Characteristic remains, however, for our author, as, in his decidedly contrasting way, for Kant, the thought thatwithout principles which at every stage transcend precise confirmation through such experience as is then accessible the organization of experience is impossible. Whether one views these principles as conventions or asa priori'forms,' they may therefore be described as hypotheses, but as hypotheses that, while lying at the basis of our actual physical sciences, at once refer to experience and help us in dealing with experience, and are yet neither confirmed nor refuted by the experiences which we possess or which we can hope to attain.
Three special instances or classes of instances, according to our author's account, may be used as illustrations of this general type of hypotheses. They are: (1) The hypothesis of the existence of continuous extensivequantain nature; (2) The principles of geometry; (3) The principles of mechanics and of the general theory of energy. In case of each of these special types of hypotheses we are at first disposed, apart from reflection, to say that wefindthe world to be thus or thus, so that, for instance, we can confirm the thesis according to which nature contains continuous magnitudes; or can prove or disprove the physical truth of the postulates of Euclidean geometry; or can confirm by definite experience the objective validity of the principles of mechanics. A closer examination reveals, according to our author, the incorrectness of all such opinions. Hypotheses of these various special types are needed; and their usefulness can be empirically shown. They are in touch with experience; and that they are not merely arbitrary conventions is also verifiable. They are nota priorinecessities; and we can easily conceive intelligent beings whose experience could be best interpreted without using these hypotheses. Yet these hypotheses arenotsubject to direct confirmation or refutation by experience. They stand then in sharp contrast to the scientific hypotheses of the other, and more frequently recognized, type,i. e., to the hypotheses which can be tested by a definite appeal to experience. To these other hypotheses our author attaches, of course, great importance. His treatment of them is full of a living appreciation of the significance of empirical investigation. But the centralproblem of the logic of science thus becomes the problem of the relation between the two fundamentally distinct types of hypotheses,i. e., between those which can not be verified or refuted through experience, and those which can be empirically tested.
The detailed treatment which M. Poincaré gives to the problem thus defined must be learned from his text. It is no part of my purpose to expound, to defend or to traverse any of his special conclusions regarding this matter. Yet I can not avoid observing that, while M. Poincaré strictly confines his illustrations and his expressions of opinion to those regions of science wherein, as special investigator, he is himself most at home, the issues which he thus raises regarding the logic of science are of even more critical importance and of more impressive interest when one applies M. Poincaré's methods to the study of the concepts and presuppositions of the organic and of the historical and social sciences, than when one confines one's attention, as our author here does, to the physical sciences. It belongs to the province of an introduction like the present to point out, however briefly and inadequately, that the significance of our author's ideas extends far beyond the scope to which he chooses to confine their discussion.
The historical sciences, and in fact all those sciences such as geology, and such as the evolutionary sciences in general, undertake theoretical constructions which relate to past time. Hypotheses relating to the more or less remote past stand, however, in a position which is very interesting from the point of view of the logic of science. Directly speaking, no such hypothesis is capable of confirmation or of refutation, because we can not return into the past to verify by our own experience what then happened. Yet indirectly, such hypotheses may lead to predictions of coming experience. These latter will be subject to control. Thus, Schliemann's confidence that the legend of Troy had a definite historical foundation led to predictions regarding what certain excavations would reveal. In a sense somewhat different from that which filled Schliemann's enthusiastic mind, these predictions proved verifiable. The result has been a considerablechange in the attitude of historians toward the legend of Troy. Geological investigation leads to predictions regarding the order of the strata or the course of mineral veins in a district, regarding the fossils which may be discovered in given formations, and so on. These hypotheses are subject to the control of experience. The various theories of evolutionary doctrine include many hypotheses capable of confirmation and of refutation by empirical tests. Yet, despite all such empirical control, it still remains true that whenever a science is mainly concerned with the remote past, whether this science be archeology, or geology, or anthropology, or Old Testament history, the principal theoretical constructions always include features which no appeal to present or to accessible future experience can ever definitely test. Hence the suspicion with which students of experimental science often regard the theoretical constructions of their confrères of the sciences that deal with the past. The origin of the races of men, of man himself, of life, of species, of the planet; the hypotheses of anthropologists, of archeologists, of students of 'higher criticism'—all these are matters which the men of the laboratory often regard with a general incredulity as belonging not at all to the domain of true science. Yet no one can doubt the importance and the inevitableness of endeavoring to apply scientific method to these regions also. Science needs theories regarding the past history of the world. And no one who looks closer into the methods of these sciences of past time can doubt that verifiable and unverifiable hypotheses are in all these regions inevitably interwoven; so that, while experience is always the guide, the attitude of the investigator towards experience is determined by interests which have to be partially due to what I should call that 'internal meaning,' that human interest in rational theoretical construction which inspires the scientific inquiry; and the theoretical constructions which prevail in such sciences are neither unbiased reports of the actual constitution of an external reality, nor yet arbitrary constructions of fancy. These constructions in fact resemble in a measure those which M. Poincaré in this book has analyzed in the case of geometry. They are constructions molded, butnotpredetermined in their details, by experience. We report facts; we let the facts speak; but we, aswe investigate, in the popular phrase, 'talk back' to the facts. We interpret as well as report. Man is not merely made for science, but science is made for man. It expresses his deepest intellectual needs, as well as his careful observations. It is an effort to bring internal meanings into harmony with external verifications. It attempts therefore to control, as well as to submit, to conceive with rational unity, as well as to accept data. Its arts are those directed towards self-possession as well as towards an imitation of the outer reality which we find. It seeks therefore a disciplined freedom of thought. The discipline is as essential as the freedom; but the latter has also its place. The theories of science are human, as well as objective, internally rational, as well as (when that is possible) subject to external tests.
In a field very different from that of the historical sciences, namely, in a science of observation and of experiment, which is at the same time an organic science, I have been led in the course of some study of the history of certain researches to notice the existence of a theoretical conception which has proved extremely fruitful in guiding research, but which apparently resembles in a measure the type of hypotheses of which M. Poincaré speaks when he characterizes the principles of mechanics and of the theory of energy. I venture to call attention here to this conception, which seems to me to illustrate M. Poincaré's view of the functions of hypothesis in scientific work.
