BOUGHT WITH THE INCOME FROM THE SAGE ENDOWMENT FUND THE GIFT OF 1S91 fll'^'=i.ci:ia .1.57^.'.^ 3777 Cornell University Library Q 175.E591914 Problems of science, 3 1924 009 271 200 Cornell University Library The original of tiiis book is in the Cornell University Library. There are no known copyright restrictions in the United States on the use of the text. http://www.archive.org/details/cu31924009271200 PROBLEMS OF SCIENCE PROBLEMS OF SCIENCE BY FEDERIGO ENRIQUES AUTHORIZED TRANSLATION BY KATHARINE ROYCE WITH AN INTRODUCTORY NOTE BY JOSIAH ROYCE PROFESSOR OF HISTORY OF PHILOSOPHY AT HARVARD UMIVBRSITY CHICAGO LONDON THE OPEN COURT PUBLISHING COMPANY 1914 copykight by The Open Court Publishing Co. 1914 TABLE OF CONTENTS. PAGE Introductory Note by Josiah Royce ix Author's Preface xiv Chapter I. — Introduction i — 32 1. The Special Problems and General Ideas of Science i 2. Science and Philosophy 3 3. The Agnostic Renunciation 4 4. The So-called Insoluble Problems S 5. The Squaring of the Circle 6 6. Perpetual Motion 8 7. Alchemy 8 8. The Problem of Knowledge 9 9. The Dangers of Language 11 10. Absolute and Relative : the Absolute in the Realm of Motion 11 11. The Absolute in the Realm of Morality - . . 12 12. What is the Absolute in a Tratiscendental Sense ? 14 13. Transcendental Procedures Considered in the Light of Physiological Psychology 15 14. Transcendental Procedures in Relation to Infinitesimal Analysis . . 15 15. The Psychological Value of the Absolute 17 16. Substance and Appearance 19 17. The Unknown 20 18. The Distinction Between Subjective and Objective According to Kant 21 19. The Distinction Between Subjective and Objective Considered Posi- tively 22 20. Subjective and Objective in the Process of Measurement 25 21. Subjective and Objective in Scientific Construction 28 220 A Critique of Positivism 29 226 Positivism and Metaphysics 30 23. Physical Positivism 33 24. Positivism in Biology 36 25. Psychological Positivism 43 26. Historical and Sociological Positivism 44 27. The Positive Ends that May Be Proposed for a Theory of Science . . 46 28. Methods : Historical, Psychological and Scientific 48 Chapter II.— Facts and Theories 53—100 I. Dreams and Reality S3 a. Illusions of the Senses 54 IV PROBLEMS OF SCIENCE. FACE 3. The Criterion of Reality SS 4. The Hypothetical Element in Reality S7 5. The Reality of the Past 59 6. Psychological Reality 61 7. The Social Aspect of Reality 62 8. Hallucinations 62 9. The Biological Value of the Belief in Reality 64 10. The Postulate of Knowledge 65 11. Common Facts and Scientific Facts 66 12. Physical Facts 67 13. Fact and Law 68 14. Astronomical Facts 7° 15. Chemical Facts 72 16. Facts of the Biological Sciences 72 17. Historical Facts 75 18. Hypothesis in its Relation to Scientific Knowledge 79 19. The Value of Scientific Knowledge 80 20. Knowledge by Means of Concepts 81 21. Empiricism and Rationalism 82 22. The Acquisition of Knowledge 83 23. Scientific Theories 85 24. The Theory of Gravitation 86 25. A Critique of Theories 90 26. The Electrostatic Theory of Poisson 91 27. The Theory of Solutions 95 28. The Economy and the Psychological Development of Theories 98 Chapter III. — The Problems of Logic 101-172 A — Pure Logic. 1. Real and Formal Logic loi 2. Verbal Forms and Symbols 104 3. Symbolic Logic and Psychological Logic 107 4. The Possibility of Formal Logic Founded Upon the Development of Mathematics 108 5. A Critique of Definition 109 6. Real Definitions and Nominal Definitions 113 7. Implicit Definition 115 8. Examples : Concepts Founded Upon Physical Data 1 17 9. Concepts Founded Upon Psychological Data 118 10. The Logical Process : The Statement of Problems 120 11. Logical Operations 121 12. Purely Logical Concepts 122 13. Elementary Types of Definition 122 14. Secondary Logical Relations and Axioms 124 15. Propositions 126 16. Given Logical Relations 126 17. Conditions which Make Concepts Possible : Logical Principles 128 18. The Compatibility of the Postulates of a Theory 130 ig. The Fundamental Principles of Arithmetic 132 TABLE OF CONTENTS, V MGE B. — The Application of Logic. 20. Fundamental Problems 135 21. Logical Representation and the Postulate of Knowledge 135 22. Substance : Matter and Energy 137 23. Cause 140 24. Recapitulation 143 25. The Actual Value of Logical Principles 143 26. The Value of Axioms : The Objective Reality of Logic 14S 27. The Limits of the Application of Logic 146 28. The Problem of Verification . .' 148 29. The Veriiication of Explicit Hypotheses 149 30. The Experience of a Finite Number of Objects 151 31. Experience of the Continuous 152 32. Conclusions as to the Interpretation of Experience iSS 33. The Postulate of Continuity and the Psychological Representation of Cause : How and Why 156 34. The Confirmation and Verification of the Implicit Hypotheses 158 35. Examples 160 36. The Present Crisis in Political Economy i6l 37. Conclusions : The Vicious Circle in Science i6s C — The Physiological Aspect of Logic. 38. Statement of the Problem 167 39. Fundamental Hypotheses 168 40. Explanation of the Axioms 170 41. Concerning the Idea of Cause 171 Chapter IV. — Geometry 173—231 A — Geometry and Reality. 1. Introduction , 173 2. Realism and Nominalism 174 3. Space and Spatial , 174 4. A Critique of Space Relations 175 5. The New Nominalism of H. Poincare 176 6. Geometry as a Part of Physics 180 7. On the Exactitude of Geometry 183 8. Space as a Concept : Abstract Geometry 184 9. Historical Hints Regarding the Non-Euclidean Geometry 186 10. The Problem of Space 190 11. Can Non-Euclidean Geometry Become an Object of Intuition? .... 195 12. Concerning Other Possible Geometries ig6 13. Non-Archimedean Geometry and the Arbitrary Nature of the Pos- tulates 197 B — The Psychological Acquisition of Geometrical Concepts. 14. Statement of the Problem 199 15. The Biological Problem of Spatial Orientation 199 16. Program of the Following Studies 202 17. Sources of our Criticism 202 18. General Observations as to the Spatial Content of Sensations 203 19. Spaces in Physiology and Space in Geometry 204 Vi PROBLEMS OF SCIENCE. PAGE 20. The Space Data of Sight and Projective Geometry 205 21. The Spatial Data of the Tactile and Muscular Sensations and Met- rical Geometry 211 22. Parallel Between the Historical Development and the Psycho- Genetic Development of the Postulates of Geometry 214 23. The Postulates of the Continuum : The Line 216 24. Postulates as to the Continuum of Two and of Three Dimensions . . 221 25. Postulates of Projective Geometry 223 26. The Postulates of Metrical Geometry 225 27. Metrical- Projective Combination: The Postulate of the Parallels . . 227 28. Conclusion 230 Chapter V. — Mechanics 232 — ^298 The Objective Significance and the Psychological Development of the Principles of Mechanics. 1. Mechanics as an Extension of Geometry 232 2. Programme 233 3. Time : Succession and Duration 234 4. Psychological Time and Physical Time 235 5. The Properties of Temporal Succession 236 6. Duration 237 7. The Postulate of the Measure of Time 241 8. Time as Independent of Place 243 9. The Historical Development and the Evidence of the Principles . . 245 10. Fundamental Concepts : . . . 249 11. Mathematical Nominalism 249 12. The Material Particle 252 13. Force 253 14. Geometrical Data of Force 255 15. Principles of Geometry in Statics 257 16. The Composition of Forces 258 17. The Foundation of Dynamics 259 18. Motion 262 19. Mass 269 20. Postulate About Mass and the Dynamic Principle of Action and Reaction 275 21. The Fundamental Laws of Motion 278 22. The Principle of Inertia Generalized 281 23. Synthetic Valuation of Principles 283 24. The Statics of the Systems : Rigid Connections 284 25. The Lever and the Inclined Plane : The Principle of Static Moments 286 26. The Principle of Virtual Work 288 27. Dynamics of the Systems : D'Alembert's Principle 290 28. Principles of Kinetic Energy and of Least Action 292 39. Verification of Dynamics 293 Chapter VI. — The Extension of Mechanics 299 ^387 A — Physics as an Extension of Mechanics. 1. The Development of the Philosophy of Mechanics 299 2. Quantity and Quality : The Cartesian Hypothesis 300 TABLE OF CONTENTS. VU PAGB 3. Examples : Weights 301 4. Quantity of Heat 302 5. The Measure of Intensity 304 6. Natural or Absolute Measure : Temperature 305 7. Recapitulation and Critique 306 8. The Two Types of Mechanism: The Cartesian and the Newtonian. 