to +the idea. After all, its assertion by the authority of respectable +teachers is sufficient evidence to begin with. In our first contact +with a set of propositions, we commence by appreciating their +importance. That is what we all do in after-life. We do not attempt, +in the strict sense, to prove or to disprove anything, unless its +importance makes it worthy of that honour. These two processes of +proof, in the narrow sense, and of appreciation do not require a rigid +separation in time. Both can be proceeded with nearly concurrently. But +in so far as either process must have the priority, it should be that +of appreciation by use. + +Furthermore, we should not endeavour to use propositions in isolation. +Emphatically I do not mean, a neat little set of experiments to +illustrate Proposition I and then the proof of Proposition I, a neat +little set of experiments to illustrate Proposition II and then the +proof of Proposition II, and so on to the end of the book. Nothing +could be more boring. Inter-related truths are utilised _en bloc_, +and the various propositions are employed in any order, and with any +reiteration. Choose some important applications of your theoretical +subject; and study them concurrently with the systematic theoretical +exposition. Keep the theoretical exposition short and simple, but +let it be strict and rigid so far as it goes. It should not be too +long for it easily to be known with thoroughness and accuracy. The +consequences of a plethora of half-digested theoretical knowledge are +deplorable. Also the theory should not be muddled up with the practice. +The child should have no doubt when it is proving and when it is +utilising. My point is that what is proved should be utilised, and that +what is utilised should--so far as is practicable--be proved. I am far +from asserting that proof and utilisation are the same thing. + +At this point of my discourse, I can most directly carry forward +my argument in the outward form of a digression. We are only just +realising that the art and science of education require a genius and +a study of their own; and that this genius and this science are more +than a bare knowledge of some branch of science or of literature. This +truth was partially perceived in the past generation; and headmasters, +somewhat crudely, were apt to supersede learning in their colleagues +by requiring left-hand bowling and a taste for football. But culture +is more than cricket, and more than football, and more than extent of +knowledge. + +Education is the acquisition of the art of the utilisation of +knowledge. This is an art very difficult to impart. Whenever a +text-book is written of real educational worth, you may be quite +certain that some reviewer will say that it will be difficult to +teach from it. Of course it will be difficult to teach from it. If it +were easy, the book ought to be burned; for it cannot be educational. +In education, as elsewhere, the broad primrose path leads to a nasty +place. This evil path is represented by a book or a set of lectures +which will practically enable the student to learn by heart all the +questions likely to be asked at the next external examination. And I +may say in passing that no educational system is possible unless every +question directly asked of a pupil at any examination is either framed +or modified by the actual teacher of that pupil in that subject. The +external assessor may report on the curriculum or on the performance +of the pupils, but never should be allowed to ask the pupil a question +which has not been strictly supervised by the actual teacher, or +at least inspired by a long conference with him. There are a few +exceptions to this rule, but they are exceptions, and could easily be +allowed for under the general rule. + +We now return to my previous point, that theoretical ideas should +always find important applications within the pupil's curriculum. This +is not an easy doctrine to apply, but a very hard one. It contains +within itself the problem of keeping knowledge alive, of preventing it +from becoming inert, which is the central problem of all education. + +The best procedure will depend on several factors, none of which can +be neglected, namely, the genius of the teacher, the intellectual type +of the pupils, their prospects in life, the opportunities offered by +the immediate surroundings of the school, and allied factors of this +sort. It is for this reason that the uniform external examination +is so deadly. We do not denounce it because we are cranks, and like +denouncing established things. We are not so childish. Also, of course, +such examinations have their use in testing slackness. Our reason of +dislike is very definite and very practical. It kills the best part of +culture. When you analyse in the light of experience the central task +of education, you find that its successful accomplishment depends on a +delicate adjustment of many variable factors. The reason is that we are +dealing with human minds, and not with dead matter. The evocation of +curiosity, of judgment, of the power of mastering a complicated tangle +of circumstances, the use of theory in giving foresight in special +cases--all these powers are not to be imparted by a set rule embodied +in one schedule of examination subjects. + +I appeal to you, as practical teachers. With good discipline, it is +always possible to pump into the minds of a class a certain quantity of +inert knowledge. You take a text-book and make them learn it. So far, +so good. The child then knows how to solve a quadratic equation. But +what is the point of teaching a child to solve a quadratic equation? +There is a traditional answer to this question. It runs thus: The +mind is an instrument, you first sharpen it, and then use it; the +acquisition of the power of solving a quadratic equation is part of +the process of sharpening the mind. Now there is just enough truth in +this answer to have made it live through the ages. But for all its +half-truth, it embodies a radical error which bids fair to stifle the +genius of the modern world. I do not know who was first responsible +for this analogy of the mind to a dead instrument. For aught I know, +it may have been one of the seven wise men of Greece, or a committee +of the whole lot of them. Whoever was the originator, there can be no +doubt of the authority which it has acquired by the continuous approval +which it has received from eminent persons. But whatever its weight of +authority, whatever the high approval which it can quote, I have no +hesitation in denouncing it as one of the most fatal, erroneous, and +dangerous conceptions ever introduced into the theory of education. The +mind is never passive; it is a perpetual activity, delicate, receptive, +responsive to stimulus. You cannot postpone its life until you have +sharpened it. Whatever interest attaches to your subject-matter, must +be evoked here and now; whatever powers you are strengthening in the +pupil, must be exercised here and now; whatever possibilities of mental +life your teaching should impart, must be exhibited here and now. That +is the golden rule of education, and a very difficult rule to follow. + +The difficulty is just this: the apprehension of general ideas, +intellectual habits of mind, and pleasurable interest in mental +achievement can be evoked by no form of words, however accurately +adjusted. All practical teachers know that education is a patient +process of the mastery of details, minute by minute, hour by hour, day +by day. There is no royal road to learning through an airy path of +brilliant generalisations. There is a proverb about the difficulty of +seeing the wood because of the trees. That difficulty is exactly the +point which I am enforcing. The problem of education is to make the +pupil see the wood by means of the trees. + +The solution which I am urging, is to eradicate the fatal disconnection +of subjects which kills the vitality of our modern curriculum. +There is only one subject-matter for education, and that is Life +in all its manifestations. Instead of this single unity, we offer +children--Algebra, from which nothing follows; Geometry, from which +nothing follows; Science, from which nothing follows; History, from +which nothing follows; a Couple of Languages, never mastered; and +lastly, most dreary of all, Literature, represented by plays of +Shakespeare, with philological notes and short analyses of plot and +character to be in substance committed to memory. Can such a list be +said to represent Life, as it is known in the midst of the living of +it? The best that can be said of it is, that it is a rapid table of +contents which a deity might run over in his mind while he was thinking +of creating a world, and had not yet determined how to put it together. + +Let us now return to quadratic equations. We still have on hand the +unanswered question. Why should children be taught their solution? +Unless quadratic equations fit into a connected curriculum, of course +there is no reason to teach anything about them. Furthermore, extensive +as should be the place of mathematics in a complete culture, I am a +little doubtful whether for many types of boys algebraic solutions of +quadratic equations do not lie on the specialist side of mathematics. +I may here remind you that as yet I have not said anything of the +psychology or the content of the specialism, which is so necessary a +part of an ideal education. But all that is an evasion of our real +question, and I merely state it in order to avoid being misunderstood +in my answer. + +Quadratic equations are part of algebra, and algebra is the +intellectual instrument which has been created for rendering clear +the quantitative aspects of the world. There is no getting out of +it. Through and through the world is infected with quantity. To talk +sense, is to talk in quantities. It is no use saying that the nation +is large,--How large? It is no use saying that radium is scarce,--How +scarce? You cannot evade quantity. You may fly to poetry and to music, +and quantity and number will face you in your rhythms and your octaves. +Elegant intellects which despise the theory of quantity, are but half +developed. They are more to be pitied than blamed. The scraps of +gibberish, which in their school-days were taught to them in the name +of algebra, deserve some contempt. + +This question of the degeneration of algebra into gibberish, both in +word and in fact, affords a pathetic instance of the uselessness of +reforming educational schedules without a clear conception of the +attributes which you wish to evoke in the living minds of the children. +A few years ago there was an outcry that school algebra was in need +of reform, but there was a general agreement that graphs would put +everything right. So all sorts of things were extruded, and graphs were +introduced. So far as I can see, with no sort of idea behind them, +but just graphs. Now every examination paper has one or two questions +on graphs. Personally, I am an enthusiastic adherent of graphs. But I +wonder whether as yet we have gained very much. You cannot put life +into any schedule of general education unless you succeed in exhibiting +its relation to some essential characteristic of all intelligent or +emotional perception. It is a hard saying, but it is true; and I do +not see how to make it any easier. In making these little formal +alterations you are beaten by the very nature of things. You are pitted +against too skilful an adversary, who will see to it that the pea is +always under the other thimble. + +Reformation must begin at the other end. First, you must make up your +mind as to those quantitative aspects of the world which are simple +enough to be introduced into general education; then a schedule of +algebra should be framed which will about find its exemplification in +these applications. We need not fear for our pet graphs, they will +be there in plenty when we once begin to treat algebra as a serious +means of studying the world. Some of the simplest applications will be +found in the quantities which occur in the simplest study of society. +The curves of history are more vivid and more informing than the dry +catalogues of names and dates which comprise the greater part of +that arid school study. What purpose is effected by a catalogue of +undistinguished kings and queens? Tom, Dick, or Harry, they are all +dead. General resurrections are failures, and are better postponed. +The quantitative flux of the forces of modern society are capable +of very simple exhibition. Meanwhile, the idea of the variable, of +the function, of rate of change, of equations and their solution, of +elimination, are being studied as an abstract science for their own +sake. Not, of course, in the pompous phrases with which I am alluding +to them here, but with that iteration of simple special cases proper to +teaching. + +If this course be followed, the route from Chaucer to the Black Death, +from the Black Death to modern Labour troubles, will connect the tales +of the mediæval pilgrims with the abstract science of algebra, both +yielding diverse aspects of that single theme, Life. I know what most +of you are thinking at this point. It is that the exact course which +I have sketched out is not the particular one which you would have +chosen, or even see how to work. I quite agree. I am not claiming that +I could do it myself. But your objection is the precise reason why a +common external examination system is fatal to education. The process +of exhibiting the applications of knowledge must, for its success, +essentially depend on the character of the pupils and the genius of the +teacher. Of course I have left out the easiest applications with which +most of us are more at home. I mean the quantitative sides of sciences, +such as mechanics and physics. + +My meaning can be illustrated by looking more closely into a special +case of this type of application. In my rough catalogue of the sort +of subjects which should form the schedule for algebra, I mentioned +Elimination. It was not put there by accident, for it covers a very +important body of thought. + +In the first place, there is the abstract process of algebraic +elimination for suitable simple cases. The pupil acquires a firm grasp +of this by the process, inevitable in education, of working an adequate +number of examples. Again, there are the graphical solutions of the +same problem. Then we consider the significance in the external world. +We consider the velocity, time, space, acceleration diagrams. We take +uniform acceleration; we eliminate "_t_" between + +_v_ = _u_ + _ƒt_, and _s_ = _ut_ + ½_ƒt_^2, + +and eliminate "_s_" between + +_v_^2 = _u_^2 + 2_ƒs_, and _s_ = _ut_ + ½_ƒt_^2. + +Then we remember that constant acceleration is a very special case, +and we consider graphical solutions or empirically given variations of +_v_ or of _ƒ_. In preference, we use those empirical formulæ +which occur in the pupil's experimental work. We compare the strong and +weak points of the algebraic and graphical solutions. + +Again, in the same connection we plot the statistics of social +phenomena against the time. We then eliminate the time between suitable +pairs. We can speculate how far we have exhibited a real casual +connection, or how far a mere temporal coincidence. We notice that +we might have plotted against the time one set of statistics for one +country and another set for another country, and thus, with suitable +choice of subjects, have obtained graphs which certainly exhibited mere +coincidence. Also other graphs exhibit obvious casual connections. We +wonder how to discriminate. And so are drawn on as far as we will. + +But in considering this description, I must beg you to remember what I +have been insisting on above. In the first place, one train of thought +will not suit all groups of children. For example, I should expect that +artisan children will want something more concrete and, in a sense, +swifter than I have set down here. Perhaps I am wrong, but that is +what I should guess. In the second place, I am not contemplating one +beautiful lecture stimulating, once and for all, an admiring class. +That is not the way in which education proceeds. No; all the time the +pupils are hard at work solving examples, drawing graphs, and making +experiments, until they have a thorough hold on the whole subject. +I am describing the interspersed explanations, the directions which +should be given to their thoughts. The pupils have got to be made to +feel that they are studying something, and are not merely executing +intellectual minuets. + +In this connection the excellence of some of the most recent text-books +on elementary algebra emanating from members of this Association, +should create an epoch in the teaching of the subject. + +Finally, if you are teaching pupils for some general examination, +the problem of sound teaching is greatly complicated. Have you ever +noticed the zig-zag moulding round a Norman arch? The ancient work is +beautiful, the modern work is hideous. The reason is, that the modern +work is done to exact measure, the ancient work is varied according +to the idiosyncrasy of the workman. Here it is crowded, and there it +is expanded. Now the essence of getting pupils through examinations +is to give equal weight to all parts of the schedule. But mankind is +naturally specialist. One man sees a whole subject, where another can +find only a few detached examples. I know that it seems contradictory +to allow for specialism in a curriculum especially designed for a +broad culture. Without contradictions the world would be simpler, and +perhaps duller. But I am certain that in education wherever you exclude +specialism you destroy life. + +We now come to the other great branch of a general mathematical +education, namely Geometry. The same principles apply. The theoretical +part should be clear-cut, rigid, short, and important. Every +proposition not absolutely necessary to exhibit the main connection +of ideas should be cut out, but the great fundamental ideas should +be all there. No omission of concepts, such as those of Similarity +and Proportion. We must remember that, owing to the aid rendered by +the visual presence of a figure, Geometry is a field of unequalled +excellence for the exercise of the deductive faculties of reasoning. +Then, of course, there follows Geometrical Drawing, with its training +for the hand and eye. + +But, like Algebra, Geometry and Geometrical Drawing must be extended +beyond the mere circle of geometrical ideas. In an industrial +neighbourhood, machinery and workshop practice form the appropriate +extension. For example, in the London Polytechnics this has been +achieved with conspicuous success. For many secondary schools I suggest +that surveying and maps are the natural applications. In particular, +plane-table surveying should lead pupils to a vivid apprehension of the +immediate application of geometric truths. Simple drawing apparatus, a +surveyor's chain, and a surveyor's compass, should enable the pupils +to rise from the survey and mensuration of a field to the construction +of the map of a small district. The best education is to be found +in gaining the utmost information from the simplest apparatus. The +provision of elaborate instruments is greatly to be deprecated. To +have constructed the map of a small district, to have considered its +roads, its contours, its geology, its climate, its relation to other +districts, the effects on the status of its inhabitants, will teach +more history and geography than any knowledge of Perkin Warbeck or +of Behren's Straits. I mean not a nebulous lecture on the subject, +but a serious investigation in which the real facts are definitely +ascertained by the aid of accurate theoretical knowledge. A typical +mathematical problem should be: Survey such and such a field, draw a +plan of it to such and such a scale, and find the area. It would be +quite a good procedure to impart the necessary geometrical propositions +without their proofs. Then, concurrently in the same term, the proofs +of the propositions would be learnt while the survey was being made. + +Fortunately, the specialist side of education presents an easier +problem than does the provision of a general culture. For this there +are many reasons. One is that many of the principles of procedure +to be observed are the same in both cases, and it is unnecessary +to recapitulate. Another reason is that specialist training takes +place--or should take place--at a more advanced stage of the pupil's +course, and thus there is easier material to work upon. But undoubtedly +the chief reason is that the specialist study is normally a study +of peculiar interest to the student. He is studying it because, for +some reason, he wants to know it. This makes all the difference. +The general culture is designed to foster an activity of mind; the +specialist course utilises this activity. But it does not do to lay too +much stress on these neat antitheses. As we have already seen, in the +general course foci of special interest will arise; and similarly in +the special study, the external connections of the subject drag thought +outwards. + +Again, there is not one course of study which merely gives general +culture, and another which gives special knowledge. The subjects +pursued for the sake of a general education are special subjects +specially studied; and, on the other hand, one of the ways of +encouraging general mental activity is to foster a special devotion. +You may not divide the seamless coat of learning. What education has +to impart is an intimate sense for the power of ideas, for the beauty +of ideas, and for the structure of ideas, together with a particular +body of knowledge which has peculiar reference to the life of the being +possessing it. + +The appreciation of the structure of ideas is that side of a cultured +mind which can only grow under the influence of a special study. I +mean that eye for the whole chess-board, for the bearing of one set of +ideas on another. Nothing but a special study can give any appreciation +for the exact formulation of general ideas, for their relations when +formulated, for their service in the comprehension of life. A mind so +disciplined should be both more abstract and more concrete. It has been +trained in the comprehension of abstract thought and in the analysis of +facts. + +Finally, there should grow the most austere of all mental qualities; I +mean the sense for style. It is an æsthetic sense, based on admiration +for the direct attainment of a foreseen end, simply and without waste. +Style in art, style in literature, style in science, style in logic, +style in practical execution have fundamentally the same æsthetic +qualities, namely, attainment and restraint. The love of a subject in +itself and for itself, where it is not the sleepy pleasure of pacing a +mental quarter-deck, is the love of style as manifested in that study. + +Here we are brought back to the position from which we started, +the utility of education. Style, in its finest sense, is the last +acquirement of the educated mind; it is also the most useful. It +pervades the whole being. The administrator with a sense for style, +hates waste; the engineer with a sense for style, economises his +material; the artisan with a sense for style, prefers good work. Style +is the ultimate morality of mind. + +But above style, and above knowledge, there is something, a vague +shape like fate above the Greek gods. That something is Power. Style +is the fashioning of power, the restraining of power. But, after all, +the power of attainment of the desired end is fundamental. The first +thing is to get there. Do not bother about your style, but solve your +problem, justify the ways of God to man, administer your province, or +do whatever else is set before you. + +Where, then, does style help? In this, with style the end is attained +without side issues, without raising undesirable inflammations. With +style you attain your end and nothing but your end. With style the +effect of your activity is calculable, and foresight is the last gift +of gods to men. With style your power is increased, for your mind is +not distracted with irrelevancies, and you are more likely to attain +your object. Now style is the exclusive privilege of the expert. +Whoever heard of the style of an amateur painter, of the style of an +amateur poet? Style is always the product of specialist study, the +peculiar contribution of specialism to culture. + +English education in its present phase suffers from a lack of definite +aim, and from an external machinery which kills its vitality. Hitherto +in this address I have been considering the aims which should govern +education. In this respect England halts between two opinions. It has +not decided whether to produce amateurs or experts. The profound change +in the world which the nineteenth century has produced is that the +growth of knowledge has given foresight. The amateur is essentially a +man with appreciation and with immense versatility in mastering a given +routine. But he lacks the foresight which comes from special knowledge. +The object of this address is to suggest how to produce the expert +without loss of the essential virtues of the amateur. The machinery +of our secondary education is rigid where it should be yielding, +and lax where it should be rigid. Every school is bound on pain of +extinction to train its boys for a small set of definite examinations. +No headmaster has a free hand to develop his general education or his +specialist studies in accordance with the opportunities of his school, +which are created by its staff, its environment, its class of boys, and +its endowments. I suggest that no system of external tests which aims +primarily at examining individual scholars can result in anything but +educational waste. + +Primarily it is the schools and not the scholars which should be +inspected. Each school should grant its own leaving certificates, based +on its own curriculum. The standards of these schools should be sampled +and corrected. But the first requisite for educational reform is the +school as a unit, with its approved curriculum based on its own needs, +and evolved by its own staff. If we fail to secure that, we simply fall +from one formalism into another, from one dung-hill of inert ideas into +another. + +In stating that the school is the true educational unit in any national +system for the safe-guarding of efficiency, I have conceived the +alternative system as being the external examination of the individual +scholar. But every Scylla is faced by its Charybdis--or, in more homely +language, there is a ditch on both sides of the road. It will be +equally fatal to education if we fall into the hands of a supervising +department which is under the impression that it can divide all schools +into two or three rigid categories, each type being forced to adopt +a rigid curriculum. When I say that the school is the educational +unit, I mean exactly what I say, no larger unit, no smaller unit. +Each school must have the claim to be considered in relation to its +special circumstances. The classifying of schools for some purposes is +necessary. But no absolutely rigid curriculum, not modified by its own +staff, should be permissible. Exactly the same principles apply, with +the proper modifications, to universities and to technical colleges. + +When one considers in its length and in its breadth the importance +of this question of the education of a nation's young, the broken +lives, the defeated hopes, the national failures, which result from the +frivolous inertia with which it is treated, it is difficult to restrain +within oneself a savage rage. In the conditions of modern life the ride +is absolute, the race which does not value trained intelligence is +doomed. Not all your heroism, not all your social charm, not all your +wit, not all your victories on land or at sea, can move back the finger +of fate. To-day we maintain ourselves. To-morrow science will have +moved forward yet one more step, and there will be no appeal from the +judgment which will then be pronounced on the uneducated. + +We can be content with no less than the old summary of educational +ideal which has been current at any time from the dawn of our +civilisation. The essence of education is that it be religious. + +Pray, what is religious education? + +A religious education is an education which inculcates duty and +reverence. Duty arises from our potential control over the course +of events. Where attainable knowledge could have changed the issue, +ignorance has the guilt of vice. And the foundation of reverence is +this perception, that the present holds within itself the complete sum +of existence, backwards and forwards, that whole amplitude of time, +which is eternity. + + + + + CHAPTER II + + TECHNICAL EDUCATION AND ITS RELATION TO SCIENCE AND LITERATURE + + (_Presidential Address to the Mathematical Association, January, + 1917_) + + +THE subject of this address is Technical Education. I wish to examine +its essential nature and also its relation to a liberal education. Such +an inquiry may help us to realise the conditions for the successful +working of a national system of technical training. It is also a very +burning question among mathematical teachers; for mathematics is +included in most technological courses. + +Now it is unpractical to plunge into such a discussion without framing +in our own minds the best ideal towards which we desire to work, +however modestly we may frame our hopes as to the result which in the +near future is likely to be achieved. + +People are shy of ideals; and accordingly we find a formulation of the +ideal state of mankind placed by a modern dramatist[1] in the mouth of +a mad priest: "In my dreams it is a country where the State is the +Church and the Church the people: three in one and one in three. It is +a commonwealth in which work is play and play is life: three in one and +one in three. It is a temple in which the priest is the worshipper and +the worshipper the worshipped: three in one and one in three. It is a +godhead in which all life is human and all humanity divine: three in +one and one in three. It is, in short, the dream of a madman." + +Now the part of this speech to which I would direct attention is +embodied in the phrase, "It is a commonwealth in which work is play +and play is life." This is the ideal of technical education. It sounds +very mystical when we confront it with the actual facts, the toiling +millions, tired, discontented, mentally indifferent, and then the +employers---- I am not undertaking a social analysis, but I shall carry +you with me when I admit that the present facts of society are a long +way off this ideal. Furthermore, we are agreed that an employer who +conducted his workshop on the principle that "work should be play" +would be ruined in a week. + +The curse that has been laid on humanity, in fable and in fact, is, +that by the sweat of its brow shall it live. But reason and moral +intuition have seen in this curse the foundation for advance. The early +Benedictine monks rejoiced in their labours because they conceived +themselves as thereby made fellow-workers with Christ. + +Stripped of its theological trappings, the essential idea remains, +that work should be transfused with intellectual and moral vision and +thereby turned into a joy, triumphing over its weariness and its pain. +Each of us will re-state this abstract formulation in a more concrete +shape in accordance with his private outlook. State it how you like, +so long as you do not lose the main point in your details. However you +phrase it, it remains the sole real hope of toiling humanity; and it +is in the hands of technical teachers, and of those who control their +spheres of activity, so to mould the nation that daily it may pass to +its labours in the spirit of the monks of old. + +The immediate need of the nation is a large supply of skilled workmen, +of men with inventive genius, and of employers alert in the development +of new ideas. + +There is one--and only one--way to obtain these admirable results. +It is by producing workmen, men of science, and employers who enjoy +their work. View the matter practically in the light of our knowledge +of average human nature. Is it likely that a tired, bored workman, +however skilful his hands, will produce a large output of first-class +work? He will limit his production, will scamp his work, and be an +adept at evading inspection; he will be slow in adapting himself to +new methods; he will be a focus of discontent, full of unpractical +revolutionary ideas, controlled by no sympathetic apprehension of the +real working of trade conditions. If, in the troubled times which may +be before us, you wish appreciably to increase the chance of some +savage upheaval, introduce widespread technical education and ignore +the Benedictine ideal. Society will then get what it deserves. + +Again, inventive genius requires pleasurable mental activity as a +condition for its vigorous exercise. "Necessity is the mother of +invention" is a silly proverb. "Necessity is the mother of futile +dodges" is much nearer to the truth. The basis of the growth of modern +invention is science, and science is almost wholly the outgrowth of +pleasurable intellectual curiosity. + +The third class are the employers, who are to be enterprising. Now +it is to be observed that it is the successful employers who are the +important people to get at, the men with business connections all over +the world, men who are already rich. No doubt there will always be a +continuous process of rise and fall of businesses. But it is futile +to expect flourishing trade, if in the mass the successful houses of +business are suffering from atrophy. Now if these men conceive their +businesses as merely indifferent means for acquiring other disconnected +opportunities of life, they have no spur to alertness. They are +already doing very well, the mere momentum of their present business +engagements will carry them on for their time. They are not at all +likely to bother themselves with the doubtful chances of new methods. +Their real soul is in the other side of their life. Desire for money +will produce hard-fistedness and not enterprise. There is much more +hope for humanity from manufacturers who enjoy their work than from +those who continue in irksome business with the object of founding +hospitals. + +Finally, there can be no prospect of industrial peace so long as +masters and men in the mass conceive themselves as engaged in a +soulless operation of extracting money from the public. Enlarged views +of the work performed, and of the communal service thereby rendered, +can be the only basis on which to found sympathetic co-operation. + +The conclusion to be drawn from this discussion is, that alike for +masters and for men a technical or technological education, which is to +have any chance of satisfying the practical needs of the nation, must +be conceived in a liberal spirit as a real intellectual enlightenment +in regard to principles applied and services rendered. In such an +education geometry and poetry are as essential as turning lathes. + +The mythical figure of Plato may stand for modern liberal education +as does that of St. Benedict for technical education. We need not +entangle ourselves in the qualifications necessary for a balanced +representation of the actual thoughts of the actual men. They are used +here as symbolic figures typical of antithetical notions. We consider +Plato in the light of the type of culture he now inspires. + +In its essence a liberal education is an education for thought and +for æsthetic appreciation. It proceeds by imparting a knowledge of +the masterpieces of thought, of imaginative literature, and of art. +The action which it contemplates is command. It is an aristocratic +education implying leisure. This Platonic ideal has rendered +imperishable services to European civilisation. It has encouraged art, +it has fostered that spirit of disinterested curiosity which is the +origin of science, it has maintained the dignity of mind in the face of +material force, a dignity which claims freedom of thought. Plato did +not, like St. Benedict, bother himself to be a fellow-worker with his +slaves; but he must rank among the emancipators of mankind. His type +of culture is the peculiar inspiration of the liberal aristocrat, the +class from which Europe derives what ordered liberty it now possesses. +For centuries, from Pope Nicholas V to the schools of the Jesuits, and +from the Jesuits to the modern headmasters of English public schools, +this educational ideal has had the strenuous support of the clergy. + +For certain people it is a very good education. It suits their type +of mind and the circumstances amid which their life is passed. But +more has been claimed for it than this. All education has been judged +adequate or defective according to its approximation to this sole type. + +The essence of the type is a large discursive knowledge of the best +literature. The ideal product of the type is the man who is acquainted +with the best that has been written. He will have acquired the chief +languages, he will have considered the histories of the rise and fall +of nations, the poetic expression of human feeling, and have read the +great dramas and novels. He will also be well grounded in the chief +philosophies, and have attentively read those philosophic authors who +are distinguished for lucidity of style. + +It is obvious that, except at the close of a long life, he will not +have much time for anything else if any approximation is to be made to +the fulfilment of this programme. One is reminded of the calculation +in a dialogue of Lucian that, before a man could be justified in +practising any one of the current ethical systems, he should have spent +a hundred and fifty years in examining their credentials. + +Such ideals are not for human beings. What is meant by a liberal +culture is nothing so ambitious as a full acquaintance with the varied +literary expression of civilised mankind from Asia to Europe, and from +Europe to America. A small selection only is required; but then, as +we are told, it is a selection of the very best. I have my doubts of +a selection which includes Xenophon and omits Confucius, but then I +have read neither in the original. The ambitious programme of a liberal +education really shrinks to a study of some fragments of literature +included in a couple of important languages. + +But the expression of the human spirit is not confined to literature. +There are the other arts, and there are the sciences. Also education +must pass beyond the passive reception of the ideas of others. Powers +of initiative must be strengthened. Unfortunately initiative does not +mean just one acquirement--there is initiative in thought, initiative +in action, and the imaginative initiative of art; and these three +categories require many subdivisions. + +The field of acquirement is large, and the individual so fleeting and +so fragmentary: classical scholars, scientists, headmasters are alike +ignoramuses. + +There is a curious illusion that a more complete culture was possible +when there was less to know. Surely the only gain was, that it was more +possible to remain unconscious of ignorance. It cannot have been a gain +to Plato to have read neither Shakespeare, nor Newton, nor Darwin. +The achievements of a liberal education have in recent times not been +worsened. The change is that its pretensions have been found out. + +My point is, that no course of study can claim any position of ideal +completeness. Nor are the omitted factors of subordinate importance. +The insistence in the Platonic culture on disinterested intellectual +appreciation is a psychological error. Action and our implication in +the transition of events amid the evitable bond of cause to effect +are fundamental. An education which strives to divorce intellectual +or æsthetic life from these fundamental facts carries with it the +decadence of civilisation. Essentially culture should be for action, +and its effect should be to divest labour from the associations of +aimless toil. Art exists that we may know the deliverances of our +senses as good. It heightens the sense-world. + +Disinterested scientific curiosity is a passion for an ordered +intellectual vision of the connection of events. But the goal of +such curiosity is the marriage of action to thought. This essential +intervention of action even in abstract science is often overlooked. No +man of science wants merely to know. He acquires knowledge to appease +his passion for discovery. He does not discover in order to know, he +knows in order to discover. The pleasure which art and science can +give to toil is the enjoyment which arises from successfully directed +intention. Also it is the same pleasure which is yielded to the +scientist and to the artist. + +The antithesis between a technical and a liberal education is +fallacious. There can be no adequate technical education which is not +liberal, and no liberal education which is not technical: that is, no +education which does not impart both technique and intellectual vision. +In simpler language, education should turn out the pupil with something +he knows well and something he can do well. This intimate union of +practice and theory aids both. The intellect does not work best in a +vacuum. The stimulation of creative impulse requires, especially in +the case of a child, the quick transition to practice. Geometry and +mechanics, followed by workshop practice, gain that reality without +which mathematics is verbiage. + +There are three main methods which are required in a national system of +education, namely, the literary curriculum, the scientific curriculum, +the technical curriculum. But each of these curricula should include +the other two. What I mean is, that every form of education should give +the pupil a technique, a science, an assortment of general ideas, and +æsthetic appreciation, and that each of these sides of his training +should be illuminated by the others. Lack of time, even for the most +favoured pupil, makes it impossible to develop fully each curriculum. +Always there must be a dominant emphasis. The most direct æsthetic +training naturally falls in the technical curriculum in those cases +when the training is that requisite for some art or artistic craft. But +it is of high importance in both a literary and a scientific education. + +The educational method of the literary curriculum is the study of +language, that is, the study of our most habitual method of conveying +to others our states of mind. The technique which should be acquired +is the technique of verbal expression, the science is the study of the +structure of language and the analysis of the relations of language +to the states of mind conveyed. Furthermore, the subtle relations of +language to feeling, and the high development of the sense organs +to which written and spoken words appeal, lead to keen æsthetic +appreciations being aroused by the successful employment of language. +Finally, the wisdom of the world is preserved in the masterpieces of +linguistic composition. + +This curriculum has the merit of homogeneity. All its various parts +are co-ordinated and play into each other's hands. We can hardly be +surprised that such a curriculum, when once broadly established, should +have claimed the position of the sole perfect type of education. Its +defect is unduly to emphasise the importance of language. Indeed the +varied importance of verbal expression is so overwhelming that its +sober estimation is difficult. Recent generations have been witnessing +the retreat of literature, and of literary forms of expression, from +their position of unique importance in intellectual life. In order +truly to become a servant and a minister of nature something more is +required than literary aptitudes. + +A scientific education is primarily a training in the art of observing +natural phenomena, and in the knowledge and deduction of laws +concerning the sequence of such phenomena. But here, as in the case of +a liberal education, we are met by the limitations imposed by shortness +of time. There are many types of natural phenomena, and to each type +there corresponds a science with its peculiar modes of observation, +and its peculiar types of thought employed in the deduction of laws. A +study of science in general is impossible in education, all that can be +achieved is the study of two or three allied sciences. Hence the charge +of narrow specialism urged against any education which is primarily +scientific. It is obvious that the charge is apt to be well-founded; +and it is worth considering how, within the limits of a scientific +education and to the advantage of such an education, the danger can be +avoided. + +Such a discussion requires the consideration of technical education. A +technical education is in the main a training in the art of utilising +knowledge for the manufacture of material products. Such a training +emphasises manual skill, and the co-ordinated action of hand and +eye, and judgment in the control of the process of construction. But +judgment necessitates knowledge of those natural processes of which the +manufacture is the utilisation. Thus somewhere in technical training +an education in scientific knowledge is required. If you minimise +the scientific side, you will confine it to the scientific experts; +if you maximise it, you will impart it in some measure to the men, +and--what is of no less importance--to the directors and managers of +the businesses. + +Technical education is not necessarily allied exclusively to science +on its mental side. It may be an education for an artist or for +apprentices to an artistic craft. In that case æsthetic appreciation +will have to be cultivated in connection with it. + +An evil side of the Platonic culture has been its total neglect of +technical education as an ingredient in the complete development +of ideal human beings. This neglect has arisen from two disastrous +antitheses, namely, that between mind and body, and that between +thought and action. I will here interject, solely to avoid criticism, +that I am well aware that the Greeks highly valued physical beauty and +physical activity. They had, however, that perverted sense of values +which is the nemesis of slave-owning. + +I lay it down as an educational axiom that in teaching you will come +to grief as soon as you forget that your pupils have bodies. This is +exactly the mistake of the post-renaissance Platonic curriculum. But +nature can be kept at bay by no pitchfork; so in English education, +being expelled from the classroom, she returned with a cap and bells in +the form of all-conquering athleticism. + +The connections between intellectual activity and the body, though +diffused in every bodily feeling, are focussed in the eyes, the ears, +the voice, and the hands. There is a co-ordination of senses and +thought, and also a reciprocal influence between brain activity and +material creative activity. In this reaction the hands are peculiarly +important. It is a moot point whether the human hand created the human +brain, or the brain created the hand. Certainly the connection is +intimate and reciprocal. Such deep-seated relations are not widely +atrophied by a few hundred years of disuse in exceptional families. + +The disuse of hand-craft is a contributory cause to the brain-lethargy +of aristocracies, which is only mitigated by sport where the concurrent +brain-activity is reduced to a minimum and the hand-craft lacks +subtlety. The necessity for constant writing and vocal exposition is +some slight stimulus to the thought-power of the professional classes. +Great readers, who exclude other activities, are not distinguished by +subtlety of brain. They tend to be timid conventional thinkers. No +doubt this is partly due to their excessive knowledge outrunning their +powers of thought; but it is partly due to the lack of brain-stimulus +from the productive activities of hand or voice. + +In estimating the importance of technical education we must rise above +the exclusive association of learning with book-learning. First-hand +knowledge is the ultimate basis of intellectual life. To a large +extent book-learning conveys second-hand information, and as such can +never rise to the importance of immediate practice. Our goal is to see +the immediate event of our lives as instances of our general ideas. +What the learned world tends to offer is one second-hand scrap of +information illustrating ideas derived from another second-hand scrap +of information. The second-handedness of the learned world is the +secret of its mediocrity. It is tame because it has never been scared +by facts. The main importance of Francis Bacon's influence does not +lie in any peculiar theory of inductive reasoning which he happened to +express, but in the revolt against second-hand information of which he +was a leader. + +The peculiar merit of a scientific education should be, that it bases +thought upon first-hand observation; and the corresponding merit of a +technical education is, that it follows our deep natural instinct to +translate thought into manual skill, and manual activity into thought. + +We are a Mathematical Association, and it is natural to ask: Where do +we come in? We come in just at this point. + +The thought which science evokes is logical thought. Now logic is of +two kinds: the logic of discovery and the logic of the discovered. + +The logic of discovery consists in the weighing of probabilities, in +discarding details deemed to be irrelevant, in divining the general +rules according to which events occur, and in testing hypotheses by +devising suitable experiments. This is inductive logic. + +The logic of the discovered is the deduction of the special events +which, under certain circumstances, would happen in obedience to the +assumed laws of nature. Thus when the laws are discovered or assumed, +their utilisation entirely depends on deductive logic. Without +deductive logic science would be entirely useless. It is merely a +barren game to ascend from the particular to the general, unless +afterwards we can reverse the process and descend from the general to +the particular, ascending and descending like the angels on Jacob's +ladder. When Newton had divined the law of gravitation he at once +proceeded to calculate the earth's attractions on an apple at its +surface and on the moon. We may note in passing that inductive logic +would be impossible without deductive logic. Thus Newton's calculations +were an essential step in his inductive verification of the great law. + +Now mathematics is nothing else than the more complicated parts of +the art of deductive reasoning, especially where it concerns number, +quantity, and space. + +In the teaching of science, the art of thought should be taught: +namely, the art of forming clear conceptions applying to first-hand +experience, the art of divining the general truths which apply, the +art of testing divinations, and the art of utilising general truths +by reasoning to more particular cases of some peculiar importance. +Furthermore, a power of scientific exposition is necessary, so that the +relevant issues from a confused mass of ideas can be stated clearly, +with due emphasis on important points. + +By the time a science, or a small group of sciences, has been taught +thus amply, with due regard to the general art of thought, we have gone +a long way towards correcting the specialism of science. The worst +of a scientific education based, as necessarily must be the case, on +one or two particular branches of science, is that the teachers under +the influence of the examination system are apt merely to stuff +their pupils with the narrow results of these special sciences. It is +essential that the generality of the method be continually brought to +light and contrasted with the speciality of the particular application. +A man who only knows his own science, as a routine peculiar to that +science, does not even know that. He has no fertility of thought, no +power of quickly seizing the bearing of alien ideas. He will discover +nothing, and be stupid in practical applications. + +This exhibition of the general in the particular is extremely difficult +to effect, especially in the case of younger pupils. The art of +education is never easy. To surmount its difficulties, especially those +of elementary education, is a task worthy of the highest genius. It is +the training of human souls. + +Mathematics, well taught, should be the most powerful instrument +in gradually implanting this generality of idea. The essence of +mathematics is perpetually to be discarding more special ideas in +favour of more general ideas, and special methods in favour of general +methods. We express the conditions of a special problem in the form of +an equation, but that equation will serve for a hundred other problems, +scattered through diverse sciences. The general reasoning is always +the powerful reasoning, because deductive cogency is the property of +abstract form. + +Here, again, we must be careful. We shall ruin mathematical education +if we use it merely to impress general truths. The general ideas are +the means of connecting particular results. After all, it is the +concrete special cases which are important. Thus in the handling of +mathematics in your results you cannot be too concrete, and in your +methods you cannot be too general. The essential course of reasoning +is to generalise what is particular, and then to particularise what is +general. Without generality there is no reasoning, without concreteness +there is no importance. + +Concreteness is the strength of technical education. I would +remind you that truths which lack the highest generality are not +necessarily concrete facts. For example, _x_ + _y_ = _y_ + _x_ is an +algebraic truth more general than 2 + 2 = 4. But "two and two make +four" is itself a highly general proposition lacking any element of +concreteness. To obtain a concrete proposition immediate intuition of a +truth concerning particular objects is requisite; for example, "these +two apples and those apples together make four apples" is a concrete +proposition, if you have direct perception or immediate memory of the +apples. + +In order to obtain the full realisation of truths as applying, and not +as empty formulæ, there is no alternative to technical education. Mere +passive observation is not sufficient. In creation only is there vivid +insight into the properties of the object thereby produced. If you +want to understand anything, make it yourself, is a sound rule. Your +faculties will be alive, your thoughts gain vividness by an immediate +translation into acts. Your ideas gain that reality which comes from +seeing the limits of their application. + +In elementary education this doctrine has long been put into practice. +Young children are taught to familiarise themselves with shapes and +colours by simple manual operations of cutting out and of sorting. But +good though this is, it is not quite what I mean. That is practical +experience before you think, experience antecedent to thought in order +to create ideas, a very excellent discipline. But technical education +should be much more than that: it is creative experience while you +think, experience which realises your thought, experience which teaches +you to co-ordinate act and thought, experience leading you to associate +thought with foresight and foresight with achievement. Technical +education gives theory, and a shrewd insight as to where theory fails. + +A technical education is not to be conceived as a maimed alternative +to the perfect Platonic culture: namely, as a defective training +unfortunately made necessary by cramped conditions of life. No +human being can attain to anything but fragmentary knowledge and a +fragmentary training of his capacities. There are, however, three main +roads along which we can proceed with good hope of advancing towards +the best balance of intellect and character: these are the way of +literary culture, the way of scientific culture, the way of technical +culture. No one of these methods can be exclusively followed without +grave loss of intellectual activity and of character. But a mere +mechanical mixture of the three curricula will produce bad results in +the shape of scraps of information never interconnected or utilised. +We have already noted as one of the strong points of the traditional +literary culture that all its parts are co-ordinated. The problem +of education is to retain the dominant emphasis, whether literary, +scientific, or technical, and without loss of co-ordination to infuse +into each way of education something of the other two. + +To make definite the problem of technical education fix attention on +two ages: one thirteen, when elementary education ends; and the other +seventeen, when technical education ends so far as it is compressed +within a school curriculum. I am aware that for artisans in junior +technical schools a three-years' course would be more usual. On the +other hand, for naval officers, and for directing classes generally, +a longer time can be afforded. We want to consider the principles +to govern a curriculum which shall land these children at the age of +seventeen in the position of having technical skill useful to the +community. + +Their technical manual training should start at thirteen, bearing a +modest proportion to the rest of their work, and should increase in +each year finally to attain to a substantial proportion. Above all +things it should not be too specialised. Workshop finish and workshop +dodges, adapted to one particular job, should be taught in the +commercial workshop, and should form no essential part of the school +course. A properly trained worker would pick them up in no time. In all +education the main cause of failure is staleness. Technical education +is doomed if we conceive it as a system for catching children young and +for giving them one highly specialised manual aptitude. The nation has +need of a fluidity of labour, not merely from place to place, but also +within reasonable limits of allied aptitudes, from one special type of +work to another special type. I know that here I am on delicate ground, +and I am not claiming that men while they are specialising on one sort +of work should spasmodically be set to other kinds. That is a question +of trade organisation with which educationalists have no concern. I am +only asserting the principles that training should be broader than the +ultimate specialisation, and that the resulting power of adaptation to +varying demands is advantageous to the workers, to the employers, and +to the nation. + +In considering the intellectual side of the curriculum we must be +guided by the principle of the co-ordination of studies. In general, +the intellectual studies most immediately related to manual training +will be some branches of science. More than one branch will, in fact, +be concerned; and even if that be not the case, it is impossible to +narrow down scientific study to a single thin line of thought. It is +possible, however, provided that we do not press the classification +too far, roughly to classify technical pursuits according to the +dominant science involved. We thus find a sixfold division, namely, +(1) Geometrical techniques, (2) Mechanical techniques, (3) Physical +techniques, (4) Chemical techniques, (5) Biological techniques, (6) +Techniques of commerce and of social service. + +By this division, it is meant that apart from auxiliary sciences +some particular science requires emphasis in the training for most +occupations. We can, for example, reckon carpentry, ironmongery, +and many artistic crafts among geometrical techniques. Similarly +agriculture is a biological technique. Probably cookery, if it includes +food catering, would fall midway between biological, physical, and +chemical sciences, though of this I am not sure. + +The sciences associated with commerce and social service would be +partly algebra, including arithmetic and statistics, and partly +geography and history. But this section is somewhat heterogeneous in +its scientific affinities. Anyhow the exact way in which technical +pursuits are classified in relation to science is a detail. The +essential point is, that with some thought it is possible to find +scientific courses which illuminate most occupations. Furthermore, the +problem is well understood, and has been brilliantly solved in many of +the schools of technology and junior technical schools throughout the +country. + +In passing from science to literature, in our review of the +intellectual elements of technical education, we note that many studies +hover between the two: for example, history and geography. They are +both of them very essential in education, provided that they are the +right history and the right geography. Also books giving descriptive +accounts of general results, and trains of thought in various sciences +fall in the same category. Such books should be partly historical +and partly expository of the main ideas which have finally arisen. +Prof. R. A. Gregory's recent book, _Discovery_, and the _Home +University Library_ series illustrate my meaning. Their value in +education depends on their quality as mental stimulants. They must not +be inflated with gas on the wonders of science, and must be informed +with a broad outlook. + +It is unfortunate that the literary element in education has rarely +been considered apart from grammatical study. The historical reason is, +that when the modern Platonic curriculum was being formed Latin and +Greek were the sole keys which rendered great literature accessible. +But there is no necessary connection between literature and grammar. +The great age of Greek literature was already past before the arrival +of the grammarians of Alexandria. Of all types of men to-day existing, +classical scholars are the most remote from the Greeks of the Periclean +times. + +Mere literary knowledge is of slight importance. The only thing that +matters is, how it is known. The facts related are nothing. Literature +only exists to express and develop that imaginative world which is our +life, the kingdom which is within us. It follows that the literary side +of a technical education should consist in an effort to make the pupils +enjoy literature. It does not matter what they know, but the enjoyment +is vital. The great English Universities, under whose direct authority +school-children are examined in plays of Shakespeare, to the certain +destruction of their enjoyment, should be prosecuted for soul-murder. + +Now there are two kinds of intellectual enjoyment: the enjoyment of +creation, and the enjoyment of relaxation. They are not necessarily +separated. A change of occupation may give the full tide of happiness +which comes from the concurrence of both forms of pleasure. The +appreciation of literature is really creation. The written word, its +music, and its associations, are only the stimuli. The vision which +they evoke is our own doing. No one, no genius other than our own, +can make our own life live. But except for those engaged in literary +occupations, literature is also a relaxation. It gives exercise to that +other side which any occupation must suppress during the working hours. +Art also has the same function in life as has literature. + +To obtain the pleasure of relaxation requires no help. The pleasure +is merely to cease doing. Some such pure relaxation is a necessary +condition of health. Its dangers are notorious, and to the greater +part of the necessary relaxation nature has affixed, not enjoyment, +but the oblivion of sleep. Creative enjoyment is the outcome of +successful effort and requires help for its initiation. Such enjoyment +is necessary for high-speed work and for original achievement. + +To speed up production with unrefreshed workmen is a disastrous +economic policy. Temporary success will be at the expense of the +nation, which, for long years of their lives, will have to support +worn-out artisans--unemployables. Equally disastrous is the +alternation of spasms of effort with periods of pure relaxation. +Such periods are the seed-times of degeneration, unless rigorously +curtailed. The normal recreation should be change of activity, +satisfying the cravings of instincts. Games afford such activity. Their +disconnection emphasises the relaxation, but their excess leaves us +empty. + +It is here that literature and art should play an essential part in +a healthily organised nation. Their services to economic production +would be only second to those of sleep or of food. I am not now talking +of the training of an artist, but of the use of art as a condition of +healthy life. It is analogous to sunshine in the physical world. + +When we have once rid our minds of the idea that knowledge is to +be exacted, there is no especial difficulty or expense involved in +helping the growth of artistic enjoyment. All school-children could +be sent at regular intervals to neighbouring theatres where suitable +plays could be subsidised. Similarly for concerts and cinema films. +Pictures are more doubtful in their popular attraction; but interesting +representations of scenes or ideas which the children have read about +would probably appeal. The pupils themselves should be encouraged +in artistic efforts. Above all the art of reading aloud should be +cultivated. The Roger de Coverley essays of Addison are perfect +examples of readable prose. + +Art and literature have not merely an indirect effect on the main +energies of life. Directly, they give vision. The world spreads wide +beyond the deliverances of material sense, with subtleties of reaction +and with pulses of emotion. Vision is the necessary antecedent to +control and to direction. In the contest of races which in its final +issues will be decided in the workshops and not on the battle-field, +the victory will belong to those who are masters of stores of trained +nervous energy, working under conditions favourable to growth. One such +essential condition is Art. + +If there had been time, there are other things which I should like +to have said: for example, to advocate the inclusion of one foreign +language in all education. From direct observation I know this to be +possible for artisan children. But enough has been put before you, +to make plain the principles with which we should undertake national +education. + +In conclusion, I recur to the thought of the Benedictines, who +saved for mankind the vanishing civilisation of the ancient world +by linking together knowledge, labour, and moral energy. Our danger +is to conceive practical affairs as the kingdom of evil, in which +success is only possible by the extrusion of ideal aims. I believe +that such a conception is a fallacy directly negatived by practical +experience. In education this error takes the form of a mean view of +technical training. Our forefathers in the dark ages saved themselves +by embodying high ideals in great organisations. It is our task, +without servile imitation, boldly to exercise our creative energies, +remembering amid discouragements that the coldest hour immediately +precedes the dawn. + + +FOOTNOTES: + +[Footnote 1: _Cf._ BERNARD SHAW: _John Bull's Other +Island_.] + + + + + CHAPTER III + + A POLYTECHNIC IN WAR-TIME + + _Address at the Prize Distribution, Borough Polytechnic Institute, + Southwark, 16th February, 1917_ + + +I WILL commence by drawing your attention to some of the satisfactory +features of the Principal's report on the work of the Institute during +the past year. It has been a year of great difficulties. Some of our +staff are serving with the colours, and our classes have been depleted. +But in spite of everything, we have done very well. First, the average +result in the examinations has been good, surprisingly good in view of +the present circumstances. The Governors attach great importance to +the maintenance of a high average result; it is the best single test +of efficiency. Again, our individual successes have been notable. We +have gained--I say _we_ because we are all one in our pleasure at +these successes--we have gained two £80 L.C.C. scholarships, nineteen +exhibitions, in addition to a first-place, and medals, prizes and +certificates. All this is very satisfactory. It tells of efficient +teaching, and of hard work and regular attendance on the part of the +students. We know that we are keeping up the standard of efficiency +which in the past has been a source of pride to every one connected +with this Institute. + +Now all this good work does not come about by itself without any one +making an effort. Such a record requires our skilled staff of teachers +and organisers. They have worked very hard during the last session +under great difficulties, in order to create the successful result +which we are here to celebrate. I know something about teaching. It +is very exacting work, and can be made successful only by continual +devotion. I am sure that I am voicing your feelings, and I know that +I am expressing those of the Governors, when I thank the ladies and +gentlemen of the staff very heartily for their services during the last +session. + +Prize-givings are always pleasant occasions. We have come here to think +about our successes, and to congratulate our students. There is no more +satisfactory Governors' Meeting in the course of the year than when we +meet on this occasion, and face our friends and tell them how pleased +we are at the successful result of their hard work. This evening I am +in a doubly happy position, for my colleagues have asked me to be their +spokesman in tendering our good wishes to the prize-winners. You have +worked hard and you have done well, and I am sure that you all deserve +your successes; they are a pleasure not only to you, but in your homes +and to your companions and fellow-students. + +Successful work here will enable you to acquire skill in your trades, +and thereby the better to earn your living. Earning a living is on the +average no bad test of service rendered to the community. A man who +has made himself skilful in his trade and has done well for himself in +his walk of life, has in general good reason to believe that he is a +citizen who has benefited his country. It is an evil day for a nation +when it loses respect for success in industry. + +But if you steer your lives by the compass which points steadily to +the North Pole of personal success, you will have missed your greatest +chances in life. The genial climate is in the south. + +What I mean is this: you must make up your mind to find the best part +of your happiness in kindly helpful relations with others. It should be +our ambition to leave our own small corner of the world a little tidier +and a little happier than when we entered it. I am well aware that +this is an old story; but old stories are sometimes true, and this is +the biggest truth in the whole world. The warm kindly feelings are the +happy feelings. The fortunate people are those whose minds are filled +with thoughts in which they forget themselves and remember others. It +is not true that nature is a mere scene of struggle in which every one +competes with his neighbour. Those communities thrive best and last +longest which are filled with a spirit of mutual help. + +The future of the country lies with you. The crown of your success +is the promise of future work, often unrecognised work, done under +discouragement, but done steadily and cheerfully. It is on you that +the country depends for the maintenance and the growth of those ideals +without which a race withers. Do not be discouraged by difficulties +which seem unsurmountable. The conditions of life which mould us all +are modified by our will, by our energy, and by the purity of our +intentions. + +If we may judge of intensity of feeling by length of memory, the +enjoyment of receiving a prize bites very deep. Across the space of +more than forty years, before many of your parents were born, or when +they were being carried about in long clothes, I can remember, as if it +were yesterday, the occasion when I received my first prize at school. +I can see the mediæval school-room, the headmaster, and my companions. +Perhaps some of you, when a generation has passed by, will remember the +scene to-night--this Stanley Gymnasium recalling the memory of Miss +Maude Stanley, who devoted to our welfare so much of her energy and +her thought--the adjoining Edric Hall associated with the name of Mr. +Edric Bayley, the Father of the Institute; Mr. Millis and Miss Smith, +the first Principal and the first Lady Superintendent, the architects +of our prosperity; Mr. Leonard Spicer, our Chairman and member of a +family and of a firm known throughout the world, and respected in +proportion as they are known. And the cause why to-night we are a small +gathering is one more reason why this assembly can never slip from your +memory. We meet at a moment when England stands in as deadly a peril as +in any previous moment of her history--such peril as when the Spanish +ships of the Armada rode in the English Channel, or when Napoleon +watched our coast across the Strait of Dover. The present danger can be +overcome only by the same courage as that which saved our freedom in +those former times. + +Therefore, to-night, in recalling the activities of the various +sections of our society which form this great Polytechnic Institute, +our thoughts go further afield. They travel by land and by sea, till +they bring before our minds the gallant band whom this Institute has +sent to the Front--more than 800 of our members are with the Colours. +What our fighting men have done for us, for the world in general, and +for the future of England, is so overwhelming that words cannot praise +them enough. I will just say one thing to you: When you read of great +deeds done in past times, of perils encountered, of adventures, of +undaunted courage, of patriotism, of self-sacrifice, of suffering +endured for noble cause, you each can say--I, too, have known such +heroes; they are among my countrymen, they are among my fellow-workers, +they are among my fellow-students and companions, they are among the +dear inmates of my home. And for those who have fallen, it is for us +to erect a monument sufficient to transmit to future ages the memory +of their sacrifice. For this purpose there is only one memorial which +can suffice, namely, the cause for