The modern science of pathology is usually regarded as dating from the earlier researches of Virchow, whose 'Cellular Pathology' was the outcome of a very careful and elaborate induction. Virchow, himself, felt a strong aversion to mere speculation. He endeavored to keep close to observation, and to relieve medical science from the control of fantastic theories, such as those of theNaturphilosophenhad been. Yet Virchow's researches were, as early as 1847, or still earlier, already under the guidance of a theoretical presupposition which he himself states as follows: "We have learned to recognize," he says, "that diseases are not autonomous organisms, that they are no entities that have entered into the body, that they are no parasites which take root in the body, butthat they merely show us the course ofthe vital processes under altered conditions" ('dasz sie nur Ablauf der Lebenserscheinungen unter veränderten Bedingungen darstellen').
The enormous importance of this theoretical presupposition for all the early successes of modern pathological investigation is generally recognized by the experts. I do not doubt this opinion. It appears to be a commonplace of the history of this science. But in Virchow's later years this very presupposition seemed to some of his contemporaries to be called in question by the successes of recent bacteriology. The question arose whether the theoretical foundations of Virchow's pathology had not been set aside. And in fact the theory of the parasitical origin of a vast number of diseased conditions has indeed come upon an empirical basis to be generally recognized. Yet to the end of his own career Virchow stoutly maintained that in all its essential significance his own fundamental principle remained quite untouched by the newer discoveries. And, as a fact, this view could indeed be maintained. For if diseases proved to be the consequences of the presence of parasites, the diseases themselves, so far as they belonged to the diseased organism, were still not the parasites, but were, as before, the reaction of the organism to theveränderte Bedingungenwhich the presence of the parasites entailed. So Virchow could well insist. And if the famous principle in question is only stated with sufficient generality, it amounts simply to saying that if a disease involves a change in an organism, and if this change is subject to law at all, then the nature of the organism and the reaction of the organism to whatever it is which causes the disease must be understood in case the disease is to be understood.
For this very reason, however, Virchow's theoretical principle in its most general formcould be neither confirmed nor refuted by experience. It would remain empirically irrefutable, so far as I can see, even if we should learn that the devil was the true cause of all diseases. For the devil himself would then simply predetermine theveränderte Bedingungento which the diseased organism would be reacting. Let bullets or bacteria, poisons or compressed air, or the devil be theBedingungento which a diseased organism reacts, the postulate that Virchowstates in the passage just quoted will remain irrefutable, if only this postulate be interpreted to meet the case. For the principle in question merely says that whatever entity it may be, bullet, or poison, or devil, that affects the organism, the disease is not that entity, but is the resulting alteration in the process of the organism.
I insist, then, that this principle of Virchow's is no trial supposition, no scientific hypothesis in the narrower sense—capable of being submitted to precise empirical tests. It is, on the contrary, a very preciousleading idea, a theoretical interpretation of phenomena, in the light of which observations are to be made—'a regulative principle' of research. It is equivalent to a resolution to search for those detailed connections which link the processes of disease to the normal process of the organism. Such a search undertakes to find the true unity, whatever that may prove to be, wherein the pathological and the normal processes are linked. Now without some such leading idea, the cellular pathology itself could never have been reached; because the empirical facts in question would never have been observed. Hence this principle of Virchow's was indispensable to the growth of his science. Yet it was not a verifiable and not a refutable hypothesis. One value of unverifiable and irrefutable hypotheses of this type lies, then, in the sort of empirical inquiries which they initiate, inspire, organize and guide. In these inquiries hypotheses in the narrower sense, that is, trial propositions which are to be submitted to definite empirical control, are indeed everywhere present. And the use of the other sort of principles lies wholly in their application to experience. Yet without what I have just proposed to call the 'leading ideas' of a science, that is, its principles of an unverifiable and irrefutable character, suggested, but not to be finally tested, by experience, the hypotheses in the narrower sense would lack that guidance which, as M. Poincaré has shown, the larger ideas of science give to empirical investigation.
I have dwelt, no doubt, at too great length upon one aspect only of our author's varied and well-balanced discussion of theproblems and concepts of scientific theory. Of the hypotheses in the narrower sense and of the value of direct empirical control, he has also spoken with the authority and the originality which belong to his position. And in dealing with the foundations of mathematics he has raised one or two questions of great philosophical import into which I have no time, even if I had the right, to enter here. In particular, in speaking of the essence of mathematical reasoning, and of the difficult problem of what makes possible novel results in the field of pure mathematics, M. Poincaré defends a thesis regarding the office of 'demonstration by recurrence'—a thesis which is indeed disputable, which has been disputed and which I myself should be disposed, so far as I at present understand the matter, to modify in some respects, even in accepting the spirit of our author's assertion. Yet there can be no doubt of the importance of this thesis, and of the fact that it defines a characteristic that is indeed fundamental in a wide range of mathematical research. The philosophical problems that lie at the basis of recurrent proofs and processes are, as I have elsewhere argued, of the most fundamental importance.
These, then, are a few hints relating to the significance of our author's discussion, and a few reasons for hoping that our own students will profit by the reading of the book as those of other nations have already done.
Of the person and of the life-work of our author a few words are here, in conclusion, still in place, addressed, not to the students of his own science, to whom his position is well known, but to the general reader who may seek guidance in these pages.