309 9. Forces Reduced to Impacts : Gravitation 311 10. The Kinetic Theory of Gases 312 11. The Theory of Elasticity : Solid Bodies 313 12. Permanent Alterations 316 13. The Mechanical Theory of Heat: Conservation of Energy 317 14. The Second Principle of Thermo-Dynamics 319 15. Irreversible Phenomena 322 16. The Mechanics of Energy 325 17. Matter and Energy 327 18. The Localization and Motion of Energy 329 19. The Elastic Explanation of Optical and Electro-Magnetic Phe- nomena 330 20. Optics 331 21. Electro-Statics 333 22. Electro-Magnetism 334 23. The Positive Content of Maxwell's Theory 336 24. Elasticity Considered as Motion 339 25. Electro-Magnetism of Bodies in Motion : Hertz's Theory 341 26. The Theory of Lorentz 345 27. A Critique : The Principle of Action and Reaction 347 28. The Principle of Relativity 349 29. Ether and Matter 352 30. The Dsmamics of the Electron : Radiation 354 31. Electrical Dynamics 3SS 32. The Electrical Explanation of Gravitation 357 33. Conclusion : General Non-Newtonian Djmamics 358 34. Physical Explanation: The Value of Mechanical Models and of Equations 363 B — The Mechanical Hypothesis and the Phenomena of Life. 35. Introduction 367 36. Preliminary Objections 368 37. Biological Determinism 369 38. Psychological Determinism and Free Will 370 39. Physicism 375 40. The Teleological Explanation 377 41. Life and the Principles of Thermo-Dynamics 381 42. The Mechanical Hypothesis and the Problems of Evolution 382 43. The Irrelevancy of the Mechanical Explanation in Biology 384 44. Conclusion 38S Index of Proper Names 389 INTRODUCTORY NOTE. THE various causes which have so long delayed the publication of the present translation of the Problems of Science, may prove to be rather aids than hindrances to the just appreciation of this very remarkable synthetic view of scientific methodology. For, as a result of these accidents, the book of Professor Enriques is offered to American readers at a time when we are better fitted than we have been during the last few years to appreciate the significance of the author's large, clear, and calm view of a wide range of per- plexing modern problems. The first edition of the Italian text of the Prohlemi della Scienza of Professor Enriques appeared in. 1906, and had already become known to a wide circle of European students, belonging to various nationalities, at the time of the International Congress of Philosophy at Heidelberg, in the late summer of 1908. At this congress I myself met the author, and undertook to do what I could towards finding an American publisher for a translation of this book. Not long after the congress, Dr. Carus, on behalf of the Open Court Publishing Company, agreed to undertake the publication of the translation. The translator completed the first draft of the manu- script by June 1909. A certain amount of revision of some of the more technical portions of the translated text remained as that part of the work which I had myself, from the outset, agreed to under- take. Moderate in quantity as this task of revision has indeed proved, it came into conflict with a great number of academic and personal duties of my own, — duties which resulted from my pre- vious engagements, and which could not at once be laid aside for the purpose of finishing my own little part of the task. Various new hindrances later intervened. In consequence of my own delay, the revised manuscript of this translation was first put in the publishers' charge as late as June 1912 ; and this American edition of the work of Enriques has since been in press. The delay has given oppor- X PROBLEMS OF SCIENCE. tunity to use the second Italian edition of the Prohlemi for the pur- pose of the revision of some passages of the translation. Since the Heidelberg Congress of Philosophy in 1908, prag- matism, which, as many readers of current discussion will remem- ber, formed the principal topic of the lively discussions of that session, has passed through its days of joyously youthful success; and is now no longer a novelty. Meanwhile, the new star of Berg- son has glowed with increasing brilliancy from year to year. "Anti- intellectualism" has become, for the time, the prevailing mood in the more popular expositions of philosophy. Mobile minds, — ^minds characterized by what James called a "dramatic" temper, — have taken a leading part in controversy. Books such as the present one may seem for the moment, to such minds, out of place. Yet precisely such moods as have been so widely represented in the general literature of popular philosophy since 1908, call for their own correction, or at all events for their own complement and supplement. What is most to be feared, at a time when discussion is so lively and when "anti-intellectualism" has gained such large and eager audiences, is not any definitive triumph of the "anti- intellectual" enthusiasms, but rather some too swift and "dramatic" reaction in the world of the ruling philosophical interests, some drastic return from the revolutionary temper of the thought of the moment to the older types of scientific orthodoxy, some renewal of the "dogmatic slumber" from which James, the Pragmatists, and Bergson, have awakened many plastic, quick-witted, but not always naturally judicial minds. At just such a moment, a book like the present work may therefore be especially useful to thoughtful students, who love patience and clear ideas quite as much as they are fond of intuitions, of brilliancy, and of "vital impetus." The work of Professor En- riques stands somewhat above and apart from those philosophical controversies which the anti-intellectual movement has inspired; for this book was prepared and published in the original Italian before those controversies assumed their latest phase. Yet the author, already prominent in the discussions of the Heidelberg Congress of 1908, has since been President of the Philosophical Congress at Bologna in 1911. Translations of his Problemi della Scienza into French and German have widely extended his influence. His book is by far the most thorough and synthetic treatment of the problems of scientific methodology which belongs to recent years, — with the sole exception of the treatment which forms part of the first two INTRODUCTORY NOTE. XI volumes of Merz's History of Thought in the Nineteenth Century. Meanwhile, owing to their widely contrasting ranges and modes of discussion, Merz's book, (which is primarily a history of science, with a treatment of methodology obligate), and the book of Enriques, (which is explicitly a scientific methodology, with numerous ref- erences to contemporary interests and controversies) : — these two books, I say, come into no sort of rivalry with each other, but supple- ment each other in a way which is all the more important because neither author can have known, I think, about the other's work until his own was substantially complete. As for the relations of the book of Enriques to the recent con- troversies to which I have just referred, the work on the Problems of Science is thoroughly "intellectual" in its tone and temper, with- out being open to any of the usual objections to "intellectualism" which are now most popular among philosophical readers. The author (himself Professor of Projective and Descriptive Geometry in the University of Bologna), approaches his "Problems" with the training of the mathematician and the logician, and with the reputation which his treatise on "Projective Geometry," and his published essays on the "Foundations of Geometry" have long since won for him. Yet this book shows no tendency to magnify overmuch the office of the geometer, or the authority of the logi- cian, or the powers of the human reason, in the interpretation of phenomena. Pragmatists will find Enriques emphasizing some of their own theses regarding what is now called the "instrumental" or the "functional" significance of thought, and of the whole scien- tific process. And this emphasis, as it appears in some of the most important general discussions (notably in the latter half of the chapter on Logic), is all the more interesting because (as we have just seen) this book,— especially in its earlier chapters, — antedates the most recent developments of pragm.atism. Yet this relatively pragmatistic element of the book of Enriques appears in a form which is both largely original, and extremely many-sided and judi- cial. Enriques views the thinking-process as indeed an "adjust- ment" to "situations," But he lays great stress upon the tendency of science to seek unity, upon the synthetic aspect of scientific theory, upon what he calls the "association" of concepts and of scien- tific "representations." And this stress upon synthesis, this sense for wholeness and for unity, gives his treatment both of the values and of the limits of scientific hypotheses and