which they died. The greatness of +England, the future of England, has been left by them to our keeping. +Guard it well. + +The greatness of a country is nothing else than the greatness of the +lives of the men and of the women who compose it. Do not look round and +think who ought to be great Englishmen--be great yourselves--you are +the people to achieve it, you who are sitting here to-night. There can +be no substitute service for this purpose. It is the collective energy +of the whole people that will be needed to fashion a new England worthy +of the sufferings which for its sake have been endured. + +A few days ago I asked a man who has worked in Egypt for many years +under Lord Kitchener, what he would pick out as the best sign of Lord +Kitchener's greatness. He answered, whatever Kitchener set himself to +do, thereby became important. Now that is the secret of it all--take +hold of your opportunities and make them important. + +Here we are in this Borough Polytechnic. What an opportunity it +represents. This Institute is a centre for social meeting, a centre +for recreation, a centre for education, a centre for discussion. We +will not sacrifice any one of our sides. They must all be part of the +greatness which we claim. Make them all first-rate. + +Consider first the social and recreative sides. For heaven's sake don't +think that you must be dull in order to be great. There is no finer +test of a nation than the way in which it fashions its amusements. +Three centuries ago after the Armada we made a good start in Southwark. +Shakespeare had his theatre here and wrote his plays to be acted in +this borough. He has walked these streets, and if you had met him in +Westminster he would, quite likely, have told you that he was going +down to the "Elephant." And even now the performances given at the +"Old Vic" are among the best in London for the purpose of seeing his +plays properly acted. What Southwark has done for the drama, she can +do for the other arts, by using this Institute as the instrument for +her energies. Why should we not be a centre for artistic enterprise--I +mean for our own art and our own enterprise, thought of by ourselves +and enjoyed by ourselves and carried through by ourselves? We shall not +always enjoy each others' creations, but the great point is to make our +own efforts. Of course all efforts require preparation and stimulus and +knowledge of what others are doing. + +At the present time--interrupted for the moment by the war--a great +revolution in the art of painting is in progress throughout the world. +Its centres are Paris and Italy and London and Munich, and its origin +in the far east, in China and Japan. There are two sides as in every +revolution, the Conservatives and the Revolutionists. Our own frescoes +in a neighbouring room represent an early stage of the movement in +London. Why should we not know all about it--obtain loans of pictures +which illustrate its phases and its cross currents, and compare these +with examples of the old style? + +But pictures are only one phase of art, and not the sort of art which +we ourselves can produce most easily. There are music, dancing, +recitation, literature, carving and modelling, and the various +decorative arts, such as embroidery, bookbinding, dress-making and +upholstery. This list, incomplete as it is, tells us two great +truths--you cannot separate art and recreation, and you cannot +separate art and business. The list includes items which we consider +as amusements, and items which we think of as business. We began with +dancing and ended with upholstery. Make them all beautiful. + +Beautiful things have dignity. Enjoy the rhythm of your dancing and +admire the beauty of your embroidery or your bookbinding. In whatever +you do, have an ideal of excellence. Any separation between art and +work is not only an error, but it is very bad business. Our brave +allies, the French, have made Paris the art centre of the world. They +have built up and maintain their large and lucrative trade in the +decorative products of France, mainly by reason of three qualities +which they possess. In the first place, they enjoy art themselves, and +reverence it. In the second place, they have a tremendous power of hard +work. And in the third place, every Frenchman, and still more every +Frenchwoman, have within them an immense fund of common sense. The +threefold secret is, Love of Art, Industry, and Common Sense. + +To make available our industry and common sense in the trades where +they are wanted, rigorous training in schools of design and technique +are necessary. We have such departments here. But all such training of +you will be a failure unless you yourselves enjoy art and beauty as +a natural recreation. A technical school of training is like a deep, +narrow well, sunk with careful labour to tap the underground river of +water which flows below the surface of our natures. But your well will +be dry unless the bright warm sun has first drawn up the vapour from +the wide ocean, and the free untrammelled breezes have carried the +clouds hither and thither, until at length they break, as it were by +chance over the distant hills and soak the land with their downpour. + +What I have said about art is a parable which applies to other +occupations and other studies. It is more than a parable; it is the +literal truth. The love of art is the love of excellence, it is the +enjoyment of the triumph of design over the shapeless products of +chance forces. An engineer, who is worth his salt, loves the beauty of +his machines, shown in their adjustment of parts and in their swift, +smooth motions. He loves also the sense of foresight and of insight +which knowledge can give him. People say that machinery and commerce +are driving beauty out of the modern world. I do not believe it. A new +beauty is being added, a more intellectual beauty, appealing to the +understanding as much as to the eye. + +The wonder of London ever takes the mind with fresh astonishment. The +city possesses parks, and palaces, and cathedrals. But no other parts +of it surpass in wonder its houses of business and its workshops and +its factories. + +In the next few years the future of the world will be decided for +centuries to come. The battles of this war are only the first part of +the contest between races, and between the ways of life for which +those races stand. We believe that England, with its various peoples +and communities scattered in islands and continents beyond the seas, +stands for ways of life infinitely precious, the way of humanity, the +way of liberty, the way of self-government, the way of good order based +on toleration and kindly feeling, the way of peaceful industry. The +final decision in this struggle will be found in the workshops of the +world. It lies in your hands. Statesmen and emperors will only register +the results which you have achieved. Your weapons will be skill, and +energy, and knowledge. You will require a sane understanding of your +own rights, and a sane understanding of the rights and the difficulties +of other classes. The greatness of England will be your greatness, and +its success your success. + +The arsenal for war is at Woolwich. This Polytechnic Institute is an +arsenal for peace, where you can find the weapons for the conquest of +your lives. + + + + + CHAPTER IV + + THE MATHEMATICAL CURRICULUM + + (_Presidential Address to the London Branch of the Mathematical + Association, 1912_) + + +THE situation in regard to education at the present time cannot find +its parallel without going back for some centuries to the break-up of +the mediæval traditions of learning. Then, as now, the traditional +intellectual outlook, despite the authority which it had justly +acquired from its notable triumphs, had grown to be too narrow for the +interests of mankind. The result of this shifting of human interest +was a demand for a parallel shifting of the basis of education, so +as to fit the pupils for the ideas which later in life would in fact +occupy their minds. Any serious fundamental change in the intellectual +outlook of human society must necessarily be followed by an educational +revolution. It may be delayed for a generation by vested interests or +by the passionate attachment of some leaders of thought to the cycle +of ideas within which they received their own mental stimulus at an +impressionable age. But the law is inexorable that education to be +living and effective must be directed to informing pupils with those +ideas, and to creating for them those capacities which will enable them +to appreciate the current thought of their epoch. + +There is no such thing as a successful system of education in a vacuum, +that is to say, a system which is divorced from immediate contact with +the existing intellectual atmosphere. Education which is not modern +shares the fate of all organic things which are kept too long. + +But the blessed word "modern" does not really solve our difficulties. +What we mean is, relevant to modern thought, either in the ideas +imparted or in the aptitudes produced. Something found out only +yesterday may not really be modern in this sense. It may belong to +some bygone system of thought prevalent in a previous age, or, what +is very much more likely, it may be too recondite. When we demand +that education should be relevant to modern thought, we are referring +to thoughts broadly spread throughout cultivated society. It is this +question of the unfitness of recondite subjects for use in general +education which I wish to make the keynote of my address this afternoon. + +It is in fact rather a delicate subject for us mathematicians. +Outsiders are apt to accuse our subject of being recondite. Let us +grasp the nettle at once and frankly admit that in general opinion it +is the very typical example of reconditeness. By this word I do not +mean difficulty, but that the ideas involved are of highly special +application, and rarely influence thought. + +This liability to reconditeness is the characteristic evil which is +apt to destroy the utility of mathematics in liberal education. So +far as it clings to the educational use of the subject, so far we +must acquiesce in a miserably low level of mathematical attainment +among cultivated people in general. I yield to no one in my anxiety to +increase the educational scope of mathematics. The way to achieve this +end is not by a mere blind demand for more mathematics. We must face +the real difficulty which obstructs its extended use. + +Is the subject recondite? Now, viewed as a whole, I think it is. +_Securus judicat orbis terrarum_--the general judgment of mankind +is sure. + +The subject as it exists in the minds and in the books of students of +mathematics _is_ recondite. It proceeds by deducing innumerable +special results from general ideas, each result more recondite than the +preceding. It is not my task this afternoon to defend mathematics as a +subject for profound study. It can very well take care of itself. What +I want to emphasise is, that the very reasons which make this science +a delight to its students are reasons which obstruct its use as an +educational instrument--namely, the boundless wealth of deductions from +the interplay of general theorems, their complication, their apparent +remoteness from the ideas from which the argument started, the variety +of methods, and their purely abstract character which brings, as its +gift, eternal truth. + +Of course, all these characteristics are of priceless value to +students; for ages they have fascinated some of the keenest intellects. +My only remark is that, except for a highly selected class, they are +fatal in education. The pupils are bewildered by a multiplicity of +detail, without apparent relevance either to great ideas or to ordinary +thoughts. The extension of this sort of training in the direction +of acquiring more detail is the last measure to be desired in the +interests of education. + +The conclusion at which we arrive is, that mathematics, if it is +to be used in general education, must be subjected to a rigorous +process of selection and adaptation. I do not mean, what is of course +obvious, that however much time we devote to the subject the average +pupil will not get very far. But that, however limited the progress, +certain characteristics of the subject, natural at any stage, must be +rigorously excluded. The science as presented to young pupils must +lose its aspect of reconditeness. It must, on the face of it, deal +directly and simply with a few general ideas of far-reaching importance. + +Now, in this matter of the reform of mathematical instruction, the +present generation of teachers may take a very legitimate pride in +its achievements. It has shown immense energy in reform, and has +accomplished more than would have been thought possible in so short a +time. It is not always recognised how difficult is the task of changing +a well-established curriculum entrenched behind public examinations. + +But for all that, great progress has been made, and, to put the matter +at its lowest, the old dead tradition has been broken up. I want to +indicate this afternoon the guiding idea which should direct our +efforts at reconstruction. I have already summed it up in a phrase, +namely, we must aim at the elimination of reconditeness from the +educational use of the subject. + +Our courses of instruction should be planned to illustrate simply a +succession of ideas of obvious importance. All pretty divagations +should be rigorously excluded. The goal to be aimed at is that the +pupil should acquire familiarity with abstract thought, should realise +how it applies to particular concrete circumstances, and should know +how to apply general methods to its logical investigation. With this +educational ideal nothing can be worse than the aimless accretion +of theorems in our text-books, which acquire their position merely +because the children can be made to learn them and examiners can set +neat questions on them. The bookwork to be learnt should all be very +important as illustrating ideas. The examples set--and let there +be as many examples as teachers find necessary--should be direct +illustrations of the theorems, either by way of abstract particular +cases or by way of application to concrete phenomena. Here it is worth +remarking that it is quite useless to simplify the bookwork, if the +examples set in examinations in fact require an extended knowledge of +recondite details. There is a mistaken idea that problems test ability +and genius, and that bookwork tests cram. This is not my experience. +Only boys who have been specially crammed for scholarships can ever +do a problem paper successfully. Bookwork properly set, not in mere +snippets according to the usual bad plan, is a far better test of +ability, provided that it is supplemented by direct examples. But this +is a digression on the bad influence of examinations on teaching. + +The main ideas which lie at the base of mathematics are not at all +recondite. They are abstract. But one of the main objects of the +inclusion of mathematics in a liberal education is to train the pupils +to handle abstract ideas. The science constitutes the first large +group of abstract ideas which naturally occur to the mind in any +precise form. For the purposes of education, mathematics consists of +the relations of number, the relations of quantity, and the relations +of space. This is not a general definition of mathematics, which, in +my opinion, is a much more general science. But we are now discussing +the use of mathematics in education. These three groups of relations, +concerning number, quantity, and space, are interconnected. + +Now, in education we proceed from the particular to the general. +Accordingly, children should be taught the use of these ideas by +practice among simple examples. My point is this: The goal should be, +not an aimless accumulation of special mathematical theorems, but the +final recognition that the preceding years of work have illustrated +those relations of number, and of quantity, and of space, which are of +fundamental importance. Such a training should lie at the base of all +philosophical thought. In fact elementary mathematics rightly conceived +would give just that philosophical discipline of which the ordinary +mind is capable. But what at all costs we ought to avoid, is the +pointless accumulation of details. As many examples as you like; let +the children work at them for terms, or for years. But these examples +should be direct illustrations of the main ideas. In this way, and +this only, can the fatal reconditeness be avoided. + +I am not now speaking in particular of those who are to be professional +mathematicians, or of those who for professional reasons require a +knowledge of certain mathematical details. We are considering the +liberal education of all students, including these two classes. This +general use of mathematics should be the simple study of a few general +truths, well illustrated by practical examples. This study should +be conceived by itself, and completely separated in idea from the +professional study mentioned above, for which it would make a most +excellent preparation. Its final stage should be the recognition of +the general truths which the work done has illustrated. As far as I +can make out, at present the final stage is the proof of some property +of circles connected with triangles. Such properties are immensely +interesting to mathematicians. But are they not rather recondite, +and what is the precise relation of such theorems to the ideal of +a liberal education? The end of all the grammatical studies of the +student in classics is to read Virgil and Horace--the greatest thoughts +of the greatest men. Are we content, when pleading for the adequate +representation in education of our own science, to say that the end of +a mathematical training is that the student should know the properties +of the nine-point circle? I ask you frankly, is it not rather a "come +down"? + +This generation of mathematical teachers has done so much strenuous +work in the way of reorganising mathematical instruction that there is +no need to despair of its being able to elaborate a curriculum which +shall leave in the minds of the pupils something even nobler than "the +ambiguous case." + +Let us think how this final review, closing the elementary course, +might be conducted for the more intelligent pupils. Partly no doubt +it requires a general oversight of the whole work done, considered +without undue detail so as to emphasise the general ideas used, and +their possibilities of importance when subjected to further study. Also +the analytical and geometrical ideas find immediate application in the +physical laboratory where a course of simple experimental mechanics +should have been worked through. Here the point of view is twofold, the +physical ideas and the mathematical ideas illustrate each other. + +The mathematical ideas are essential to the precise formulation of the +mechanical laws. The idea of a precise law of nature, the extent to +which such laws are in fact verified in our experience, and the rôle of +abstract thought in their formulation, then become practically apparent +to the pupil. The whole topic of course requires detailed development +with full particular illustration, and is not suggested as requiring +merely a few bare abstract statements. + +It would, however, be a grave error to put too much emphasis on the +mere process of direct explanation of the previous work by way of +final review. My point is, that the latter end of the course should +be so selected that in fact the general ideas underlying all the +previous mathematical work should be brought into prominence. This may +well be done by apparently entering on a new subject. For example, +the ideas of quantity and the ideas of number are fundamental to +all precise thought. In the previous stages they will not have been +sharply separated; and children are, rightly enough, pushed on to +algebra without too much bother and quantity. But the more intelligent +among them at the end of their curriculum would gain immensely by a +careful consideration of those fundamental properties of quantity in +general which lead to the introduction of numerical measurement. This +is a topic which also has the advantage that the necessary books are +actually to hand. Euclid's fifth book is regarded by those qualified +to judge as one of the triumphs of Greek mathematics. It deals with +this very point. Nothing can be more characteristic of the hopelessly +illiberal character of the traditional mathematical education than the +fact that this book has always been omitted. It deals with ideas, +and therefore was ostracised. Of course a careful selection of the +more important propositions and a careful revision of the argument are +required. This also is to hand in the publications of my immediate +predecessor in the office of president, Prof. Hill. Furthermore, in Sir +T. L. Heath's complete edition of Euclid, there is a full commentary +embodying most of what has been said and thought on the point. Thus +it is perfectly easy for teachers to inform themselves generally +on the topic. The whole book would not be wanted, but just the few +propositions which embody the fundamental ideas. The subject is not +fit for backward pupils; but certainly it could be made interesting to +the more advanced class. There would be great scope for interesting +discussion as to the nature of quantity, and the tests which we should +apply to ascertain when we are dealing with quantities. The work +would not be at all in the air, but would be illustrated at every +stage by reference to actual examples of cases where the quantitative +character is absent, or obscure, or doubtful, or evident. Temperature, +heat, electricity, pleasure and pain, mass and distance could all be +considered. + +Another idea which requires illustration is that of functionality. +A function in analysis is the counterpart of a law in the physical +universe, and of a curve in geometry. Children have studied the +relations of functions to curves from the first beginning of their +study of algebra, namely in drawing graphs. Of recent years there has +been a great reform in respect to graphs. But at its present stage it +has either gone too far or not far enough. It is not enough merely +to draw a graph. The idea behind the graph--like the man behind the +gun--is essential in order to make it effective. At present there is +some tendency merely to set the children to draw curves, and there to +leave the whole question. + +In the study of simple algebraic functions and of trigonometrical +functions we are initiating the study of the precise expression of +physical laws. Curves are another way of representing these laws. The +simple fundamental laws--such as the inverse square and the direct +distance--should be passed under review, and the applications of the +simple functions to express important concrete cases of physical laws +considered. I cannot help thinking that the final review of this topic +might well take the form of a study of some of the main ideas of the +differential calculus applied to simple curves. There is nothing +particularly difficult about the conception of a rate of change; and +the differentiation of a few powers of _x_, such as _x_^2, +_x_^3, etc., could easily be effected; perhaps by the aid of +geometry even sin _x_ and cos _x_ could be differentiated. +If we once abandon our fatal habit of cramming the children with +theorems which they do not understand, and will never use, there will +be plenty of time to concentrate their attention on really important +topics. We can give them familiarity with conceptions which really +influence thought. + +Before leaving this topic of physical laws and mathematical functions, +there are other points to be noticed. The fact that the precise law is +never really verified by observation in its full precision is capable +of easy illustration and of affording excellent examples. Again, +statistical laws, namely laws which are only satisfied on the average +by large numbers, can easily be studied and illustrated. In fact a +slight study of statistical methods and their application to social +phenomena affords one of the simplest examples of the application of +algebraic ideas. + +Another way in which the students' ideas can be generalised is by the +use of the History of Mathematics, conceived not as a mere assemblage +of the dates and names of men, but as an exposition of the general +current of thought which occasioned the subjects to be objects of +interest at the time of their first elaboration. The use of the History +of Mathematics is to be considered at a later stage of our proceedings +this afternoon. Accordingly I merely draw attention to it now, to +point out that perhaps it is the very subject which may best obtain the +results for which I am pleading. + +We have indicated two main topics, namely general ideas of quantity +and of laws of nature, which should be an object of study in the +mathematical curriculum of a liberal education. But there is another +side to mathematics which must not be overlooked. It is the chief +instrument for discipline in logical method. + +Now, what is logical method, and how can any one be trained in it? + +Logical method is more than the mere knowledge of valid types of +reasoning and practice in the concentration of mind necessary to +follow them. If it were only this, it would still be very important; +for the human mind was not evolved in the bygone ages for the sake of +reasoning, but merely to enable mankind with more art to hunt between +meals for fresh food supplies. Accordingly few people can follow close +reasoning without considerable practice. + +More than this is wanted to make a good reasoner, or even to enlighten +ordinary people with knowledge of what constitutes the essence of the +art. The art of reasoning consists in getting hold of the subject at +the right end, of seizing on the few general ideas which illuminate +the whole, and of persistently marshalling all subsidiary facts round +them. Nobody can be a good reasoner unless by constant practice he +has realised the importance of getting hold of the big ideas and of +hanging on to them like grim death. For this sort of training geometry +is, I think, better than algebra. The field of thought of algebra is +rather obscure, whereas space is an obvious insistent thing evident to +all. Then the process of simplification, or abstraction, by which all +irrelevant properties of matter, such as colour, taste, and weight, are +put aside is an education in itself. Again, the definitions and the +propositions assumed without proof illustrate the necessity of forming +clear notions of the fundamental facts of the subject-matter and of the +relations between them. All this belongs to the mere prolegomena of the +subject. When we come to its development, its excellence increases. The +learner is not initially confronted with any symbolism which bothers +the memory by its rules, however simple they may be. Also, from the +very beginning the reasoning, if properly conducted, is dominated by +well-marked ideas which guide each stage of development. Accordingly +the essence of logical method receives immediate exemplification. + +Let us now put aside for the moment the limitations introduced by +the dullness of average pupils and the pressure on time due to other +subjects, and consider what geometry has to offer in the way of a +liberal education. I will indicate some stages in the subject, +without meaning that necessarily they are to be studied in this +exclusive order. The first stage is the study of _congruence_. +Our perception of congruence is in practice dependent on our judgments +of the invariability of the intrinsic properties of bodies when their +external circumstances are varying. But however it arises, congruence +is in essence the correlation of two regions of space, point by point, +so that all homologous distances and all homologous angles are equal. +It is to be noticed that the definition of the equality of lengths and +angles is their congruence, and all tests of equality, such as the use +of the yard measure, are merely devices for making immediate judgments +of congruence easy. I make these remarks to suggest that apart from the +reasoning connected with it, congruence, both as an example of a larger +and very far-reaching idea and also for its own sake, is well worthy +of attentive consideration. The propositions concerning it elucidate +the elementary properties of the triangle, the parallelogram, and the +circle, and of the relations of two planes to each other. It is very +desirable to restrict the proved propositions of this part within the +narrowest bounds, partly by assuming redundant axiomatic propositions, +and partly by introducing only those propositions of absolutely +fundamental importance. + +The second stage is the study of similarity. This can be reduced to +three or four fundamental propositions. Similarity is an enlargement of +the idea of congruence, and, like that idea, is another example of a +one-to-one correlation of points of spaces. Any extension of study of +this subject might well be in the direction of the investigation of one +or two simple properties of similar and similarly situated rectilinear +figures. The whole subject receives its immediate applications in plans +and maps. It is important, however, to remember that trigonometry is +really the method by which the main theorems are made available for use. + +The third stage is the study of the elements of trigonometry. This is +the study of the periodicity introduced by rotation and of properties +preserved in a correlation of similar figures. Here for the first time +we introduce a slight use of the algebraic analysis founded on the +study of number and quantity. The importance of the periodic character +of the functions requires full illustration. The simplest properties of +the functions are the only ones required for the solution of triangles, +and the consequent applications to surveying. The wealth of formulæ, +often important in themselves, but entirely useless for this type of +study, which crowd our books should be rigorously excluded, except +so far as they are capable of being proved by the pupils as direct +examples of the bookwork. + +This question of the exclusion of formulæ is best illustrated by +considering this example of Trigonometry, though of course I may well +have hit on an unfortunate case in which my judgment is at fault. A +great part of the educational advantage of the subject can be obtained +by confining study to Trigonometry of one angle and by exclusion of the +addition formulæ for the sine and cosine of the sum of two angles. The +functions can be graphed, and the solution of triangles effected. Thus +the aspects of the science as (1) embodying analytically the immediate +results of some of the theorems deduced from congruence and similarity, +(2) as a solution of the main problem of surveying, (3) as a study of +the fundamental functions required to express periodicity and wave +motion, will all be impressed on the pupils' minds both by bookwork and +example. + +If it be desired to extend this course, the addition formulæ should +be added. But great care should be taken to exclude specialising +the pupils in the wealth of formulæ which comes in their train. By +"exclude" is meant that the pupils should not have spent time or energy +in acquiring any facility in their deduction. The teacher may find +it interesting to work a few such examples before a class. But such +results are not among those which learners need retain. Also, I would +exclude the whole subject of circumscribed and inscribed circles both +from Trigonometry and from the previous geometrical courses. It is +all very pretty, but I do not understand what its function is in an +elementary non-professional curriculum. + +Accordingly, the actual bookwork of the subject is reduced to very +manageable proportions. I was told the other day of an American college +where the students are expected to know by heart ninety formulæ or +results in Trigonometry alone. We are not quite so bad as that. In +fact, in Trigonometry we have nearly approached the ideal here sketched +out as far as our elementary courses are concerned. + +The fourth stage introduces Analytical Geometry. The study of graphs in +algebra has already employed the fundamental notions, and all that is +now required is a rigorously pruned course on the straight line, the +circle, and the three types of conic sections, defined by the forms of +their equations. At this point there are two remarks to be made. It is +often desirable to give our pupils mathematical information which we do +not prove. For example, in co-ordinate geometry, the reduction of the +general equation of the second degree is probably beyond the capacities +of most of the type of students whom we are considering. But that need +not prevent us from explaining the fundamental position of conics, as +exhausting the possible types of such curves. + +The second remark is to advocate the entire sweeping away of +geometrical conics as a separate subject. Naturally, on suitable +occasions the analysis of analytical geometry will be lightened by +the use of direct deduction from some simple figure. But geometrical +conics, as developed from the definition of a conic section by the +focus and directrix property, suffers from glaring defects. It is +hopelessly recondite. The fundamental definition of a conic, _SP_ = _e_ +· _PM_, usual in this subject at this stage, is thoroughly bad. It is +very recondite, and has no obvious importance. Why should such curves +be studied at all, any more than those defined by an indefinite number +of other formulæ? But when we have commenced the study of the Cartesian +methods, the equations of the first and second degrees are naturally +the first things to think about. + +In this ideal course of Geometry, the fifth stage is occupied with +the elements of Projective Geometry. The general ideas of cross ratio +and of projection are here fundamental. Projection is yet a more +general instance of that one-to-one correlation which we have already +considered under congruence and similarity. Here again we must avoid +the danger of being led into a bewildering wealth of detail. + +The intellectual idea which projective geometry is to illustrate is +the importance in reasoning of the correlation of all cases which +can be proved to possess in common certain identical properties. The +preservation of the projective properties in projection is the one +important educational idea of the subject. Cross ratio only enters +as the fundamental metrical property which is preserved. The few +propositions considered are selected to illustrate the two allied +processes which are made possible by this procedure. One is proof +by simplification. Here the simplification is psychological and not +logical--for the general case is logically the simplest. What is meant +is: Proof by considering the case which is in fact the most familiar to +us, or the easiest to think about. The other procedure is the deduction +of particular cases from known general truths, as soon as we have a +means of discovering such cases or a criterion for testing them. + +The projective definition of conic sections and the identity of the +results obtained with the curves derived from the general equation +of the second degree are capable of simple exposition, but lie on +the border-line of the subject. It is the sort of topic on which +information can be given, and the proofs suppressed. + +The course of geometry as here conceived in its complete ideal--and +ideals can never be realised--is not a long one. The actual amount +of mathematical deduction at each stage in the form of bookwork is +very slight. But much more explanation would be given, the importance +of each proposition being illustrated by examples, either worked out +or for students to work, so selected as to indicate the fields of +thought to which it applies. By such a course the student would gain an +analysis of the leading properties of space, and of the chief methods +by which they are investigated. + +The study of the elements of mathematics, conceived in this spirit, +would constitute a training in logical method together with an +acquisition of the precise ideas which lie at the base of the +scientific and philosophical investigations of the universe. Would it +be easy to continue the excellent reforms in mathematical instruction +which this generation has already achieved, so as to include in the +curriculum this wider and more philosophic spirit? Frankly, I think +that this result would be very hard to achieve as the result of +single individual efforts. For reasons which I have already briefly +indicated, all reforms in education are very difficult to effect. But +the continued pressure of combined effort, provided that the ideal is +really present in the minds of the mass of teachers, can do much, and +effects in the end surprising modification. Gradually the requisite +books get written, still more gradually the examinations are reformed +so as to give weight to the less technical aspects of the subject, and +then all recent experience has shown that the