Jules Henri Poincaré was born at Nancy, in 1854, the son of a professor in the Faculty of Medicine at Nancy. He studied at the École Polytechnique and at the École des Mines, and later received his doctorate in mathematics in 1879. In 1883 he began courses of instruction in mathematics at the École Polytechnique; in 1886 received a professorship of mathematical physics in the Faculty of Sciences at Paris; then became member of the Academy of Sciences at Paris, in 1887, and devoted his life to instruction and investigation in the regions of pure mathematics, of mathematical physics and of celestial mechanics. His list of published treatises relating tovarious branches of his chosen sciences is long; and his original memoirs have included several momentous investigations, which have gone far to transform more than one branch of research. His presence at the International Congress of Arts and Science in St. Louis was one of the most noticeable features of that remarkable gathering of distinguished foreign guests. In Poincaré the reader meets, then, not one who is primarily a speculative student of general problems for their own sake, but an original investigator of the highest rank in several distinct, although interrelated, branches of modern research. The theory of functions—a highly recondite region of pure mathematics—owes to him advances of the first importance, for instance, the definition of a new type of functions. The 'problem of the three bodies,' a famous and fundamental problem of celestial mechanics, has received from his studies a treatment whose significance has been recognized by the highest authorities. His international reputation has been confirmed by the conferring of more than one important prize for his researches. His membership in the most eminent learned societies of various nations is widely extended; his volumes bearing upon various branches of mathematics and of mathematical physics are used by special students in all parts of the learned world; in brief, he is, as geometer, as analyst and as a theoretical physicist, a leader of his age.
Meanwhile, as contributor to the philosophical discussion of the bases and methods of science, M. Poincaré has long been active. When, in 1893, the admirableRevue de Métaphysique et de Moralebegan to appear, M. Poincaré was soon found amongst the most satisfactory of the contributors to the work of that journal, whose office it has especially been to bring philosophy and the various special sciences (both natural and moral) into a closer mutual understanding. The discussions brought together in the present volume are in large part the outcome of M. Poincaré's contributions to theRevue de Métaphysique et de Morale. The reader of M. Poincaré's book is in presence, then, of a great special investigator who is also a philosopher.
For a superficial observer, scientific truth is beyond the possibility of doubt; the logic of science is infallible, and if the scientists are sometimes mistaken, this is only from their mistaking its rules.
"The mathematical verities flow from a small number of self-evident propositions by a chain of impeccable reasonings; they impose themselves not only on us, but on nature itself. They fetter, so to speak, the Creator and only permit him to choose between some relatively few solutions. A few experiments then will suffice to let us know what choice he has made. From each experiment a crowd of consequences will follow by a series of mathematical deductions, and thus each experiment will make known to us a corner of the universe."
Behold what is for many people in the world, for scholars getting their first notions of physics, the origin of scientific certitude. This is what they suppose to be the rôle of experimentation and mathematics. This same conception, a hundred years ago, was held by many savants who dreamed of constructing the world with as little as possible taken from experiment.
On a little more reflection it was perceived how great a place hypothesis occupies; that the mathematician can not do without it, still less the experimenter. And then it was doubted if all these constructions were really solid, and believed that a breath would overthrow them. To be skeptical in this fashion is still to be superficial. To doubt everything and to believe everything are two equally convenient solutions; each saves us from thinking.
Instead of pronouncing a summary condemnation, we ought therefore to examine with care the rôle of hypothesis; we shall then recognize, not only that it is necessary, but that usually it islegitimate. We shall also see that there are several sorts of hypotheses; that some are verifiable, and once confirmed by experiment become fruitful truths; that others, powerless to lead us astray, may be useful to us in fixing our ideas; that others, finally, are hypotheses only in appearance and are reducible to disguised definitions or conventions.
These last are met with above all in mathematics and the related sciences. Thence precisely it is that these sciences get their rigor; these conventions are the work of the free activity of our mind, which, in this domain, recognizes no obstacle. Here our mind can affirm, since it decrees; but let us understand that while these decrees are imposed uponourscience, which, without them, would be impossible, they are not imposed upon nature. Are they then arbitrary? No, else were they sterile. Experiment leaves us our freedom of choice, but it guides us by aiding us to discern the easiest way. Our decrees are therefore like those of a prince, absolute but wise, who consults his council of state.
Some people have been struck by this character of free convention recognizable in certain fundamental principles of the sciences. They have wished to generalize beyond measure, and, at the same time, they have forgotten that liberty is not license. Thus they have reached what is callednominalism, and have asked themselves if the savant is not the dupe of his own definitions and if the world he thinks he discovers is not simply created by his own caprice.[1]Under these conditions science would be certain, but deprived of significance.
If this were so, science would be powerless. Now every day we see it work under our very eyes. That could not be if it taught us nothing of reality. Still, the things themselves are not what it can reach, as the naïve dogmatists think, but only the relations between things. Outside of these relations there is no knowable reality.
Such is the conclusion to which we shall come, but for that we must review the series of sciences from arithmetic and geometry to mechanics and experimental physics.
What is the nature of mathematical reasoning? Is is really deductive, as is commonly supposed? A deeper analysis shows us that it is not, that it partakes in a certain measure of the nature of inductive reasoning, and just because of this is it so fruitful. None the less does it retain its character of rigor absolute; this is the first thing that had to be shown.
Knowing better now one of the instruments which mathematics puts into the hands of the investigator, we had to analyze another fundamental notion, that of mathematical magnitude. Do we find it in nature, or do we ourselves introduce it there? And, in this latter case, do we not risk marring everything? Comparing the rough data of our senses with that extremely complex and subtile concept which mathematicians call magnitude, we are forced to recognize a difference; this frame into which we wish to force everything is of our own construction; but we have not made it at random. We have made it, so to speak, by measure and therefore we can make the facts fit into it without changing what is essential in them.
Another frame which we impose on the world is space. Whence come the first principles of geometry? Are they imposed on us by logic? Lobachevski has proved not, by creating non-Euclidean geometry. Is space revealed to us by our senses? Still no, for the space our senses could show us differs absolutely from that of the geometer. Is experience the source of geometry? A deeper discussion will show us it is not. We therefore conclude that the first principles of geometry are only conventions; but these conventions are not arbitrary and if transported into another world (that I call the non-Euclidean world and seek to imagine), then we should have been led to adopt others.