theories, an original and a very notable character. In his view of the work and of the Xil PROBLEMS OF SCIENCE. uses of natural science, Enriques stands in strong contrast to the original or Comtean type of "Positivism"; for he greatly empha- sizes both the "objective aspect" and the significance of constructive scientific theories. As a methodologist, Enriques also finds a posi- tive value in many "hypotheses" of such a type that Ostwald's well- known maxims of scientific method would condemn them in ad- vance. Nor does Enriques agree with Mach's or with Pearson's limitation of the business of science to the simple "description" of physical phenomena. Yet, despite this fondness and this respect for synthesis and for the "association" of various scientific concepts and "modes of representation," Enriques has as sincere an aversion to what he takes to be genuinely "metaphysical" constructions as has any posi- tivist; as vigorous a hostility to the "transcendental" and to the "absolute" as is cultivated by any philosopher of our "Chicago School" ; and as clear, if not as vehement, a respect for the relation between thought and will as is expressed by any Pragmatist. What sets Enriques most apart from most of the thinkers, — pragmatists, positivists, relativists, — with whom one would be most likely to associate him, — or on occasion to confound him, — is a certain judicial temper, a breadth of view, a fondness for synthesis, an exactness of intellectual training, a love of the comparative study of his topic, — in brief a spirit which is as rare as it is requisite in a man who is to prove a thoroughly good methodologist. En- riques certainly does not, as a philosopher, blindly overrate the work or the powers of the intellect. On the contrary, he emphasizes the imperfection, the relativity, the tentative and inadequate character of all scientific and theoretical construction. Yet he is neither scep- tic, nor anti-intellectualist. He does justice to the "instrumental" function of thought. But he is certainly no mere "instrumentalist." For the stress which he lays upon the "objective aspect" of even the most highly theoretical portions of scientific theory; and his insistence upon the tendency of science towards a genuine and irrevocable progress, not merely in its mutable and transient control of special experiences, but in its total view of nature, — these ten- dencies in Enriques seem to exclude any interpretation of his phi- losophy of science as a mere "instrumentalism." For Enriques, the "absolute" is no object for science. But what is won, in a scientific way, is won, and the whole tendency of the scientific attainment of truth is to be not a dealing with what is merely mutable, but an irreversible progress towards a survey of the unity of the real, — INTRODUCTORY NOTE. XUl a grasping of real "invariants," and of wholes. These are theses that have a prominent place in the extremely careful, far-seeing, critical, and constructive methodology which constitutes this wealthy and well-wrought book. Where so much is offered, it is hard to select what the reader should most consider. Personally I have taken very special interest in the treatment which Professor Enriques gives to the Principles of Geometry, — a topic which he has made especially his own, and which (as here discussed) will appeal not only to students of the logic of mathematics, but to psychologists interested in those aspects of the problem of space which especially concern their own work. The concluding chapter, dealing as it does with a wide range of highly technical physical problems and theories, is at once the most difficult (both for the translator and for the reader) and the most characteristic of the book. Here the synthetic tendencies of our author, — his wide outlook, his fairness of judgment, his careful comparisons, his bringing together of matters which are, for most readers, hopelessly far apart, — all tend to show what this book is, — a treatise on methodology such as we have long needed, and have here at length before us in English. May the work of the President of the last Congress of Philosophy serve to quicken as well as to nourish interest both in science and in methodology. May it aid us in treating more judiciously, more broadly, and more exactly, the current controversies concerning the office and the scope of the human intellect. And above all may it foster that spirit of unity in thoughtful research which its author has so well illustrated, — that spirit namely which tends to unite the work, not only of various sciences, but of various nations. December, 1, 1913. JOSIAH ROYCE. PREFACE. A TRAIN of thought which gradually came to maturity during the ten years from 1890 to 1900 has resulted in a critical study of certain problems relating to the logical and psychological devel- opment of scientific knowledge. These problems are here entitled: "Problems of Science." The plan of the work may be said to have been settled, with the exception of the last chapter, ever since the year 1901. In that year I began to state my views of the subject in various lectures and conferences. The formal arrangement of the material has been but slightly retouched since that time. It is rather difficult to state bow the general spirit of my treat- ment is related to the philosophical distinctions current in the schools. I should like to characterize this spirit as at once critical and posi- tive: because I really think that I have interpreted in a clearer and more scientific way, and have reconciled (without eclectic com- promises) certain speculative tendencies by which my thought was prompted at the outset. But I do not conceal from myself the fact that the ideas set forth in this book are profoundly different from those which are current under the name of critical positivism. The reading of the first, or introductory, chapter alone is sufficient to prove this fact. The arguments that are developed follow the headings of the various parts of the work, which are recapitulated in the index. The connection between these widely different topics consists in a general view of scientific procedure, which I have sought to explain by means of an inductive exposition, reenforced by many examples. The analysis of what constitutes reality is developed in Chap- ter II into a critique of facts and of theories, so handled as to dis- tinguish between the positive content of science on the one hand, and its subjective aspects on the other. From this analysis arise two classes of problems which are PREFACE. XV successively studied: (1) the problems connected with the logical transformation of concepts, regarded both as a psychological devel- opment and as an instrument of knowledge (Chap. Ill) ; and (2) those problems which refer to the significance and to the acquisition of the more general concepts of space, time, force, motion etc. (Chaps. IV, V). The theoretical questions of physics are examined in Chap. VI, in connection with a critique of the theory of mechanics; and this critique ends with certain observations as to the extension of mechan- ical explanations into the field of the phenomena of life. The idea of science which I have formed is not here explicitly developed in harmony with a general system of philosophy. It is not a part of my plan to examine the relations between cognition and volition further than is required by the very definition of science. And so I postulate the value of science, and every further effort to appreciate what that value is will be excluded from this critique. But that does not imply that I consider knowledge as an end in itself. It is plain that "science for the sake of science" is a formula devoid of social content. And on the other hand it is plain that cognition can offer to volition only the means to work with, but not the ends, and that it is absurd to seek in science the guiding principles of life. But let us keep in mind that the will to be scientific, outside of its utility, offers in itself a significant moral standard. For such a will recognizes and affirms the True as independent of fear or of desire, and in this way the consciousness, as well as the power, of a will capable of looking beyond the transitory ends of the pres- ent towards a more lofty future progress, promotes the full develop- ment of the human being. My faith in this philosophy of science has led me from the fields of geometry, where thought rests quietly in the security of acquired facts, to discuss the building up of a science of knowledge which may become the common possession of the studious and may tend to unify the various domains of knowledge in one synthetic view of the cognitive methods. Hence the plan of a work, which, by bringing together so many diverse objects and problems, oversteps the boundaries of custom set by oUr scientific public, and which must thus excite a natural distrust. To offset this I have merely the consciousness of my fifteen XVI PROBLEMS OF SCIENCE. years work. But where indeed learning and talent prove themselves too unequal to the arduous task, my mind is buoyed up by the hope that the labor will not have been in vain if the vision of the unit/ of science thus evoked may avail to unite in a common brotherhood, and for a lofty end, the efforts of our youth, which tend in the vari- ous lines of study to the conquest of truth. May this vision rise above the differences of matter and of technique, as I progress. And may it above all so light the way for Italians, that the work of renewal begun by our elders may be continued through the full and harmonious development of our national genius. F. E. I. INTRODUCTION. 