majority of teachers are +only too ready to welcome any practicable means of rescuing the subject +from the reproach of being a mechanical discipline. + + + + + CHAPTER V + + THE PRINCIPLES OF MATHEMATICS IN RELATION TO ELEMENTARY TEACHING + + (_International Congress of Mathematicians, Cambridge, August, + 1912_) + + +WE are concerned not with the advanced teaching of a few specialist +mathematical students, but with the mathematical education of the +majority of boys in our secondary schools. Again these boys must be +divided into two sections: one section consists of those who desire to +restrict their mathematical education; the other section is made up of +those who will require some mathematical training for their subsequent +professional careers, either in the form of definite mathematical +results or in the form of mathematically trained minds. + +I shall call the latter of these two sections the mathematical section, +and the former the non-mathematical section. But I must repeat that by +the mathematical section is meant that large number of boys who desire +to learn more than the minimum amount of mathematics. Furthermore, most +of my remarks about these sections of boys apply also to elementary +classes of our University students. + +The subject of this paper is the investigation of the place which +should be occupied by modern investigations respecting mathematical +principles in the education of both of these sections of school-boys. + +To find a foothold from which to start the inquiry, let us ask why the +non-mathematical section should be taught any mathematics at all beyond +the barest elements of arithmetic. What are the qualities of mind which +a mathematical training is designed to produce when it is employed as +an element in a liberal education? + +My answer, which applies equally to both sections of students, is +that there are two allied forms of mental discipline which should be +acquired by a well-designed mathematical course. These two forms though +closely allied are perfectly distinct. + +The first form of discipline is not in its essence logical at all. +It is the power of clearly grasping abstract ideas, and of relating +them to particular circumstances. In other words, the first use of +mathematics is to strengthen the power of abstract thought. I repeat +that in its essence this has nothing to do with logic, though as a +matter of fact a logical discipline is the best method of producing the +desired effect. It is not the philosophical theory of abstract ideas +which is to be acquired, but the habit and the power of using them. +There is one and only way of acquiring the habit and the power of using +anything, that is by the simple common-place method of habitually using +it. There is no other short cut. If in education we desire to produce +a certain conformation of mind, we must day by day, and year by year, +accustom the students' minds to grow into the desired structural shape. +Thus to teach the power of grasping abstract ideas and the habit of +using them, we must select a group of such ideas, which are important +and are also easy to think about because they are clear and definite. + +The fundamental mathematical truths concerning geometry, ratio, +quantity, and number, satisfy these conditions as do no others. Hence, +the fundamental universal position held by mathematics as an element of +a liberal education. + +But what are the fundamental mathematical truths concerning geometry, +quantity and number? + +At this point we come to the great question of the relation between +the modern science of the principles of mathematics and a mathematical +education. + +My answer to the question as to these fundamental mathematical truths +is, that in any absolute sense there are none. There is no unique +small body of independent primitive unproved propositions which are +the necessary starting points of all mathematical reasoning on these +subjects. In mathematical reasoning the only absolute necessary +pre-suppositions are those which make logical deduction possible. +Between these absolute logical truths and so-called fundamental truths +concerning geometry, quantity and number, there is a whole new world of +mathematical subjects concerning the logic of propositions, of classes, +and of relations. + +But this subject is too abstract to form an elementary training ground +in the difficult art of abstract thought. + +It is for this reason that we have to make a compromise and start from +such obvious general ideas as naturally occur to all men when they +perceive objects with their senses. + +In geometry, the ideas elaborated by the Greeks and presented by Euclid +are, roughly speaking, those adapted for our purpose, namely, ideas +of volumes, surfaces, lines, of straightness and of curvature, of +intersection and of congruence, of greater and less, of similarity, +shape, and scale. In fact, we use in education those general ideas of +spatial properties which must be habitually present in the mind of any +one who is to observe the world of phenomena with understanding. + +Thus we come back to Plato's opinion that for a liberal education, +geometry, as he knew it, is the queen of sciences. + +In addition to geometry, there remain the ideas of quantity, ratio, +and of number. This in practice means, elementary algebra. Here the +prominent ideas are those of "any number," in other words, the use +of the familiar _x_, _y_, _z_, and of the dependence of variables on +each other, or otherwise, the idea of functionality. All this is to be +gradually acquired by the continual use of the very simplest functions +which we can devise: of linear functions, graphically represented +by straight lines; of quadratic functions, graphically represented +by parabolas; and of those simple implicit functions, graphically +represented by conic sections. Thence, with good fortune and a willing +class, we can advance to the ideas of rates of increase, still +confining ourselves to the simplest possible cases. + +I wish here emphatically to remind you that both in geometry and in +algebra a clear grasp of these general ideas is not what the pupil +starts from, it is the goal at which he is to arrive. The method of +progression is continual practice in the consideration of the simplest +particular cases, and the goal is not philosophical analysis but the +power of use. + +But how is he to practise himself in their use? He cannot simply sit +down and think of the relation _y_ = _x_ + 1, he must employ +it in some simple obvious way. + +This brings us to the second power of mind which is to be produced by +a mathematical training, namely, the power of logical reasoning. Here +again, it is not the knowledge of the philosophy of logic which it is +essential to teach, but the habit of thinking logically. By logic, I +mean deductive logic. + +Deductive logic is the science of certain relations, such as +implication, etc., between general ideas. When logic begins, definite +particular individual things have been banished. I cannot relate +logically this thing to that thing, for example this pen to that pen, +except by the indirect way of relating some general idea which applies +to this pen to some general idea which applies to that pen. And the +individualities of the two pens are quite irrelevant to the logical +process. This process is entirely concerned with the two general ideas. +Thus the practice of logic is a certain way of employing the mind +in the consideration of such ideas; and an elementary mathematical +training is in fact nothing else but the logical use of the general +ideas respecting geometry and algebra which we have enumerated above. +It has therefore, as I started this paper by stating, a double +advantage. It makes the mind capable of abstract thought, and it +achieves this object by training the mind in the most important kind of +abstract thought, namely, deductive logic. + +I may remind you that other choices of a type of abstract thought +might be made. We might train the children to contemplate directly the +beauty of abstract moral ideas, in the hope of making them religious +mystics. The general practice of education has decided in favour of +logic, as exemplified in elementary mathematics. + +We have now to answer the further question, what is the rôle of +logical precision in the teaching of mathematics? Our general answer +to the implied question is obvious: logical precision is one of the +two objects of the teaching of mathematics, and it is the only weapon +by which the teaching of mathematics can achieve its other object. To +teach mathematics is to teach logical precision. A mathematical teacher +who has not taught that has taught nothing. + +But having stated this thesis in this unqualified way, its meaning must +be carefully explained; for otherwise its real bearing on the problem +will be entirely misunderstood. + +Logical precision is the faculty to be acquired. It is the quality of +mind which it is the object of the teaching to impart. Thus the habit +of reading great literature is the goal at which a literary education +aims. But we do not expect a child to start its first lesson by reading +for itself Shakespeare. We recognise that reading is impossible till +the pupil has learnt its alphabet and can spell, and then we start it +with books of one syllable. + +In the same way, a mathematical education should grow in logical +precision. It is folly to expect the same careful logical analysis at +the commencement of the training as would be appropriate at the end. +It is an entire misconception of my thesis to construe it as meaning +that a mathematical training should assume in the pupil a power of +concentrated logical thought. My thesis is in fact the exact opposite, +namely, that this power cannot be assumed, and has got to be acquired, +and that a mathematical training is nothing else than the process of +acquiring it. My whole groundwork of assumption is that this power does +not initially exist in a fully developed state. Of course like every +other power which is acquired, it must be developed gradually. + +The various stages of development must be guided by the judgment and +the genius of the teacher. But what is essential is, that the teacher +should keep clearly in his mind that it is just this power of logical +precise reasoning which is the whole object of his efforts. If his +pupils have in any measure gained this, they have gained all. + +We have not yet, however, fully considered this part of our subject. +Logical precision is the full realisation of the steps of the argument. +But what are the steps of the argument? The full statement of all the +steps is far too elaborate and difficult an operation to be introduced +into the mathematical reasoning of an educational curriculum. Such a +statement involves the introduction of abstract logical ideas which +are very difficult to grasp, because there is so rarely any need to +make them explicit in ordinary thought. They are therefore not a fit +subject-ground for an elementary education. + +I do not think that it is possible to draw any theoretical line +between those logical steps which form a theoretically full logical +investigation, and those which are full enough for most practical +purposes, including that of education. The question is one of +psychology, to be solved by a process of experiment. The object to be +attained is to gain that amount of logical alertness which will enable +its possessors to detect fallacy and to know the types of sound logical +deduction. The objects of going further are partly philosophical, and +also partly to lay bare abstract ideas whose investigation is in itself +important. But both these objects are foreign to education. + +My opinion is, that, on the whole, the type of logical precision handed +down to us by the Greek mathematicians is, roughly speaking, what we +want. In geometry, this means the sort of precision which we find in +Euclid. I do not mean that we should use his famous _Elements_ as +a text-book, nor that here and there a certain compression in his mode +of exposition is not advisable. All this is mere detail. What I do mean +is, that the sort of logical transition which he made explicit, we +should make explicit, and that the sort of transition which he omits, +we should omit. + +I doubt, however, whether it is desirable to plunge the student into +the full rigour of euclidean geometry without some mitigation. It is +for this reason that the modern habit, at least in England, of laying +great stress in the initial stages on training the pupil in simple +constructions from numerical data is to be praised. It means that after +a slight amount of reasoning on the euclidean basis of accuracy, the +mind of the learner is relieved by doing the things in various special +cases, and noting by rough measurements that the desired results are +actually attained. It is important, however, that the measurements +be not mistaken for the proofs. Their object is to make the beginner +apprehend what the abstract ideas really mean. + +Again in algebra, the notation and the practical use of the symbols +should be acquired in the simplest cases, and the more theoretical +treatment of the symbolism reserved to a suitable later stage. My +rule would be initially to learn the meaning of the ideas by a crude +practice in simple ways, and to refine the logical procedure in +preparation for an advance to greater generality. In fact the thesis of +my paper can be put in another way thus, the object of a mathematical +education is, to acquire the powers of analysis, of generalisation, +and of reasoning. The two processes of analysis and generalisation were +in my previous statement put together as the power of grasping abstract +ideas. + +But in order to analyse and to generalise, we must commence with the +crude material of ideas which are to be analysed and generalised. +Accordingly it is a positive error in education to start with the +ultimate products of this process, namely the ideas in their refined +analysed and generalised forms. We are thereby skipping an important +part of the training, which is to take the ideas as they actually exist +in the child's mind, and to exercise the child in the difficult art of +civilising them and clothing them. + +The schoolmaster is in fact a missionary, the savages are the ideas in +the child's mind; and the missionary shirks his main task if he refuse +to risk his body among the cannibals. + +At this point I should like to turn your attention to those pupils +forming the mathematical section. There is an idea, widely prevalent, +that it is possible to teach mathematics of a relatively advanced +type--such as differential calculus, for instance--in a way useful to +physicists and engineers without any attention to its logic or its +theory. + +This seems to me to be a profound mistake. It implies that a merely +mechanical knowledge without understanding of ways of arriving at +mathematical results is useful in applied science. It is of no use +whatever. The results themselves can all be found stated in the +appropriate pocket books and in other elementary works of reference. No +one when applying a result need bother himself as to why it is true. He +accepts it and applies it. What is of supreme importance in physics and +in engineering is a mathematically trained mind, and such a mind can +only be acquired by a proper mathematical discipline. + +I fully admit that the proper way to start such a subject as the +differential calculus is to plunge quickly into the use of the notation +in a few absurdly simple cases, with a crude explanation of the idea +of rates of increase. The notation as thus known can then be used by +the lecturers in the Physical and Engineering Laboratories. But the +mathematical training of the applied scientists consists in making +these ideas precise and the proofs accurate. + +I hope that the thesis of this paper respecting the position of logical +precision in the teaching of mathematics has been rendered plain. The +habit of logical precision with its necessary concentration of thought +upon abstract ideas is not wholly possible in the initial stages of +learning. It is the ideal at which the teacher should aim. Also logical +precision, in the sense of logical explicitness, is not an absolute +thing: it is an affair of more or less. Accordingly the quantity of +explicitness to be introduced at each stage of progress must depend +upon the practical judgment of the teacher. Lastly, in a sense, the +instructed mind is less explicit; for it travels more quickly over a +well-remembered path, and may save the trouble of putting into words +trains of thought which are very obvious to it. But on the other +hand it atones for this rapidity by a concentration on every subtle +point where a fallacy can lurk. The habit of logical precision is the +instinct for the subtle difficulty. + + + + + CHAPTER VI + + THE ORGANISATION OF THOUGHT + + (_Presidential Address to Section A, British Association, Newcastle, + 1916_) + + +THE subject of this address is the organisation of thought, a topic +evidently capable of many diverse modes of treatment. I intend more +particularly to give some account of that department of logical science +with which some of my own studies have been connected. But I am +anxious, if I can succeed in so doing, to handle this account so as to +exhibit the relation with certain considerations which underlie general +scientific activities. + +It is no accident that an age of science has developed into an age +of organisation. Organised thought is the basis of organised action. +Organisation is the adjustment of diverse elements so that their +mutual relations may exhibit some predetermined quality. An epic poem +is a triumph of organisation, that is to say, it is a triumph in the +unlikely event of its being a good epic poem. It is the successful +organisation of multitudinous sounds of words, associations of words, +pictorial memories of diverse events and feelings ordinarily occurring +in life, combined with a special narrative of great events: the whole +so disposed as to excite emotions which, as defined by Milton, are +simple, sensuous, and passionate. The number of successful epic poems +is commensurate, or rather, is inversely commensurate, with the obvious +difficulty of the task of organisation. + +Science is the organisation of thought. But the example of the epic +poem warns us that science is not any organisation of thought. It is +an organisation of a certain definite type which we will endeavour to +determine. + +Science is a river with two sources, the practical source and the +theoretical source. The practical source is the desire to direct our +actions to achieve predetermined ends. For example, the British nation, +fighting for justice, turns to science, which teaches it the importance +of compounds of nitrogen. The theoretical source is the desire to +understand. Now I am going to emphasise the importance of theory in +science. But to avoid misconception I most emphatically state that I +do not consider one source as in any sense nobler than the other, or +intrinsically more interesting. I cannot see why it is nobler to strive +to understand than to busy oneself with the right ordering of one's +actions. Both have their bad sides; there are evil ends directing +actions, and there are ignoble curiosities of the understanding. + +The importance, even in practice, of the theoretical side of science +arises from the fact that action must be immediate, and takes place +under circumstances which are excessively complicated. If we wait for +the necessities of action before we commence to arrange our ideas, in +peace we shall have lost our trade, and in war we shall have lost the +battle. Success in practice depends on theorists who, led by other +motives of exploration, have been there before, and by some good chance +have hit upon the relevant ideas. By a theorist I do not mean a man who +is up in the clouds, but a man whose motive for thought is the desire +to formulate correctly the rules according to which events occur. A +successful theorist should be excessively interested in immediate +events, otherwise he is not at all likely to formulate correctly +anything about them. Of course, both sources of science exist in all +men. + +Now, what is this thought organisation which we call science? The +first aspect of modern science which struck thoughtful observers was +its inductive character. The nature of induction, its importance, and +the rules of inductive logic have been considered by a long series of +thinkers, especially English thinkers: Bacon, Herschel, J. S. Mill, +Venn, Jevons, and others. I am not going to plunge into an analysis +of the process of induction. Induction is the machinery and not the +product, and it is the product which I want to consider. When we +understand the product we shall be in a stronger position to improve +the machinery. + +First, there is one point which it is necessary to emphasise. There +is a tendency in analysing scientific processes to assume a given +assemblage of concepts applying to nature, and to imagine that the +discovery of laws of nature consists in selecting by means of inductive +logic some one out of a definite set of possible alternative relations +which may hold between the things in nature answering to these obvious +concepts. In a sense this assumption is fairly correct, especially +in regard to the earlier stages of science. Mankind found itself in +possession of certain concepts respecting nature--for example, the +concept of fairly permanent material bodies--and proceeded to determine +laws which related the corresponding percepts in nature. But the +formulation of laws changed the concepts, sometimes gently by an added +precision, sometimes violently. At first this process was not much +noticed, or at least was felt to be a process curbed within narrow +bounds, not touching fundamental ideas. At the stage where we now +are, the formulation of the concepts can be seen to be as important +as the formulation of the empirical laws connecting the events in the +universe as thus conceived by us. For example, the concepts of life, +of heredity, of a material body, of a molecule, of an atom, of an +electron, of energy, of space, of time, of quantity, and of number. I +am not dogmatising about the best way of getting such ideas straight. +Certainly it will only be done by those who have devoted themselves to +a special study of the facts in question. Success is never absolute, +and progress in the right direction is the result of a slow, gradual +process of continual comparison of ideas with facts. The criterion of +success is that we should be able to formulate empirical laws, that is, +statements of relations, connecting the various parts of the universe +as thus conceived, laws with the property that we can interpret the +actual events of our lives as being our fragmentary knowledge of this +conceived interrelated whole. + +But, for the purpose of science, what is the actual world? Has science +to wait for the termination of the metaphysical debate till it can +determine its own subject-matter? I suggest that science has a much +more homely starting-ground. Its task is the discovery of the relations +which exist within that flux of perceptions, sensations, and emotions +which forms our experience of life. The panorama yielded by sight, +sound, taste, smell, touch, and by more inchoate sensible feelings, +is the sole field of activity. It is in this way that science is the +thought organisation of experience. The most obvious aspect of this +field of actual experience is its disorderly character. It is for +each person a _continuum_, fragmentary, and with elements not +clearly differentiated. The comparison of the sensible experiences of +diverse people brings its own difficulties. I insist on the radically +untidy, ill-adjusted character of the fields of actual experience from +which science starts. To grasp this fundamental truth is the first +step in wisdom, when constructing a philosophy of science. This fact +is concealed by the influence of language, moulded by science, which +foists on us exact concepts as though they represented the immediate +deliverances of experience. The result is, that we imagine that we have +immediate experience of a world of perfectly defined objects implicated +in perfectly defined events which, as known to us by the direct +deliverance of our senses, happen at exact instants of time, in a space +formed by exact points, without parts and without magnitude: the neat, +trim, tidy, exact world which is the goal of scientific thought. + +My contention is, that this world is a world of ideas, and that its +internal relations are relations between abstract concepts, and that +the elucidation of the precise connection between this world and the +feelings of actual experience is the fundamental question of scientific +philosophy. The question which I am inviting you to consider is this: +How does exact thought apply to the fragmentary, vague _continua_ +of experience? I am not saying that it does not apply: quite the +contrary. But I want to know how it applies. The solution I am +asking for is not a phrase, however brilliant, but a solid branch of +science, constructed with slow patience, showing in detail how the +correspondence is effected. + +The first great steps in the organisation of thought were due +exclusively to the practical source of scientific activity, without +any admixture of theoretical impulse. Their slow accomplishment was +the cause and also the effect of the gradual evolution of moderately +rational beings. I mean the formation of the concepts of definite +material objects, of the determinate lapse of time, of simultaneity, +of recurrence, of definite relative position, and of analogous +fundamental ideas, according to which the flux of our experience is +mentally arranged for handy reference: in fact, the whole apparatus of +commonsense thought. Consider in your mind some definite chair. The +concept of that chair is simply the concept of all the interrelated +experiences connected with that chair--namely, of the experience of the +folk who made it, of the folk who sold it, of the folk who have seen +it or used it, of the man who is now experiencing a comfortable sense +of support, combined with our expectations of an analogous future, +terminated finally by a different set of experiences when the chair +collapses and becomes firewood. The formation of that type of concept +was a tremendous job, and zoologists and geologists tell us that it +took many tens of millions of years. I can well believe it. + +I now emphasise two points. In the first place, science is rooted in +what I have just called the whole apparatus of commonsense thought. +That is the _datum_ from which it starts, and to which it +must recur. We may speculate, if it amuses us, of other beings in +other planets who have arranged analogous experiences according to +an entirely different conceptual code--namely, who have directed +their chief attention to different relations between their various +experiences. But the task is too complex, too gigantic, to be revised +in its main outlines. You may polish up commonsense, you may contradict +it in detail, you may surprise it. But ultimately your whole task is to +satisfy it. + +In the second place, neither commonsense nor science can proceed with +their task of thought organisation without departing in some respect +from the strict consideration of what is actual in experience. Think +again of the chair. Among the experiences upon which its concept is +based, I included our expectations of its future history. I should +have gone further and included our imagination of all the possible +experiences which in ordinary language we should call perceptions of +the chair which might have occurred. This is a difficult question, and +I do not see my way through it. But, at present, in the construction of +a theory of space and of time there seem insuperable difficulties if we +refuse to admit ideal experiences. + +This imaginative perception of experiences, which, if they occurred, +would be coherent with our actual experiences, seems fundamental in our +lives. It is neither wholly arbitrary, nor yet fully determined. It +is a vague background which is only made in part definite by isolated +activities of thought. Consider, for example, our thoughts of the +unseen flora of Brazil. + +Ideal experiences are closely connected with our imaginative +reproduction of the actual experiences of other people, and also +with our almost inevitable conception of ourselves as receiving our +impressions from an external complex reality beyond ourselves. It +may be that an adequate analysis of every source and every type of +experience yields demonstrative proof of such a reality and of its +nature. Indeed, it is hardly to be doubted that this is the case. The +precise elucidation of this question is the problem of metaphysics. One +of the points which I am urging in this address is, that the basis of +science does not depend on the assumption of any of the conclusions of +metaphysics; but that both science and metaphysics start from the same +given groundwork of immediate experience, and in the main proceed in +opposite directions on their diverse tasks. + +For example, metaphysics inquires how our perceptions of the chair +relate us to some true reality. Science gathers up these perceptions +into a determinate class, adds to them ideal perceptions of analogous +sort, which under assignable circumstances would be obtained, and this +single concept of that set of perceptions is all that science needs; +unless indeed you prefer that thought find its origin in some legend of +those great twin brethren, the Cock and Bull. + +My immediate problem is to inquire into the nature of the texture of +science. Science is essentially logical. The nexus between its concepts +is a logical nexus, and the grounds for its detailed assertions +are logical grounds. King James said, "No bishops, no king." With +greater confidence we can say, "No logic, no science." The reason for +the instinctive dislike which most men of science feel towards the +recognition of this truth is, I think, the barren failure of logical +theory during the past three or four centuries. We may trace this +failure back to the worship of authority, which in some respects +increased in the learned world at the time of the Renaissance. Mankind +then changed its authority, and this fact temporally acted as an +emancipation. But the main fact, and we can find complaints[2] of it +at the very commencement of the modern movement, was the establishment +of a reverential attitude towards any statement made by a classical +author. Scholars became commentators on truths too fragile to bear +translation. A science which hesitates to forget its founders is lost. +To this hesitation I ascribe the barrenness of logic. Another reason +for distrust of logical theory and of mathematics is the belief that +deductive reasoning can give you nothing new. Your conclusions are +contained in your premises, which by hypothesis are known to you. + +In the first place this last condemnation of logic neglects the +fragmentary, disconnected character of human knowledge. To know one +premise on Monday, and another premise on Tuesday, is useless to you +on Wednesday. Science is a permanent record of premises, deductions, +and conclusions, verified all along the line by its correspondence +with facts. Secondly, it is untrue that when we know the premises we +also know the conclusions. In arithmetic, for example, mankind are not +calculating boys. Any theory which proves that they are conversant with +the consequences of their assumptions must be wrong. We can imagine +beings who possess such insight. But we are not such creatures. +Both these answers are, I think, true and relevant. But they are +not satisfactory. They are too much in the nature of bludgeons, +too external. We want something more explanatory of the very real +difficulty which the question suggests. In fact, the true answer is +embedded in the discussion of our main problem of the relation of logic +to natural science. + +It will be necessary to sketch in broad outline some relevant features +of modern logic. In doing so I shall try to avoid the profound general +discussions and the minute technical classifications which occupy the +main part of traditional logic. It is characteristic of a science in +its earlier stages--and logic has become fossilised in such a stage--to +be both ambitiously profound in its aims and trivial in its handling of +details. + +We can discern four departments of logical theory. By an analogy +which is not so very remote I will call these departments or sections +the arithmetic section, the algebraic section, the section of +general-function theory, the analytical section. I do not mean that +arithmetic arises in the first section, algebra in the second section, +and so on; but the names are suggestive of certain qualities of +thought in each section which are reminiscent of analogous qualities +in arithmetic, in algebra, in the general theory of a mathematical +function, and in the mathematical analysis of the properties of +particular functions. + +The first section--namely, the arithmetic stage--deals with the +relations of definite propositions to each other, just as arithmetic +deals with definite numbers. Consider any definite proposition; call +it "_p_." We conceive that there is always another proposition which +is the direct contradictory to "_p_"; call it "not-_p_." When we have +got two propositions, _p_ and _q_, we can form derivative propositions +from them, and from their contradictories. We can say, "At last one of +_p_ or _q_ is true, and perhaps both." Let us call this proposition +"_p_ or _q_." I may mention as an aside that one of the greatest living +philosophers has stated that this use of the word "or"--namely, "_p_ or +_q_" in the sense that either or both may be true--makes him despair of +exact expression. We must brave his wrath, which is unintelligible to +me. + +We have thus got hold of four new propositions, namely, "_p_ or _q_," +and "not-_p_ or _q_," and "_p_ or not-_q_," and "not-_p_ or not-_q_." +Call these the set of disjunctive derivatives. There are, so far, +in all eight propositions, _p_, not-_p_, _q_, not-_q_, and the four +disjunctive derivatives. Any pair of these eight propositions can be +taken, and substituted for _p_ and _q_ in the foregoing treatment. +Thus each pair yields eight propositions, some of which may have been +obtained before. By proceeding in this way we arrive at an unending set +of propositions of growing complexity, ultimately derived from the two +original propositions _p_ or _q_. Of course, only a few are important. +Similarly we can start from three propositions, _p_, _q_, _r_, or +from four propositions, _p_, _q_, _r_, _s_, and so on. Any one of the +propositions of these aggregates may be true or false. It has no other +alternative. Whichever it is, true or false, call it the "truth-value" +of the proposition. + +The first section of logical inquiry is to settle what we know of the +truth-values of these propositions, when we know the truth-values of +some of them. The inquiry, so far as it is worth while carrying it, +is not very abstruse, and the best way of expressing its results is a +detail which I will not now consider. This inquiry forms the arithmetic +stage. + +The next section of logic is the algebraic stage. Now, the difference +between arithmetic and algebra is, that in arithmetic definite +numbers are considered, and in algebra symbols--namely, letters--are +introduced which stand for any numbers. The idea of a number is also +enlarged. These letters, standing for any numbers, are called sometimes +variables and sometimes parameters. Their essential characteristic is +that they are undetermined, unless, indeed, the algebraic conditions +which they satisfy implicitly determine them. Then they are sometimes +called unknowns. An algebraic formula with letters is a blank form. It +becomes a determinate arithmetic statement when definite numbers are +substituted for the letters. The importance of algebra is a tribute to +the study of form. Consider now the following proposition-- + + The specific heat of mercury is 0·033. + +This is a definite proposition which, with certain limitations, is +true. But the truth-value of the proposition does not immediately +concern us. Instead of mercury put a mere letter which is the name of +some undetermined thing: we get-- + + The specific heat of _x_ is 0·033. + +This is not a proposition; it has been called by Russell a +propositional function. It is the logical analogy of an algebraic +expression. Let us write ƒ(_x_) for any propositional function. + +We could also generalise still further, and say, + + The specific heat of _x_ is _y_. + +We thus get another propositional function, F(_x_, _y_), +of two arguments _x_ and _y_, and so on for any number of +arguments. + +Now, consider ƒ(_x_). There is the range of values of _x_, +for which ƒ(_x_) is a proposition, true or false. For values of +_x_ outside this range, ƒ(_x_) is not a proposition at all, +and is neither true nor false. It may have vague suggestions for us, +but it has no unit meaning of definite assertion. For example, + + The