In mechanics we should be led to analogous conclusions, and should see that the principles of this science, though more directly based on experiment, still partake of the conventional character of the geometric postulates. Thus far nominalism triumphs; but now we arrive at the physical sciences, properly so called. Here the scene changes; we meet another sort of hypotheses and we see their fertility. Without doubt, at first blush, the theories seem to us fragile, and the history of science proves to us how ephemeral they are; yet they do not entirely perish,and of each of them something remains. It is this something we must seek to disentangle, since there and there alone is the veritable reality.
The method of the physical sciences rests on the induction which makes us expect the repetition of a phenomenon when the circumstances under which it first happened are reproduced. Ifallthese circumstances could be reproduced at once, this principle could be applied without fear; but that will never happen; some of these circumstances will always be lacking. Are we absolutely sure they are unimportant? Evidently not. That may be probable, it can not be rigorously certain. Hence the important rôle the notion of probability plays in the physical sciences. The calculus of probabilities is therefore not merely a recreation or a guide to players of baccarat, and we must seek to go deeper with its foundations. Under this head I have been able to give only very incomplete results, so strongly does this vague instinct which lets us discern probability defy analysis.
After a study of the conditions under which the physicist works, I have thought proper to show him at work. For that I have taken instances from the history of optics and of electricity. We shall see whence have sprung the ideas of Fresnel, of Maxwell, and what unconscious hypotheses were made by Ampère and the other founders of electrodynamics.
The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting? If, on the contrary, all the propositions it enunciates can be deduced one from another by the rules of formal logic, why is not mathematics reduced to an immense tautology? The syllogism can teach us nothing essentially new, and, if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill so many volumes are nothing but devious ways of sayingAisA?
Without doubt, we can go back to the axioms, which are at the source of all these reasonings. If we decide that these can not be reduced to the principle of contradiction, if still less we see in them experimental facts which could not partake of mathematical necessity, we have yet the resource of classing them among synthetica priorijudgments. This is not to solve the difficulty, but only to baptize it; and even if the nature of synthetic judgments were for us no mystery, the contradiction would not have disappeared, it would only have moved back; syllogistic reasoning remains incapable of adding anything to the data given it: these data reduce themselves to a few axioms, and we should find nothing else in the conclusions.
No theorem could be new if no new axiom intervened in its demonstration; reasoning could give us only the immediatelyevident verities borrowed from direct intuition; it would be only an intermediary parasite, and therefore should we not have good reason to ask whether the whole syllogistic apparatus did not serve solely to disguise our borrowing?
The contradiction will strike us the more if we open any book on mathematics; on every page the author will announce his intention of generalizing some proposition already known. Does the mathematical method proceed from the particular to the general, and, if so, how then can it be called deductive?
If finally the science of number were purely analytic, or could be analytically derived from a small number of synthetic judgments, it seems that a mind sufficiently powerful could at a glance perceive all its truths; nay more, we might even hope that some day one would invent to express them a language sufficiently simple to have them appear self-evident to an ordinary intelligence.
If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from the syllogism.
The difference must even be profound. We shall not, for example, find the key to the mystery in the frequent use of that rule according to which one and the same uniform operation applied to two equal numbers will give identical results.
All these modes of reasoning, whether or not they be reducible to the syllogism properly so called, retain the analytic character, and just because of that are powerless.
The discussion is old; Leibnitz tried to prove 2 and 2 make 4; let us look a moment at his demonstration.
I will suppose the number 1 defined and also the operationx+ 1 which consists in adding unity to a given numberx.
These definitions, whatever they be, do not enter into the course of the reasoning.
I define then the numbers 2, 3 and 4 by the equalities
(1) 1 + 1 = 2; (2) 2 + 1 = 3; (3) 3 + 1 = 4.
In the same way, I define the operationx+ 2 by the relation:
(4)x+ 2 = (x+ 1) + 1.
That presupposed, we have
2 + 1 + 1 = 3 + 1(Definition 2),3 + 1 = 4(Definition 3),2 + 2 = (2 + 1) + 1(Definition 4),
whence
2 + 2 = 4 Q.E.D.
It can not be denied that this reasoning is purely analytic. But ask any mathematician: 'That is not a demonstration properly so called,' he will say to you: 'that is a verification.' We have confined ourselves to comparing two purely conventional definitions and have ascertained their identity; we have learned nothing new.Verificationdiffers from true demonstration precisely because it is purely analytic and because it is sterile. It is sterile because the conclusion is nothing but the premises translated into another language. On the contrary, true demonstration is fruitful because the conclusion here is in a sense more general than the premises.
The equality 2 + 2 = 4 is thus susceptible of a verification only because it is particular. Every particular enunciation in mathematics can always be verified in this same way. But if mathematics could be reduced to a series of such verifications, it would not be a science. So a chess-player, for example, does not create a science in winning a game. There is no science apart from the general.
It may even be said the very object of the exact sciences is to spare us these direct verifications.
Let us, therefore, see the geometer at work and seek to catch his process.
The task is not without difficulty; it does not suffice to open a work at random and analyze any demonstration in it.
We must first exclude geometry, where the question is complicated by arduous problems relative to the rôle of the postulates, to the nature and the origin of the notion of space. For analogous reasons we can not turn to the infinitesimal analysis.We must seek mathematical thought where it has remained pure, that is, in arithmetic.
A choice still is necessary; in the higher parts of the theory of numbers, the primitive mathematical notions have already undergone an elaboration so profound 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 precisely in the demonstration of the most elementary theorems the authors of the classic treatises have shown the least precision and rigor. We must not impute this to them as a crime; they have yielded to a necessity; beginners are not prepared for real mathematical rigor; they would see in it only useless and irksome subtleties; it would be a waste of time to try prematurely to make them more exacting; they must pass over rapidly, but without skipping stations, the road traversed slowly by the founders of the science.
Why is so long a preparation necessary to become habituated to this perfect rigor, which, it would seem, should naturally impress itself upon all good minds? This is a logical and psychological problem well worthy of study.
But we shall not take it up; it is foreign to our purpose; all I wish to insist on is that, not to fail of our purpose, we must recast the demonstrations of the most elementary theorems and give them, not the crude form in which they are left, so as not to harass beginners, but the form that will satisfy a skilled geometer.