1. THE SPECIAL PROBLEMS AND GENERAL IDEAS OF SCIENCE A DOUBLE fatality hangs over one who has consecrated his days to science. ' If he would contribute to the advancement of science, he must prepare himself first of all by a patient study of the thousands of details which constitute its technique; he must learn the results ob- tained by numberless laborers whose researches tend toward the same aim. He must master their conceptions and subject them to a new criticism. This work so engrosses the attention of the investigator that he has little time left for casting a glance over the branches of science which are developing beside him. Yet this necessity also weighs upon his soul 1 If on the one hand he ought to study special prob- lems, on the other, he cannot exempt himself from considering the ends set before special research by rising to a general outlook which shall command the view of a broader scientific basis. This double necessity causes a conflict of tendencies, and this conflict in our system of production results in a loss of time and of work from which the intellectual world suffers. Most investigators, if they are not rightly directed, shut them- selves up in a narrow circle and fall into a blind empiricism. Others lose themselves in the region of confused generalities, while a few finer spirits find the way for themselves, and often must win again by fresh efforts that which they should have a right to expect as the outcome of the completed work of their companions in labor. But the age of heroes, that of Descartes or Leibniz, whose genius opened all the doors of science, seems closed forever! The conquests of the past weigh upon the present and upon the future. And if it is permissible to hope that a happier use of our intellectual powers may put an end to the confusion of to-day, 2 PROBLEMS OF SCIENCE. Still it is neither credible nor desirable that we should return to a state of affairs in which science was the exclusive task of a few superior men. We must recognize that a thousand forces united may succeed in raising those masses of rock which weigh so heavily upon the shoulders of the giant. Our need is to perfect the organization of that work which ought to be accomplished, under free conditions, through a fitting scientific education. To this end it is necessary that all enlightened men, in whatever special branch of study they are respectively work- ing, should be conscious of the unity of the aims of science. They will then give a helping hand each to the other, and will aid one another in cordial understanding. The isolated efforts of individuals will be replaced by the more fruitful work of learned societies.^ Nor will superior minds lack a place in such organizations of productive work. Rather, freed from the need of hampering their abilities as investigators by the acquisition of an over-minute learn- ing, they will be able to profit more freely by the advantages of this community, and in consequence to make themselves more useful to it. They will become organizers, connecting the various researches to the general ends of science, of which they will then be able to gain a wider and more precise view. But while we thus imagine the plan of an ideal organization for scientific production, are we not merely indulging in a poetic dream ? Certainly we must not conceal the difficulties which present themselves upon our road; but these cannot rob us of our faith in progress, which must indeed be slow, but which must lead, in each domain, to superior forms of social life. The end for which we ought to strive to-day is a scientific edu- cation, which shall enable the workers in any field whatsoever to understand better how the object of their own research is subordi- nated to more general problems. We need to awaken in people's minds the feeling of a larger harmony in which apparent contradictions are reconciled. Nothing is so dangerous as to shut oneself within a circle from which, with rigid logic, everything is banished that does not fit in with the results of a narrow experience ! * The signs are not wanting that the need of scientific association is becom- ing more and more acutely felt. I will cite as authoritative testimony that of E. Picard in his admirable report upon the general condition of the sciences published by the Paris Exposition in 1900. INTRODUCTION. 2. SCIENCE AND PHILOSOPHY. The aims thus sketched are very dearly connected with the influence which philosophy ought to exert upon science. For, in sum, philosophy is the expression of an impulse which in the order- / ing of the various branches of knowledge, urges us, although in various ways, towards unity and generality. But if philosophy does not accomplish its office as completely as is needful, the reason should be sought in that state of things whence arose, at the beginning of the past century, the sad strife which still divides the philosophers from the men of science. We do not here purpose to search into the causes of this strife, since it would not now be helpful to renew the old recriminations, when on either hand are now to be seen the signs of a happy reconciliation. Let us merely observe that the judgment of students of science ought calmly to correct that lack of clearness and precision which accompanies certain nebulous expressions of speculative thought. But such a correction loses all its efficacy in case it includes in common condemnation one who only conceals the emptiness of his purposes beneath obscure language, and one who, being perhaps the victim of some unavoidable error of method, still aims to find unity in multiplicity and to separate the determinable from the in- determinate. And the value of criticism is still more diminished, when, not satisfied with blaming the philosopher, it turns against philosophy itself and contrasts with the variability of the latter the solid con- structions of the edifice of science. This accusation can be heeded only by one who has not comprehended that philosophic thought / does not necessarily require the solution of particular and well-'" defined problems. Philosophy represents rather a tendency of the human intellect — a tendency which determines, so to speak, the style of the edifice of science, so that this edifice takes on diverse forms as it grows. The severe judgment of scientific men, of which we have al- ready spoken, has been most especially directed against that philos- ophy which, under the influence of modern tendencies, attempted to scale the loftiest summits of abstraction, during the first half of the last century. Fortified by the decisive condemnation of Auguste Comte, posi- tive science entertains the belief that there is in this philosophy no movement of ideas, but only a vain battle of words. Such an opin- 4 PROBLEMS OF SCIENCE. ion does not confine itself to combating the metaphysics of the modern systems as a vicious manner of treating certain problems, but it even goes so far as to deny the very existence of the problems referred to by such speculations. Thus it not seldom happens that the reader of Comte, ignoring strangely enough his constructive work in his exposition of the gen- eral results of the sciences — a. work the freshness of which compels admiration after sixty years — that the reader, I say, accords the larger favor to what is negative in this philosophy. For Comte metes out equal condemnation to everything that was the object of classical inquiry. To him it seems without examination that all this labor has been spent in vain, because its aim was to make known that which never will be knowable in any sense whatever. 