specific heat of water is 0·033 + +is a proposition which is false; and-- + + The specific heat of virtue is 0·033 + +is, I should imagine, not a proposition at all; so that it is neither +true nor false, though its component parts raise various associations +in our minds. This range of values, for which ƒ(_x_) has sense, is +called the "type" of the argument _x_. + +But there is also a range of values of _x_ for which ƒ(_x_) +is a true proposition. This is the class of those values of the +argument which _satisfy_ ƒ(_x_). This class may have no +members, or, in the other extreme, the class may be the whole type of +the arguments. + +We thus conceive two general propositions respecting the indefinite +number of propositions which share in the same logical form, that is, +which are values of the same propositional function. One of these +propositions is, + + ƒ(_x_) yields a true proposition for each value of _x_ of + the proper type; + +the other proposition is, + + There is a value of _x_ for which ƒ(_x_) is true. + +Given two, or more, propositional functions ƒ(_x_) and ϕ(_x_) +with the same argument _x_, we form derivative propositional +functions, namely, + +ƒ(_x_) or ϕ(_x_), ƒ(_x_) or not-ϕ(_x_), + +and so on with the contradictories, obtaining, as in the arithmetical +stage, an unending aggregate of propositional functions. Also each +propositional function yields two general propositions. The theory +of the interconnection between the truth-values of the general +propositions arising from any such aggregate of propositional functions +forms a simple and elegant chapter of mathematical logic. + +In this algebraic section of logic the theory of types crops up, as we +have already noted. It cannot be neglected without the introduction of +error. Its theory has to be settled at least by some safe hypothesis, +even if it does not go to the philosophic basis of the question. This +part of the subject is obscure and difficult, and has not been finally +elucidated, though Russell's brilliant work has opened out the subject. + +The final impulse to modern logic comes from the independent discovery +of the importance of the logic variable by Frege and Peano. Frege went +further than Peano, but by an unfortunate symbolism rendered his work +so obscure that no one fully recognised his meaning who had not found +it out for himself. But the movement has a large history reaching back +to Leibniz and even to Aristotle. Among English contributors are De +Morgan, Boole, and Sir Alfred Kempe; their work is of the first rank. + +The third logical section is the stage of general-function theory. +In logical language, we perform in this stage the transition from +intension to extension, and investigate the theory of denotation. Take +the propositional function, ƒ(_x_). There is the class, or range +of values for _x_, whose members satisfy ƒ(_x_). But the same +range may be the class whose members satisfy another propositional +function ϕ(_x_). It is necessary to investigate how to indicate +the class by a way which is indifferent as between the various +propositional functions which are satisfied by any member of it, and of +it only. What has to be done is to analyse the nature of propositions +about a class--namely, those propositions whose truth-values depend on +the class itself and not on the particular meaning by which the class +is indicated. + +Furthermore, there are propositions about alleged individuals +indicated by descriptive phrases: for example, propositions about "the +present King of England," who does exist, and "the present Emperor +of Brazil," who does not exist. More complicated, but analogous, +questions involving propositional functions of two variables involve +the notion of "correlation," just as functions of one argument involve +classes. Similarly functions of three arguments yield three-cornered +correlations, and so on. This logical section is one which Russell has +made peculiarly his own by work which must always remain fundamental. +I have called this the section of functional theory, because its +ideas are essential to the construction of logical denoting functions +which include as a special case ordinary mathematical functions, such +as sine, logarithm, etc. In each of these three stages it will be +necessary gradually to introduce an appropriate symbolism, if we are to +pass on to the fourth stage. + +The fourth logical section, the analytic stage, is concerned with the +investigation of the properties of special logical constructions, +that is, of classes and correlations of special sorts. The whole of +mathematics is included here. So the section is a large one. In fact, +it is mathematics, neither more nor less, but it includes an analysis +of mathematical ideas not hitherto included in the scope of that +science, nor, indeed, contemplated at all. The essence of this stage is +construction. It is by means of suitable constructions that the great +framework of applied mathematics, comprising the theories of number, +quantity, time, and space, is elaborated. + +It is impossible, even in brief outline, to explain how mathematics +is developed from the concepts of class and correlation, including +many-cornered correlations, which are established in the third section. +I can only allude to the headings of the process, which is fully +developed in the work, _Principia Mathematica_, by Mr. Russell +and myself. There are in this process of development seven special +sorts of correlations which are of peculiar interest. The first sort +comprises one-to-many, many-to-one, and one-to-one correlations. +The second sort comprises serial relations, that is, correlations +by which the members of some field are arranged in serial order, so +that, in the sense defined by the relation, any member of the field +is either before or after any other member. The third class comprises +inductive relations, that is, correlations on which the theory of +mathematical induction depends. The fourth class comprises selective +relations, which are required for the general theory of arithmetic +operations, and elsewhere. It is in connection with such relations that +the famous multiplicative axiom arises for consideration. The fifth +class comprises vector relations, from which the theory of quantity +arises. The sixth class comprises ratio relations, which interconnect +number and quantity. The seventh class comprises three-cornered and +four-cornered relations which occur in geometry. + +A bare enumeration of technical names, such as the above, is not very +illuminating, though it may help to a comprehension of the demarcations +of the subject. Please remember that the names are technical names, +meant, no doubt, to be suggestive, but used in strictly defined +senses. We have suffered much from critics who consider it sufficient +to criticise our procedure on the slender basis of a knowledge of +the dictionary meanings of such terms. For example, a one-to-one +correlation depends on the notion of a class with only one member, and +this notion is defined without appeal to the concept of the number one. +The notion of diversity is all that is wanted. Thus the class α has +only one member, if (1) the class of values of _x_ which satisfies +the propositional function, + + _x_ is not a member of α, + +is not the whole type of relevant values of _x_, and if (2) the +propositional function, + + _x_ and _y_ are members of α, and _x_ is diverse from + _y_ + +is false, whatever be the values of _x_ and _y_ in the +relevant type. + +Analogous procedures are obviously possible for higher finite cardinal +members. Thus, step by step, the whole cycle of current mathematical +ideas is capable of logical definition. The process is detailed and +laborious, and, like all science, knows nothing of a royal road of +airy phrases. The essence of the process is, first, to construct +the notion in terms of the forms of propositions, that is, in terms +of the relevant propositional functions, and secondly, to prove the +fundamental truths which hold about the notion by reference to the +results obtained in the algebraic section of logic. + +It will be seen that in this process the whole apparatus of special +indefinable mathematical concepts, and special _a priori_ +mathematical premises, respecting number, quantity, and space, has +vanished. Mathematics is merely an apparatus for analysing the +deductions which can be drawn from any particular premises, supplied +by commonsense, or by more refined scientific observation, so far as +these deductions depend on the forms of the propositions. Propositions +of certain forms are continually occurring in thought. Our existing +mathematics is the analysis of deductions which concern those forms and +in some way are important, either from practical utility or theoretical +interest. Here I am speaking of the science as it in fact exists. A +theoretical definition of mathematics must include in its scope any +deductions depending on the mere forms of propositions. But, of course +no one would wish to develop that part of mathematics which in no sense +is of importance. + +This hasty summary of logical ideas suggests some reflections. The +question arises, How many forms of propositions are there? The +answer is, An unending number. The reason for the supposed sterility +of logical science can thus be discerned. Aristotle founded the +science by conceiving the idea of the form of a proposition, and by +conceiving deduction as taking place in virtue of the forms. But he +confined propositions to four forms, now named A, I, E, O. So long as +logicians were obsessed by this unfortunate restriction, real progress +was impossible. Again, in their theory of form, both Aristotle and +subsequent logicians came very near to the theory of the logical +variable. But to come very near to a true theory, and to grasp its +precise application, are two very different things, as the history of +science teaches us. Everything of importance has been said before by +somebody who did not discover it. + +Again, one reason why logical deductions are not obvious is, that +logical form is not a subject which ordinarily enters into thought. +Commonsense deduction probably moves by blind instinct from concrete +proposition to concrete proposition, guided by some habitual +association of ideas. Thus commonsense fails in the presence of a +wealth of material. + +A more important question is the relation of induction, based on +observation, to deductive logic. There is a tradition of opposition +between adherents of induction and of deduction. In my view, it +would be just as sensible for the two ends of a worm to quarrel. +Both observation and deduction are necessary for any knowledge worth +having. We cannot get at an inductive law without having recourse to a +propositional function. For example, take the statement of observed +fact, + + This body is mercury, and its specific heat is 0·033. + +The propositional function is formed, + + Either _x_ is not mercury, or its specific heat is 0·033. + +The inductive law is the assumption of the truth of the general +proposition, that the above propositional function is true for every +value of _x_ in the relevant type. + +But it is objected that this process and its consequences are so simple +that an elaborate science is out of place. In the same way, a British +sailor knows the salt sea when he sails over it. What, then, is the use +of an elaborate chemical analysis of sea-water? There is the general +answer, that you cannot know too much of methods which you always +employ; and there is the special answer, that logical forms and logical +implications are not so very simple, and that the whole of mathematics +is evidence to this effect. + +One great use of the study of logical method is not in the region of +elaborate deduction, but to guide us in the study of the formation of +the main concepts of science. Consider geometry, for example. What are +the points which compose space? Euclid tells us that they are without +parts and without magnitude. But how is the notion of a point derived +from the sense-perceptions from which science starts? Certainly points +are not direct deliverances of the senses. Here and there we may see or +unpleasantly feel something suggestive of a point. But this is a rare +phenomenon, and certainly does not warrant the conception of space as +composed of points. Our knowledge of space properties is not based on +any observations of relations between points. It arises from experience +of relations between bodies. Now a fundamental space-relation between +bodies is that one body may be part of another. We are tempted to +define the "whole and part" relation by saying that the points occupied +by the part are some of the points occupied by the whole. But "whole +and part" being more fundamental than the notion of "point," this +definition is really circular and vicious. + +We accordingly ask whether any other definition of "spatial whole and +part" can be given. I think that it can be done in this way, though, +if I be mistaken, it is unessential to my general argument. We have +come to the conclusion that an extended body is nothing else than the +class of perception of it by all its percipients, actual or ideal. Of +course, it is not any class of perceptions, but a certain definite sort +of class which I have not defined here, except by the vicious method +of saying that they are perceptions of body. Now, the perceptions of a +part of a body are among the perceptions which compose the whole body. +Thus two bodies _a_ and _b_ are both classes of perceptions; and _b_ is +part of _a_ when the class which is _b_ is contained in the class which +is _a_. It immediately follows from the logical form of this definition +that if _b_ is part of _a_, and _c_ is part of _b_, then _c_ is part of +_a_. Thus the relation "whole to part" is transitive. Again, it will +be convenient to allow that a body is part of itself. This is a mere +question of how you draw the definition. With this understanding, the +relation is reflexive. Finally, if _a_ is part of _b_, and _b_ is part +of _a_, then _a_ and _b_ must be identical. These properties of "whole +and part" are not fresh assumptions, they follow from the logical form +of our definition. + +One assumption has to be made if we assume the ideal infinite +divisibility of space. Namely, we assume that every class of +perceptions which is an extended body contains other classes of +perceptions which are extended bodies diverse from itself. This +assumption makes rather a large draft on the theory of ideal +perceptions. Geometry vanishes unless in some form you make it. The +assumption is not peculiar to my exposition. + +It is then possible to define what we mean by a point. A point is the +class of extended objects which, in ordinary language, contain that +point. The definition, without presupposing the idea of a point, is +rather elaborate, and I have not now time for its statement. + +The advantage of introducing points into geometry is the simplicity +of the logical expression of their mutual relations. For science, +simplicity of definition is of slight importance, but simplicity of +mutual relations is essential. Another example of this law is the way +physicists and chemists have dissolved the simple idea of an extended +body, say of a chair, which a child understands, into a bewildering +notion of a complex dance of molecules and atoms and electrons and +waves of light. They have thereby gained notions with simpler logical +relations. + +Space as thus conceived is the exact formulation of the properties of +the apparent space of the commonsense world of experience. It is not +necessarily the best mode of conceiving the space of the physicist. +The one essential requisite is that the correspondence between the +commonsense world in its space and the physicists' world in its space +should be definite and reciprocal. + +I will now break off the exposition of the function of logic in +connection with the science of natural phenomena. I have endeavoured +to exhibit it as the organising principle, analysing the derivation +of the concepts from the immediate phenomena, examining the structure +of the general propositions which are the assumed laws of nature, +establishing their relations to each other in respect to reciprocal +implications, deducing the phenomena we may expect under given +circumstances. + +Logic, properly used, does not shackle thought. It gives freedom, and +above all, boldness. Illogical thought hesitates to draw conclusions, +because it never knows either what it means, or what it assumes, or +how far it trusts its own assumptions, or what will be the effect of +any modification of assumptions. Also the mind untrained in that part +of constructive logic which is relevant to the subject in hand will be +ignorant of the sort of conclusions which follow from various sorts of +assumptions, and will be correspondingly dull in divining the inductive +laws. The fundamental training in this relevant logic is, undoubtedly, +to ponder with an active mind over the known facts of the case, +directly observed. But where elaborate deductions are possible, this +mental activity requires for its full exercise the direct study of the +abstract logical relations. This is applied mathematics. + +Neither logic without observation, nor observation without logic, can +move one step in the formation of science. We may conceive humanity +as engaged in an internecine conflict between youth and age. Youth is +not defined by years but by the creative impulse to make something. The +aged are those who, before all things, desire not to make a mistake. +Logic is the olive branch from the old to the young, the wand which in +the hands of youth has the magic property of creating science. + + +FOOTNOTES: + +[Footnote 2: _E.g._ in 1551 by Italian schoolmen; cf. Sarpi's +_History of the Council of Trent_, under that date.] + + + + + CHAPTER VII + + THE ANATOMY OF SOME SCIENTIFIC IDEAS + + + _I. Fact_ + +THE characteristic of physical science is, that it ignores all +judgments of value: for example, æsthetic or moral judgments. It is +purely matter-of-fact, and this is the sense in which we must interpret +the sonorous phrase, "Man, the servant and the minister of Nature." + +The sphere of thought which is thus left is even then too wide for +physical science. It would include Ontology, namely, the determination +of the nature of what truly exists; in other words, Metaphysics. From +an abstract point of view this exclusion of metaphysical inquiry +is a pity. Such an inquiry is a necessary critique of the worth of +science, to tell us what it all comes to. The reasons for its careful +separation from scientific thought are purely practical; namely, +because we can agree about science--after due debate--whereas in +respect to metaphysics debate has hitherto accentuated disagreement. +These characteristics of science and metaphysics were unexpected in the +early days of civilised thought. The Greeks thought that metaphysics +was easier than physics, and tended to deduce scientific principles +from _a priori_ conceptions of the nature of things. They were +restrained in this disastrous tendency by their vivid naturalism, their +delight in first-hand perception. Mediæval Europe shared the tendency +without the restraint. It is possible that some distant generations +may arrive at unanimous conclusions on ontological questions, whereas +scientific progress may have led to ingrained opposing veins of +thought which can neither be reconciled nor abandoned. In such times +metaphysics and physical science will exchange their rôles. Meanwhile +we must take the case as we find it. + +But a problem remains. How can mankind agree about science without +a preliminary determination of what really is? The answer must be +found in an analysis of the facts which form the field of scientific +activity. Mankind perceives, and finds itself thinking about its +perceptions. It is the thought that matters and not that element of +perception which is not thought. When the immediate judgment has been +formed--Hullo, red!--it does not matter if we can imagine that in other +circumstances--in better circumstances, perhaps--the judgment would +have been--Hullo, blue!--or even--Hullo, nothing! For all intents and +purposes, at the time it was red. Everything else is hypothetical +reconstruction. The field of physical science is composed of these +primary thoughts, and of thoughts about these thoughts. + +But--to avoid confusion--a false simplicity has been introduced above +into the example given of a primary perceptive thought. "Hullo, red!" +is not really a primary perceptive thought, though it often is the +first thought which finds verbal expression even silently in the mind. +Nothing is in isolation. The perception of red is of a red object in +its relations to the whole content of the perceiving consciousness. + +Among the most easily analysed of such relations are the space +relations. Again the red object is in immediate perception nothing else +than a red object. It is better termed an "object of redness." Thus a +better approximation to an immediate perceptive judgment is, "Hullo, +object of redness there!" But, of course, in this formulation other +more complex relations are omitted. + +This tendency towards a false simplicity in scientific analysis, to +an excessive abstraction, to an over-universalising of universals, +is derived from the earlier metaphysical stage. It arises from the +implicit belief that we are endeavouring to qualify the real with +appropriate adjectives. In conformity with this tendency we think, +"this real thing is red." Whereas our true goal is to make explicit +our perception of the apparent in terms of its relations. What we +perceive is redness related to other apparents. Our object is the +analysis of the relations. + +One aim of science is the harmony of thought, that is, to secure +that judgments which are logical contraries should not be +thought-expressions of consciousness. Another aim is the extension of +such harmonised thought. + +Some thoughts arise directly from sense-presentation, and are part of +the state of consciousness which is perception. Such a thought is, "An +object of redness is there." But in general the thought is not verbal, +but is a direct apprehension of qualities and relations within the +content of consciousness. + +Amid such thoughts there can be no lack of harmony. For direct +apprehension is in its essence unique, and it is impossible to +apprehend an object as both red and blue. Subsequently it may be judged +that if other elements of the consciousness had been different, the +apprehension would have been of a blue object. Then--under certain +circumstances--the original apprehension will be called an error. But +for all that the fact remains, there was an apprehension of a red +object. + +When we speak of sense-presentation, we mean these primary thoughts +essentially involved in its perception. But there are thoughts about +thoughts, and thoughts derived from other thoughts. These are secondary +thoughts. At this point it is well explicitly to discriminate between +an actual thought-expression, namely, a judgment actually made, and a +mere proposition which is a hypothetical thought-expression, namely, +an imagined possibility of thought-expression. Note that the actual +complete thought-content of the consciousness is explicitly neither +affirmed or denied. It is just what _is_ thought. Thus, to think +"two and two make four" is distinct from affirming that two and two +make four. In the first case the proposition is the thought-expression, +in the second case the affirmation of the proposition is the +thought-expression, and the proposition has been degraded to a mere +proposition, namely, to a hypothetical thought-expression which is +reflected upon. + +A distinction is sometimes made between facts and thoughts. So far as +physical science is concerned, the facts are thoughts, and thoughts +are facts. Namely, the facts of sense-presentation as they affect +science are those elements in the immediate apprehensions which are +thoughts. Also, actual thought-expressions, primary or secondary, are +the material facts which science interprets. + +The distinction that facts are given, but thoughts are free, is +not absolute. We can select and modify our sense-presentation, so +that facts--in the narrower sense of immediate apprehension of +sense-presentation--are to some degree subject to volition. Again, our +stream of thought-expression is only partially modified by explicit +volition. We can choose our physical experience, and we find ourselves +thinking; namely, on the one hand there is selection amid the dominant +necessity of sense, and on the other hand, the thought-content of +consciousness (so far as secondary thoughts are concerned) is not +wholly constituted by the selection of will. + +Thus, on the whole there is a large primary region of secondary +thought, as well as of the primary thoughts of sense-presentation, +which is given in type. That is the way in which we do think of things, +not wholly from any abstract necessity, so far as we know, but because +we have inherited the method from an environment. It is the way we find +ourselves thinking, a way which can only be fundamentally laid aside by +an immense effort, and then only for isolated short periods of time. +This is what I have called the "whole apparatus of commonsense thought." + +It is this body of thought which is assumed in science. It is a way +of thinking rather than a set of axioms. It is, in fact, the set of +concepts which commonsense has found useful in sorting out human +experience. It is modified in detail, but assumed in gross. The +explanations of science are directed to finding conceptions and +propositions concerning nature which explain the importance of these +common sense notions. For example, a chair is a common sense notion, +molecules and electrons explain our vision of chairs. + +Now science aims at harmonising our reflective and derivative thoughts +with the primary thoughts involved in the immediate apprehension of +sense-presentation. It also aims at producing such derivative thoughts, +logically knit together. This is scientific theory; and the harmony to +be achieved is the agreement of theory with observation, which is the +apprehension of sense-presentation. + +Thus there is a twofold scientific aim: (1) the production of theory +which agrees with experience; and (2) the explanation of commonsense +concepts of nature, at least in their main outlines. This explanation +consists in the preservation of the concepts in a scientific theory of +harmonised thought. + +It is not asserted that this is what scientists in the past meant to +achieve, or thought that they could achieve. It is suggested as the +actual result of scientific effort, so far as that effort has had +any measure of success. In short, we are here discussing the natural +history of ideas and not volitions of scientists. + + + _II. Objects_ + +We perceive things in space. For example, among such things are dogs, +chairs, curtains, drops of water, gusts of air, flames, rainbows, +chimes of bells, odours, aches and pains. There is a scientific +explanation of the origin of these perceptions. This explanation +is given in terms of molecules, atoms, electrons, and their mutual +relations, in particular of their space-relations, and waves of +disturbance of these space-relations which are propagated through +space. The primary elements of the scientific explanation--molecules, +etc.--are not the things directly perceived. For example, we do not +perceive a wave of light; the sensation of sight is the resultant +effect of the impact of millions of such waves through a stretch of +time. Thus the object directly perceived corresponds to a series of +events in the physical world, events which are prolonged through +a stretch of time. Nor is it true that a perceived object always +corresponds to the same group of molecules. After a few years we +recognise the same cat, but we are thereby related to different +molecules. + +Again, neglecting for a moment the scientific explanation, the +perceived object is largely the supposition of our imagination. When +we recognised the cat, we also recognised that it was glad to see +us. But we merely heard its mewing, saw it arch its back, and felt it +rubbing itself against us. We must distinguish, therefore, between the +many direct objects of sense, and the single indirect object of thought +which is the cat. + +Thus, when we say that we perceived the cat and understood its +feelings, we mean that we heard a sense-object of sound, that we saw a +sense-object of sight, that we felt a sense-object of touch, and that +we thought of a cat and imagined its feelings. + +Sense-objects are correlated by time-relations and space-relations. +Three simultaneous sense-objects which are also spatially coincident, +are combined by thought into the perception of one cat. Such +combination of sense-objects is an instinctive immediate judgment in +general without effort of reasoning. Sometimes only one sense-object +is present. For example, we hear mewing and say there must be a cat +in the room. The transition from the sense-object to the cat has +then been made, by deliberate ratiocination. Even the concurrence of +sense-objects may provoke such a self-conscious effort. For example, in +the dark we feel something, and hear mewing from the same place, and +think, Surely this is a cat. Sight is more bold; when we see a cat, we +do not think further. We identify the sight with the cat, whereas the +cat and the mew are separate. But such immediate identification of +a sight object and an object of thought may lead to error; the birds +pecked at the grapes of Apelles. + +A single sense-object is a complex entity. The sight-object of a +tile on the hearth may remain unchanged as we watch it in a steady +light, remaining ourselves unchanged in position. Even then it is +prolonged in time, and has parts in space. Also it is somewhat +arbitrarily distinguished from a larger whole of which it forms part. +But the glancing fire-light and a change in our position alters the +sight-object. We judge that the tile thought-object remains unchanged. +The sight-object of the coal on the fire gradually modifies, though +within short intervals it remains unchanged. We judge that the coal +thought-object is changing. The flame is never the same, and its shape +is only vaguely distinguishable. + +We conclude that a single self-identical sight-object is already a +phantasy of thought. Consider the unchanging sight-object of the tile, +as we remain still in a steady light. Now a sense-object perceived at +one time is a distinct object from a sense-object seen at another time. +Thus the sight of the tile at noon is distinct from its sight at 12.30. +But there is no such thing as a sense-object at an instant. As we stare +at the tile, a minute, or a second, or a tenth of a second, has flown +by: essentially there is a duration. There is a stream of sight, and +we can distinguish its parts. But the parts also are streams, and +it is only in thought that the stream separates into a succession of +elements. The stream may be "steady" as in the case of the unchanging +sight-tile, or may be "turbulent" as in the case of the glancing +sight-flame. In either case a sight-object is some arbitrarily small +part of the stream. + +Again, the stream which forms the succession of sight-tiles is merely a +distinguishable part of the whole stream of sight-presentation. + +So, finally, we conceive ourselves each experiencing a complete +time-flux (or stream) of sense-presentation. This stream is +distinguishable into parts. The grounds of distinction are differences +of sense--including within that term, differences of types of sense, +and differences of quality and of intensity within the same type +of sense--and differences of time-relations, and differences of +space-relations. Also the parts are not mutually exclusive and exist in +unbounded variety. + +The time-relation between the parts raises the questions of memory and +recognition, subjects too complex for discussion here. One remark must +be made. If it be admitted, as stated above, that we live in durations +and not in instants, namely, that the present essentially occupies +a stretch of time, the distinction between memory and immediate +presentation cannot be quite fundamental; for always we have with us +the fading present as it becomes the immediate past. This region of our +consciousness is neither pure memory nor pure immediate presentation. +Anyhow, memory is also a presentation in consciousness. + +Another point is to be noted in connection with memory. There is no +directly perceived time-relation between a present event and a past +event. The present event is only related to the memory of the past +event. But the memory of a past event is itself a present element +in consciousness. We assert the principle that directly perceived +relations can only exist between elements of consciousness, both in +that present during which the perception occurs. All other relations +between elements of perception are inferential constructions. It thus +becomes necessary to explain how the time stream of events establishes +itself in thought, and how the apparent world fails to collapse into +one single present. The solution of the difficulty is arrived at +by observing that the present is itself a duration, and therefore +includes directly perceived time-relations between events contained +within it. In other words we put the present on the same footing as +the past and the future in respect to the inclusion within it of +antecedent and succeeding events, so that past, present, and future +are in this respect exactly analogous ideas. Thus there will be two +events _a_ and _b_, both in the same present, but the event +_a_ will be directly perceived to precede the event _b_. +Again time flows on, and the event _a_ fades into the past, and +in the new present duration events _b_ and _c_ occur, event +_b_ preceding event _c_, also in the same present duration +there is the memory of the time-relation between _a_ and _b_. +Then by an inferential construction the event _a_ in the past +precedes the event _c_ in the present. By proceeding according to +this principle the time-relations between elements of consciousness, +not in the same present, are established. The method of procedure +here explained is a first example of what we will call the Principle +of Aggregation. This is one of the fundamental principles of mental +construction according to which our conception of the external physical +world is constructed. Other examples will later on be met with. + +The space-relations between the parts are confused and fluctuating, +and in general lack determinate precision. The master-key by which +we confine our attention to such parts as possess mutual relations +sufficiently simple for our intellects to consider is the principle of +convergence to simplicity with diminution of extent. We will call it +the "principle of convergence." This principle extends throughout the +whole field of sense-presentation. + +The first application of the principle occurs in respect to time. +The shorter the stretch of time, the simpler are the aspects of the +sense-presentation contained within it. The perplexing effects of +change are diminished and in many cases can be neglected. Nature has +restricted the acts of thought which endeavour to realise the content +of the present, to stretches of time sufficiently short to secure this +static simplicity over the greater part of the sense-stream. + +Spatial relations become simplified within the approximately static +sense-world of the short time. A further simplicity is gained by +partitioning this static world into parts of restricted space-content. +The various parts thus obtained have simpler mutual space-relations, +and again the principle of convergence holds. + +Finally, the last simplicity is obtained by partitioning the +parts, already restricted as to space and time, into further parts +characterised by homogeneity in type of sense, and homogeneity in +quality and intensity of sense. These three processes of restriction +yield, finally, the sense-objects which have been mentioned above. Thus +the sense-object is the result of an active process of discrimination +made in virtue of the principle of convergence. It is the result of +the quest for simplicity of relations within the complete stream of +sense-presentation. + +The thought-objects of perception are instances of a fundamental +law of nature, the law of objective stability. It is the law of the +coherence of sense-objects. This law of stability has an application +to time and an application to space; also it must be applied in +conjunction with that other law, the principle of convergence to +simplicity from which sense-objects are derived. + +Some composite partial streams of sense-presentation can be +distinguished with the following characteristics: (1) the +time-succession of sense-objects, belonging to a single sense, involved +in any such a composite partial stream, is composed of very similar +objects whose modifications increase only gradually, and thus forms +a homogeneous component stream within the composite stream; (2) the +space-relations of those sense-objects (of various senses) of such +a composite stream which are confined within any sufficiently short +time are identical so far as they are definitely apprehended, and thus +these various component streams, each homogeneous, "cohere" to form the +whole composite partial stream; (3) there are other sense-presentations +occurring in association with that composite partial stream which +can be determined by rules derived from analogous composite partial +streams, with other space and time relations, provided that the analogy +be sufficiently close. Call these the "associated sense-presentations." +A partial stream of this sort, viewed as a whole, is here called a +"first crude thought-object of perception." + +For example, we look at an orange for half a minute, handle it, and +smell it, note its position in the fruit-basket, and then turn away. +The stream of sense-presentation of the orange during that half-minute +is a first crude thought-object of perception. Among the associated +sense presentations are those of the fruit-basket which we conceive as +supporting the orange. + +The essential ground of the association of sense-objects of +various types, perceived within one short duration, into a first +crude thought-object of perception is the coincidence of their +space-relations, that is, in general an approximate coincidence of +such relations perhaps only vaguely apprehended. Thus coincident +space-relations associate sense-objects into a first crude +thought-object, and diverse space-relations dissociate sense-objects +from aggregation into a first crude thought-object. In respect to +some groups of sense-objects the association may be an immediate +judgment devoid of all inference, so that the primary perceptual +thought is that of the first crude thought-object, and the separate +sense-objects are the result of reflective analysis acting on memory. +For example sense-objects of sight and sense-objects of touch are +often thus primarily associated and only secondarily dissociated in +thought. But sometimes the association is wavering and indeterminate, +for example, that between the sound-object of the mew of the cat and +the sight-object of the cat. Thus to sum up, the partial stream of +sense-perceptions coalesces into that first crude thought-object of +perception which is the momentary cat because the sense-perceptions +belonging to this stream are in the same place, but equally it would +be true to say that they are in the same place because they belong +to the same momentary cat. This analysis of the complete stream of +sense-presentation in any small present duration into a variety of +first crude thought-objects only partially fits the facts; for one +reason because many sense-objects, such as sound for instance, have +vague and indeterminate space-relations, for example vaguely those +space-relations which we associate with our organs of sense and also +vaguely those of the origin from which (in the scientific explanation) +they proceed. + +The procedure by which the orange of half a minute is elaborated into +the orange in the ordinary sense of the term involves in addition the +two principles of aggregation and of hypothetical sense-presentation. + +The principle of aggregation, as here employed, takes the form that +many distinct first crude thought-objects of perception are conceived +as one thought-object of perception, if the many partial streams +forming these objects are sufficiently analogous, if their times of +occurrence are distinct, and if the associated sense-presentations are +sufficiently analogous. + +For example, after leaving the orange, in five minutes we return. +A new first crude thought-object of perception presents itself to +us, indistinguishable from the half-minute orange we previously +experienced; it is in the same fruit-basket. We aggregate the two +presentations of an orange into the same orange. By such aggregations +we obtain "second crude thought-objects of perception." But however far +we can proceed with aggregation of this type, the orange is more than +that. For example, what do we mean when we say, The orange is in the +cupboard, if Tom has not eaten it? + +The world of present fact is more than a stream of sense-presentation. +We find ourselves with emotions, volitions, imaginations, conceptions, +and judgments. No factor which enters into consciousness is by itself +or even can exist in isolation. We are analysing certain relations +between sense-presentation and other factors of consciousness. +Hitherto we have taken into account merely the factors of concept and +judgment. Imagination is necessary to complete the orange, namely, +the imagination of hypothetical sense-presentations. It is beside +the point to argue whether we ought to have such imaginations, or to +discuss what are the metaphysical truths concerning reality to which +they correspond. We are here only concerned with the fact that such +imaginations exist and essentially enter into the formation of the +concepts of the thought-objects of perception which are the first +data of science. We conceive the orange as a permanent collection of +sense-presentations existing as if they were an actual element in our +consciousness, which they are not. The orange is thus conceived as +in the cupboard with its shape, odour, colour, and other qualities. +Namely, we imagine hypothetical possibilities of sense-presentation, +and conceive their want of actuality in our consciousness as immaterial +to their existence in fact. The fact which is essential for science is +our conception; its meaning in regard to the metaphysics of reality is +of no scientific importance, so far as physical science is concerned. + +The orange completed in this way is the thought-object of perception. + +It must be remembered that the judgments and concepts arising in the +formation of thought-objects of perception are in the main instinctive +judgments, and instinctive concepts, and are not concepts and judgments +consciously sought for and consciously criticised before adoption. +Their adoption is facilitated by and interwoven with the expectation of +the future in which the hypothetical passes into the actual, and also +with the further judgment of the existence of other consciousnesses, +so that much that is hypothetical to one consciousness is judged to be +actual to others. + +The thought-object of perception is, in fact, a device to make plain +to our reflective consciousness relations which hold within the +complete stream of sense-presentation. Concerning the utility of this +weapon there can be no question; it is the rock upon which the whole +structure of commonsense thought is erected. But when we consider the +limits of its application the evidence is confused. A great part of our +sense-presentation can be construed as perception of various persistent +thought-objects. But hardly at any time can the sense-presentations +be construed wholly in that way. Sights lend themselves easily to +this construction, but sight can be baffled: for example, consider +reflections in looking-glasses, apparently bent sticks half in and half +out of water, rainbows, brilliant patches of light which conceal the +object from which they emanate, and many analogous phenomena. Sound +is more difficult; it tends largely to disengage itself from any such +object. For example, we see the bell, but we hear the sound which +comes from the bell; yet we also say that we hear the bell. Again, a +toothache is largely by itself, and is only indirectly a perception +of the nerve of the tooth. Illustrations to the same effect can be +accumulated from every type of sensation. + +Another difficulty arises from the fact of change. The thought-object +is conceived as one thing, wholly actual at each instant. But since +the meat has been bought it has been cooked, the grass grows and then +withers, the coal burns in the fire, the pyramids of Egypt remain +unchanged for ages, but even the pyramids are not wholly unchanged. The +difficulty of change is merely evaded by affixing a technical Latin +name to a supposed logical fallacy. A slight cooking leaves the meat +the same object, but two days in the oven burns it to a cinder. When +does the meat cease to be? Now the chief use of the thought-object +is the concept of it as one thing, here and now, which later can be +recognised, there and then. This concept applies sufficiently well to +most things for short times, and to many things for long times. But +sense-presentation as a whole entirely refuses to be patient of the +concept. + +We have now come to the reflective region of explanation, which is +science. + +A great part of the difficulty is at once removed by applying the +principle of convergence to simplicity. We habitually make our +thought-objects too large; we should think in smaller parts. For +example, the Sphinx has changed by its nose becoming chipped, but by +proper inquiry we could find the missing part in some private house +of Western Europe or Northern America. Thus, either part, the rest +of the Sphinx or the chip, regains its permanence. Furthermore, we +enlarge this explanation by conceiving parts so small that they can +only be observed under the most favourable circumstances. This is a +wide extension of the principle of convergence in its application to +nature; but it is a principle amply supported by the history of exact +observation. + +Thus, change in thought-objects of perception is largely explained as +a disintegration into smaller parts, themselves thought-objects of +perception. The thought-objects of perception which are presupposed in +the common thought of civilised beings are almost wholly hypothetical. +The material universe is largely a concept of the imagination which +rests on a slender basis of direct sense-presentation. But none the +less it is a fact; for it is a fact that actually we imagine it. Thus +it is actual in our consciousness just as sense-presentation also is +actual there. The effort of reflective criticism is to make these two +factors in our consciousness agree where they are related, namely, +to construe our sense-presentation as actual realisation of the +hypothetical thought-objects of perception. + +The wholesale employment of purely hypothetical thought-objects of +perception enables science to explain some of the stray sense-objects +which cannot be construed as perceptions of a thought-object of +perception: for example, sounds. But the phenomena as a whole defy +explanation on these lines until a further fundamental step is taken, +which transforms the whole concept of the material universe. Namely, +the thought-object of perception is superseded by the thought-object of +science. + +The thought-objects of science are molecules, atoms, and electrons. The +peculiarity of these objects is that they have shed all the qualities +which are capable of direct sense-representation in consciousness. +They are known to us only by their associated phenomena, namely, +series of events in which they are implicated are represented in our +consciousness by sense-presentations. In this way, the thought-objects +of science are conceived as the causes of sense-representation. The +transition from thought-objects of perception to thought-objects of +science is decently veiled by an elaborate theory concerning primary +and secondary qualities of bodies. + +This device, by which sense-presentations are represented in thought +as our perception of events in which thought-objects of science are +implicated, is the fundamental means by which a bridge is formed +between the fluid vagueness of sense and the exact definition of +thought. In thought a proposition is either true or false, an entity +is exactly what it is, and relations between entities are expressible +(in idea) by definite propositions about distinctly conceived entities. +Sense-perception knows none of these things, except by courtesy. +Accuracy essentially collapses at some stage of inquiry. + + + _III. Time and Space_ + +_Recapitulation._--Relations of time and relations of space +hold between sense-objects of perception. These sense-objects are +distinguished as separate objects by the recognition of either (1) +differences of sense-content, or (2) time-relations between them +other than simultaneity, or (3) space-relations between them other +than coincidence. Thus sense-objects arise from the recognition of +contrast within the complete stream of sense-presentation, namely, from +the recognition of the objects as related terms, by relations which +contrast them. Differences of sense-content are infinitely complex in +their variety. Their analysis under the heading of general ideas is the +unending task of physical science. Time-relations and space-relations +are comparatively simple, and the general ideas according to which +their analysis should proceed are obvious. + +This simplicity of time and space is perhaps the reason why thought +chooses them as the permanent ground for objectival distinction, +throwing the various sense-objects thus obtainable into one heap, as +a first crude thought-object of perception, and thence, as described +above, obtaining a thought-object of perception. Thus a thought-object +of perception conceived as in the present of a short duration is a +first crude thought-object of perception either actual or hypothetical. +Such a thought-object of perception, confined within a short duration, +takes on the space-relations of its component sense-objects within that +same duration. Accordingly thought-objects of perception, conceived in +their whole extents, have to each other the time-relationships of their +complete existences, and within any small duration have to each other +the space-relationships of their component sense-objects which lie +within that duration. + +Relations bind together: thus thought-objects of perception are +connected in time and in space. The genesis of the objectival analysis +of sense-presentation is the recognition of sense-objects as distinct +terms in time-relations and space-relations: thus thought-objects of +perception are separated by time and by space. + +_Whole and Part._--A sense-object is part of the complete stream +of presentation. This concept of being a part is merely the statement +of the relation of the sense-object to the complete sense-presentation +for that consciousness. Also a sense-object can be part of another +sense-object. It can be a part in two ways, namely, a part in time +and a part in space. It seems probable that both these concepts of +time-part and space-part are fundamental; that is, are concepts +expressing relations which are directly presented to us, and are not +concepts about concepts. In that case no further definition of the +actual presentation is possible. It may even then be possible to +define an adequate criterion of the occurrence of such a presentation. +For example, adopting for the moment a realist metaphysic as to the +existence of the physical world of molecules and electrons, the vision +of a chair as occurring for some definite person at some definite +time is essentially indefinable. It is his vision, though each of us +guesses that it must be uncommonly like our vision under analogous +circumstances. But the existence of the definable molecules and +waves of light in certain definable relations to his bodily organs +of sense, his body also being in a certain definable state, forms an +adequate criterion of the occurrence of the vision, a criterion which +is accepted in Courts of Law and for physical science is tacitly +substituted for the vision. + +The connection between the relations "whole and part" and "all and +some" is intimate. It can be explained thus so far as concerns directly +presented sense-objects. Call two sense-objects "separated" if there +is no third sense-object which is a part of both of them. Then an +object A is composed of the two objects B and C, if (1) B and C are +both parts of A, (2) B and C are separated, and (3) there is no part of +A which is separated both from B and from C. In such a case the class +α which is composed of the two objects B and C is often substituted +in thought for the sense-object A. But this process presupposes the +fundamental relation "whole and part." Conversely the objects B and C +may be actual sense-objects, but the sense-object A which corresponds +to the class α may remain hypothetical. For example, the round world +on which we live remains a conception corresponding to no single +sense-object at any time presented in any human being's consciousness. + +It is possible, however, that some mode of conceiving the +whole-and-part relation between extended objects as the all-and-some +relation of logical classes can be found. But in this case the +extended objects as here conceived cannot be the true sense-objects +which are present to consciousness. For as here conceived a part of a +sense-object is another sense-object of the same type; and therefore +one sense-object cannot be a class of other sense-objects, just as +a tea-spoon cannot be a class of other tea-spoons. The ordinary way +in thought by which whole-and-part is reduced to all-and-some is by +the device of points, namely, the part of an object occupies some +of the points occupied by the whole object. If any one holds that +in his consciousness the sense-presentation is a presentation of +point-objects, and that an extended object is merely a class of such +point-objects collected together in thought, then this ordinary method +is completely satisfactory. We shall proceed on the assumption that +this conception of directly perceived point-objects has no relation to +the facts. + +In the preceding address on "The Organisation of Thought," another mode +is suggested. But this method would apply only to the thought-object +of perception, and has no reference to the primary sense-objects here +considered. Accordingly it must reckon as a subordinate device for a +later stage of thought. + +Thus the point-object in time and the point-object in space, and the +double point-object both in time and space, must be conceived as +intellectual constructions. The fundamental fact is the sense-object, +extended both in time and space, with the fundamental relation of +whole-to-part to other such objects, and subject to the law of +convergence to simplicity as we proceed in thought through a series of +successively contained parts. + +The relation whole-to-part is a temporal or spatial relation, and +is therefore primarily a relation holding between sense-objects of +perception, and it is only derivatively ascribed to the thought-objects +of perception of which they are components. More generally, space and +time relations hold primarily between sense-objects of perception and +derivatively between thought-objects of perception. + +_Definition of Points._--The genesis of points of time and of +space can now be studied. We must distinguish (1) sense-time and +sense-space, and (2) thought-time of perception and thought-space of +perception. + +Sense-time and sense-space are the actually observed time-relations +and space-relations between sense-objects. Sense-time and sense-space +have no points except, perhaps, a few sparse instances, sufficient +to suggest the logical idea; also, sense-time and sense-space are +discontinuous and fragmentary. + +Thought-time of perception and thought-space of perception are the time +and space relations which hold between thought-objects of perception. +Thought-time of perception and thought-space of perception are each +continuous. By "continuous" is here meant that all thought-objects of +perception have to each other a time (or space) relation. + +The origin of points is the effort to take full advantage of +the principle of convergence to simplicity. In so far as this +principle does not apply, a point is merely a cumbrous way of +directing attention to a set of relations between a certain set of +thought-objects of perception, which set of relations, though actual so +far as a thought-object is actual, is (under this supposition) of no +particular importance. Thus the proved importance in physical science +of the concepts of points in time and points in space is a tribute to +the wide applicability of this principle of convergence. + +Euclid defines a point as without parts and without magnitude. In +modern language a point is often described as an ideal limit by +indefinitely continuing the process of diminishing a volume (or +area). Points as thus conceived are often called convenient fictions. +This language is ambiguous. What is meant by a fiction? If it means +a conception which does not correspond to any fact, there is some +difficulty in understanding how it can be of any use in physical +science. For example, the fiction of a red man in a green coat +inhabiting the moon can never be of the slightest scientific service, +simply because--as we may presume--it corresponds to no fact. By +calling the concept of points a convenient fiction, it must be meant +that the concept does correspond to some important facts. It is, then, +requisite, in the place of such vague allusiveness, to explain exactly +what are the facts to which the concept corresponds. + +We are not much helped by explaining that a point is an ideal limit. +What is a limit? The idea of a limit has a precise meaning in the +theory of series, and in the theory of the values of functions; but +neither of these meanings apply here. It may be observed that, before +the ordinary mathematical meanings of limit had received a precise +explanation, the idea of a point as a limit might be considered as +one among other examples of an idea only to be apprehended by direct +intuition. This view is not now open to us. Thus, again, we are +confronted with the question: What are the precise properties meant +when a point is described as an ideal limit? The discussion which now +follows is an attempt to express the concept of a point in terms of +thought-objects of perception related together by the whole-and-part +relation, considered either as a time-relation or as a space-relation. +If it is so preferred, it may be considered that the discussion is +directed towards a precise elucidation of the term "ideal limit" as +often used in this connection. + +The subsequent explanations can be made easier to follow by a small +piece of symbolism: Let _aEb_ mean that "_b_ is part of _a_." We need +not decide whether we are talking of time-parts or space-parts, but +whichever choice is supposed to be made must be conceived as adhered to +throughout any connected discussion. The symbol _E_ may be considered +as the initial letter of "encloses," so we read "_aEb_" as "_a_ +encloses _b_." Again the "field of _E_" is the set of things which +either enclose or are enclosed, _i. e._ everything "_a_," which is such +that _x_ can be found so that either _aEx_ or _xEa_. A member of the +field of _E_ is called "an enclosure-object." + +Now, we assume that this relation of whole-to-part, which in the future +we will call "enclosure," always satisfies the conditions in that the +relation _E_ is (1) transitive, (2) asymmetrical, and (3) with its +domain including its converse domain. + +These four conditions deserve some slight consideration; only the first +two of them embody hypotheses which enter vitally into the reasoning. + +Condition (1) may be stated as the condition that _aEb_ and _bEc_ +always implies _aEc_. The fact that an entity _b_ can be found such +that _aEb_ and _bEc_ may be conceived as a relation between _a_ and +_c_. It is natural to write _E_^2 for this relation. Thus the condition +is now written: If _aE_^2_c_, then _aEc_. This can be still otherwise +expressed by saying that the relation _E_^2 implies, whenever it holds, +that the relation _E_ also holds. + +Condition (2) is partly a mere question of trivial definition, and +partly a substantial assumption. The asymmetrical relation (_E_) is +such that _aEb_ and _bEa_ can never hold simultaneously. This property +splits up into two parts: (1) that no instance of _aEb_ and _bEa_ and +"_a_ diverse from _b_," can occur, and (2) that _aEa_ cannot occur. +The first part is a substantial assumption, the second part (so far as +we are concerned) reduces to the trivial convention that we shall not +consider an object as part of itself, but will confine attention to +"proper parts." + +Condition (3) means that _aEb_ always implies that _c_ can be +found such that _bEc_. This condition, taken in conjunction with +the fact that we are only considering proper parts, is the assertion of +the principle of the indefinite divisibility of extended objects, both +in space and in time. + +An indivisible part will lack duration in time, and extension in +space, and is thus an entity of essentially a different character +to a divisible part. If we admit such indivisibles as the only true +sense-objects, our subsequent procedure is an unnecessary elaboration. + +It will be found that a fourth condition is necessary owing to +logical difficulties connected with the theory of an infinite number +of choices. It will not be necessary for us to enter further on +this question, which involves difficult considerations of abstract +logic. The outcome is, that apart from hypothesis we cannot prove the +existence of the sets, each containing an infinite number of objects, +which are here called points, as will be explained immediately. + +Now consider a set of enclosure objects which is such that (1) +of any two of its members one encloses the other, and (2) there +is no member which is enclosed by all the others, and (3) there +is no enclosure-object, not a member of the set which is enclosed +by every member of the set. Call such a set a "convergent set of +enclosure-objects." As we pass along the series from larger to smaller +members, evidently we converge towards an ideal simplicity to any +degree of approximation to which we like to proceed, and the series as +a whole embodies the complete ideal along that route of approximation. +In fact, to repeat, the series is a _route of approximation_. + +We have now to inquire if the principle of convergence to simplicity +may be expected to yield the same type of simplicity for every such +convergent route. The answer is, as we might expect, namely, that this +depends upon the nature of the properties which are to be simplified. + +For example, consider the application to time. Now, time is +one-dimensional; so when this property of one-dimensionality has been +expressed by the proper conditions, not here stated, a convergent set +of enclosure-objects must, considered as a route of approximation, +exhibit the properties of one unique instant of time, as ordinarily +conceived by the euclidean definition. Accordingly, whatever simplicity +is to be achieved by the application to time of the principle of +convergence to simplicity must be exhibited among the properties of +any such route of approximation. + +For space, different considerations arise. Owing to its multiple +dimensions, we can show that different convergent sets of +enclosure-objects, indicating different routes of approximation, may +exhibit convergence to different types of simplicity, some more complex +than others. + +For example, consider a rectangular box of height _h_ ft., breadth +_b_ ft., and thickness _c_ ft. Now, keep _h_ and _b_ constant, and +let the central plane (height _h_, breadth _b_) perpendicular to the +thickness be fixed, then make _c_ diminish indefinitely. We thus obtain +a convergent series of an indefinitely large number of boxes, and there +is no smallest box. Thus this convergent series exhibits the route of +approximation towards the type of simplicity expressed as being a plane +area of height _h_, breadth _b_, and no thickness. + +Again, by keeping the central line of height _h_ fixed, and by making +_b_ and _c_ diminish indefinitely, the series converges to the segment +of a straight line of length _h_. + +Finally, by keeping only the central point fixed, and by making _h_, +_b_, and _c_ diminish indefinitely, the series converges to a point. + +Furthermore, we have introduced as yet no concept which would prevent +an enclosure-object being formed of detached fragments in space. Thus +we can easily imagine a convergent set which converges to a number of +points in space. For example, each object of the set might be formed of +two not overlapping spheres of radius _r_, with centres _A_ and _B_. +Then by diminishing _r_ indefinitely, and keeping _A_ and _B_ fixed, we +have convergence to the pair of points _A_ and _B_. + +It remains now to consider how those convergent sets which converge to +a single point can be discriminated from all the other types of such +sets, merely by utilising concepts founded on the relation of enclosure. + +Let us name convergent sets by Greek letters; by proceeding "forward" +along any such set let us understand the process of continually passing +from the larger to the smaller enclosure-objects which form the set. + +The convergent set α will be said to "cover" the convergent set β, if +every member of α encloses some members of β. We notice that if an +enclosure-object _x_ encloses any member (_y_) of β, then every member +of the "tail-end" of β, found by proceeding forward along β from _y_, +must be enclosed by _x_. Thus if α covers β, every member of α encloses +every member of the tail-end of β, starting from the largest member of +β which is enclosed by that member of α. + +It is possible for each of two convergent sets to cover the other. For +example, let one set (α) be a set of concentric spheres converging to +their centre _A_, and the other set (β) be a set of concentric +cubes, similarly situated, converging to the same centre _A_. Then +α and β will each cover the other. + +Let two convergent sets which are such that each covers the other be +called "equal." + +Then it is a sufficient condition to secure that a convergent set +α possesses the point type of convergence, if every convergent set +covered by it is also equal to it, namely, α is a convergent set with +the punctual type of convergence, if "α covers β" always implies that β +covers α. + +It can easily be seen by simple examples that the other types of +convergence to surfaces or lines or sets of points cannot possess this +property. Consider, for example, the three convergent sets of boxes in +the preceding illustration, which converge respectively to a central +plane, a central line in the central plane, and the central point in +the central line. The first set covers the second and third sets, and +the second set covers the third set, but no two of the sets are equal. + +It is a more difficult question to determine whether the condition here +indicated as sufficient to secure the punctual type of convergence +is also necessary. The question turns on how far thought-objects +of perception possess exact boundaries prior to the elaboration of +exact mathematical concepts of space. If they are to be conceived as +possessing such exact boundaries, then convergent sets converging to +points on such boundaries must be allowed for. The procedure necessary +for the specification of the complete punctual condition becomes then +very elaborate,[3] and will not be considered here. + +But such exact determination as is involved in the conception of +an exact spatial boundary does not seem to belong to the true +thought-object of perception. The ascription of an exact boundary +really belongs to the transition stage of thought as it passes from +the thought-object of perception to the thought-object of science. +The transition from the sense-object immediately presented to the +thought-object of perception is historically made in a wavering +indeterminate line of thought. The definite stages here marked out +simply serve to prove that a logically explicable transition is +possible. + +We accordingly assume that the condition laid down above to secure the +punctual convergence of a convergent set of enclosure-objects is not +only sufficient, but necessary. + +It can be proved that, if two convergent sets of enclosure-objects are +both equal to a third convergent set, they are equal to each other. +Consider now any punctual convergent set (α). We want to define the +"point" to which α is a route of approximation in a way which is +neutral between α and all the convergent sets which are equal to α. +Each of these sets is a route of approximation to the same "point" +as α. This definition is secured if we define the point as the class +formed by all the enclosure-objects which belong either to α or to +any convergent set which is equal to α. Let _P_ be this class +of enclosure-objects. Then any convergent set (β) which consists of +enclosure-objects entirely selected from members of the class _P_ +must be a route of approximation to the same "point" as does the +original punctual set α; namely, provided that we choose a small enough +enclosure-object in β, we can always find a member of α which encloses +it; and provided that we choose a small enough enclosure-object in +α, we can always find a member of β which encloses it. Thus _P_ +only includes convergent sets of the punctual type, and the route +of approximation indicated by any two convergent sets selected from +_P_ converges to identical results. + +_The Uses of Points._--The sole use of points is to facilitate the +employment of the principle of Convergence to Simplicity. By this +principle some simple relations in appropriate circumstances become +true, when objects are considered which are sufficiently restricted in +time or in space. The introduction of points enables this principle to +be carried through to its ideal limit. For example, suppose _g_ (_a_, +_b_, _c_) represents some statement concerning three enclosure-objects, +_a_, _b_, _c_, which may be true if the objects are sufficiently +restricted in extent. Let _A_, _B_, _C_ be three given points, +then we define _g_ (_A_, _B_, _C_) to mean that _whatever_ three +enclosure-objects _a_, _b_, _c_ are chosen, such that _a_ is a member +of _A_, _b_ of _B_, and _c_ of _C_, it is _always possible_ to find +three other members of _A_, _B_, _C_, namely, _x_ a member of _A_, _y_ +of _B_, and _z_ of _C_, such that _aEx_, _bEy_, _cEz_, and _g_ (_x_, +_y_, _z_). So by going far enough down in the tail-ends of _A_, _B_, +_C_ we can always secure three objects _x_, _y_, _z_ for which _g_ +(_x_, _y_, _z_) is true. + +For example, let _g_ (_A_, _B_, _C_) mean "_A_, _B_, _C_ are three +points in a linear row." This must be construed to mean that whatever +three objects _a_, _b_, _c_ we choose, members of _A_, _B_, _C_ +respectively, we can always find three objects _x_, _y_, _z_, also +members of _A_, _B_, _C_ respectively, and such that _a_ encloses _x_, +_b_ encloses _y_, _c_ encloses _z_, and also such that _x_, _y_, _z_ +are in a linear row. + +Sometimes a double convergence is necessary, namely, a convergence of +conditions as well as a convergence of objects. For example, consider +the statement, "the points _A_ and _B_ are two feet apart." Now, the +exact statement "two feet apart" does not apply to objects. For objects +_x_ and _y_ we must substitute the statement, "the distance between +_x_ and _y_ lies between the limits (2 ± _e_) feet." Here _e_ is some +number, less than two, which we have chosen for this statement. Then +the points _A_ and _B_ are two feet apart; if, _however we choose the +number e_, whatever enclosure-objects _a_ and _b_, members of _A_ and +_B_ respectively, we consider, we can always find enclosure-objects _x_ +and _y_, members of _A_ and _B_ respectively, such that _a_ encloses +_x_ and _b_ encloses _y_, and also such that the distance between _x_ +and _y_ lies between the limits (2 ± _e_) feet. It is evident, since +_e_ can be chosen as small as we please, that this statement exactly +expresses the condition that _A_ and _B_ are two feet apart. + +_Straight Lines and Planes._--But the problem of the intellectual +construction of straight lines and planes is not yet sufficiently +analysed. We have interpreted the meaning of the statement that three +or more points are collinear, and can similarly see how to interpret +the meaning of the statement that four or more points are coplanar, +in either case deriving the exact geometrical statements from vaguer +statements respecting extended objects. + +This procedure only contemplates groups of finite numbers of points. +But straight lines and planes are conceived as containing infinite +numbers of points. This completion of lines and planes is obtained +by a renewed application of the principle of aggregation, just as +a set of first crude thought-objects of perception are aggregated +into one complete thought-object of perception. In this way repeated +judgments of the collinearity of sets of points are finally, when +certain conditions of interlacing are fulfilled, aggregated in the +single judgment of all the points of the groups as forming one whole +collinear group. Similarly for judgments of coplanarity. This process +of logical aggregation can be exhibited in its exact logical analysis. +But it is unnecessary here to proceed to such details. Thus we conceive +our points as sorted into planes and straight lines, concerning which +the various axioms of geometry hold. These axioms, in so far as they +essentially require the conception of points, are capable