Definition of Addition.—I suppose already defined the operationx+ 1, which consists in adding the number 1 to a given numberx.
This definition, whatever it be, does not enter into our subsequent reasoning.
We now have to define the operationx+a, which consists in adding the numberato a given numberx.
Supposing we have defined the operation
x+ (a− 1),
the operationx+awill be defined by the equality
(1)x+a= [x+ (a− 1)] + 1.
We shall know then whatx + ais when we know whatx+ (a− 1) is, and as I have supposed that to start with we knew whatx+ 1 is, we can define successively and 'by recurrence' the operationsx+ 2,x+ 3, etc.
This definition deserves a moment's attention; it is of a particular nature which already distinguishes it from the purely logical definition; the equality (1) contains an infinity of distinct definitions, each having a meaning only when one knows the preceding.
Properties of Addition.—Associativity.—I say that
a+ (b + c) = (a + b) +c.
In fact the theorem is true forc= 1; it is then written
a+ (b+ 1) = (a + b) + 1,
which, apart from the difference of notation, is nothing but the equality (1), by which I have just defined addition.
Supposing the theorem true forc= γ, I say it will be true forc= γ + 1.
In fact, supposing
(a + b) + γ =a+ (b+ γ),
it follows that
[(a + b) + γ] + 1 = [a+ (b+ γ)] + 1
or by definition (1)
(a + b) + (γ + 1) =a+ (b+ γ + 1) =a+ [b+ (γ + 1)],
which shows, by a series of purely analytic deductions, that the theorem is true for γ + 1.
Being true forc= 1, we thus see successively that so it is forc= 2, forc= 3, etc.
Commutativity.—1º I say that
a+ 1 = 1 +a.
The theorem is evidently true fora= 1; we canverifyby purely analytic reasoning that if it is true fora= γ it will be true fora= γ + 1; for then
(γ + 1) + 1 = (1 + γ) + 1 = 1 + (γ + 1);
now it is true fora= 1, therefore it will be true fora= 2, fora= 3, etc., which is expressed by saying that the enunciated proposition is demonstrated by recurrence.
2º I say that
a+b=b+a.
The theorem has just been demonstrated forb= 1; it can be verified analytically that if it is true forb= β, it will be true forb= β + 1.
The proposition is therefore established by recurrence.
Definition of Multiplication.—We shall define multiplication by the equalities.
(1)a× 1 =a.
(2)a×b= [a× (b− 1)] +a.
Like equality (1), equality (2) contains an infinity of definitions; having defined a × 1, it enables us to define successively:a× 2,a× 3, etc.
Properties of Multiplication.—Distributivity.—I say that
(a+b) ×c= (a×c) + (b×c).
We verify analytically that the equality is true forc= 1; then that if the theorem is true forc= γ, it will be true forc= γ + 1.
The proposition is, therefore, demonstrated by recurrence.
Commutativity.—1º I say that
a× 1 = 1 ×a.
The theorem is evident fora= 1.
We verify analytically that if it is true fora= α, it will be true fora= α + 1.
2º I say that
a×b=b×a.
The theorem has just been proven forb= 1. We could verify analytically that if it is true forb= β, it will be true forb= β + 1.
Here I stop this monotonous series of reasonings. But this very monotony has the better brought out the procedure which is uniform and is met again at each step.
This procedure is the demonstration by recurrence. We first establish a theorem forn= 1; then we show that if it is true ofn− 1, it is true ofn, and thence conclude that it is true for all the whole numbers.
We have just seen how it may be used to demonstrate the rules of addition and multiplication, that is to say, the rules of the algebraic calculus; this calculus is an instrument of transformation, which lends itself to many more differing combinations than does the simple syllogism; but it is still an instrument purely analytic, and incapable of teaching us anything new. If mathematics had no other instrument, it would therefore be forthwith arrested in its development; but it has recourse anew to the same procedure, that is, to reasoning by recurrence, and it is able to continue its forward march.
If we look closely, at every step we meet again this mode of reasoning, either in the simple form we have just given it, or under a form more or less modified.
Here then we have the mathematical reasoningpar excellence, and we must examine it more closely.
The essential characteristic of reasoning by recurrence is that it contains, condensed, so to speak, in a single formula, an infinity of syllogisms.
That this may the better be seen, I will state one after another these syllogisms which are, if you will allow me the expression, arranged in 'cascade.'
These are of course hypothetical syllogisms.
The theorem is true of the number 1.
Now, if it is true of 1, it is true of 2.
Therefore it is true of 2.
Now, if it is true of 2, it is true of 3.
Therefore it is true of 3, and so on.
We see that the conclusion of each syllogism serves as minor to the following.
Furthermore the majors of all our syllogisms can be reduced to a single formula.
If the theorem is true ofn− 1, so it is ofn.
We see, then, that in reasoning by recurrence we confine ourselves to stating the minor of the first syllogism, and the general formula which contains as particular cases all the majors.
This never-ending series of syllogisms is thus reduced to a phrase of a few lines.
It is now easy to comprehend why every particular consequence of a theorem can, as I have explained above, be verified by purely analytic procedures.
If instead of showing that our theorem is true of all numbers, we only wish to show it true of the number 6, for example, it will suffice for us to establish the first 5 syllogisms of our cascade; 9 would be necessary if we wished to prove the theorem for the number 10; more would be needed for a larger number; but, however great this number might be, we should always end by reaching it, and the analytic verification would be possible.
And yet, however far we thus might go, we could never rise to the general theorem, applicable to all numbers, which alone can be the object of science. To reach this, an infinity of syllogisms would be necessary; it would be necessary to overleap an abyss that the patience of the analyst, restricted to the resources of formal logic alone, never could fill up.
I asked at the outset why one could not conceive of a mind sufficiently powerful to perceive at a glance the whole body of mathematical truths.
