 Introduction to Science and Hypothesis This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer, visit LibriVox.org. This reading by Karl Manchester 2008. Science and Hypothesis by Henri Plancaret. Introduction by Judd Lama It is to be hoped that as a consequence of the present active scrutiny of our educational aims and methods, and of the resulting encouragement of the study of modern languages, we shall not remain as a nation so much isolated from ideas and tendencies in continental thought and literature as we have been in the past. As things are however, the translation of this book is doubtless required at any rate it brings vividly before us an instructive point of view. Though some of Monsieur Plancaret's chapters have been collected from well-known treatises written several years ago, and indeed are sometimes in detail not quite up to date, besides occasionally suggesting the suspicion that his views may possibly have been modified in the interval, yet their publication in a compact form has excited a warm welcome in this country. It must be confessed that the English language hardly lends itself as a perfect medium for the rendering of the delicate shades of suggestion and illusion characteristic of Monsieur Plancaret's play around his subject. Notwithstanding the excellence of the translation, loss in this respect is inevitable. There has been of late a growing trend of opinion prompted in part by general philosophical views in the direction that the theoretical constructions of physical science are largely factitious, that instead of presenting a valid image of the relation of things on which further progress can be based, they are still little better than a mirage. The best method of abating this skepticism is to become acquainted with the real scope and modes of application of conceptions which, in the popular language of superficial exposition and even in the unguarded and playful paradox of their authors intended only for the instructed eye, often look bizarre enough. But how much advantage will accrue if men of science become their own epistemologists and show to the world by critical exposition in non-technical terms of the results and methods of their constructive work, that more than mere instinct is involved in it? The community has indeed a right to expect as much as this. It would be hard to find anyone better qualified for this kind of exposition, either from the profundity of his own mathematical achievements or from the extent and freshness of his interest in the theories of physical science than the author of this book. If an appreciation might be ventured on as regards the later chapters, they are perhaps intended to present the stern logical analyst quizzing the cultivator of physical ideas as to what he is driving at and whether he expects to go rather than any responsible attempt towards a settled confession of faith. Thus, when Monsieur Poincaré allows himself for a moment to indulge in a process of evaporation of the principle of energy, he is content to sum up and to leave the matter there for his readers to think it out. Though hardly necessary in the original French, it may not now be superfluous to point out that independent reflection and criticism on the part of the reader are tacitly implied here as elsewhere. An interesting passage is the one devoted to Maxwell's theory of the functions of the aether and the comparison of the close-knit theories of the classical French mathematical physicists with the somewhat loosely connected corpus of ideas by which Maxwell, the interpreter and successor of Faraday, has posthumously recast the whole face of physical science. How many times has that theory been rewritten since Maxwell's day and yet how little has it been altered in essence except by further developments in the problem of moving bodies from the form in which he left it? If, as Monsieur Poincaré remarks, the French instinct for precision and lucid demonstration sometimes finds itself ill at ease with physical theories of the British school, he as readily admits and indeed fully appreciates the advantages on the other side. Our own mental philosophers have been shocked at the point of view indicated by the proposition hazarded by Laplace that a sufficiently developed intelligence if it were made acquainted with the positions and motions of the atoms at any instant could predict all future history. No amount of demure suffices sometimes that this is not a conception universally entertained in physical science. It was not so even in Laplace's own day. From the point of view of the study of the evolution of the sciences there are a few episodes more instructive than the collision between Laplace and Young with regard to the theory of capillarity. The precise and intricate mathematical analysis of Laplace starting from fixed preconceptions regarding atomic forces which were to remain intact in the development of the argument came into contrast with the tentative mobile intuitions of Young. Yet the latter was able to grasp by sheer direct mental force the fruitful though partial analogies of this recondite class of phenomena with more familiar operations of nature and to form a direct picture of the way things interacted such as could only have been illustrated quite possibly damaged or obliterated by premature effort to translate it into elaborate analytical formulas. The apposue of Young were apparently devoid of all cogency to Laplace. While Young expressed, doubtless into extreme away, his sense of the inanity of the array of mathematical logic of his rival. The subsequent history involved the nemesis that the fabric of Laplace was taken down and reconstructed in the next generation by Poisson. While the modern of the subject turns at any rate in England to neither of those expositions for illumination, but rather finds in the partial and succinct indications of Young the best starting point for further effort. It seems however hard to accept entirely the distinction suggested between the methods of cultivating theoretical physics in the two countries. To mention only two transcendent names which stand at the very front of the greatest developments of physical science of the last century, Carnot and Fresnel, their procedure was certainly not on the lines thus described. Possibly it is not devoid of significance that each of them attained his first effective recognition from the British school. It may in fact be maintained that the part played by mechanical and such like theories analogies if you will, is an essential one. The reader of this book will appreciate that human mind has need of many instruments of comparison and discovery besides the unrelenting logic of the infinitesimal calculus. The dynamical basis which underlies the objects of our most frequent experience has now been systemised into a great calculus of exact thought and traces of new real relationships may come out more vividly when considered in terms of our familiar acquaintance with dynamical systems than when formulated under the paler shadow of more analytical abstractions. It is even possible for a constructive physicist to conduct his mental operations entirely by dynamical images though Helmholtz as well as our author seems to class a predilection in this direction as a British trait. A time arrives when as in other subjects ideas have crystallised out into distinctness. Their exact verification and development forms a problem in mathematical physics but whether the mathematical analogies still survive or new terms are now introduced devoid of all naive mechanical basis it matters essentially little. The precise determination of the relations of things in the rational scheme of nature in which we find ourselves is the fundamental task and for its fulfilment in any direction advantage has to be taken of our knowledge impartial of new aspects and types of relationship which may have become familiar perhaps in quite different fields nor can it be forgotten that the most fruitful and fundamental conceptions of abstract pure mathematics itself have often been suggested from these mechanical ideas of flux and force where the play of intuition is our most powerful guide. The study of the historical evolution of physical theories is essential to understanding of their import. It is in the mental workshop of a Fresnel, a Kelvin or a Helmholtz that profound ideas of the deep things of nature are struck out and assumed form when pondered over and paraphrased by philosophers we see them react on the conduct of life it is the business of criticism to polish them gradually to the common measure of human understanding oppressed though we are with the necessity of being specialists if we are to know anything thoroughly in these days of accumulated detail we may at any rate profitably study the historical evolution of knowledge over a field wider than our own the aspect of the subject which has here been dwelt on is that scientific progress considered historically is not a strictly logical process and does not proceed by syllogisms new ideas emerge dimly into intuition come into consciousness from nobody knows where and become the material on which the mind operates forging them gradually into consistent doctrine which can be welded onto existing domains of knowledge but this process is never complete a crude connection can always be pointed to by a logician as an indication of the imperfection of human constructions if intuition plays a part which is so important it is surely necessary that we should possess a firm grasp of its limitations in Monsieur Poincaré's earlier chapters the reader can gain very pleasantly a vivid idea of the various and highly complicated ways of docketing our perceptions of the relations of external things all equally valid that were open to the human race to develop strange to say they never tried any of them and satisfied with the very remarkable practical fitness of the scheme of geometry and dynamics that came naturally to hand did not consciously trouble themselves about the possible existence of others until recently still more recently has it been found that the good Bishop Berkeley's logical jibes against the Newtonian ideas of fluxions and limiting ratios cannot be adequately appeased in the rigorous mathematical conscience until our apparent continuities are resolved mentally into discrete aggregates which we only partially apprehend the irresistible impulse to atomize everything thus proves to be not merely a disease of the physicist a deeper origin in the nature of knowledge itself is suggested everywhere want of absolute exact adaptation can be detected if pains are taken between the various constructions that result from our mental activity and the impressions which give rise to them the bluntness of our unaided sensual perceptions which are the source in part of the intuitions of the race is well brought out in this connection by Monsieur Poincaré is there real contradiction harmony usually proves to be recovered by