3. THE AGNOSTIC RENUNCIATION. It has been justly observed that the present tendency in dealing with such matters results from an especially modem attitude of mind towards certain traditional questions. We have hoped that a sort of general pacification would result, "for our whole social life, from that critical or dogmatic agnosticism to which all ways of speculation led during the last century. A little more than thirty years ago, an illustrious physiologist (Du Bois Reymond) proclaimed the concept of such an agnosticism, summing it up in an eternal I gnordbimus which weighs upon the science of to-day. And more recently a movement of thought, which constitutes a singular retrogression in the history of civilization, has arisen with "the bankruptcy of science" as its war-cry; a cry purposely chosen to signify the plain principle that knowledge cannot set a standard -^ for the will. Not in vain was aroused the phantasm of a reaUty which must remain forever inaccessible to any kind of scientific determination. But it is not a part of our plan to examine here, under such an aspect, the consequences of the philosophic renunciation expressed in the affirmation of the unknowable. Let it suffice to observe that a happy reaction is at work at present against this cowardice of the modern spirit, and is already clearly shown in various fields of science. Truly those who have had the audacity to set limits to human knowledge have not always been so prudent as to keep to the vague region of indefinite things. The bounds thus set have mostly been INTRODUCTION. 5 surpassed, in an unforeseen way, and it has become plain that it is no more legitimate to affirm our future ignorance about any subject whatever, than it is to believe ourselves possessed of a knowledge not yet acquired. Let us cite a single instructive example, calling to mind how spectrum analysis succeeded in disproving after a few years the statement of Auguste Comte which denied that astronomy could penetrate the mystery of the chemical constitution of the heavenly bodies. For the rest the surest proofs upon which agnostic conclusions attempt to lean, offer slight resistance to the progress of criticism; so that they may be compared to certain fortresses, terrible machines of war, which no superior force could beat down, but which a hand- ful of men may easily overcome by making a detour, without joining battle at all. Just so does science proceed, going around the difficulty which blocks its road! Science would never have reached its present state, had it not been constantly changing the form and the statement of its prob- lems, adapting the scope of its researches to the changed condi- tions of thought. This kind of procedure seems so general, in every branch of the knowable, that a skeptical mind, considering things under a special aspect, might well smile at a progress which was never permitted to follow a straight line. But since, none the less, he who thinks of things in their en- tirety, must recognize that there has been progress, it is plainly seen that scientific questions include something essential, apart from the special way in which they are conceived in a particular epoch by the scholars who study such problems. The search for this essen- tial aspect which is concealed in every question, is the office of the true philosophical spirit, which is not satisfied with pausing on the surface of things. 4. THE SO-CALLED INSOLUBLE PROBLEMS. In a broad sense there are no insoluble problems, since every problem corresponds to a feeling, often obscure, which may be satisfied by the discovery of some new fact that increases our power over the external world. There are only problems not yet suitably expressed, and idle discussions, void of sense, in which through defect of method, we at times wander far from our actual goal, be- 6 PROBLEMS OF SCIENCE. cause we do not know how to set it dearly before us, and so are seeking an answer to ill formulated questions. This is what the history of science clearly shows us, and from that history we will try to draw some instructive examples. 5. THE SQUARING OF THE CIRCLE. One need not be versed in geometry, mechanics or chemistry, to have heard of certain famous problems, such as the squaring of the circle, perpetual motion, and that changing of metals into gold, for the sake of which medieval investigators so wearied themselves in their search for the philosopher's stone. Such problems, which are commonly quoted to prove the weak- ness of the human mind humbled before insurmountable difficulties, offer food for more interesting reflections, by which our faith in scientific thought may be confirmed. The problem of squaring the circle is the most noted of the three enigmas that the Greek geometricians have left as an heirloom to their successors. The trisection of the angle and the duplication of the cube had already received, in modern times, a sufficient solution, but the squaring of the circle completely defied the principal analysts of the past century. Only twenty-four year ago (1882) this difficulty was conquered! But the manner in which such a result was reached, and the very sense of the solution obtained, have the greatest interest in relation to our purpose. "To square the circle" means, for one who has not an exact no- tion of the problem, "To construct a square containing the same area as a specified circle." That such a square exists, reasons of continuity may easily demonstrate, since the side of the square itself may be easily constructed, when we have a segment equal to the length of the circumference. This observation suffices to assure us that the proposed problem is not absolutely impossible. Yet all efforts, renewed almost without pause during twenty centuries, have of necessity been doomed to come to naught in face of the insufficiency of the means which were expected to do the work. Nor would any intellectual superiority have given the key to the enigma, had not a new critical method cleared up the old con- cepts relating to the solution of geometric problems. The ruler and the compels were the only instruments used by the Euclidean geometry in its constructions. And although it would INTRODUCTION. / not be out of place to suppose that the Greeks themselves had sus- pected that the means used were insufficient for the solution of the three problems which remained for them insoluble, yet they lacked the power to make themselves sure of the matter by means of anal- ysis. The matter has been put in a new light for us since Descartes founded analytic geometry. Then appeared the true sense of the question upon which so many efforts had been spent in vain: "Operating upon the diameter of a circle, by means of the ruler and compass, can one construct the side of a square having the same area as a circle, or (which comes to the same thing) can one construct a segment equal to the length of the circumference?" Thus the word "construct" assumed a meaning determined with reference to certain instruments (ruler and compass) which were exclusively to be used; and thus the proposed problem ap- peared in a new aspect. If the length of the circumference is to be constructed in the manner pointed out, the number ir, which expresses its relation to the diameter, must possess certain well-determined analytic prop- erties. Hence it becomes a precise question to learn whether such properties belong to it. And in this form we see a priori how the problem admits of an affirmative or a negative answer. The question was resolved, in 1882, by the work of Lindemann, who happily succeeded in extending to a larger field of numbers the methods wisely thought out by Hermitc in his study of the number e, the base of the Napierean logarithms. The answer is negative. We should not then seek to square the circle by means of Euclidean constructions, since the solution of the problem is in that sense impossible. But as we have already pointed out, that impossibility exists only in relation to the instruments specified. Since the solution exists, it cannot be in the absolute sense im- possible to obtain it. What is required is, then, to think out a suit- able instrument which shall be capable of furnishing the solution by meeting all the practical needs. From considering the question in this new aspect we may say that a satisfactory solution of the problem has been given by the integraph of Abdank Abakanowicz, commonly used for the com- putation of areas. 8 PROBLEMS OF SCIENCE. 