of being +exhibited as the outcome of vaguer, less exact judgments respecting the +relations of extended objects. + +_Empty Space._--It must be observed that the points, hitherto +defined, necessarily involve thought-objects of perception, and lie +within the space-extension occupied by such objects. It is true that +such objects are largely hypothetical, and that we can bring into our +hypotheses sufficient objects to complete our lines and planes. But +every such hypothesis weakens the connection between our scientific +concept of nature and the actual observed facts which are involved in +the actual sense-presentations. + +Occam's razor, _Entia non multiplicanda præter necessitatem_, +is not an arbitrary rule based on mere logical elegancy. Nor is its +application purely confined to metaphysical speculation. I am ignorant +of the precise reason for its metaphysical validity, but its scientific +validity is obvious, namely, every use of hypothetical entities +diminishes the claim of scientific reasoning to be the necessary +outcome of a harmony between thought and sense-presentation. As +hypothesis increases, necessity diminishes. + +Commonsense thought also supports this refusal to conceive of all +space as essentially depending on hypothetical objects which fill it. +We think of material objects as filling space, but we ask whether any +objects exist between the Earth and the Sun, between the stars, or +beyond the stars. For us, space is there; the only question is whether +or not it be full. But this form of question presupposes the meaning of +empty space, namely, of space not containing hypothetical objects. + +This brings up a wider use of the concept of points, necessitating +a wider definition. Hitherto we have conceived points as indicating +relations of enclosure between objects. We thus arrive at what now +we will term "material points." But the idea of points can now be +transformed so as to indicate the possibilities of external relations +not those of enclosure. This is effected by an enlargement of the +concept of ideal points, already known to geometers. + +Define "material lines" to be complete collinear classes of collinear +points. Consider now the set of material lines which contain a certain +material point. Call such a set of lines an ideal point. This set of +lines indicates a possibility of position, which is in fact occupied +by that material point common to all the material lines. So this ideal +point is an occupied ideal point. Now consider a set of three material +lines, such that any two are coplanar, but not the whole three, and +further consider the complete set of material lines such that each is +coplanar with each of the three material lines first chosen. The axioms +which hold for the material lines will enable us to prove that any two +lines of this set are coplanar. Then the whole set of lines, including +the three original lines, forms an ideal point, according to the +definition in its full generality. Such an ideal point may be occupied. +In that case there is a material point common to all the lines of the +set, but it may be unoccupied. Then the ideal point merely indicates +a possibility of spatial relations which has not been realised. It is +the point of empty space. Thus the ideal points, which may or may not +be occupied, are the points of geometry viewed as an applied science. +These points are distributed into straight lines and planes. But any +further discussion of this question will lead us into the technical +subject of the axioms of geometry and their immediate consequences. +Enough has been said to show how geometry arises according to the +relational theory of space. + +Space as thus conceived is the thought-space of the material world. + + + _IV. Fields of Force_ + +The thought-objects of science are conceived as directly related to +this thought-space. Their spatial relations are among those indicated +by the points of the thought-space. Their emergence in science has +been merely a further development of processes already inherent in +commonsense thought. + +Relations within the complete sense-presentation were represented +in thought by the concept of thought-objects of perception. All +sense-presentation could not be represented in this way; also the +change and disappearance of thought-objects occasioned confusion +of thought. A reduction to order of this confusion was attempted +by the concepts of permanent matter with primary and secondary +qualities. Finally, this has issued in the secondary qualities being +traced as perception of events generated by the objects, but--as +perceived--entirely disconnected with them. Also the thought-objects +of perception have been replaced by molecules and electrons and +ether-waves, until at length it is never the thought-object of science +which is perceived, but complicated series of events in which they are +implicated. If science be right, nobody ever perceived a thing, but +only an event. The result is, that the older language of philosophy +which still survives in many quarters is now thoroughly confusing +when brought into connection with the modern concepts of science. +Philosophy--that is, the older philosophy--conceives the thing as +directly perceived. According to scientific thought, the ultimate thing +is never perceived, perception essentially issuing from a series of +events. It is impossible to reconcile the two points of view. + +The advantage of the modern scientific concept is that it is enabled +to "explain" the fluid vague outlines of sense-presentation. The +thought-object of perception is now conceived as a fairly stable +state of motion of a huge group of molecules, constantly changing, +but preserving a certain identity of characteristics. Also stray +sense-objects, not immediately given as part of a thought-object of +perception, are now explicable: the dancing light-reflection, the +vaguely heard sound, the smell. In fact, the perceived events of +the scientific world have the same general definition and lack of +definition, and the same general stability and lack of stability, +as the sense-objects of the complete sense-presentation or as the +thought-objects of perception. + +The thought-objects of science, namely, molecules, atoms, and +electrons, have gained in permanence. The events are reduced to changes +in space-configuration. The laws determining these changes are the +ultimate laws of nature. + +The laws of change in the physical universe proceed on the assumption +that the preceding states of the universe determine the character of +the change. Thus, to know the configurations and events of the universe +up to and including any instant would involve sufficient data from +which to determine the succession of events throughout all time. + +But in tracing the antecedents of events, commonsense thought, dealing +with the world of thought-objects of perception, habitually assumes +that the greater number of antecedent events can be neglected as +irrelevant. Consideration of causes is restricted to a few events +during a short preceding interval. Finally, in scientific thought it +has been assumed that the events in an arbitrarily small preceding +duration are sufficient. Thus physical quantities and their successive +differential co-efficients up to any order at the instant, but with +their limiting values just before that instant, are on this theory +sufficient to determine the state of the universe at all times after +the instant. More particular laws are assumed. But the search for +them is guided by this general principle. Also it is assumed that the +greater number of events in the physical universe are irrelevant to +the production of any particular effect, which is assumed to issue +from relatively few antecedents. These assumptions have grown out of +the experience of mankind. The first lesson of life is to concentrate +attention on few factors of sense-presentations, and on still fewer of +the universe of thought-objects of perception. + +The principle by which--consciously or unconsciously--thought has been +guided is that in searching for particular causes, remoteness in time +and remoteness in space are evidences of comparative disconnection +of influence. The extreme form of this principle is the denial of +any action at a distance either in time or space. The difficulty in +accepting this principle in its crude form is, that since there are no +contiguous points, only coincident bodies can act on each other. I can +see no answer to this difficulty--namely, either bodies have the same +location and are thus coincident, or they have different locations and +are thus at a distance and do not act on each other. + +This difficulty is not evaded by the hypothesis of an ether, +continuously distributed. For two reasons: in the first place, the +continuity of the ether does not avoid the dilemma; and secondly, the +difficulty applies to time as well as to space, and the dilemma would +prove that causation producing change is impossible, namely, no changed +condition could be the result of antecedent circumstances. + +On the other hand, a direct interaction between two bodies separated +in space undoubtedly offends the conception of distance as implying +physical disconnection as well as spatial relation. There is no logical +difficulty in the assumption of action at a distance as in the case +of its denial, but it is contradictory to persistent assumptions of +that apparatus of commonsense thought which it is the main business +of science to harmonise with sense-presentation, employing only the +minimum of modification. + +Modern science is really unconcerned with this debate. Its +(unacknowledged) conceptions are really quite different, though the +verbal explanations retain the form of a previous epoch. The point of +the change in conception is that the old thought-object of science was +conceived as possessing a simplicity not belonging to the material +universe as a whole. It was secluded within a finite region of space, +and changes in its circumstances could only arise from forces which +formed no essential part of its nature. An ether was called into +existence to explain the active relations between these passive +thought-objects. The whole conception suffers from the logical +difficulties noted above. Also no clear conception can be formed of +the sense in which the ether is explanatory. It is to possess a type +of activity denied to the original thought-object, namely, it carries +potential energy, whereas the atom possessed only kinetic energy, +the so-called potential energy of an atom belonging really to the +surrounding ether. The truth is, that ether is really excepted from the +axiom "no action at a distance," and the axiom thereby is robbed of all +its force. + +The modern thought-object of science--not yet explicitly +acknowledged--has the complexity of the whole material universe. In +physics, as elsewhere, the hopeless endeavour to derive complexity +from simplicity has been tacitly abandoned. What is aimed at is not +simplicity, but persistence and regularity. In a sense regularity +is a sort of simplicity. But it is the simplicity of stable mutual +relations, and not the simplicity of absence of types of internal +structure or of type of relation. This thought-object fills all space. +It is a "field"; that is to say, it is a certain distribution of +scalar and vector quantities throughout space, these quantities having +each its value for each point of space at each point of time, being +continuously distributed throughout space and throughout time, possibly +with some exceptional discontinuities. The various types of quantity +which form the field have fixed relations to each other at each point +of time and space. These relations are the ultimate laws of nature. + +For example, consider an electron. There is a scalar distribution of +electricity, which is what is ordinarily called the electron. This +scalar distribution has a volume-density ρ at the time _t_ at any +point (_x_, _y_, _z_). Thus ρ is a function of (_x_, _y_, _z_, _t_), +which is zero except within a restricted region. Furthermore, at +any time _t_, as an essential adjunct, there is a continuous space +distribution at each point of the two vectors (_X_, _Y_, _Z_), which is +the electric force, and (α, β, γ), which is the magnetic force. Lastly, +individuality is ascribed to the scalar electric distribution, so that +in addition to its conservation of quantity--involved in the assumed +laws--it is also possible to assign the velocities with which the +various individual parts of the distribution are moving. Let (_u_, _v_, +_w_) be this velocity at (_x_, _y_, _z_, _t_). + +This whole scheme of scalar and vector quantities, namely, ρ, (_X_, +_Y_, _Z_), (α, β, γ), (_u_, _v_, _w_) is interconnected by the +electromagnetic laws. It follows from these laws that the electron, in +the sense of the scalar distribution ρ, is to be conceived as at each +instant propagating from itself an emanation which travels outwards +with the velocity of light _in vacuo_, and from which (_X_, _Y_, +_Z_) and (α, β, γ) can be calculated, so far as they are due to that +electron. Thus the field, at any time, due to the electron as a whole +depends on the previous history of the electron, the nearer to the +electron the more recent being the relevant history. The whole scheme +of such a field is one single thought-object of science: the electron +and its emanations form one essential whole, namely one thought-object +of science, essentially complex and essentially filling all space. The +electron proper, namely, the scalar distribution ρ, is the focus of the +whole, the essential focal property being that the field at any instant +is completely determined by the previous history of the focus and of +its space relations through all previous time. But the field and the +focus are not independent concepts, they are essentially correlated in +one organised unity, namely, they are essentially correlated terms in +the field of one relation in virtue of which the entities enter into +our thoughts. + +The fields of a group of electrons are superposed according to the +linear law for aggregation, namely, pure addition for analogous scalar +quantities and the parallelogram law for analogous vectors. The changes +in motion of each electron depend entirely on the resultant field in +the region it occupies. Thus a field can be viewed as a possibility of +action, but a possibility which represents an actuality. + +It is to be noted that the two alternative views of causation are +here both included. The complete field within any region of space +depends on the past histories of all the electrons, histories extending +backwards in proportion to their distances. Also this dependence can +be conceived as a transmission. But viewing the cause which effects +changes on the electron within that region, it is solely that field +within the region, which field is coincident with that electron both in +time and in space. + +This process of conceiving the actuality underlying a possibility is +the uniform process by which regularity and permanence is introduced +into scientific thought, namely, we proceed from the actuality of the +fact to the actuality of possibility. + +In conformity with this principle, propositions are the outgrowth +from actual thought-expressions, thought-objects of perceptions from +crude sense-objects, hypothetical thought-objects of perception from +actual thought-objects of perception, material points from hypothetical +infinite suites of hypothetical thought-objects of perception, +ideal points from material points, thought-objects of science from +thought-objects of perception, fields of electrons from actual mutual +reactions of actual electrons. + +The process is a research for permanence, uniformity, and simplicity +of logical relation. But it does not issue in simplicity of internal +structure. Each ultimate thought-object of science retains every +quality attributed to the whole scientific universe, but retains them +in a form characterised by permanence and uniformity. + + + _V. Conclusion_ + +We commenced by excluding judgments of worth and ontological judgments. +We conclude by recalling them. Judgments of worth are no part of +the texture of physical science, but they are part of the motive of +its production. Mankind have raised the edifice of science, because +they have judged it worth while. In other words, the motives involve +innumerable judgments of value. Again, there has been conscious +selection of the parts of the scientific field to be cultivated, and +this conscious selection involves judgments of value. These values may +be æsthetic, or moral, or utilitarian, namely, judgments as to the +beauty of the structure, or as to the duty of exploring the truth, or +as to utility in the satisfaction of physical wants. But whatever the +motive, without judgments of value there would have been no science. + +Again, ontological judgments were not excluded by reason of any lack +of interest. They are in fact presupposed in every act of life: in our +affections, in our self-restraints, and in our constructive efforts. +They are presupposed in moral judgments. The difficulty about them is +the absence of agreement as to the method of harmonising the crude +judgments of commonsense. + +Science does not diminish the need of a metaphysic. Where this need is +most insistent is in connection with what above has been termed "the +actuality underlying a possibility." A few words of explanation may +render the argument clearer, although they involve a rash approach +to metaphysical heights which it is not the purpose of this paper to +explore. + +The conception of subject and object in careless discussion covers +two distinct relations. There is the relation of the whole perceiving +consciousness to part of its own content, for example, the relation +of a perceiving consciousness to an object of redness apparent to it. +There is also the relation of a perceiving consciousness to an entity +which does not exist in virtue of being part of the content of that +consciousness. Such a relation, so far as known to the perceiving +consciousness, must be an inferred relation, the inference being +derived from an analysis of the content of the perceiving consciousness. + +The bases for such inferences must be elements in consciousness +directly known as transcending their immediate presentation in +consciousness. Such elements are universal logical truths, moral and +æsthetic truths, and truths embodied in hypothetical propositions. +These are the immediate objects of perception which are other than the +mere affections of the perceiving subject. They have the property of +being parts of the immediate presentations for individual subjects and +yet more than such parts. All other existence is inferred existence. + +In this chapter we are more directly concerned with truths embodied +in hypothetical propositions. Such truths must not be confused with +any doubtfulness which attaches to our judgments of the future course +of natural phenomena. A hypothetical proposition, like a categorical +judgment, may or may not be doubtful. Also like a categorial judgment, +it expresses a fact. This fact is twofold: as a presentation in +consciousness, it is just this hypothetical judgment; as expressing a +categorical fact, it states a relation which lies beyond consciousness, +holding between entities thereby inferred. + +But this metaphysical analysis, short though it be, is probably wrong, +and at the best will only command very partial assent. Certainly; +and this admission brings out the very point which I wished to make. +Physical science is based on elements of thought, such as judgments +registering actual perceptions, and judgments registering hypothetical +perceptions which under certain circumstances would be realised. +These elements form the agreed content of the apparatus of commonsense +thought. They require metaphysical analysis; but they are among the +data from which metaphysics starts. A metaphysic which rejects them +has failed, in the same way as physical science has failed when it is +unable to harmonise them into its theory. + +Science only renders the metaphysical need more urgent. In itself +it contributes little directly to the solution of the metaphysical +problem. But it does contribute something, namely, the exposition of +the fact that our experience of sensible apparent things is capable of +being analysed into a scientific theory, a theory not indeed complete, +but giving every promise of indefinite expansion. This achievement +emphasises the intimate relation between our logical thought and the +facts of sensible apprehension. Also the special form of scientific +theory is bound to have some influence. In the past false science +has been the parent of bad metaphysics. After all, science embodies +a rigorous scrutiny of one part of the whole evidence from which +metaphysicians deduce their conclusions. + + +FOOTNOTES: + +[Footnote 3: Cf. _Révue de Métaphysique et de Morale_, May 1916, +where this question is dealt with by the author at the end of an +article, "La théorie relationniste de l'espace."] + + + + + CHAPTER VIII + + SPACE, TIME, AND RELATIVITY + + (_Paper read to Section A at the Manchester Meeting of the British + Association, 1915, and later before the Aristotelian Society_) + + +FUNDAMENTAL Problems concerning space and time have been considered +from the standpoints created by many different sciences. The object of +this paper is the humble one of bringing some of these standpoints into +relation with each other. This necessitates a very cursory treatment of +each point of view. + +Mathematical physicists have evolved their theory of relativity to +explain the negative results of the Morley-Michelson experiment and of +the Trouton experiment. Experimental psychologists have considered the +evolution of spatial ideas from the crude sense-data of experience. +Metaphysicians have considered the majestic uniformity of space and +time, without beginning and without end, without boundaries, and +without exception in the truths concerning them; all these qualities +the more arresting to our attention from the confused accidental +nature of the empirical universe which is conditioned by them. +Mathematicians have studied the axioms of geometry, and can now deduce +all that is believed to be universally true of space and of time by the +strictest logic from a limited number of assumptions. + +These various lines of thought have been evolved with surprisingly +little interconnection. Perhaps it is as well. The results of science +are never quite true. By a healthy independence of thought perhaps we +sometimes avoid adding other people's errors to our own. But there can +be no doubt that the normal method of cross-fertilising thought is by +considering the same, or allied problems to our own, in the form which +they assume in other sciences. + +Here I do not propose to enter into a systematic study of these various +chapters of science. I have neither the knowledge nor the time. + +First, let us take the ultimate basis of any theory of relativity. +All space measurement is from stuff in space to stuff in space. The +geometrical entities of empty space never appear. The only geometrical +properties of which we have any direct knowledge are properties of +those shifting, changeable appearances which we call things in space. +It is the sun which is distant, and the ball which is round, and the +lamp-posts which are in linear order. Wherever mankind may have got its +idea of an infinite unchangeable space from, it is safe to say that it +is not an immediate deliverance of direct observation. + +There are two antagonistic philosophical ways of recognising this +conclusion. + +One is to affirm that space and time are conditions for sensible +experience, that without projection into space and time sensible +experience would not exist. Thus, although it may be true to say that +our knowledge of space and time is given in experience, it is not true +to say that it is deduced from experience in the same sense that the +Law of Gravitation is so deduced. It is not deduced, because in the act +of experiencing we are necessarily made aware of space as an infinite +given whole, and of time as an unending uniform succession. This +philosophical position is expressed by saying that space and time are +_a priori_ forms of sensibility. + +The opposed philosophical method of dealing with the question is +to affirm that our concepts of time and space are deductions from +experience, in exactly the same way as the Law of Gravitation is such +a deduction. If we form exact concepts of points, lines and surfaces, +and of successive instants of time, and assume them to be related as +expressed by the axioms of geometry and the axioms for time, then we +find that we have framed a concept which, with all the exactness of +which our observations are capable, expresses the facts of experience. + +These two philosophic positions are each designed to explain a +certain difficulty. The _a priori_ theory explains the absolute +universality ascribed to the laws of space and time, a universality +not ascribed to any deduction from experience. The experiential theory +explains the derivation of the space-time concepts without introducing +any other factors beyond those which are admittedly present in framing +the other concepts of physical science. + +But we have not yet done with the distinctions which in any discussion +of space or time must essentially be kept in mind. Put aside the +above question as to how these space-time concepts are related to +experience--What are they when they are formed? + +We may conceive of the points of space as self-subsistent entities +which have the indefinable relation of being occupied by the ultimate +stuff (matter, I will call it) which is there. Thus, to say that the +sun is _there_ (wherever it is) is to affirm the relation of +occupation between the set of positive and negative electrons which +we call the sun and a certain set of points, the points having an +existence essentially independent of the sun. This is the absolute +theory of space. The absolute theory is not popular just now, but it +has very respectable authority on its side--Newton, for one--so treat +it tenderly. + +The other theory is associated with Leibniz. Our space concepts are +concepts of relations between things in space. Thus there is no such +entity as a self-subsistent point. A point is merely the name for some +peculiarity of the relations between the matter which is, in common +language, said to be in space. + +It follows from the relative theory that a point should be definable in +terms of the relations between material things. So far as I am aware, +this outcome of the theory has escaped the notice of mathematicians, +who have invariably assumed the point as the ultimate starting ground +of their reasoning. Many years ago I explained some types of ways in +which we might achieve such a definition, and more recently have added +some others. Similar explanations apply to time. Before the theories +of space and time have been carried to a satisfactory conclusion on +the relational basis, a long and careful scrutiny of the definitions +of points of space and instants of time will have to be undertaken, +and many ways of effecting these definitions will have to be tried and +compared. This is an unwritten chapter of mathematics, in much the same +state as was the theory of parallels in the eighteenth century. + +In this connection I should like to draw attention to the analogy +between time and space. In analysing our experience we distinguish +events, and we also distinguish things whose changing relations form +the events. If I had time it would be interesting to consider more +closely these concepts of events and of things. It must suffice now to +point out that things have certain relations to each other which we +consider as relations between the space extensions of the things; for +example, one space can contain the other, or exclude it, or overlap +it. A point in space is nothing else than a certain set of relations +between spatial extensions. + +Analogously, there are certain relations between events which we +express by saying that they are relations between the temporal +durations of these events, that is, between the temporal extensions of +the events. [The durations of two events A and B may one precede the +other, or may partially overlap, or may one contain the other, giving +in all six possibilities.] The properties of the extension of an event +in time are largely analogous to the extension of an object in space. +Spatial extensions are expressed by relations between objects, temporal +extensions by relations between events. + +The point in time is a set of relations between temporal extensions. +It needs very little reflection to convince us that a point in time +is no direct deliverance of experience. We live in durations, and not +in points. But what community, beyond the mere name, is there between +extension in time and extension in space? In view of the intimate +connection between time and space revealed by the modern theory of +relativity, this question has taken on a new importance. + +I have not thought out an answer to this question. I suggest, however, +that time and space embody those relations between objects on which +depends our judgment of their externality to ourselves. Namely, +location in space and location in time both embody and perhaps +necessitate a judgment of externality. This suggestion is very vague, +and I must leave it in this crude form. + + + _Diverse Euclidean Measure Systems_ + +Turning now to the mathematical investigations on the axioms of +geometry, the outcome, which is most important for us to remember, +is the great separation which it discloses between non-metrical +projective geometry, and metrical geometry. Non-metrical projective +geometry is by far the more fundamental. Starting with the concepts +of points, straight lines, and planes (of which not all three need +be taken as indefinable), and with certain very simple non-metrical +properties of these entities--such as, for instance, that two points +uniquely determine a straight line--nearly the whole of geometry can +be constructed. Even quantitative coordinates can be introduced, to +facilitate the reasoning. But no mention of distance, area, or volume, +need have been introduced. Points will have an order on the line, but +order does not imply any settled distance. + +When we now inquire what measurements of distance are possible, we find +that there are different systems of measurement all equally possible. +There are three main types of system: any system of one type gives +Euclidean geometry, any system of another type gives Hyperbolic (or +Lobatchewskian) geometry, any system of the third type gives Elliptic +geometry. Also different beings, or the same being if he chooses, +may reckon in different systems of the same type, or in systems of +different types. Consider the example which will interest us later. +Two beings, A and B, agree to use the same three intersecting lines +as axes of _x_, _y_, _z_. They both employ a system of measurement of +the Euclidean type, and (what is not necessarily the case) agree as to +the plane at infinity. That is, they agree as to the lines which are +parallel. Then with the usual method of rectangular Cartesian axes, +they agree that the coordinates of P are the lengths ON, NM, MP. So +far all is harmony. A fixes on the segment OU_{1}, on O_x_, as being +the unit length, and B on the segment OV_{1}, on O_x_. A calls his +coordinates (_x_, _y_, _z_), and B calls them (X, Y, Z). + +Then it is found [since both systems are Euclidean] that, whatever +point P be taken, + +X = β_x_, Y = γ_y_, Z = δ_z_. [β ≠ γ ≠ δ.] + +They proceed to adjust their differences, and first take the +_x_-coordinates. Obviously they have taken different units of +length along O_x_. The length OU_{1}, which A calls one unit, B +calls β units. B changes his unit length to OU_{1}, from its original +length OV_{1}, and obtains X = _x_. But now, as he must use the +same unit for all his measurements, his other coordinates are altered +in the same ratio. Thus we now have + +X = _x_, Y = γ_y_/β, Z = δ_z_/β. + +The fundamental divergence is now evident. A and B agree as to their +units along O_x_. They settled that by taking along that axis a +given segment OU_{1} as having the unit length. But they cannot agree +as to what segment along O_y_ is equal to OU_{1}. A says it is +OU_{2}, and B that it is OU_{2}′. Similarly for lengths along OZ. + +The result is that A's spheres + +_x_^2 + _y_^2 + _z_^2 = _r_^2, + +are B's ellipsoids, + + X^2 + β^2Y^2/γ^2 + β^2Z^2/δ^2 = _r_^2, +_i. e._ X^2/β^2 + Y^2/γ^2 + Z^2/δ^2 = _r_^2/β^2. + +Thus the measurement of angles by the two is hopelessly at variance. + +If β ≠ γ ≠ δ, there is one, and only one, set of common rectangular +axes at O, namely that from which they started. If γ = δ, but β ≠ γ, +then there are a singly infinite number of common rectangular axes +found by rotating the axes round O_x_. This is, for us, the +interesting case. The same phenomena are reproduced by transferring to +any parallel axes. + +The root of the difficulty is, that A's measuring rod, which for him is +a rigid invariable body, appears to B as changing in length when turned +in different directions. Similarly all measuring rods, satisfactory +to A, violate B's immediate judgment of invariability, and change +according to the same law. There is no way out of the difficulty. +Two rods ρ and σ coincide whenever laid one on the other; ρ is held +still, and both men agree that it does not change. But σ is turned +round. A says it is invariable, B says it changes. To test the matter, +ρ is turned round to measure it, and exactly fits it. But while A is +satisfied, B declares that ρ has changed in exactly the same way as did +σ. Meanwhile B has procured two material rods satisfactory to him as +invariable, and A makes exactly the same objections. + +We shall say that A and B employ diverse Euclidean metrical systems. + +The most extraordinary fact of human life is that all beings seem to +form their judgments of spatial quantity according to the same metrical +system. + + + _Relativity in Modern Physics_ + +Owing to the fact that points of space are incapable of direct +recognition, there is a difficulty--apart from any abstract question +of the nature of space--in deciding on the motion to be ascribed to +any body. Even if there be such a thing as absolute position, it is +impossible in practice to decide directly whether a body's absolute +position has changed. All spatial measurement is relative to matter. + +Newton's laws of motion in their modern dress evade this difficulty +by asserting that a framework of axes of coordinates can be defined +by their relations to matter such that, assuming these axes to be at +rest, and all velocities to be measured relatively to them, the laws +hold. The same expedient has to be employed for time, namely, the laws +hold when the measurement of the flow of time is made by the proper +reference to periodic events. Thus the laws assert that the framework +and the natural clock adapted for their use have been successfully +found. + +But, if one framework will do, an infinity of others serve equally +well; namely, not only--as is of course the case--all those at rest +relatively to the first framework, but also all those which move +without relative rotation with uniform velocity relatively to the +first. This whole set of frameworks is on a level in respect to +Newton's laws. We will call them Dynamical frameworks. + +Now, suppose there are two observers, A and B. They agree in their +non-metrical projective geometry, _e.g._, what A calls a straight +line so does B. They also both apply a Euclidean metrical system of +measurement to this space. Their two metrical systems also agree in +having the same plane at infinity, that is, lines which are parallel +for A are also parallel for B. Furthermore, they have both successfully +applied Newton's laws to the movement of matter, and agree in having +the same sets of dynamical axes. But the framework (among these sets) +which A chooses to regard as at rest is different from the frame (among +the same sets) which B so regards. + +Without alteration of their respective judgment of rest, they choose +their co-ordinate axes so that the origins (O for A, and O′ for B) are +in relative motion along OO′, which is the axis of _x_ for both. + +Further, since OO′ is the line of symmetry of their diverse Euclidean +systems, we assume that the two measure-systems agree for planes +perpendicular to OO′, _i.e._, we assume a symmetry round OO′. +Then if, for A at O, the distance OO′ be ξ, the relations at any +instant between A's coordinates (_x_, _y_, _z_) and B's +coordinates (_x′_, _y′_, _z′_) for the same point P are +given by + +_x′_ = β(_x_ - ξ), _y′_ = _y_, _z′_ = _z_. + +Also, according to A's clock, O′ is moving forward with a uniform +velocity _v_, and we measure A's time from the instant of the +coincidence of O and O′. + +Thus + +ξ = _vt_, + +and + +_x′_ = β(_x_ - _vt_), _y′_ = _y_, _z′_ = _z_. + +We now consider B's clock, and ask for the most general supposition +which is consistent with the fact that their judgments as to