shifting our attitude to the phenomena all experience leads us to interpret the totality of things as a consistent cosmos undergoing evolution the naturalists will say in the large scale workings of which we are interested spectators and explorers while of the inner relations and ramifications we only apprehend dim glimpses when our formulation of experience is imperfect or even paradoxical we learn to attribute the fault to our point of view and to expect that future adaptation will put it right but truth resides in a deep well we forget to the bottom only while deriving enjoyment and insight from Monsieur Poincaré's socratic exposition of the limitations of the human outlook on the universe let us beware of counting limitation as imperfection and drifting into an inadequate conception of the wonderful fabric of human knowledge J. Lama End of introduction Preface to Science and Hypothesis This is a LibriVox recording All LibriVox recordings are in the public domain For more information or to volunteer, please visit LibriVox.org Recording by Peter Eastman Science and Hypothesis by Henri Poincaré Author's Preface To the superficial observer scientific truth is unassailable The logic of science is infallible and if scientific men sometimes make mistakes it is because they have not understood the rules of the game Mathematical truths are derived from a few self-evident propositions by a chain of flawless reasonings They are imposed not only on us but on nature itself By them the creator is fettered as it were choice is limited to a relatively small number of solutions A few experiments, therefore, will be sufficient to enable us to determine what choice he has made From each experiment a number of consequences will follow by a series of mathematical deductions and in this way each of them will reveal to us a corner of the universe This to the minds of most people and to students who are getting their first ideas of physics is the origin of certainty in science This is what they take to be the role of experiment and mathematics and thus too it was understood a hundred years ago by many men of science who dreamed of constructing the world with the aid of the smallest possible amount of material borrowed from experiment But upon more mature reflection the position held by hypothesis was seen It was recognized that it is as necessary to the experimenter as it is to the mathematician and then the doubt arose if all these constructions are built on solid foundations The conclusion was drawn that a breath would bring them to the ground This skeptical attitude does not escape the charge of superficiality To doubt everything or to believe everything are two equally convenient solutions Both dispense with the necessity of reflection Instead of a summary condemnation we should examine with the utmost care the role of hypothesis We shall then recognize not only that it is necessary but that in most cases it is legitimate We shall also see that there are several kinds of hypotheses that some are verifiable and when once confirmed by experiment become truths of great fertility that others may be useful to us in fixing our ideas And finally that others are hypotheses only in appearance and reduced to definitions or to conventions in disguise The latter can be met with especially in mathematics and in the sciences to which it is applied From them indeed the sciences derive their rigor Such conventions are the result of the unrestricted activity of the mind which in this domain recognizes no obstacle For here the mind may affirm because it lays down its own laws But let us clearly understand that while these laws are imposed on our science which otherwise could not exist they are not imposed on nature Are they then arbitrary? No For if they were they would not be fertile Experience leaves us our freedom of choice but it guides us by helping us to discern the most convenient path to follow Our laws are therefore like those of an absolute monarch who is wise and consults great Some people have been struck by this characteristic of free convention which may be recognized in certain fundamental principles of the sciences Some have set no limits to their generalizations and at the same time they have forgotten that there is a difference between liberty and the purely arbitrary so that they are compelled to end in what is called nominalism They have asked if the savant is not the dupe of his own definitions and if the world he thinks he has discovered is not simply the creation of his own caprice Under these conditions science would retain its certainty but would not attain its object and would become powerless Now we daily see what science is doing for us This could not be unless it taught us something about reality The aim of science is not things themselves as the dogmatists in their simplicity imagine, but the relations between things Outside those relations there is no reality knowable Such is the conclusion to which we are led But to reach that conclusion we must pass and review the series of sciences from arithmetic and geometry to mechanics and experimental physics What is the nature of mathematical reasoning Is it really deductive as is commonly supposed Careful analysis shows us that it is nothing of the kind that it participates to some extent in the nature of inductive reasoning and for that reason it is fruitful But nonetheless does it retain its character of absolute rigor and this is what must first be shown When we know more of this instrument which is placed in the hands of the investigator by mathematics we have then to analyze another fundamental idea that of mathematical magnitude Do we find it in nature or have we ourselves introduced it and if the latter be the case are we not running a risk of coming to incorrect conclusions all round comparing the rough data of our senses with that extremely complex and subtle conception which mathematicians call magnitude we are compelled to recognize a divergence The framework into which we wish to make everything fit is one of our own construction But we did not construct it at random we constructed it by measurement so to speak and that is why we can fit the facts into it and find their essential qualities Space is another framework which we impose on the world Whence are the first principles of geometry derived Are they imposed on us by logic? Lobachevsky by inventing non-Euclidean geometries has shown that this is not the case Is space revealed to us by our senses No, for the space revealed to us by our senses is absolutely different from the space of geometry Is geometry derived from experience? Careful discussion will give the answer No We therefore conclude that the principles of geometry are only conventions But, these conventions are not arbitrary and if transported into another world which I shall call the non-Euclidean world and which I shall endeavour to describe we shall find ourselves compelled to adopt more of them In mechanics we shall be led to analogous conclusions and we shall see that the principles of this science, although more directly based on experience still share the conventional character of the geometrical postulates So far nominalism triumphs But we now come to the physical sciences properly so called and here the scene changes We meet with hypotheses of another kind and we fully grasp how fruitful they are No doubt at the outset theories seem unsound and the history of science shows us how ephemeral they are But they do not entirely perish and if each of them some traces still remain It is these traces which we must try to discover because in them and in them alone is the true reality The method of the physical sciences is based upon the induction which leads us to expect the recurrence of a phenomenon when the circumstances which give rise to it are repeated If all the circumstances could be simultaneously reproduced this principle could be fearlessly applied But this never happens Some of the circumstances will always be missing Are we absolutely certain that they are unimportant? Evidently not It may be probable but it cannot be rigorously certain Hence the importance of the role that is played in the physical sciences by the law of probability The calculus of probabilities is therefore not merely a recreation or a guide to the Bachara player and we must thoroughly examine the principles on which it is based In this connection I have but very incomplete results to lay before the reader for the vague instinct which enables us to determine probability almost defies analysis After a study of the conditions under which the work of the physicist is carried on I have thought it best to show him at work For this purpose I have taken instances from the history of optics and of electricity We shall thus see how the ideas of Fresnel and Maxwell took their eyes and what unconscious hypotheses were made by Ampere and the other founders of electrodynamics and of author's preface Chapter one of Science and Hypothesis This is a LibriVox recording All LibriVox recordings are in the public domain For more information or to volunteer please visit LibriVox.org Recording by Ashwin Jain Science and Hypothesis by Henry Poincaré Chapter one of the Nature of Mathematical Reasoning The very possibility of mathematical science seems an insoluble contradiction If the science is only deductive in appearance from whence is derived that perfect trigger which is challenged by none If on the contrary all the propositions which it enunciates may be derived in order by the rules of formal logic How is it that mathematics is not reduced to a gigantic tautology? The syllogism can teach us nothing essentially new and if everything must spring from the principle of identity then everything should be capable of being reduced to that principle Are we then to admit that the enunciations of all theorems with which so many volumes are filled are only indirect ways of saying that A is A No doubt we may refer back to axioms which are a source of all these reasonings If it is felt that they cannot be reduced to the principle of contradiction If we decline to see them in any more than experimental facts which have no part or lot in mathematical necessity there is still one resource left to us We may class them among a priori synthetic views but this is no solution to the difficulty it is merely giving it a name and even if the nature of the synthetic views had no longer for us any mystery the contradiction would not have disappeared it would have only been shrugged so logistic reasoning remains incapable of adding anything to the data that are given it the data are reduced to axioms and this is all we should find in the conclusions No theorem can be new unless a new axiom intervenes in its demonstration reasoning can only give us immediately evident truths borrowed from direct intuition it would only be an intermediary to the data site should we not therefore have reason for asking if this logistic apparatus serves to disguise what we have