6. PERPETUAL MOTION. An impossibility even more absolute than squaring the circle seems to frustrate the desires of those who for centuries have been seeking perpetual motion. But the scientific problem connected with this research, far from proving to be insoluble, has led to the dis- covery of a leading principle of nature. If the matter is considered in its broader aspect, the requirement of perpetual motion reveals itself as that of a particular relation between the dynamic elements capable of generating the motion of a machine, and the work done by the machine. Now such a relation is established by the principle of the con- servation of energy, through which precisely perpetual motion, as it is commonly understood, turns out to be impossible. Meanwhile it was observed that if, independently of the in- equality of temperatures, one could succeed in transforming heat into mechanical work, a new form of perpetual motion might be reached. We should then have a machine which would utilize, with- out being obliged to stop, the work done by a body constantly cool- ing off below the temperature of the environment. But to the new demand, suggested by the search for such a perpetual motion of the second species, the second principle of thermo-dynamics answers, in the field of physical experiments, by denying the possibility of such a transformation. We shall be led to realize the great positive fruitfulness of this principle when we consider that in connection with the first principle of the conser- vation of energy, the new principle gives to us, by the theorem of Carnot, a way of estimating the work done by heat-machines. 7. ALCHEMY. We have seen how the problem of perpetual motion ultimately reaches an element which is quantitatively invariable in the trans- formation of energy. Instead, a qualitative irreducibility^ of matter, in relation to our means of experiment, is the result to which the researches of the alchemists would lead. Thus the problem of "changing base metals into gold" seemed to reappear in a much more general form whence modern chemistry has arisen. The manifold changes of matter, appearing at first in the guise ' The relative character of this irreducibility has been brought to light by the recent researches of Ramsay which seem to prove the transformation of radium into helium and of copper into lithium. INTRODUCTION. V of the miraculous, had struck the fancy of the early investigators, to whom no change in the constitution of bodies could have seemed impossible. But when the criticism of observed facts made it pos- sible to catch a glimpse of "the law" which dominated the varieties of phenomena, the problem of chemistry began to assume its true scientific aspect, rising to the general investigation of the relations and conditions which rule the transformation of matter. This may be called the new statement of the problem which was hidden in the minds of the alchemists, in so far at least as we take account of the obscure scientific spirit by which their researches were prompted. If, on the other hand, the growth of modern science is considered, one cannot fail to recognize how the very wealth so longed for by the ancient scholars has been surpassed. For the industrial applications of chemistry give, in our times, far greater and more valuable returns than the riches of Midas, which the transformation of metals into gold would have given. 8. THE PROBLEM OF KNOWLEDGE. No greed for riches has driven philosophers to devote their whole energies to the problems of reality and of knowledge. If they had succeeded in their endeavors, one sole prize might seem to have awaited them; that is, the prize of reconquering, after having passed through philosophic skepticism, that solid and simple faith of men, which is beyond and above all criticism. But precisely the most logical minds, starting upon this road, seemed to reach the very opposite result. Nothing guaranteed the existence of this asserted reality, which it is not granted us to attain by any means. The idea alone is true, and the ego remains secure as the ruler of a world which crumbles around it. A marvelous conclusion, to which indeed one might easily make the answer that Diogenes made to Zeno, who claimed to prove that motion does not exist: The cynic, rising from the ground where he sat, silently began to walk. Just so does positivism answer metaphysical idealism, pointing to the facts which science has collected. To the pride of spirit, which believes itself sole lord of a world of dreams, and would discover all laws within itself, science replies by showing a reality which spreads about and beyond us, and so escapes the vain claim that it should be subject to our sentiments or to our will. But really jests are but blunt weapons against philosophers. 10 PROBLEMS OF SCIENCE. And he who contents himself with laughing deserves perhaps to be reminded of the proverb "risus ahundat in ore stultorum." In dealing with men of high powers, it surely is wiser to try to understand them, and to draw profit even from their errors. For an absurd conclusion cannot stop the process of thought; and an error in which one sees only the ridiculous, is an opportunity for instruction lost. How can we doubt that of which every man is most certain even from his earliest infancy? We cannot understand this save by going back, in our imagi- nation, to that almost forgotten age when dreams are confused with reality, and the image reflected in a glass seems as real as the per- son standing before it. Since truth and error enter our minds by the door which the senses open to knowledge, we are soon obliged to be on our guard against illusions. Man's determination not to be deceived is precisely the origin of the problem of knowledge. The question is always and only this : to learn to know and to grasp reality in the midst of a thousand causes of error which tend to vitiate our observation. It is needful then to establish a relative distinction. By losing sight of such a relativity in order to follow the mirage of a fantastic absolute, meta- physical idealism has brought us directly back to our starting-point, namely the confusing of dreams with real things. Whether we give to the one the name of the other, or vice versa, there is no essential difference between the two cases. It sometimes happens in Alpine excursions that one who does not know just how far off is the goal, thinks he is about to reach it, while he is mounting the rocky summit of some height facing it. But once scaled, a new valley opens unexpectedly to view. One must descend with care. After a fatiguing walk of several hours he finds himself perhaps no higher than when he started. But the time and labor have not been wasted, since even if the summit seems now more distant, in the enlarged horizon, in reality we have come nearer to our goal by surmounting an obstacle which had concealed it. We must not then lose courage, nor give up in a moment of weakness. Let us then begin our struggle anew, with a right good will! And if the slope is steep, if great crevasses hidden by snow open under our feet, let us climb with care, holding firmly to one another. Let us tie ourselves to the rope and give each other a hand! INTRODUCTION. 11 This metaphor may apply to metaphysical idealism, which, climbing the rocky hill of a fantastic absolute, finds itself confronted by a deep valley, beyond which glows the reality to be reconquered. We are truly nearer to this real goal, now that we are enlightened as to the relative nature of the question. Then let all well-disposed men, undaunted by a moment's dis- comfort, unite to conquer, with renewed efforts, the new difficulty which appears before our eyes! 9. THE DANGERS OF LANGUAGE. We must, above all, avoid the errors of the past. Therefore we should take notice that language, which we use to express our thoughts, is, in the last analysis, a system of symbolic representations of things. Since language furnishes a process of schematizing, rising by degrees to the expression of more general facts, it allows us to reason about abstract ideas, very far from the immediate real- ity which appeals to our senses. But the use of this powerful instrument, which comes to the aid of our mental weakness, is not without its dangers. Taking flight toward the lofty realms of thought, we run the risk of for- getting the meaning of words, which become void of sense as soon as they cease to represent things. Having reached this point, nothing is easier than to use symbols formally, while the development of thought tending toward generality, no longer finds any check in the concrete world to which it remains foreign. If then you would not lose yourself in a dream devoid of sense, you should not forget the supreme condition of positivity, by means of which the critical judgment must affirm or deny, in the last analysis, facts either particular or general. 