the fact +of uniform motion are in agreement. + +We do not assume that events in various parts of space which A +considers to be simultaneous are so considered by B. But we assume that +at any point P, with coordinates (_x_, _y_, _z_) for A, there is a +determinate relation between B's time T and _x_, _y_, _z_, _t_. + +Put + +T = ƒ(_x_, _y_, _z_, _t_). + +Write + +P = δT/δ_x_, Q = δT/δ_y_, R = δT/δ_z_, S = δT/δ_t_. + +Now suppose that the point P is moving, and that (_u_{1}, +_u_{2}, _u_{3}) is its set of component velocities along +the axes according to A's "space and clock" system, and (U_{1}, U_{2}, +U_{3}) is its set of component velocities according to B's "space and +clock" system. Then by mere differentiation it follows after a short +mathematical deduction that + +U_{1} = {(_d_β/_dt_)(_x_ - _vt_) + β(_u_{1} - _v_)}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}, +U_{2} = _u_{2}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}, +U_{3} = _u_{3}/{P_u_{1} + Q_u_{2} + R_u_{3} + S}. + +But we have assumed that, whatever the direction of the resultant +velocity (_u_{1}, _u_{2}, _u_{3}), the velocities (U_{1}, U_{2}, U_{3}) +and (_u_{1}, _u_{2}, _u_{3}) are both uniform when either is uniform. + +Hence it is easily proved that β, P, Q, R, S are independent of the +coordinates (_x_, _y_, _z_) and of the time _t_. In other words, they +are constant. + +Hence we obtain + +U_{1} = β(_u_{1} - _v_)/{P_u_{1} + Q_u_{2} + R_u_{3} + S}, + +and + +T = P_x_ + Q_y_ + R_z_ + S_t_. + +But we assumed that OO′, _i.e._, O_x_, is an axis of +symmetry. It follows from this assumption that + +Q = R = 0. + +We thus obtain the simplified results + +T = P_x_ + S_t_, } +U_{1} = β(_u_{1} - _v_)/(P_u_{1} + S), } (I) +U_{2} = _u_{2}/(P_u_{1} + S), } +U_{3} = _u_{3}/(P_u_{1} + S). } + +Here we remember that (_u_{1}, _u_{2}, _u_{3}) are +the velocities of any particle according to A's "space and clock" +system, and that (U_{1}, U_{2}, U_{3}) are the velocities of the same +point according to B's "space and clock" system. We have obtained the +most general relations consistent with the facts that (1) they both +employ Euclidean systems, related as described above, and (2) they +agree in their judgments on the uniformity of velocity. + +We now compare their judgments on the magnitudes of velocities. + +Let the magnitude of the velocity of P be V according to A's judgment, +and V′ according to B's’ judgment. + +Then + + V^2 = _u_{1}^2 + _u_{2}^2 + _u_{3}^2, +V′^2 = U_{1}^2 + U_{2}^2 + U_{3}^2. + +Also we can put + +_u_{1} = _l_V, _u_{2} = _m_V, _u_{3} = _n_V, + +where (_l_, _m_, _n_) have nothing to do with the magnitude V, but +simply depend on the direction of motion. In fact (_l_, _m_, _n_) are +the "direction cosines" of the velocity according to A's judgment. By +substituting in the above equation for V^2 we see that + +_l_^2 + _m_^2 + _n_^2 = 1. + +Now, substituting for (_u_{1}, _u_{2}, _u_{3}) in +the equations (I) above, and squaring and adding, and eliminating +_m_^2 + _n_^2 by the relation just found, we at once find + +V′^2 = ((β^2 - 1)V^2_l_^2 - 2β^2V_vl_ + β^2_v_^2 + V^2)/(PV_l_ + S)^2. + +It is thus seen that in general the relation of V′ to V depends on the +direction cosine _l_. Now _l_ is the cosine of the angle +which the direction of the velocity V makes with O_x_, according +to A's judgment. + +The meaning of this relation is, that if A discharges, from guns at +the point P, shells with a given muzzle velocity V according to his +judgment, B will consider that their muzzle velocities are different +from each other, except in the case of pairs of guns equally inclined +to the axis OO′. Instances of this type of diversity of judgment can +be noted any day by any one who looks out of the window of a railway +carriage, and forgets that he is travelling. + +Now, suppose the velocity V′ bears a relation to the velocity V, which +is independent of _l_. Then _l_ must disappear from the above +formula. There are two conditions to be satisfied + +One condition is + +V^2 = β^2_v_^2/(β^2 - 1), + +or in a more convenient form + +β^2 = 1/(1 - _v_^2/V^2). + +The meaning of this condition is, that there is one, and only one, +muzzle velocity V (according to A's judgment), namely, the muzzle +velocity given by the above formula, which can have the property that +B will judge that all the guns are firing in their diverse directions +with one common muzzle velocity. + +Let us now suppose that V has this peculiar value: that is, if we look +on this value V as known, we must suppose that β is given by the second +of the above formulæ. + +The other condition allows P and S to be put in the forms + +P = -β_v_/λV^2, S = β/λ, + +where + +V′ = λV. + +Thus we have the bundle of formulæ + +β^2 = 1/(1 - _v_^2/V^2), +T = β{_t_ - _vx_/V^2}/λ, +V′ = λV. + +The value which we give to λ is purely a matter for the adjustment of +units. If we want A and B to agree in their judgments of the magnitude +of this peculiar muzzle-velocity, we put λ = 1. + +We then get the formulæ usually adopted, namely + +β^2 = 1/(1 - _v_^2/V^2), } +T = β{_t_ - _vx_/V^2}, } (II) +V′ = V. } + +But if we prefer that A and B should reckon (according to A's judgment) +in the same units of time, we put λ = β, and obtain + +β^2 = 1/(1 - _v_^2/V^2), +T = (_t_ - _vx_)/V^2, +V′ = βV. + +But A and B are in any case in such hopeless difficulties over their +comparisons of time-judgments that the detail of using the same units +does not help them much. Accordingly the formulæ marked (II) are those +used. Thus A and B agree in their judgments as to the magnitude of one +special velocity V, whatever may be the direction in which the entity +possessing it is moving. + +In order to reach this measure of agreement, they have to disagree as +to their space judgments and their time judgments. The root cause of +their disagreement is their diverse judgment as to which axis system is +to be taken at rest for the purpose of measuring velocities. + +Before discussing the nature of the disagreement disclosed in formula +(II), let us ask why we should bring these difficulties on our heads by +supposing that two people in relative motion, who both (for the purpose +of measuring velocities) assume that they are at rest, should agree in +their judgments in respect to this special velocity V. + +Such an agreement has no counterpart in any of our obvious judgments +made from railway carriages. Surely we can wait till the contingency +occurs before discussing the confusion which it creates. + +But the contingency has occurred. It occurs when we consider the +velocity of light. Perhaps I may venture to remind a philosophical +society that light moves so very quickly that it is difficult to +consider its velocity at all. So we need not be surprised that this +peculiar fact concerning its velocity is not more obvious. + +Now V being the velocity of light, unless _v_ is large, _v_/V +(and still more _v_^2/V^2) will be quite inappreciable. The only +velocity ready to hand which is big enough to give _v_/V an +appreciable value is the velocity of the earth in its orbit. + +Many diverse experiments have been made, and they all agree in +concluding that a man who assumes the earth to be at rest will find by +measurement that the velocity of light is the same in all directions. +Furthermore, when the same man turns his attention to interstellar or +interplanetary phenomena, and assumes the sun to be at rest, he will +again find the velocity of light to be the same in all directions. +These are well-attested experiments made at long intervals of time. + +This is the exact contingency contemplated above. + +Again the velocity of light _in vacuo_ has recently taken on a +new dignity. It used to be one among other wave velocities such as +the velocity of sound in air, or in water, or the velocity of surface +waves in water. But Clerk Maxwell discovered that all electromagnetic +influences are propagated with the velocity of light, and now modern +physical science half suspects that electromagnetic influences are +the only physical influences which relate the changes in the physical +world. Accordingly the velocity of light becomes the fundamental +natural velocity, and experiment shows that our judgment of its +magnitude is not affected by our choice of the framework at rest, so +long as we keep to a set of dynamical axes. These experiments on light +have been confirmed by other electromagnetic experiments not involving +light. + +Thus we are driven to equations (II), where V is the velocity of light. + +The first conclusion to be drawn from equations (II) is that two people +who make different choices of bodies at rest will disagree as to +their measuring rods in the way described above. There is no peculiar +difficulty about that. The only wonder is that all people agree so well +in their judgments as to metrical systems. A mathematical angel would +naturally expect incarnate men to be in violent disagreement on this +subject. + +But the case of time is different. For simplicity of statement we speak +of A as at O, and B as at O′. We remember that O′ is moving relatively +to O with velocity _v_ in direction OO′. Suppose A and B are +looking in this direction; and they both measure their time from the +instant when they met, as O′ passed over O. Then we have + +T = ((_t_ - _vx_)/V^2)/((1 - _v_^2)/V^2). + +Now, suppose we consider all the events all over space which A +considers to have happened simultaneously at the time _t_. The +events of this set which occurred anywhere on a plane perpendicular to +OO′ at a distance _x_ in front of O (according to A's reckoning), +will have occurred according to B's reckoning at the time T as given +above. Let us fix our attention on the fact that B does not consider +all these events to be simultaneous. For let T_{1}, and T_{2} be B's +times for such events on planes _x__{1}, and _x__{2}. Then + +T_{1} - T_{2} = _v_(_x__{2} - _x__{1})/(V^2 - _v_^2). + +Thus if _x__{2} be greater than _x__{1}, T_{2} is less than +T_{1}. Thus B judges the more distant events in front of him to have +happened earlier than the nearer events in front of him, and _vice +versa_ for the events behind. This disturbance of the judgment of +simultaneity is the fundamental fact. Obviously the measurement of time +intervals is a detail compared to simultaneity. A may think a sermon +long, and B may think it short, but at least they should both agree +that it stopped when the clock hand pointed at the hour. The worst of +the matter is that so far as any test can be applied there is no method +of discriminating between the validities of their judgments. + +Thus we are confronted with two distinct concepts of the common world, +A's space-time concept, and B's space-time concept. Who is right? It is +no use staying for an answer. We must follow the example of the wise +old Roman, and pass on to other things. + +Thus estimates of quantity in space and time, and, to some extent, even +estimates of order, depend on the individual observer. But what are +the crude deliverances of sensible experience, apart from that world +of imaginative reconstruction which for each of us has the best claim +to be called our real world? Here the experimental psychologist steps +in. We cannot get away from him. I wish we could, for he is frightfully +difficult to understand. Also, sometimes his knowledge of the +principles of mathematics is rather weak, and I sometimes suspect---- +No, I will not say what I sometimes think: probably he, with equal +reason, is thinking the same sort of thing of us. + +I will, however, venture to summarise conclusions, which are, I +believe, in harmony with the experimental evidence, both physical and +psychological, and which are certainly suggested by the materials for +that unwritten chapter in mathematical logic which I have already +commended to your notice. The concepts of space and time and of +quantity are capable of analysis into bundles of simpler concepts. In +any given sensible experience it is not necessary, or even usual, that +the whole complete bundle of such concepts apply. For example, the +concept of externality may apply without that of linear order, and the +concept of linear order may apply without that of linear distance. + +Again, the abstract mathematical concept of a space-relation +may confuse together distinct concepts which apply to the given +perceptions. For example, linear order in the sense of a linear +projection from the observer is distinct from linear order in the sense +of a row of objects stretching across the line of sight. + +Mathematical physics assumes a given world of definitely related +objects, and the various space-time systems are alternative ways of +expressing those relations as concepts in a form which also applies to +the immediate experience of observers. + +Yet there must be one way of expressing the relations between objects +in a common external world. Alternative methods can only arise as the +result of alternative standpoints; that is to say, as the result of +leaving something added by the observer sticking (as it were) in the +universe. + +But this way of conceiving the world of physical science, as composed +of hypothetical objects, leaves it as a mere fairy tale. What is really +actual are the immediate experiences. The task of deductive science +is to consider the concepts which apply to these data of experience, +and then to consider the concepts relating to these concepts, and +so on to any necessary degree of refinement. As our concepts become +more abstract, their logical relations become more general, and less +liable to exception. By this logical construction we finally arrive +at conceptions, (i) which have determinate exemplifications in the +experience of the individuals, and (ii) whose logical relations have +a peculiar smoothness. For example, conceptions of mathematical time, +of mathematical space, are such smooth conceptions. No one lives in +"an infinite given whole," but in a set of fragmentary experiences. +The problem is to exhibit the concepts of mathematical space and time +as the necessary outcome of these fragments by a process of logical +building up. Similarly for the other physical concepts. This process +builds a common world of conceptions out of fragmentary worlds of +experience. The material pyramids of Egypt are a conception, what is +actual are the fragmentary experiences of the races who have gazed on +them. + +So far as science seeks to rid itself of hypothesis, it cannot go +beyond these general logical constructions. For science, as thus +conceived, the divergent time orders considered above present no +difficulty. The different time systems simply register the different +relations of the mathematical construct to those individual +experiences (actual or hypothetical) which could exist as the crude +material from which the construct is elaborated. + +But after all it should be possible so to elaborate the mathematical +construct so as to eliminate specific reference to particular +experiences. Whatever be the data of experience, there must be +something which can be said of them as a whole, and that something is +a statement of the general properties of the common world. It is hard +to believe that with proper generalisation time and space will not be +found among such properties. + + + _Commentary added on reading the Paper before the Aristotelian + Society_ + +The first six pages of the paper consist of a summary of ideas which +ought to be in our minds while considering problems of time and space. +The ideas are mostly philosophical, and the summary has been made by +an amateur in that science; so there is no reason to ascribe to it any +importance except that of a modest reminder. There are only two points +in this summary to which I would draw attention. + +On pp. 192 and 193 there occurs-- + +"Wherever mankind ... unending uniform succession." + +If I understand Kant rightly--which I admit to be very +problematical--he holds that in the act of experience we are aware +of space and time as ingredients necessary for the occurrence of +experience. I would suggest--rather timidly--that this doctrine should +be given a different twist, which in fact turns it in the opposite +direction--namely, that in the act of experience we perceive a whole +formed of related differentiated parts. The relations between these +parts possess certain characteristics, and time and space are the +expressions of some of the characteristics of these relations. Then the +generality and uniformity which are ascribed to time and space express +what may be termed the uniformity of the texture of experience. + +The success of mankind--modest though it is--in deducing uniform laws +of nature is, so far as it goes, a testimony that this uniformity of +texture goes beyond those characteristics of the data of experience +which are expressed as time and space. Time and space are necessary +to experience in the sense that they are characteristics of our +experience; and, of course, no one can have our experience without +running into them. I cannot see that Kant's deduction amounts to much +more than saying that "what is, is"--true enough, but not very helpful. + +But I admit that what I have termed the "uniformity of the texture of +experience" is a most curious and arresting fact. I am quite ready to +believe that it is a mere illusion; and later on in the paper I suggest +that this uniformity does not belong to the immediate relations of the +crude data of experience, but is the result of substituting for them +more refined logical entities, such as relations between relations, or +classes of relations, or classes of classes of relations. By this means +it can be demonstrated--I think--that the uniformity which must be +ascribed to experience is of a much more abstract attenuated character +than is usually allowed. This process of lifting the uniform time and +space of the physical world into the status of logical abstractions has +also the advantage of recognising another fact, namely, the extremely +fragmentary nature of all direct individual experience. + +My point in this respect is that fragmentary individual experiences +are all that we know, and that all speculation must start from these +_disjecta membra_ as its sole datum. It is not true that we are +directly aware of a smooth running world, which in our speculations we +are to conceive as given. In my view the creation of the world is the +first unconscious act of speculative thought; and the first task of a +self-conscious philosophy is to explain how it has been done. + +There are roughly two rival explanations. One is to assert the world +as a postulate. The other way is to obtain it as a deduction, not a +deduction through a chain of reasoning, but a deduction through a chain +of definitions which, in fact, lifts thought on to a more abstract +level in which the logical ideas are more complex, and their relations +are more universal. In this way the broken limited experiences sustain +that connected infinite world in which in our thoughts we live. There +are three more remarks while on this point I wish to make-- + +(i) The fact that immediate experience is capable of this deductive +superstructive must mean that it itself has a certain uniformity of +texture. So this great fact still remains. + +(ii) I do not wish to deny the world as a postulate. Speaking without +prejudice, I do not see how in our present elementary state of +philosophical advance we can get on without middle axioms, which, in +fact, we habitually assume. + +My position is, that by careful scrutiny we should extrude such +postulates from every part of our organised knowledge in which it is +possible to do without them. + +Now, physical science organises our knowledge of the relations between +the deliverances of our various senses. I hold that in this department +of knowledge such postulates, though not entirely to be extruded, can +be reduced to a minimum in the way which I have described. + +I have not the slightest knowledge of theories respecting our emotions, +affections, and moral sentiments, and I can well believe that in +dealing with them further postulates are required. And in practice I +recognise that we all make such postulates, uncritically. + +(iii) The next paragraphs on pp. 193 and 194 are as follows-- + +"The opposed philosophical method ... physical science." + +It will be noted that, in the light of what has just been stated, the +first of these paragraphs (which, I hope, faithfully expresses the +experimental way of approaching the problem) really obscures the point +which I have been endeavouring to make. The phrase, "If we form the +exact concepts of points, etc.," is fatally ambiguous as between the +method of postulating entities with assigned relations, and the method +of forming logical constructions, and thus reaching points, etc., as +the result of a chain of definitions. + +Turning now to pp. 194-195, we come to the following paragraphs-- + +"The other theory ... eighteenth century." + +We note again that the relational theory of space from another point +of view brings us back to the idea of the fundamental space-entities +as being logical constructs from the relations between things. The +difference is, that this paragraph is written from a more developed +point of view, as it implicitly assumes the things in space, and +conceives space as an expression of certain of their relations. +Combining this paragraph with what has gone before, we see that the +suggested procedure is first to define "things" in terms of the data of +experience, and then to define space in terms of the relations between +things. + +This procedure is explicitly assumed in the next short paragraph: "In +this connection ... from the events." + +The gist of the remaining paragraphs of this section is contained in +the paragraph at the bottom of p. 196: "The point in time ... new +importance." + +The sentence, "We live in durations, and not in points," can be +amplified by the addition, "We live in space-extensions and not in +space-points." + +It must be noted that "whole and part" as applied to extensions in +space or time must be different from the "all and some" of logic, +unless we admit points to be the fundamental entities. For "spatial +whole and spatial part" can only mean "all and some" if they really +mean "all the points and some of the points." But if extensions and +their relations are more fundamental than points, this interpretation +is precluded. I suggest that "spatial whole and spatial part" is +intimately connected with the fundamental relation between things from +which our space ideas spring. + +The relation of space whole to space part has many formal properties +which are identical with the properties of "all and some." Also when +points have been defined, we can replace it by the conception of "all +the points and some of the points." But the confusion between the two +relations is fatal to sound views on the subject. + + + _Diverse Euclidean Measure Systems_ + +The next section deals with the measure systems applicable to space. + +A measure system is a group of congruent transformations of space +into itself. Consider a rigid body occupying all space. Let this body +be moved in any way so that the particles of the body which occupied +points P_{1}, P_{2}, P_{3}, etc., now occupy points Q_{1}, Q_{2}, +Q_{3}, etc. Then any point P_{1} in space is uniquely related to the +corresponding point Q_{1} in space by a one-to-one transformation with +certain characteristics. By the aid of these transformations we can +achieve the definition of distance in a way which definitely determines +the distance between any two points, provided that we can define what +we mean by a congruent transformation without introducing the idea of +distance. If we introduced the idea of distance, we should simply say +that a congruent transformation is one which leaves all distances +unchanged, _i. e._, if P_{1}, P_{2} are transformed into Q_{1}, +Q_{2} then the distance P_{1}P_{2} is equal to the distance Q_{1}Q_{2}. + +But mathematicians have succeeded in defining congruent transformations +without any reference to distance. + +There are alternative groups of such congruent transformations, and +each group gives a different measure system for space. The distance +P_{1}P_{2} may equal the distance Q_{1}Q_{2} for one measure system, +and will not equal it for another measure system. All these different +measure systems are on the same level, equally applicable. A being +with a strong enough head could think of them all at once as applying +to space. The result so far as it interests us in respect to the +theory of relativity is explained on pp. 197-200, ending with "The +most extraordinary fact ... same metrical system." This final sentence +bears on Poincaré's assertion that the measure system adopted is +purely "conventional." I presume that by "conventional" a certain +arbitrariness of choice is meant; and in that case, I must express +entire dissent. It is true that within the circle of geometrical ideas +there is no means of giving any preference to any one measure system, +and any one is as good as any other. But it is not true that if we +look at a normal carriage wheel, and at an oval curve one foot broad +and ten feet long, we experience any arbitrariness of judgment in +deciding which has approximately the form of a circle. Accordingly to +Poincaré the choice between them, as representing a circle, is entirely +conventional. + +Again, we equally form immediate judgments as to whether a body is +approximately rigid. We know that a paving stone is rigid, and that a +concertina is not rigid. This again necessitates a determinate measure +system, selected from among the others. + +Accordingly we conclude that (i) each being does, in fact, employ a +determinate measure system, which remains the same, except possibly +for very small variations, and (ii) the measure systems of different +human beings agree, to within the limits of our observations. These +conclusions are not the less extraordinary because no plain man has +ever doubted them. + +It is an interesting subject to investigate exactly what are the +fundamental uniformities of experience which necessitate this +conclusion. It is not so easy as it looks, since we have to divest +ourselves of all aid of scientific hypothesis if our conclusions are to +be demonstrative. + + + _Relativity in Modern Physics_ + +Pp. 201-202, "Owing to the fact ... which B so regards." + +The fundamental formulæ for the theory of relativity are the relations +between diverse co-ordinate systems given on p. 203, and formulæ II +at the bottom of p. 207. The general explanation of one method in +which these formulæ arise--namely, Einstein's method--is given on +pp. 201-211. Namely, we seek the condition that for all dynamical +axes the velocity of light should be the same, and the same in all +directions. It should be noted that the experiments which, so far as +they go, confirm these formulæ, can also be explained in another way +which makes the theory of relativity unnecessary. We need only ascribe +to the ether a certain property of contraction in the direction of +motion, and the thing is done. So no one need be bludgeoned into +accepting the rather bizarre doctrine of relativity, nor indeed any +other scientific generalisation. The good old homely ether, which we +all know, can in this case serve the purpose. Just as an author of +genius, if he lives long enough, survives the inevitable accusation +of immorality, so the ether by dint of persistence has outlived all +reputation of extravagance. But if we detach ourselves from the glib +phraseology concerning it, the scientific ether is uncommonly like the +primitive explanation of the soul, as a little man inside us, which can +sometimes be caught escaping in the form of a butterfly. As soon as the +ether has to be patched up with special properties to explain special +experiments, its scientific use is problematical, and its philosophic +use is _nil_. + +Philosophically the ether seems to me to be an ambitious attempt +to give a complete explanation of the physical universe by making +an elephant stand on a tortoise. Scientifically it has a perfectly +adequate use by veiling the extremely abstract character of scientific +generalisations under a myth, which enables our imaginations to work +more freely. I am not advocating the extrusion of ether from our +scientific phraseology, even though at special points we have to +abandon it. + +But the key to the reasons why it is worth while to consider seriously +the doctrine of relativity is to be found on pp. 209, 210: "Again the +velocity of light ... not involving light." Namely, we have begun to +suspect that all physical influences require time for their propagation +in space. This generalisation is a long way from being proved. +Gravitation stands like a lion in the path. But if it be the case, then +all idea of an immediate presentation to us of an aspect of the world +as it in fact is, must necessarily be abandoned. What we perceive at +any instant is already ancient history, with the dates of the various +parts hopelessly mixed. + +We must add to this the difficulty of determining what is at rest +and what is in motion, and the further difficulty of determining a +definite uniform flow of time. It is no use discussing this matter +as though, but for the silly extravagant doctrine of relativity, +everything would be plain sailing. It isn't. You may be quite sure +that when, after prolonged study, you endeavour to give the simplest +explanation of a grave difficulty, you will be accused of extravagance. +I have no responsibility for the doctrine of relativity, and hold no +brief for it, but it has some claim to be considered as a comparatively +simple way out of a scientific maze. + +In the first place, we use the Newtonian dynamical sets of axes, +and the Newtonian clock to extricate ourselves partially from the +difficulties of rest, motion, and time. These have proved capable +of scientific determination within the limits of our experimental +accuracy. Thus the only thing left over is the choice of the axes +at rest, which is a completely indeterminate problem on Newtonian +principles. + +Again, so far as we can at present guess by adopting the theory +that all metrical influence is electromagnetic, all influences +are propagated with the velocity of light _in vacuo_. This +electromagnetic hypothesis is by no means established, but it gives +the simplest of all possible results in respect to the propagation of +influence, which we therefore adopt. + +But what dynamical axes are we taking as at rest? Now our practical +choice gives a range of relative velocities small compared to that of +light. So except for certain refined experiments it does not matter. +There are two possibilities-- + +(i) We may assume that one set of axes are at rest, and that the others +will show traces of motion in respect to the velocity of light; or-- + +(ii) That the velocity of light is the same in all directions whichever +be the dynamical axes assumed. + +The first supposition is negatived by experiment, and hence we are +driven to the second supposition; which immediately lands us in the +whole theory of relativity. + +But if we will not have this theory we must reject the earlier +supposition that the velocity of light _in vacuo_ is the same in +all directions. This we do, in fact, by assuming an ether, and assuming +a certain law for its modification. Then we, in fact, adopt the first +supposition so far as to hold that there are dynamical axes specially +at rest, namely, at rest relatively to the undisturbed ether. Then an +assumed law for the modification of the ether so alters the velocity of +light that we explain why no dynamical axes show traces of motion. + +I wish now to go back to the point which I made a few minutes ago, +that what we perceive at any instant is ancient history with its dates +hopelessly mixed. In the earlier part of my comments I emphasised the +point that our only data as to the physical world are our sensible +perceptions. We must not slip into the fallacy of assuming that we are +comparing a given world with given perceptions of it. The physical +world is in some general sense of the term a deduced concept. + +Our problem is, in fact, to fit the world to our perceptions, and not +our perceptions to the world. + + + PRINTED IN GREAT BRITAIN BY RICHARD CLAY & SONS, LIMITED, + BRUNSWICK ST., STAMFORD ST., S.E., AND BUNGAY, SUFFOLK. + + + + +From WILLIAMS & NORGATE'S LIST + + + =A SPIRITUAL PILGRIMAGE.= By the Rev. R. J. CAMPBELL. + (The Rev. R. J. Campbell's own Story of his Religious Life.) 2nd + Impression. Demy 8vo. =7s. 6d.= net. + + "This is eminently a book for which to be thankful--simple, + straightforward, kindly. Neither does it lack the saving grace of + humour."--_Guardian._ + + "Mr. Campbell devotes a chapter to his re-ordination, and it must be + admitted that his defence of this act is effective."--_Methodist + Times._ + + =EDINBURGH.= By the Right Hon. Sir HERBERT MAXWELL, + Bart., D.C.L., Author of "Life of the Duke of Wellington," "Scottish + Gardens," etc. President of the Society of Antiquaries of Scotland, + 1910-13, and Chairman of the Royal Commission on Scottish Historical + Monuments. With a Coloured Frontispiece and 64 pages of Illustrations + of the past and present city. Medium 8vo. =10s. 6d.= net. + + A History of the Scottish Capital from the earliest times to the + nineteenth century. + + =FURTHER PAGES OF MY LIFE.= By the Right Rev. W. BOYD + CARPENTER, K.C.V.O., D.D., D.C.L., Formerly Bishop of Ripon, + Author of "Some Pages of My Life," "The Witness of Religious + Experience," etc. With Illustrations. Medium 8vo. =10s. 6d.= net. + + Reflections mingled with intimate reminiscences and recollections of + Royal personages and of men eminent in many spheres. + + =SOME PAGES OF MY LIFE.= By the Right Rev. W. BOYD + CARPENTER. New and Cheaper Edition. Large post 8vo. =5s.= + net. + + All who have listened to his eloquent preaching will read with delight + his musing on what life has brought the author. + + =RAPHAEL MELDOLA=, Hon. D.Sc. (Oxon.), Hon. LL.D. (St. And.), + F.R.S., Professor of Chemistry in the City and Guilds of London + Technical College. Reminiscences of his worth and work, by those who + knew him, together with a Chronological List of his Publications + (1868-1915). Edited by JAMES MARCHANT. With a Preface by the + Right Hon. LORD MOULTON, K.C.B., F.R.S. With a Portrait. + Crown 8vo. =5s.= net. + + "The book is one of the most delightful biographies of scientific + men that have been published during recent years."--_Chemist and + Druggist._ + + =THE MANUFACTURE OF HISTORICAL MATERIAL.= An Elementary Study in + the Sources of Story. By J. W. JEUDWINE, LL.B. (Camb.), of + Lincoln's Inn, Barrister-at-Law, Author of "The First Twelve Centuries + of British Story," and other works. Crown 8vo. =6s.= net. + + A consideration of the successive phases of Historical Research; it + explains necessary processes through which all the material has to + pass before it is placed before us as history. + + + LONDON: WILLIAMS AND NORGATE + + + + + SOME BOOKS ON RUSSIA AND POLAND + + +With a Coloured Frontispiece, 12 Photogravure Plates, 28 Illustrations + in the text, and 8 Maps. Demy 8vo. =7s. 6d.= net (postage + =6d.= extra). + + + =A Thousand Years of Russian History= + + By SONIA E. HOWE. + + "We can recommend the volume as an excellent, careful, and + well-written history of a great nation."--_Daily Telegraph._ + + + Demy 8vo. Cloth, =6s.= net. + + =The False Dmitri= + + A Russian Romance and Tragedy. Described by British Eye-witnesses, + 1604-1612. + + Edited with a Preface by SONIA E. HOWE. + + Illustrated by reproduction of contemporary portraits. + + "With more than a whiff of the atmosphere Hakluyt got into his famous + volume."--_Northern Whig._ + + "Of special interest at the present time."--_Oxford Chronicle._ + + +With 16 Plates and 28 Illustrations in the text. Demy 8vo. =7s. +6d.= net. + + =Some Russian Heroes, Saints and Sinners= + + A portrait gallery of outstanding figures in Russian history who were + typical of their times and their country. + + By SONIA E. HOWE. + + "It gives a vivid study of Russia ... and interesting book."--_Land + and Water._ + + + Fcap. 8vo. Cloth, =1s. 3d.= net. In full leather presentation + binding, =2s. 6d.= net. + + =An Outline of Russian Literature= + + By the Hon. MAURICE BARING, + + Author of "With the Russians in Manchuria," "A Year in Russia," "The + Russian People," etc. + + ∵ Gives a clear and interesting account of the growth of Russian + literature. + + +Fcap. 8vo. Cloth, =1s. 3d.= net. In full leather binding, =2s. +6d.= net. + + =Poland= + + By W. ALISON PHILLIPS, M.A., + + Lecky Professor of Modern History in the University of Dublin. + + ∵ A concise survey of the History of Poland up to 1915. + + + LONDON: WILLIAMS AND NORGATE + 14 HENRIETTA STREET, COVENT GARDEN, W.C. + + + + + =TRANSCRIBER’S NOTES= + +Simple typographical errors have been silently corrected; unbalanced +quotation marks were remedied when the change was obvious, and +otherwise left unbalanced. + +Punctuation, hyphenation, and spelling were made consistent when a +predominant preference was found in the original book; otherwise they +were not changed. + + + +*** END OF THE PROJECT GUTENBERG EBOOK 77011 *** |