borrowed the contradiction will strike us the more if we open any book on mathematics on every page the author announces his intention of generalizing some proposition already known does the mathematical method proceed from the particular to the general and if so we are all deductive finally if the science of number were mainly analytical or could be analytically derived from a few synthetic intuitions it seems that a sufficiently powerful mind could with a single glance perceive all its truths nay one might even hope that someday a language will be invented simple enough for these truths even if these consequences are challenged it must be granted that the mathematical reason has of itself a kind of creative virtue and is therefore to be distinguished from the syllogism the difference must be profound we shall not for instance find the key to the mystery in the frequent use the rule by which the same uniform operation applied to two equal numbers will give identical results in modes of reasoning whether or not reducible to the syllogism properly so called retain the analytical character and I so fact lose their power the argument is an old one let us see how 11its tried to show that 2 and 2 make 4 assume the number 1 to be defined and also the operation x plus 1 that is the adding of unity to a given number x these definitions whatever they may be do not enter into the subsequent reasoning I next define the numbers 2, 3, 4 by the equalities equality 1 1 plus 1 equals to 2 equality 2 2 plus 1 equals to 3 equality 3 plus 1 equals to 4 and in number 4 the same way I define the operation x plus 2 by the relation relation 4 x plus 2 equals to x plus 1 plus 1 given this we have 2 plus 2 equals to 2 plus 1 plus 1 2 plus 1 plus 1 equals to 3 plus 1 3 plus 1 equals to 4 2 plus 2 equals to 4 QED it cannot be denied that this reasoning is purely analytical but if we ask a mathematician he will reply this is not a demonstration properly so called it is a verification we have confined ourselves to bringing together one or other to purely conventional definitions and we have verified their identity nothing new has been learned verification differs from proof precisely because it is analytical and because it leads to nothing it leads to nothing because the conclusion is nothing but the premises translated into another language a real proof on the other hand is fruitful because the conclusion is in a sense more general than the premises the quality 2 plus 2 equals to 4 can be verified because it is particular each individual initiation in mathematics may be always verified in the same way but if mathematics could be reduced to a series of such verifications it would not be a science a chess player for instance does not create a science by winning a piece there is no science but the science of the general it may even be said that the object of exact science is to dispense with these direct verifications 3 let us now see the geometry at work and try to surprise some of his methods the task is not without difficulty it is not enough to open a book at random and analyze any proof which we may come across first of all geometry must be excluded or the question becomes complicated by difficult problems relating to the role of the postulates the nature and origin of the idea of space for analogous reasons we cannot avail ourselves of the infinitesimal calculus we must seek mathematical thought where it has been remained pure that is in arithmetic but we still have to choose in the higher parts of the theory of numbers the primitive mathematical ideas have already undergone so profound an elaboration that it becomes difficult to analyze them it is therefore at the beginning of arithmetic that we must expect to find the explanation we seek but it happens that it precisely in the proofs of the most elementary theorems that the authors of classic treatises have displayed the least precision and rigor we may not impute this to them as a crime they have obeyed a necessity beginners are not prepared for real mathematical rigor they would see in it nothing but empty tedious subtleties it would be waste of time to try to make them more exciting they have to pass rapidly and without stopping over the road which was trodden slowly by the founders of the science why so long a preparation necessary to habituate oneself to this perfect rigor which it would seem should naturally be imposed on all minds this is a logical and psychological problem which is well worth three of study but we shall not dwell on it it is far into our subject all I wish to insist on is that we shall fail in our purpose unless we reconstruct the proofs of the elementary theorems and give them not the rough form in which they are left so as not to vary the beginner but the form which will satisfy the skill geometry definition of addition I assume that the operation x plus 1 has been defined it consists in adding the number 1 to a given number x whatever may be said of this definition it does not enter into the subsequent reasoning we now have to define the operation x plus a it consists in adding the number a to any given number x suppose that we have defined the operation x plus a minus 1 the operation x plus a will be defined by the equality x plus a equals to x plus a minus 1 plus 1 we shall know what x plus a is when we know what x plus a minus 1 is and as I have assumed that to start with we know what x plus 1 is we can define successfully and by recurrence the operations x plus 2, x plus 3 etc this definition deserves a moment attention it is of a particular nature distinguishes it even at this stage from the purely logical definition the quality 1 in fact contains an infinite number of distinct definitions each having only one meaning when we know the meaning of its predecessor properties of addition associative I say that a plus b plus c equals to a plus b plus c in fact the theorem is true for c equals to 1 it may then be written a plus b plus 1 equals to a plus b plus 1 which remembering the difference of notation is nothing but the equality 1 by which I have just defined addition I assume the theorem true for c equals to y I say that it will be true for c equals to y plus 1 let a plus b plus equals to a plus b plus it follows that a plus b plus 1 equals to a plus b plus plus 1 or by definition 1 a plus b plus plus 1 equals to a plus b plus plus 1 equals to a plus b plus plus 1 We shows by a series of purely analytical deductions that the theorem is true for plus 1. Being true for c equals to 1, we see that it is successfully true for s, c equals to 2, c equals to 3, etc., commutative, 1. Let us say that a plus 1 equals to 1 plus a. The theorem is evidently true for a equals to 1. We can verify by purely analytical reasoning that if it is true for a equals to y, it will be true for a equals to y plus 1. Now it is true for a equals to 1 and therefore is true for a equals to 2, a equals to 3 and so on. This is what is meant by saying that the proof is demonstrated by recurrence. I say that a plus b equals to b plus a. The theorem has just been shown to hold root for b equals to 1 and it may be verified analytically that if it is true for b equals to beta, it will be true for b equals to beta plus 1. The proposition is thus established by recurrence. Definition of multiplication. We shall define multiplication by the equalities 1, a cross 1 equals to a 2, a cross b equals to a cross b minus 1 plus a. Both of these include an infinite number of definitions. Having defined a cross 1, it enables us to define in succession a cross 2, a cross 3 and so on. Properties of multiplication. Distributive. I'll say that a plus b cross c equals to a cross c plus b cross c. We can verify analytically that the theorem is true for c equals to 1. Then if it is true for c equals to y, it will be true for c equals to y plus 1. The proposition is then proved by recurrence. Commutative. 1. I say that a cross 1 equals to 1 cross a. The theorem is obvious for a equals to 1. We can verify analytically that if it is true for a equals to y, it will be true for a equals to y plus 1. I say that a cross b equals to b cross a. The theorem has just been proved for b equals to 1. We can verify analytically that if it is true for b equals to y, it will be true for b equals to y plus 1. 4. This monotonous series of reasoning may now be slayed aside, but their very monotony brings vividly to light the process which is uniform and is met again at every step. The process is proved by recurrence. We first show that a theorem is true for n equals to 1. We then show that if it is true for n minus 1, it is true for it. And we conclude that it is true for all integers. We have now seen how it may be used for the proof of the rules of addition and multiplication, that is to say, for the rules of algebraical calculus. This calculus is an instrument of transformation which lends itself to many different combinations than the simple syllogism, but it is still a purely analytical instrument and is incapable of teaching us anything new. If mathematics had no other instrument, it would immediately be arrested in its development, but it has to close anew to the same process, that is, to reasoning by recurrence, and it can continue its forward march. Then if we look carefully, we find this mode of reasoning at every step, either under the simple form which we have just given to it, or under a more or less modified form. It is therefore mathematical reasoning by excellence, and we must examine it closer. 