The answer is now easy; a chess-player is able to combine four moves, five moves, in advance, but, however extraordinary he may be, he will never prepare more than a finite number of them; if he applies his faculties to arithmetic, he will not be able to perceive its general truths by a single direct intuition; to arrive at the smallest theorem he can not dispense with the aid of reasoning by recurrence, for this is an instrument which enables us to pass from the finite to the infinite.
This instrument is always useful, for, allowing us to overleap at a bound as many stages as we wish, it spares us verifications, long, irksome and monotonous, which would quickly become impracticable. But it becomes indispensable as soon as we aim at the general theorem, to which analytic verification would bring us continually nearer without ever enabling us to reach it.
In this domain of arithmetic, we may think ourselves very far from the infinitesimal analysis, and yet, as we have just seen, the idea of the mathematical infinite already plays a preponderant rôle, and without it there would be no science, because there would be nothing general.
The judgment on which reasoning by recurrence rests can be put under other forms; we may say, for example, that in an infinite collection of different whole numbers there is always one which is less than all the others.
We can easily pass from one enunciation to the other and thus get the illusion of having demonstrated the legitimacy of reasoning by recurrence. But we shall always be arrested, we shall always arrive at an undemonstrable axiom which will be in reality only the proposition to be proved translated into another language.
We can not therefore escape the conclusion that the rule of reasoning by recurrence is irreducible to the principle of contradiction.
Neither can this rule come to us from experience; experience could teach us that the rule is true for the first ten or hundred numbers; for example, it can not attain to the indefinite series of numbers, but only to a portion of this series, more or less long but always limited.
Now if it were only a question of that, the principle of contradiction would suffice; it would always allow of our developing as many syllogisms as we wished; it is only when it is a question of including an infinity of them in a single formula, it is only before the infinite that this principle fails, and there too, experience becomes powerless. This rule, inaccessible to analytic demonstration and to experience, is the veritable type of the synthetica priorijudgment. On the other hand, we can not think of seeing in it a convention, as in some of the postulates of geometry.
Why then does this judgment force itself upon us with an irresistible evidence? It is because it is only the affirmation of the power of the mind which knows itself capable of conceiving the indefinite repetition of the same act when once this act is possible. The mind has a direct intuition of this power, and experience can only give occasion for using it and thereby becoming conscious of it.
But, one will say, if raw experience can not legitimatize reasoning by recurrence, is it so of experiment aided byinduction? We see successively that a theorem is true of the number 1, of the number 2, of the number 3 and so on; the law is evident, we say, and it has the same warranty as every physical law based on observations, whose number is very great but limited.
Here is, it must be admitted, a striking analogy with the usual procedures of induction. But there is an essential difference. Induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us. Mathematical induction, that is, demonstration by recurrence, on the contrary, imposes itself necessarily because it is only the affirmation of a property of the mind itself.
Mathematicians, as I have said before, always endeavor togeneralizethe propositions they have obtained, and, to seek no other example, we have just proved the equality:
a+ 1 = 1 +a
and afterwards used it to establish the equality
a+b=b+a
which is manifestly more general.
Mathematics can, therefore, like the other sciences, proceed from the particular to the general.
This is a fact which would have appeared incomprehensible to us at the outset of this study, but which is no longer mysterious to us, since we have ascertained the analogies between demonstration by recurrence and ordinary induction.
Without doubt recurrent reasoning in mathematics and inductive reasoning in physics rest on different foundations, but their march is parallel, they advance in the same sense, that is to say, from the particular to the general.
Let us examine the case a little more closely.
To demonstrate the equality
a+ 2 = 2 +a
it suffices to twice apply the rule
(1)a+ 1 = 1 +a
and write
(2)a+ 2 =a+ 1 + 1 = 1 +a+ 1 = 1 + 1 +a= 2 +a.
The equality (2) thus deduced in purely analytic way from the equality (1) is, however, not simply a particular ease of it; it is something quite different.
We can not therefore even say that in the really analytic and deductive part of mathematical reasoning we proceed from the general to the particular in the ordinary sense of the word.
The two members of the equality (2) are simply combinations more complicated than the two members of the equality (1), and analysis only serves to separate the elements which enter into these combinations and to study their relations.
Mathematicians proceed therefore 'by construction,' they 'construct' combinations more and more complicated. Coming back then by the analysis of these combinations, of these aggregates, so to speak, to their primitive elements, they perceive the relations of these elements and from them deduce the relations of the aggregates themselves.
This is a purely analytical proceeding, but it is not, however, a proceeding from the general to the particular, because evidently the aggregates can not be regarded as more particular than their elements.
Great importance, and justly, has been attached to this procedure of 'construction,' and some have tried to see in it the necessary and sufficient condition for the progress of the exact sciences.
Necessary, without doubt; but sufficient, no.
For a construction to be useful and not a vain toil for the mind, that it may serve as stepping-stone to one wishing to mount, it must first of all possess a sort of unity enabling us to see in it something besides the juxtaposition of its elements.
Or, more exactly, there must be some advantage in considering the construction rather than its elements themselves.
What can this advantage be?
Why reason on a polygon, for instance, which is always decomposable into triangles, and not on the elementary triangles?
It is because there are properties appertaining to polygons of any number of sides and that may be immediately applied to any particular polygon.
Usually, on the contrary, it is only at the cost of the mostprolonged exertions that they could be found by studying directly the relations of the elementary triangles. The knowledge of the general theorem spares us these efforts.
A construction, therefore, becomes interesting only when it can be ranged beside other analogous constructions, forming species of the same genus.
If the quadrilateral is something besides the juxtaposition of two triangles, this is because it belongs to the genus polygon.
Moreover, one must be able to demonstrate the properties of the genus without being forced to establish them successively for each of the species.
To attain that, we must necessarily mount from the particular to the general, ascending one or more steps.
The analytic procedure 'by construction' does not oblige us to descend, but it leaves us at the same level.
We can ascend only by mathematical induction, which alone can teach us something new. Without the aid of this induction, different in certain respects from physical induction, but quite as fertile, construction would be powerless to create science.
Observe finally that this induction is possible only 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 game do not resemble one another.