10. ABSOLUTE AND RELATIVE: THE ABSOLUTE IN THE REALM OF MOTION. These observations throw a strong light on the classic argu- ments with which men have attempted to prove the existence of something absolute, which must eternally escape our knowledge. We find in language the word "absolute" opposed to "relative." The word has a significance easy to infer from the use ordinarily made of it, in regard to any subject whatever. If we are driving in a carriage, we see the trees passing before our eyes, and we say that they are moving with relation to us ; but, 12 PROBLEMS OF SCIENCE. in the absolute sense, the trees are staying still, and it is the carriage in which we are traveling that moves. This absolute in its turn becomes a relative thing, if considered from the point of view of astronomy. The trees are carried by the earth, which revolves around the sun. But the sun itself seems also endowed with its proper motion ^ with respect to the distant stars, whose reciprocal variations appear negligible for a limited period of time. Nevertheless observations made during centuries have shown that even these stars, improperly called "fixed," are moving, with regard to one another, changing their relative distances to a degree which we must judge to be enormous, if we realize the variation of angles according to which they are seen from a point so distant as our earth. In conclusion, the motion conceived as absolute in a certain group of facts, appears as relative in a wider field ; it is an absolute subject to degrees, corresponding to the need of seeking for our science a more fixed point of support. I have merely wished to cite an example, without pushing the discussion to the limits that we might reach, in the actual state of our knowledge. We shall have further occasion to take up the problem: What is the most absolute sense that we can attribute to motion? But it is always a question of giving to the word "absolute" a more ex- tended relative meaning, such as may better satisfy known mechan- ical relations when taken as a whole. 11. THE ABSOLUTE IN THE REALM OF MORALITY. Let us choose a second example, in a completely diiferent field of thought. He who desires an end ought to desire at least some of the means which lead to that end. In this sense the feeling of ought appears amongst the determining motives of all continued volition, be it good or bad. This kind of duty toward oneself is joined with similar duties toward society. These duties, however acquired by the suggestive influence of others, still can be understood only as duties relating to an implicitly accepted end, even if it be accepted through the will of a social group rather than through that of the individual. Now morality distinguishes between such duties, contrasting the "absolute duties" with the "relative." It admits that the diffi- culties of accomplishment may excuse the non-fulfilment of the INTRODUCTION. 13 latter, but on the contrary it decrees that such a reason does not avail to excuse any one from fulfilling the former. Why ? Because the absolute duties in question relate to general ends, the permanent observance of which has for human society a value above any passing sacrifice or hardship however great. But the value of this end, which has an absolute significance in regard to certain motives of action, appears in its turn as relative when compared with other ends of the same order. The duty which demands its fulfilment, notwithstanding the sacrifice or the injury of the agent, does not equally require the subordination of other duties. The end does not justify the means. And the moral con- flict can be reconciled only by a comparative judgment of the ideals in question and of their subordination to a higher ideal. But there is no ideal which may not be surpassed, and the loft- iest end, which within a certain social group, and at a certain epoch has an absolute sense, becomes relative in a broader com- parison with the morals of different peoples who have different modes of life. But this objection may be raised : Is not justice the ideal which absolutely cannot be surpassed ? In fact always and for all stages of social evolution, the human idea of justice expresses the highest synthesis of appreciative judgments. But cannot these judgments themselves always be extended to a larger circle of relations? And does it not therefore follow that their general and abstract expression can never be regarded as complete? The absolute value of morality, then, signifies nothing but a larger relativity. Such a conclusion is imposed upon whoever scientifically conceives morality as a fact, regardless of all possible considerations of advantage or disadvantage which could be con- nected with this result. But the dreaded danger of evil result does not exist for him who keeps before his eyes the preeminence of general ethical ideals over the motives of individual actions. For this is the only thing of practical importance expressed by such an absolute. Meanwhile the statement that morality is relative, tends in itself to raise the standards of our judgments and conduct, above all in the relations of different peoples, in different conditions of life. Nothing is more unjust than to extend the canons of our morality to men unlike ourselves; and the absurd pretense of imposing our rules upon them in the name of a natural superiority, ought simply to make a philosopher laugh, if the practical consequences did not suggest sadder reflections to his mind. 14 PROBLEMS OF SCIENCE. 12. WHAT IS THE ABSOLUTE IN A TRANSCENDENTAL SENSE? In the above examples, the absolute, in agreement with the com- mon use of language, appears as signifying a deeper relativity, and one more distant from us. It is true, in such cases, that a vague consciousness tells us the word is not used in its strict sense. But we are not usually engaged in defining such a sense, in the larger extension which is thus dimly implied. Let us now see what the word has come to mean to the philos- ophers. Since there are degrees of relativity, we may claim to reach the end of an infinite series, in order to attain something which shall no longer have any relative aspect, and which may then properly receive the name of absolute. An infinite series of degrees come to an end? It is evident that the proposition is self - contradictory. But this manifest ab- surdity does not yield in presence of an illusion that is deeply rooted in the human mind. There is a peculiar advantage in making use of symbols, in that we may represent the term occupying in a given series any place whatever, without successively completing in our thoughts the operations needful to reach that place. Thus, for example, we can reason about the number 164, 792, 843 without counting one after another all the units of which it is composed. Or in the same way, we may undertake arithmetical calculations concerning 2^""*", without following out the thousand successive multiplications in- dicated by the symbol. In these cases the mind accomplishes, so to speak, in a shorter way, by virtue of logically established relations, a series of possible operations, which would merely require more time. And the thought pauses upon a well-defined object in the series itself. But the habit which is thus contracted of substituting a sym- bol for the operations which constitute the real definition, gives rise to the illusion that the thing may be defined by the symbol; and therefore that it should suffice to denote by a word the last term of an infinite series, in order that an object should correspond to this word. Nevertheless the symbol is in this case devoid of sense, since the transcendental operation is impossible which we at- tempt to indicate by means of it. Abbreviated procedures suffice for the notion of performing more rapidly a finite number of opera- INTRODUCTION. 15 tions, never an infinite number. Infinity cannot be exhausted in however long a time, or however much we toil at abbreviating or grouping together its terms. 13. TRANSCENDENTAL PROCEDURES CONSIDERED IN THE LIGHT OF PHYSIOLOGICAL PSYCHOLOGY. We can state this last proposition in a more exact form, which may put in clearer light the illegitimacy of the transcendental procedures mentioned above. Physiological psychology in fact allows us to measure the time required by an act of thought. And since in every case there is a minimum of duration, we could not possibly grant to the human mind the faculty of accomplishing infinite acts, in any determinate time whatsoever. 