5. The essential characteristic of reasoning by recurrence is that it contains condensed, so to speak, in a single formula and an infinite number of syllogisms. We shall see this more clearly if we enunciate the syllogisms one after another. They follow one another, if one may use the expression in a cascade. The following are the hypothetical syllogisms. The theorem is true for the number one. Now, if it is true for one, it is true for true. Therefore, it is true of two. Now, if it is true of two, it is true of three. Hence, it is true of three, and so on. We see that the conclusion of each syllogism serves as the minor of its successor. Further, the measures of all our syllogisms may be reduced to a single form. If the theorem is true for n-1, it is true for n. We see then that in reasoning by recurrence, we can find ourselves to the enunciation of the minor of the four syllogism, and the general formula which contains as particular cases all the measures. This unending series of syllogisms is thus reduced to a phrase of a few lines. It is now easy to understand why every particular consequence of a theorem may as I have explained above be verified by purely analytical process. If, instead of proving that our theorem is true for all numbers, we only wish to show that it is true for the number six, for instance. It will be enough to establish the first five syllogism in our cascade. We shall require nine if we wish to prove it for the number ten. For a greater number, we shall require more still. But however great the number may be, we shall always reach it, and the analytical verification will always be possible. But however far we went, we should never reach the general theorem applicable to all numbers, which alone is the object of science. To reach it, we should require an infinite number of syllogisms, and we should have to cross an abyss with the patience of the analyst. Extracted to the resources of formal logic, we'll never succeed in crossing. I asked at the outset why we cannot conceive of a mind powerful enough to see at a glance the whole body of mathematical truth. The answer is now easy. A chess player can combine four or five moves ahead. But however extraordinary a player may be, he cannot prepare for more than a finite number of moves. If he applies his faculties to arithmetic, he cannot conceive its general truths by a direct intuition alone. To prove even the smallest theorem, he must use reasoning by recurrence. For that is the only instrument which enables us to pass from the finite to the infinite. This instrument is always useful, for it enables us to leap over as many stages as we wish. It frees us from the necessity of long tedious and monotonous verifications, which would rapidly become impracticable. Then when we take in hand the general theorem, it becomes indispensable for otherwise we should ever be approaching the analytical verification without ever actually reaching it. In this domain of arithmetic, we may think ourselves very far from the infinitesimal analysis where the idea of mathematical infinity is already playing a preponderating part. And without it there would be no science at all, because there would be nothing general. Six. The views upon which reasoning by recurrence is based may be exhibited in other forms. We may say, for instance, that in any finite collection of different integers, there is always one which is smaller than any other. We may readily pass from one enunciation to another, and thus give ourselves the illusion of having proved the reasoning by recurrence is legitimate. But we shall always be brought to a full stop. We shall always come to an indemonstrable axiom, which will at bottom be but the preposition we had to prove translated into another language. We cannot therefore escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction. Nor can the rule come to us from experiment. Experiment may teach us that the rule is true for the first 10 or the first 100 numbers, for instance. It will not bring us to the indefinite series of numbers, but only to a more or less long, but always limited portion of the series. Now, if that were all that is in question, the principle of contradiction would be sufficient. It would only enable us to develop as many syllogisms as we wished. It is only when it is a question of a single formula to embrace an infinite number of syllogisms that this principle breaks down, and there too experiment is powerless to aid. This rule, inaccessible to analytical proof and to experiment, is the exact type of the a priori, synthetic intuition. On the other hand, we cannot see it in a convention as in the case of the postulates of geometry. Why then is this view imposed upon us with such an arrestable weight of evidence? It is because it is only the affirmation of the power of the mind which knows it can conceive of the indefinite repetition of the same act, that the act is once possible. The mind has a direct intuition of this power, and experiment can only be for it an opportunity of using it, and thereby of becoming conscious of it. But it will be said, if the legitimacy of reasoning by recurrence cannot be established by experiment alone, is it so with experiment aided by induction? We see successively that a theorem is true of the number one, of the number two, of the number three, and so on. The law is manifest, we say, and it is so on the same ground that every physical law is true, which is based on a very large but limited number of observations. It cannot escape our notice that there is a striking analogy with the usual processes of induction, but an essential difference exists. Induction applied to the physical sciences is always uncertain, because it is based on the belief in a general order of the universe, an order which is external to us. Mathematical induction, that is, proof by recurrence, is on the contrary, necessarily imposed on us, because it is only the affirmation of a property of the mind itself, seven. Mathematicians, as I have said before, always endeavor to generalize the propositions they have obtained. To seek no further example, we have just shown the equality a plus one equals to one plus a, and we then used it to establish the equality a plus b equals to b plus a, which is obviously more general. Mathematics may, therefore, like the other sciences, proceed from the particular to the general. This is a fact which might otherwise have appeared incomprehensible to us at the beginning of this study, but which has no longer anything mysterious about it, since we have ascertained the analogies between proof by recurrence and ordinary induction. No doubt mathematical recurrent reasoning and physical inductive reasoning are based on different foundations, but they move in parallel lines and in the same direction, namely from the particular to the general. Let us examine the case a little more closely. To prove the equality a plus two equals to two plus a, we need only apply the rule a plus one equals to one plus a twice and right. a plus two equals to a plus one plus one equals to one plus a plus one equals to one plus one plus a equals to two plus a two. The quality thus deduced by purely analytical means is not, however, a simple particular case. It is something quite different. We are not, therefore, even say in the really analytical and deductive part of mathematical reasoning that we proceed from the general to the particular in the ordinary sense of the words. The two sides of the equality two are merely more complicated combinations than the two sides of the quality one. An analysis only serves to separate the elements which enter into these combinations and to study their relations. Mathematicians therefore proceed by construction. They construct more complicated combinations. When they analyze these combinations, these aggregates, so to speak, into their primitive elements, they see the relations of the elements and deduce the relations of the aggregates themselves. The process is purely analytical, but is not a passing from the general to the particular, for the aggregates obviously cannot be regarded as more particular than their elements. Great importance has been rightly attached to this process of construction in some claim to see in it the necessary and sufficient condition of the progress of the exact sciences. Necessary no doubt but not sufficient for a construction to be useful and not mere waste of mental effort for it to serve as a stepping stone to hire things. It must first of all possess a kind of unity enabling us to see something more than the juxtaposition of its elements or more accurately. There must be some advantage in considering the construction rather than the elements themselves. What can this advantage be? Why reason on a polygon, for instance, which is always decomposable into triangles and not on elementary triangles? It is because there are properties of polygons of any number of sites and they can be immediately applied to any particular kind of polygons. In most cases it is only after long efforts that those properties can be discovered by directly studying the relations of elementary triangles. If the quadrilateral is anything more than the juxtaposition of two triangles, it is because it is of the polygon type. A construction only becomes interesting when it can be placed side by side with other analogous constructions for forming species of the same genus. To do this we must necessarily go back from the particular to the general ascending one or more steps. The analytical process by construction does not compare us to descent but it leaves us at the same level. We can only ascend by mathematical induction or from it alone we can learn something new. Without the aid of this induction, which in certain respects differ from what is as fruitful as physical induction, construction will be powerless to create science. Let me observe in conclusion that this induction is only possible if the same operation can be repeated indefinitely. That is why the theory of chess can never become a science. For the different moves of the same piece are limited and do not resemble each other. End of chapter 1, recording by Ashwin Jain. Chapter 2 of Science and Hypothesis. This is a LibriVox recording. All LibriVox recordings are in the public domain. For more information or to volunteer please visit LibriVox.org recording by Anosimum. Science and Hypothesis by Henry Poincaré. Chapter 2. Mathematical Magnitude and Experiment. If you want to know what the mathematicians mean by continuum it is useless to appeal to geometry. The geometry is always seeking more or less to represent to himself the figures he is studying but his representations are only instruments to him. He uses space in his geometry just as he uses chalk and further too much importance must not be attached to accidents which are often nothing more than the whiteness of the chalk. The pure analyst has not to dread this pitfall. He has disengaged mathematics from all extraneous elements and he is in a position to answer our question. Tell me exactly what this continuum is about which mathematicians reason. Many analysts who reflect on their art have already done so. Monsieur Tanari for instance in his introduction à la théorie des fonctions d'une variable. Let us start with the integers. Between any two consecutive sets intercalate one or more intermediary sets and then between these sets others again and so on indefinitely. We thus get an unlimited number of terms and these will be the numbers which we call fractional, rational or commensurable. But this is not yet all between these terms which be it marked are already infinite in number other terms are intercalate and these are called irrational or incommensurable. Before going any further let me make a preliminary remark. The continuum thus conceived is no longer a collection of individuals arranged in a certain order. Infinite in number it is true but external the one to the other. This is not the ordinary conception in which it is supposed that between the elements of the continuum exists an intimate connection making of it one whole in which the point has no existence previous to the line but the line does exist previous to the point. Multiplicity alone subsists unity has disappeared. The continuum is unity in multiplicity according to the celebrated formula. The analysts have even less reason to define their continuum as they do since it is always on this that they reason when they