To learn what mathematicians understand by a continuum, one should not inquire of geometry. The geometer always seeks to represent to himself more or less the figures he studies, but his representations are for him only instruments; in making geometry he uses space just as he does chalk; so too much weight should not be attached to non-essentials, often of no more importance than the whiteness of the chalk.
The pure analyst has not this rock to fear. He has disengaged the science of mathematics from all foreign elements, and can answer our question: 'What exactly is this continuum about which mathematicians reason?' Many analysts who reflect on their art have answered already; Monsieur Tannery, for example, in hisIntroduction à la théorie des fonctions d'une variable.
Let us start from the scale of whole numbers; between two consecutive steps, intercalate one or more intermediary steps, then between these new steps still others, and so on indefinitely. Thus we shall have an unlimited number of terms; these will be the numbers called fractional, rational or commensurable. But this is not yet enough; between these terms, which, however, are already infinite in number, it is still necessary to intercalate others called irrational or incommensurable. A remark before going further. The continuum so conceived is only a collection of individuals ranged in a certain order, infinite in number, it is true, butexteriorto one another. This is not the ordinary conception, wherein is supposed between the elements of the continuum a sort of intimate bond which makes of them a whole, where the point does not exist before the line, but the line before the point. Of the celebrated formula, 'the continuum is unity in multiplicity,' only the multiplicity remains, the unity has disappeared. The analysts are none the less right in defining their continuum as they do, for they always reason on just this as soon as they pique themselves on their rigor. But this isenough to apprise us that the veritable mathematical continuum is a very different thing from that of the physicists and that of the metaphysicians.
It may also be said perhaps that the mathematicians who are content with this definition are dupes of words, that it is necessary to say precisely what each of these intermediary steps is, to explain how they are to be intercalated and to demonstrate that it is possible to do it. But that would be wrong; the only property of these steps which is used in their reasonings[2]is that of being before or after such and such steps; therefore also this alone should occur in the definition.
So how the intermediary terms should be intercalated need not concern us; on the other hand, no one will doubt the possibility of this operation, unless from forgetting that possible, in the language of geometers, simply means free from contradiction.
Our definition, however, is not yet complete, and I return to it after this over-long digression.
Definition of Incommensurables.—The mathematicians of the Berlin school, Kronecker in particular, have devoted themselves to constructing this continuous scale of fractional and irrational numbers without using any material other than the whole number. The mathematical continuum would be, in this view, a pure creation of the mind, where experience would have no part.
The notion of the rational number seeming to them to present no difficulty, they have chiefly striven to define the incommensurable number. But before producing here their definition, I must make a remark to forestall the astonishment it is sure to arouse in readers unfamiliar with the customs of geometers.
Mathematicians study not objects, but relations between objects; the replacement of these objects by others is therefore indifferent to them, provided the relations do not change. The matter is for them unimportant, the form alone interests them.
Without recalling this, it would scarcely be comprehensible that Dedekind should designate by the nameincommensurable numbera mere symbol, that is to say, something very differentfrom the ordinary idea of a quantity, which should be measurable and almost tangible.
Let us see now what Dedekind's definition is:
The commensurable numbers can in an infinity of ways be partitioned into two classes, such that any number of the first class is greater than any number of the second class.
It may happen that among the numbers of the first class there is one smaller than all the others; if, for example, we range in the first class all numbers greater than 2, and 2 itself, and in the second class all numbers less than 2, it is clear that 2 will be the least of all numbers of the first class. The number 2 may be chosen as symbol of this partition.
It may happen, on the contrary, that among the numbers of the second class is one greater than all the others; this is the case, for example, if the first class comprehends all numbers greater than 2, and the second all numbers less than 2, and 2 itself. Here again the number 2 may be chosen as symbol of this partition.
But it may equally well happen that neither is there in the first class a number less than all the others, nor in the second class a number greater than all the others. Suppose, for example, we put in the first class all commensurable numbers whose squares are greater than 2 and in the second all whose squares are less than 2. There is none whose square is precisely 2. Evidently there is not in the first class a number less than all the others, for, however near the square of a number may be to 2, we can always find a commensurable number whose square is still closer to 2.
In Dedekind's view, the incommensurable number
√2 or (2)½
is nothing but the symbol of this particular mode of partition of commensurable numbers; and to each mode of partition corresponds thus a number, commensurable or not, which serves as its symbol.
But to be content with this would be to forget too far the origin of these symbols; it remains to explain how we have been led to attribute to them a sort of concrete existence, and, besides,does not the difficulty begin even for the fractional numbers themselves? Should we have the notion of these numbers if we had not previously known a matter that we conceive as infinitely divisible, that is to say, a continuum?
The Physical Continuum.—We ask ourselves then if the notion of the mathematical continuum is not simply drawn from experience. If it were, the raw data of experience, which are our sensations, would be susceptible of measurement. We might be tempted to believe they really are so, since in these latter days the attempt has been made to measure them and a law has even been formulated, known as Fechner's law, according to which sensation is proportional to the logarithm of the stimulus.
But if we examine more closely the experiments by which it has been sought to establish this law, we shall be led to a diametrically opposite conclusion. It has been observed, for example, that a weightAof 10 grams and a weightBof 11 grams produce identical sensations, that the weightBis just as indistinguishable from a weightCof 12 grams, but that the weightAis easily distinguished from the weightC. Thus the raw results of experience may be expressed by the following relations:
A=B,B=C,A which may be regarded as the formula of the physical continuum. But here is an intolerable discord with the principle of contradiction,
and the need of stopping this 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 that experience has given
the occasion. We can not believe that two quantities equal to a third are
not equal to one another, and so we are led to suppose thatAis
different fromBandBfromC, but that the imperfection of our
senses has not permitted of our distinguishing them. Creation of the Mathematical Continuum.—First Stage.So far it would suffice, in accounting for the facts, to intercalate
betweenAandBa few terms, which would remain discrete.