are particularly proud of their rigor. It is enough to warn the reader that the real mathematical continuum is quite different from that of the physicists and from that of the metaphysicians. It may also be said perhaps that mathematicians who are contented with this definition are the dupes of words that the nature of each of these sets should be precisely indicated that it should be explained how they are to be intercalated and that it should be shown how it is possible to do it. This however would be wrong. The only property of the sets which comes into the reasoning is that of proceeding or succeeding these or those other sets. This alone should therefore intervene in the definition. So we need not concern ourselves with the manner in which the sets are intercalated and no one will doubt the possibility of the operation if he only remembers that possible in the language of geometries simply means exempt from contradiction but our definition is not yet complete and we come back to it after this rather long digression. Definition of incommensurables. The mathematicians of the Berlin school and Kronecker in particular have devoted themselves to constructing this continuous skill of irrational and fractional numbers without using any other materials than the integer. The mathematical continuum from this point of view will be a pure creation of the mind in which experiment would have no part. The idea of rational number not seeming to present to them any difficulty they have confined their attention mainly to defining incommensurable numbers. But before reproducing their definition here I must make an observation that will allay the astonishment which this will not fail to provoke in readers who are but little familiar with the habits of geometries. Mathematicians do not study objects but the relations between objects. To them it is a matter of indifference if these objects are replaced by others provided that the relations do not change. Matter does not engage their attention they are interested by form alone. If we did not remember it we could hardly understand that Kronecker gives the name of incommensurable number to a simple symbol that is to say something very different from the idea we think we ought to have of a quantity which should be measurable and almost tangible. Let us see now what is Kronecker's definition. Commensurable numbers may be divided into classes in an infinite number of ways subject to the condition that any number whatever of the first class is greater than any number of the second. It may happen that among the numbers of the first class there is one which is smaller than all the rest. If for instance we arrange in the first class all the numbers greater than two and two itself and in the second class all the numbers smaller than two it is clear that two will be the smallest of all the numbers of the first class. The number two may therefore be chosen as the symbol of this division. It may happen on the contrary that in the second class there is one which is greater than all the rest. This is what takes place for example if the first class comprises all the numbers greater than two and if in the second are all the numbers less than two and two itself. Here again the number two might be chosen as the symbol of this division. But it may equally well happen that we can find neither in the first class a number smaller than all the rest nor in the second class a number greater than all the rest. Suppose for instance we place in the first class all the numbers whose squares are greater than two and in the second all the numbers whose squares are smaller than two. We know that in neither of them is a number whose square is equal to two. Evidently there will be in the first class no number which is smaller than all the rest for however near the square of a number may be to two we can always find a commensurable whose square is still nearer to two. From Koeniger's point of view the incommensurable number two is nothing but the symbol of this particular method of division of commensurable numbers and to each mode of repetition corresponds in this way a number commensurable or not which serves as a symbol. But to be satisfied with this would be to forget the origin of these symbols. It remains to explain how we have been led to attribute to them a kind of concrete existence and on the other hand does not the difficulty begin with fractions. Should we have the notion of these numbers if we did not previously know a matter which we conceive as infinitely divisible that is a continuum the physical continuum we are next led to ask if the idea of the mathematical continuum is not simply drawn from experiment if that be so the rough data of experiment which are our sensations could be measured we might indeed be tempted to believe that this is so for in recent times there has been an attempt to measure them and the law has even been formulated known as Fechten's law according to which sensation is proportional to the logarithm of the stimulus but if we examine the experiments by which the endeavor has been made to establish this law we shall be led to a diametrically opposite conclusion. It has for instance been observed that a weight a of 10 grams and a weight b of 11 grams produced identical sensations that the weight b could no longer be distinguished from a weight c of 12 grams but that the weight a was readily distinguished from the weight c thus the rough results of the experiments may be expressed by the following relations a is b b is c a is smaller than c which may be regarded as the formula of the physical continuum but here is an intolerable disagreement with the law of contradiction and the necessity of banishing this disagreement has compelled us to invent the mathematical continuum we are therefore forced to conclude that this notion has been created entirely by the mind but it is experiment that has provided the opportunity we cannot believe that two quantities which are equal to a third are not equal to one another and we are thus led to suppose that a is different from b and b from c and that if we have not been aware of this it is due to the imperfections of our senses the creation of the mathematical continuum first stage so far it would suffice in order to account for facts to intercalate between a and b a small number of terms which would remain discrete what happens now if we have recourse to some instrument to make up for the weakness of our senses if for example we use a microscope such terms as a and b which before were indistinguishable from one another appear now to be distinct but between a and b which are distinct is intercalated another new term d which we can distinguish neither from a nor from b although we may use the most delicate methods the rough results of our experiments will always present the characters of the physical continuum with a contradiction which is inherent in it we only escape from it by incessantly intercalating new terms between the terms already distinguished and this operation must be pursued indefinitely we might conceive that it would be possible to stop if we could imagine an instrument powerful enough to decompose the physical continuum into discrete elements just as the telescope resolves the Milky Way into stars but this we cannot imagine it is always with our senses that we use our instruments it is with the eye that we observe the image magnified by the microscope and this image must therefore always retain the characters of visual sensation and therefore those of the physical continuum nothing distinguishes a length directly observed from half that length doubled by the microscope the hole is homogeneous to the part and there is a fresh contradiction or rather there would be one if the number of the terms were supposed to be finite it is clear that the part containing less terms than the whole cannot be similar to the whole the contradiction ceases as soon as the number of terms is regarded as infinite there is nothing for example to prevent us from regarding the aggregate of integers as similar to the aggregate of even numbers which is however only a part of it in fact to each integer corresponds another even number which is its double but it is not only to escape this contradiction contained in the empiric data that the mind is led to create the concept of a continuum formed of an indefinite number of terms here everything takes place just as in the series of the integers we have the faculty of conceiving that a unit may be added to a collection of units thanks to experiment we have had the opportunity of exercising this faculty and are conscious of it but from this fact we feel that our power is unlimited and that we can count indefinitely although we have never had to count more than a finite number of objects in the same way as soon as we have intercalated terms between two consecutive terms of a series we feel that this operation may be continued without limit and that so to speak there is no intrinsic reason for stopping as an abbreviation i may give the name of a mathematical continuum of the first order to every aggregate of terms formed after the same law as the scale of commensurable numbers if then we intercalate new sets according to the laws of incommensurable numbers we obtain what may be called a continuum of the second order second stage we have only taken our first step we have explained the origin of continuums of the first order we must now see why this is not sufficient and why the incommensurable numbers had to be invented if we try to imagine a line it must have the characters of the physical continuum that is to say our representation must have a certain breadth two lines will therefore appear to us under the form of two narrow bands and if we are content with this rough image it is clear that where two lines cross they must have some common part but the pure geometry makes one further effort without entirely renouncing the aid of his senses he tries to imagine a line without breadth and a point without size this he can do only by imagining a line at the limit towards which tends a band that is growing thinner and thinner and the point as a limit to which which is tending an area that is growing smaller and smaller our two bands however narrow they may be will always have a common area the smaller they are the smaller it will be and its limit is what the geometry calls a point this is why it is said that the two lines which cross must have a common point and this truth seems intuitive but a contradiction would be implied if we conceived of lines as continuums of the first order that is the lines traced by the geometry should only give us points the coordinates of which are rational numbers the contradiction would be manifest if we were for instance to assert the existence of lines and circles it is clear in fact that if the points whose coordinates are commensurable were alone regarded as real the in-circle of the square and the diagonal of the square would not intersect since the coordinates of the point of intersection are incommensurable even then we should have only certain incommensurable numbers and not all these numbers but let us imagine a line divided into two half-rays demitoids each of these half-rays will appear to our minds as a band of a certain breadth these bands will fit close together because there must be no interval between them the common part will appear to us to be a point which will still remain as we imagine the bands to become thinner and thinner so that we admit as an intuitive truth that if a line be divided into two half-rays the common frontier of these half-rays is a point here we recognize the conception of chronicle in which an incommensurable number was regarded as the common frontier of two classes of rational numbers such as the origin of the continuum of the second order which is the mathematical continuum properly so called summary to sum up the mind has the faculty of creating symbols and it is thus that it has constructed the mathematical continuum which is only a particular system of symbols the only limit to its power is a necessity of avoiding all contradiction but the mind only makes use of it when experiment gives a reason for it in the case with which we are concerned the reason is given by the idea of the physical continuum drawn from the rough data of the senses but this idea leads to a series of contradictions from each of which in turn we must be freed in this way we are forced to imagine a more and more complicated system of symbols that on which we shall dwell is not merely exempt from internal contradiction it was so already at all the steps we've taken but it is no longer in contradiction with the various propositions which are called intuitive and which are derived from more or less elaborate empirical notions measurable magnitude so far we have not spoken of the measure of magnitudes we can tell if any one of them is greater than any other but we cannot say that it is two or three times as large so far I've only considered the order in which the terms are arranged but that is not sufficient for most applications we must learn how to compare the interval which separates any two terms on this condition alone with a continuum become measurable and the operations of arithmetic be applicable this can only be done by the aid of a new and special convention and this convention is that in such a case the interval between the terms a and b is equal to the interval which separates c and d for instance we started with the integers and between two consecutive sets we intercalated n intermediary sets by convention we now assume these new sets to be equidistant this is one of the ways of defining the addition of two magnitudes for if the interval a b is by definition equal to the interval c d the interval a d will by definition be the sum of the intervals a b and a c this definition is very largely but not altogether arbitrary it must satisfy certain conditions the commutative and associative laws of addition for instance but provided the definition we choose satisfies these laws the choice is indifferent and we need not state it precisely remarks we are now in a position to discuss several important questions one is the creative power of the mind exhausted by the creation of the mathematical continuum the answer is in the negative and this is shown in a very striking manner by the work of Dupont Raymond we know that mathematicians distinguish between infinitismals of different orders and that infinitismals of the second order are infinitely small not only absolutely so but also in relation to those of the first order it's not difficult to imagine infinitismals of fractional or even of irrational order and here once more we find the mathematical continuum which has been dealt with in the preceding pages further there are infinitismals which are infinitely small with reference to those of the first order and infinitely large with respect to the order one plus e however small e may be here then are new terms intercalated in our series and if i may be permitted to revert to the terminology used in the preceding pages a terminology which is very convenient although it has not been consecrated by usage i shall say that we've created a kind of continuum of the third order it is an easy matter to go further but it is idle to do so for we would only be imagining symbols without any possible application and no one will dream of doing that this continuum the third order to which we are led by the consideration of the different orders of infinitismals is in itself of but little use and hardly worth quoting geometers look on it as a mere curiosity the mind only uses its creative faculty when experiment requires it two when we are once in possession of the conception of the mathematical continuum are we protected from contradictions analogous to those which gave it birth no and the following is an instance he is a savant indeed who will not take it as evident that every curve has a tangent and in fact if we think of a curve and a straight line as two narrow bands we can always arrange them in such a way that they have a common part without intersecting suppose now that the breadth of the bands diminishes indefinitely the common part will still remain and in the limit so to speak the two lines will have a common point although they do not intersect that is they will touch the geometry who reasons in this way is only doing what we have done when we proved that two lines which intersect have a common point and his intuition might also seem to be quite legitimate but this is not the case we can show that there are curves which have no tangent if we define such a curve as an analytical continuum of the second order no doubt some sacrifice analogous to those we have discussed above would enable us to get rid of this contradiction but as the latter is only met with in very exceptional cases we need not trouble to do so instead of endeavouring to reconcile intuition and analysis we're content to sacrifice one of them and as analysis must be flawless intuition must go to the wall the physical continuum of several dimensions we have discussed above the physical continuum as it is derived from the immediate evidence of our senses or if the reader prefers from the rough results of fechness experiments i've shown that these results are summed up in the contradictory formulae a is b b is c and a is smaller than c let us now see how this notion is generalized and how from it may be derived the concept of continuums of several dimensions consider any two aggregates of sensations we can either distinguish between them or we cannot just as in fechness experiments the weight of 10 grams could be distinguished from the weight of 12 grams but not from the weight of 11 grams this is all that is required to construct the continuum of several dimensions let us call one of these aggregates of sensation an element it will be in a measure analogous to the point of the mathematicians but will not be however the same thing we cannot say that our element has no size for we cannot distinguish it from its immediate neighbors and it is thus surrounded by a kind of fog if the astronomical comparison may be allowed our elements would be like nebulae whereas the mathematical points would be like stars if this be granted a system of elements will form a continuum if we can pass from any one of them to any other by a series of consecutive elements such that each cannot be distinguished from its predecessor this linear series is to the line of the mathematicians what the isolated element was to the point before going further i must explain what is meant by a cut let us consider a continuum see and remove from it certain of its elements which for a moment we shall regard as no longer belonging to the continuum we shall call the aggregate of elements thus removed a cut by means of this cut the continuum c will be subdivided into several distinct continuums the aggregate of elements which remain will cease to form a single continuum there will then be on c two elements a and b which we must look upon as belonging to two distinct continuums and we see that this must be so because it will be impossible to find a linear series of consecutive elements of c each of the elements indistinguishable from the preceding the first being a and the last b unless one of the elements of this series is indistinguishable from one of the elements of the cut it may happen on the contrary that the cut may not be sufficient to subdivide the continuum c to classify the physical continuums we must first of all ascertain the nature of the cuts which must be made in order to subdivide them if a physical continuum c may be subdivided by a cut reducing to a finite number of elements all distinguishable the one from the other and therefore forming neither one continuum nor several continuums we shall call c a continuum of one dimension if on the contrary c can only be subdivided by cuts which are themselves continuums we shall say that c is of several dimensions if the cuts are continuums of one dimension then we shall say that c has two dimensions if cuts of two dimensions are sufficient we shall say that c is of three dimensions and so on thus the notion of the physical continuum of several dimensions is defined thanks to the very simple fact that two aggregates of sensations may be distinguishable or indistinguishable the mathematical continuum of several dimensions the conception of the mathematical continuum of n dimensions may be led up to quite naturally by a process similar to that which we discussed at the beginning of this chapter a point of such a continuum is defined by a system of its distinct magnitudes which we call its coordinates the magnitudes need not always be measurable there is for instance one branch of geometry independent of the measure of magnitudes in which we are only concerned with knowing for example if on a curve a b c the point b is between the points a and c and in which it is immaterial whether the arc a b is equal to or twice the arc b c this branch is called analysis cytos it contains quite a large body of doctrine which has attracted the attention of the greatest geometers and from which i derived one from another a whole series of remarkable theorems what distinguishes these theorems from those of ordinary geometry is that they are purely qualitative they are still true if the figures are copied by an unskillful draftsman with the result that the proportions are distorted and the straight lines are placed by lines which are more or less curved as soon as measurement is introduced into the continuum we've just defined the continuum becomes space and geometry is born but the discussion of this is reserved for part two end of chapter two chapter three of science and hypothesis this is a LibriVox recording all LibriVox recordings are in the public domain for more information or to volunteer please visit LibriVox.org recording