What happens now if we have recourse to some instrument tosupplement the feebleness of our senses, if, for example, we
make use of a microscope? Terms such asAandB, before indistinguishable,
appear now distinct; but betweenAandB, now become
distinct, will be intercalated a new term,D, that we can
distinguish neither fromAnor fromB. Despite the employment
of the most highly perfected methods, the raw results of our
experience will always present the characteristics of the physical
continuum with the contradiction which is inherent in it. We shall escape it only by incessantly intercalating new terms
between the terms already distinguished, and this operation must
be continued indefinitely. We might conceive the stopping of
this operation if we could imagine some instrument sufficiently
powerful to decompose the physical continuum into discrete elements,
as the telescope resolves the milky way into stars. But
this we can not imagine; in fact, it is with the eye we observe the
image magnified by the microscope, and consequently this image
must always retain the characteristics of visual sensation and
consequently those of the physical continuum. Nothing distinguishes a length observed directly from the
half of this length doubled by the microscope. The whole is
homogeneous with the part; this is a new contradiction, or
rather it would be if the number of terms were supposed finite;
in fact, it is clear that the part containing fewer terms than the
whole could not be similar to the whole. The contradiction ceases when the number of terms is regarded
as infinite; nothing hinders, for example, considering the aggregate
of whole numbers as similar to the aggregate of even numbers,
which, however, is only a part of it; and, in fact, to each
whole number corresponds an even number, its double. But it is not only to escape this contradiction contained in the
empirical data that the mind is led to create the concept of a
continuum, formed of an indefinite number of terms. All happens as in the sequence of whole numbers. We have
the faculty of conceiving that a unit can be added to a collection
of units; thanks to experience, we have occasion to exercise this
faculty and we become conscious of it; but from this moment
we feel that our power has no limit and that we can count indefinitely,
though we have never had to count more than a finite
number of objects. Just so, as soon as we have been led to intercalate means
between two consecutive terms of a series, we feel that this operation
can be continued beyond all limit, and that there is, so to
speak, no intrinsic reason for stopping. As an abbreviation, let me call a mathematical continuum
of the first order every aggregate of terms formed according to
the same law as the scale of commensurable numbers. If we
afterwards intercalate new steps according to the law of formation
of incommensurable numbers, we shall obtain what we
will call a continuum of the second order. Second Stage.—We have made hitherto only the first stride;
we have explained the origin of continua of the first order; but it
is necessary to see why even they are not sufficient and why the
incommensurable numbers had to be invented. If we try to imagine a line, it must have the characteristics
of the physical continuum, that is to say, we shall not be able
to represent it except with a certain breadth. Two lines then
will appear to us under the form of two narrow bands, and, if
we are content with this rough image, it is evident that if the
two lines cross they will have a common part. But the pure geometer makes a further effort; without entirely
renouncing the aid of the senses, he tries to reach the concept of
the line without breadth, of the point without extension. This
he can only attain to by regarding the line as the limit toward
which tends an ever narrowing band, and the point as the limit
toward which tends an ever lessening area. And then, our two
bands, however narrow they may be, will always have a common
area, the smaller as they are the narrower, and whose limit will
be what the pure geometer calls a point. This is why it is said two lines which cross have a point in
common, and this truth seems intuitive. But it would imply contradiction if lines were conceived as
continua of the first order, that is to say, if on the lines traced
by the geometer should be found only points having for coordinates
rational numbers. The contradiction would be manifest
as soon as one affirmed, for example, the existence of straights
and circles. It is clear, in fact, that if the points whose coordinates arecommensurable were alone regarded as real, the circle inscribed
in a square and the diagonal of this square would not intersect,
since the coordinates of the point of intersection are incommensurable. That would not yet be sufficient, because we should get in this
way only certain incommensurable numbers and not all those
numbers. But conceive of a straight line divided into two rays. Each
of these rays will appear to our imagination as a band of a certain
breadth; these bands moreover will encroach one on the
other, since there must be no interval between them. The common
part will appear to us as a point which will always remain
when we try to imagine our bands narrower and narrower, so
that we admit as an intuitive truth that if a straight is cut into
two rays their common frontier is a point; we recognize here the
conception of Dedekind, in which an incommensurable number
was regarded as the common frontier of two classes of rational
numbers. Such is the origin of the continuum of the second order, which
is the mathematical continuum properly so called. Résumé.—In recapitulation, 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.
Its power is limited only by the necessity of avoiding all contradiction;
but the mind only makes use of this faculty if experience
furnishes it a stimulus thereto. In the case considered, this stimulus was the notion of the
physical continuum, drawn from the rough data of the senses.
But this notion leads to a series of contradictions from which it
is necessary successively to free ourselves. So we are forced to
imagine a more and more complicated system of symbols. That
at which we stop is not only exempt from internal contradiction
(it was so already at all the stages we have traversed), but
neither is it in contradiction with various propositions called intuitive,
which are derived from empirical notions more or less
elaborated. Measurable Magnitude.—The magnitudes we have studied
hitherto are notmeasurable; we can indeed say whether a givenone of these magnitudes is greater than another, but not whether
it is twice or thrice as great. So far, I have only considered the order in which our terms
are ranged. But for most applications that does not suffice. We
must learn to compare the interval which separates any two
terms. Only on this condition does the continuum become a
measurable magnitude and the operations of arithmetic applicable. This can only be done by the aid of a new and specialconvention.
We willagreethat in such and such a case the interval
comprised between the termsAandBis equal to the interval
which separatesCandD. For example, at the beginning of our
work we have set out from the scale of the whole numbers and we
have supposed intercalated between two consecutive stepsnintermediary steps; well, these new steps will be by convention
regarded as equidistant. This is a way of defining the addition of two magnitudes, because
if the intervalABis by definition equal to the intervalCD,
the intervalADwill be by definition the sum of the intervalsABandAC. This definition is arbitrary in a very large measure. It is not
completely so, however. It is subjected to certain conditions
and, for example, to the rules of commutativity and associativity
of addition. But provided the definition chosen satisfies these
rules, the choice is indifferent, and it is useless to particularize it.
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