by Leon Meyer science and hypothesis by Henri Poincaré chapter three non-nuclidean geometries every conclusion presumes premises these premises are either self-evident and need no demonstration or can be established only if based on other propositions and as we cannot go back in this way to infinity every deductive science and geometry in particular must rest upon a certain number of indemonstrable axioms all treatisees of geometry begin therefore with the annunciation of these axioms but there is a distinction to be drawn between them some of these for example things which are equal to the same thing are equal to one another are not propositions in geometry but propositions in analysis i look upon them as analytical a priori intuitions and they concern me no further but i must insist on other axioms which are special to geometry of these most treatisees explicitly enunciate three one only one line can pass through two points two a straight line is the shortest distance between two points three through one point only one parallel can be drawn to a given straight line although we generally dispense with proving the second of these axioms it would be possible to deduce it from the other two and from those much more numerous axioms which are implicitly admitted without annunciation as i shall explain further on for a long time a proof of the third axiom known as euclid's postulate was sought in vain it is impossible to imagine the efforts that have been spent in pursuit of this chimera finally at the beginning of the 19th century and almost simultaneously two scientists a russian and a bulgarian lobachevsky and boyoui showed irrefutably that this proof is impossible they have nearly rid us of inventors of geometries without a postulate and ever since the accadémie des sciences receives only about one or two new demonstrations a year but the question was not exhausted and it was not long before a great step was taken by the celebrated memoir of riemann entitled uber de hypotasen velky der geometrie som grundeligen this little work has inspired most of the recent treatises to which i shall later on refer and among which i may mention those of beltrami and helmholz the geometry of lobachevsky if it were possible to deduce euclid's postulate from the several axioms it is evident that by rejecting the postulate and retaining the other axioms we should be led to contradictory consequences it would be therefore impossible to found on those premises a coherent geometry now this is precisely what lobachevsky has done he assumes at the outset that several parallels may be drawn through a point to a given straight line and he retains all the other axioms of euclid from these hypotheses he deduces a series of theorems between which it is impossible to find any contradiction and he constructs the geometry as impeccable in its logic as euclidean geometry the theorems are very different however from those to which we are accustomed and at first will be found a little disconcerting for instance the sum of the angles of a triangle is always less than two right angles and the difference between that sum and two right angles is proportional to the area of the triangle it is impossible to construct a figure similar to a given figure but of different dimensions if the circumference of a circle be divided into its equal parts and tangents be drawn at the points of intersection then its tangents will form a polygon if the radius of the circle is small enough but if the radius is large enough they will never meet we need not multiply these examples lobachevsky's propositions have no relation to those of euclid but they are nonetheless logically interconnected remont's geometry let us imagine to ourselves a world only peopled with beings of no thickness and suppose that these infinitely flat animals are all in one in the same plane from which they cannot emerge let us further admit that this world is sufficiently distant from other worlds to be withdrawn from their influence and while we are making these hypotheses it will not cost as much to endow these beings with reasoning power and to believe them capable of making a geometry in that case they will certainly attribute to space only two dimensions but now suppose that these imaginary animals while remaining without thickness have the form of a spherical and not of a plane figure and are all on the same sphere from which they cannot escape what kind of a geometry will they construct it is clear that they will attribute to space only two dimensions the straight line to them will be the shortest distance from one point on the sphere to another that is to say an arc of a great circle in a word their geometry will be spherical geometry what they will call space will be the sphere on which they are confined and on which take place all the phenomena with which they are acquainted their space will therefore be unbounded since on a sphere one may always walk forward without ever being brought to a stop and yet it will be finite the end will never be found but the complete tour can be made well Riemann's geometry is spherical geometry extended to three dimensions to construct it the german mathematician had first of all to throw overboard not only Euclid's postulate but also the first axiom that only one line can pass through two points on a sphere through two given points we can in general draw only one great circle which as we have just seen would be to our imaginary beings a straight line but there was one exception if the two given points are at the ends of a diameter an infinite number of great circles can be drawn through them in the same way in Riemann's geometry at least in one of its forms through two points only one straight line can in general be drawn but there are exceptional cases in which through two points an infinite number of straight lines can be drawn so there is a kind of opposition between the geometries of Riemann and Lobachevsky for instance the sum of the angles of a triangle is equal to two right angles in Euclid's geometry less than two right angles and that of Lobachevsky and greater than two right angles and that of Riemann the number of parallel lines that can be drawn through a given point to a given line is one in Euclid's geometry none in Riemann's and an infinite number in the geometry of Lobachevsky let us add that Riemann's space is finite although unbounded in the sense which we have above attached to these words spaces with constant curvature one objection however remains possible there was no contradiction between the theorems of Lobachevsky and Riemann but however numerous are the other consequences that these geometries have deduced from their hypotheses they had to arrest their course before they exhausted them all for the number would be infinite and who can say that if they had carried their deductions further they would not have eventually reached some contradiction this difficulty does not exist for Riemann's geometry provided it is limited to two dimensions as we have seen the two-dimensional geometry of Riemann in fact does not differ from spherical geometry which is only a branch of ordinary geometry and is therefore outside all contradiction Beltrami by showing that Lobachevsky's two-dimensional geometry was only a branch of ordinary geometry has equally refuted the objection as far as it is concerned this is the course of his argument let us consider any figure whatever on a surface imagine this figure to be traced on a flexible and inextensible canvas applied to the surface in such a way that when the canvas is displaced and deformed the different lines of the figure change their form without changing their length as a rule this flexible and inextensible figure cannot be displaced without leaving the surface but there are certain surfaces for which such a movement would be possible if we resume the comparison that we made just now and imagine beings without thickness living on one of these surfaces they will regard as possible the motion of a figure all the lines of which remain of a constant length such a movement would appear absurd on the other hand to animals without thickness living on the surface of variable curvature these surfaces of constant curvature are of two kinds the curvature of some is positive and they may be deformed so as to be applied to a sphere the geometry of these surfaces is therefore reduced to spherical geometry namely Riemann's the curvature of others is negative Beltrami has shown that the geometry of these surfaces is identical with that of Lobachevsky thus the two-dimensional geometries of Riemann and Lobachevsky are connected with Euclidean geometry interpretation of non Euclidean geometries thus vanishes the objection so far as two-dimensional geometries are concerned it would be easy to extend Beltrami's reasoning to three-dimensional geometries and minds which do not recoil before space of four dimensions will see no difficulty in it but such minds are few in number I prefer then to proceed otherwise let us consider a certain plane which I shall call the fundamental plane and let us construct a kind of dictionary by making a double series of terms written in two columns and corresponding each to each just as in ordinary dictionaries the words in two languages which have the same signification correspond to one another space the portion of space situated above the fundamental plane plane sphere cutting orthogonally the fundamental plane line circle cutting orthogonally the fundamental plane sphere sphere circle circle angle angle distance between two points logarithm of the anharmonic ratio of these two points and of the intersection of the fundamental plane with the circle passing through these two points and cutting it orthogonally etc let us now take Lobachevsky's theorems and translate them by the aid of this dictionary as we would translate a German text with the aid of a German French dictionary we shall then obtain the theorems of ordinary geometry for instance Lobachevsky's theorem the sum of the angles of a triangle is less than two right angles may be translated thus if a curvilinear triangle has for its sides arcs of circles which if produced would cut orthogonally the fundamental plane the sum of the angles of this curvilinear triangle will be less than two right angles thus however far the consequences of Lobachevsky's hypotheses are carried they will never lead to a contradiction in fact if two of Lobachevsky's theorems were contradictory the translations of these two theorems made by the aid of our dictionary would be contradictory also but these translations are theorems of ordinary geometry and no one doubts that ordinary geometry is exempt from contradiction whence is the certainty derived and how far is it justified that is a question upon which I cannot enter here but it is a very interesting question and I think not insoluble nothing therefore is left of the objection I formulated above but this is not all Lobachevsky's geometry being susceptible of a concrete interpretation ceases to be a useless logical exercise and may be applied I have no time here to deal with these applications nor with what her Klein and myself have done by using them in the integration of linear equations further this interpretation is not unique and several dictionaries may be constructed analogous to that above which will enable us by a simple translation to convert Lobachevsky's theorems into the theorems of ordinary geometry implicit axioms are the axioms implicitly enunciated in our textbooks the only foundation of geometry we may be assured of the contrary when we see that when they are abandoned one after another there are still left standing some propositions which are common to the geometries of Euclid Lobachevsky and Riemann these propositions must be based on premises that geometers admit without enunciation it is interesting to try and extract them from the classical proofs John Stuart Mill asserted that every definition contains an axiom because by defining we implicitly affirm the existence of the object defined that is going rather too far it is but rarely in mathematics that a definition is given without following it up by the proof of the existence of the object defined and when this is not done it is generally because the reader can easily supply it and it must not be forgotten that the word existence has not the same meaning when it refers to a mathematical entity as when it refers to a material object a mathematical entity exists provided there is no contradiction implied in its definition either in itself or with the propositions previously admitted but if the observation of John Stuart Mill cannot be applied to all definitions it is nonetheless true for some of them a plane is sometimes defined in the following manner the plane is a surface such that the line which joins any two points upon it lies wholly on that surface now there is obviously a new axiom concealed in this definition it is true we might change it and that would be preferable but then we should have to annunciate the axiom explicitly other definitions may give rise to no less important reflections such as for example that of the equality of two figures two figures are equal when they can be superposed to superpose them one of them must be displaced until it coincides with the other but how must it be displaced if we ask that question no doubt we should be told that it ought to be done without deforming it and as an invariable solid is displaced the vicious circle would then be evident as a matter of fact this definition defines nothing it has no meaning to a being living in a world in which there are only fluids if it seems clear to us it is because we are accustomed to the properties of natural solids which do not much differ from those of the ideal solids all of whose dimensions are invariable however imperfect as it may be this definition implies an axiom the possibility of the motion of an invariable figure is not a self-evident truth at least it is only so in the application to Euclid's postulate and not as an analytical a priori intuition would be moreover when we study the definitions and the proofs of geometry we see that we are compelled to admit without proof not only the possibility of this motion but also some of its properties this first arises in the definition of the straight line many defective definitions have been given but the true one is that which is understood in all the proofs in which the straight line intervenes it may happen that the motion of an invariable figure may be such that all the points of a line belonging to the figure are motionless while all the points situated outside that line are in motion such a line would be called a straight line we have deliberately in this annunciation separated the definition from the axiom which it implies many proofs such as those of the cases of the equality of triangles of the possibility of drawing a perpendicular from a point to a straight line assume propositions the annunciations of which are dispensed with for they necessarily imply that it is possible to move a figure in space in a certain way the fourth geometry among these explicit axioms there is one which seems to me to deserve some attention because when we abandon it we can construct a fourth geometry as coherent as those of Euclid Lobochevsky and Riemann to prove that we can always draw a perpendicular at a point A to a straight line AB we consider a straight line AC movable about the point A and initially identical with the fixed straight line AB we then can make it turn about the point A until it lies in AB produced thus we assume two propositions first that such a rotation is possible and then that it may continue until the two lines align the one in the other produced if the first point is conceded and the second rejected we are led to a series of theorems even stranger than those of Lobochevsky and Riemann but equally free from contradiction I shall give only one of these theorems and I shall not choose the least remarkable of them a real straight line may be perpendicular to itself Lee's theorem the number of axioms implicitly introduced into classical proofs is greater than necessary and it would be interesting to reduce them to a minimum it may be asked in the first place if this reduction is possible if the number of necessary axioms and that of imaginable geometries is not infinite a theorem due to sophist lee is of weighty importance in this discussion it may be enunciated in the following manner suppose the following premises are admitted one space has in dimensions two the movement of an invariable figure is possible three p conditions are necessary to determine the position of this figure in space the number of geometries compatible with these premises will be limited I may even add that if n is given a superior limit can be assigned to p if therefore the possibility of the movement is granted we can only invent a finite and even a rather restricted number of three-dimensional geometries Riemann's geometries however this result seems contradicted by Riemann for that scientist constructs an infinite number of geometries and that to which his name is usually attached is only a particular case of them all depends he says on the manner in which the length of a curve is defined now there is an infinite number of ways of defining this length and each of them may be the starting point of a new geometry that is perfectly true but most of these definitions are incompatible with the movement of a variable figure such as we assume to be possible in lee's theorem these geometries of Riemann so interesting on various grounds can never be therefore purely analytical and would not lend themselves to proofs analogous to those of Euclid on the nature of axioms most mathematicians regard lobachevsky's geometry as a mere logical curiosity some of them have however gone further if several geometries are possible they say is it certain that our geometry is the one that is true experiment no doubt teaches us that the sum of the angles of a triangle is equal to two right angles but this is because the triangles we deal with are too small according to lobachevsky the difference is proportional to the area of the triangle and will not this become sensible when we operate on much larger triangles and when our measurements become more accurate Euclid's geometry would thus be a provisory geometry now to discuss this view we must first of all ask ourselves what is the nature of geometrical axioms are they synthetic a priori intuitions as contafirmed they would then be imposed upon us with such a force that we could not conceive of the contrary proposition nor could we build upon it a theoretical edifice there would be no non Euclidean geometry to convince ourselves of this let us take a true synthetic a priori intuition the following for instance which played an important part in the first chapter if a theorem is true for the number one and if it has been proved that it is true of n plus one provided it is true of n it will be true for all positive integers let us next try to get rid of this and while rejecting this proposition let us construct a false arithmetic analogous to non Euclidean geometry we shall not be able to do it we shall be even tempted at the outset to look upon these intuitions as analytical besides to take up again our fiction of animals without thickness we can scarcely admit that these beings if their minds are like ours would adopt the Euclidean geometry which would be contradicted by all their experience ought we then to conclude that the axioms of geometry are experimental truths but we do not make experiments on ideal lines or ideal circles we can only make them on material objects on what therefore would experiments serving as a foundation for geometry be based the answer is easy we have seen above that we constantly reason as if the geometrical figures behaved like solids what geometry would borrow from experiment would be therefore the properties of these bodies the properties of light and its propagation in a straight line have also given rise to some of the propositions of geometry and in particular to those of projective geometry so that from that point of view one would be tempted to say that metrical geometry is the study of solids and projective geometry that of light but difficulty remains and is unsurmountable if geometry were an experimental science it would not be an exact science it would be subjected to continual revision nay it would from that day forth be proved to be erroneous for we know that no rigorously invariable solid exists the geometrical axioms are therefore neither synthetic a priori intuitions nor experimental facts they are conventions our choice among all possible conventions is guided by experimental facts but it remains free and is only limited by the necessity of avoiding every contradiction and thus it is that postulates may remain rigorously true even when the experimental laws which have determined their adoption are only approximate in other words the axioms of geometry i do not speak of those of arithmetic are only definitions in disguise what then are we to think of the question is euclidean geometry true it has no meaning we might as well ask if the metric system is true and if the old weights and measures are false if cartesian coordinates are true and polar coordinates false one geometry cannot be more true than another it can only be more convenient now euclidean geometry is and will remain the most convenient first because it is the simplest and it is not so only because of our mental habits or because of the kind of direct intuition that we have of euclidean space it is the simplest in itself just as a polynomial of the first degree is simpler than a polynomial of the second degree secondly because it sufficiently agrees with the properties of natural solids those bodies which we can compare and measure by means of our senses end of chapter three