14. TRANSCENDENTAL PROCEDURES IN RELATION TO INFINI- TESIMAL ANALYSIS. We need not then be surprised that the opposite supposition, used as the basis of a vicious mode of definition, leads to number- less absurdities Infinitesimal analysis forms a field in which such absurdities were most plainly manifest before these transcendental processes of reasoning were happily banished from that field. The critique of the concepts of iniinite and of infinitesimal, of series and of limit, seems to form the most fitting preparation for thoroughly compre- hending the sense of the foregoing observations. The first thing pointed out by such a critique is that the word "infinite" cannot be applied to any given number or quantity, but denotes only a mode of increase of a variable quantity, which may be susceptible of receiving a higher value than any preassigned constant value. This is exactly expressed by saying that the infinite has no actual sense according to the acceptation of Leibniz,^ but only a potential or genetic sense. The same holds good for the infinitesimals. The importance of this way of conceiving things consists in recognizing the ab- surdity of expecting to define a number, by means of an iniinite series, as the last term of that series. It may be that outside of 'It is true that this statement ought to be modified with regard to the systems of non-Archimedean numbers, lately constructed in different ways, by Veronese, Levi-Civita, Hilbert etc. We may however leave these con- structions aside. It is sufficient to say that they do not imply the use of transcendental processes of definition. 16 PROBLEMS OF SCIENCE. the series there is really to be found a number which constitutes its limitj which the terms of the aforesaid series continue to ap- proach; but the existence of this limit only expresses in substance a property of the mode of variation of the terms of another series, built up from the diflferences between the limit-number and the terms of the original series. Therefore the existence of the limit cannot be proved by the tnere fact of the existence of the series. In other words, the limit cannot be defined exclusively by means of that series, but only by comparing the series to something which is independently given from outside. The practical value of this statement is well known by this time to all those who understand infinitesimal analysis, since in- finite algorithms generally give rise to series having no limits, and reasoning as if these expressions were capable of defining such a limit leads to the most singular absurdities. For example the series obtained by an infinite process of sum- mation can appear not only as converging towards a limit but as divergent or indeterminate. To these two last categories belong respectively the series 1-1 + 1-1+.... But the use of the series in our calculations would permit us to prove the equality of any two numbers whatsoever! This suffices ; for even in the handling of convergent series, we must not forget that they are only conventional means of repre- senting a number that is their limit and that does not belong to them. If we were to forget this fact, we should indeed be led to admit a transcendental element, and to regard the series as the sum of an infinite number of terms. In the latter case we should feel authorized to work upon these according to the properties of a sum, interchanging, for example, the order of the terms. But in this manner, for example, from the convergent series 1-% + %-% + %..., one may form at will series converging towards different limits, and also diverging or indeterminate series. "To eliminate all transcendental processes of definition or of rea- soning": that is the essential condition for understanding the in- finitesimal calculus — a condition which Cesaro states at the be- ginning of his admirable lectures, thus warning the reader to banish from his mind all metaphysical ideas! INTRODUCTION. 17 To the teachings which have been aiForded us by the infini- tesimal analysis, let us add some which spring from the modern theory of assemblages. Here the process of transcendental definition has presented itself in the very construction of certain "assemblages" taken as the "totality of the infinite multitude of entities which have certain specified characteristics." Among the examples brought to light by the studies of Cantor, Du Bois Reymond, etc. let us choose one very simple one to which Russell has lately called the attention of geometers. We may construct in several ways an assemblage s^ [a], composed of whatever entities you will oj, a^, oj — ^but subject to the rule that J is not itself contained among its own elements — an assemblage (such then that no one of the entities a coincides with s. ) Now let us define 2 as the assemblage of all the assemblages s to which the given property belongs. %^[s]. The first condition is that 5 must not be found among the elements s otherwise there would be a contradiction in the property supposed by the s. But if, on the other hand, 2 is found outside of the assemblage [s], this assemblage does not exhaust all pos- sible .y which possess the aforesaid properties. The contradiction shows that the concept of % is illusory, and thus makes plain the vicious nature of the transcendental process by which S was defined. 15. THE PSYCHOLOGICAL VALUE OF THE ABSOLUTE. But the logical analysis which reveals the vice of the transcen- dental definition, does not exhaust the question of the absolute. How otherwise could we explain the place held by the absolute among the beliefs that are bound up with the deepest sentiments of the human mind? In what way could an error in the statement of a problem give value to a symbol which we have seen to be devoid of sense ? In order to answer such questions we had better go back to the considerations concerning "the Absolute in Morals." We have al- ready noticed the peculiar character of the human will, in that it is 18 PROBLEMS OF SCIENCE. eminently progressive, that it arranges, namely, the ends it proposes in a hierarchy, so that the nearer end is subordinated to the more distant. The stability of such a hierarchy demands that at every mo- ment the higher aim should act upon the will as a sufficiently strong motive against the motives of the moment which might tend to turn it aside or to weaken its determination. And this auto-sugges- tive force is given by the trials which the will itself has made of its own firmness, and is shown in the resulting consciousness which the will has of not changing. ' One sees then how the progressive extension of this hierarchy of ends and its stability constitute two contradictory psychological requirements for the human will. A new higher end cannot come to modify the ends already accepted by the will, without diminishing its faith in its own consistency and force. When, in the life of the individual or of society, the outlook over a field of purposes enlarges rapidly, there follows one of those crit- ical periods which are characterized by the disorganization of the will. This state of impotence, annulling the human personality for the moment, usually contains within itself its own corrective, be- cause it destroys the interest in the search for new aims. Progress being thus stopped, all the discordant motives striving within the troubled mind finally engage in one supreme battle; and when a determination is declared victorious, by seeming able to subordinate the others, the mind leans upon that with all the energy of the reac- tion which the need of escaping from a sad condition gives. Any man passing from childhood to youth, or from youth to manhood, usually passes through such a critical period as that just described, and comes forth from it by his own courage, or by the help of others. Just so in certain historical moments, analogous crises of the will are produced in the social order. We have then periods of revolutionary disorganization which follow an over-rapid progress and bring it to a close by exalting unduly the concept of that author- ity which had established itself generally. The psychological value of the absolute is inherent in the con- ditions of progress above described. And it would be easy to furnish historical proofs. The illusion of transcending the infinite series of ends answers to the need of seeking beyond an end of agonizing doubts, by ending with imperative authority the conflicts of the will. INTRODUCTION. 19 In this sense the absolute is a condition which sets the spirit free, rather than a guiding idea, and through it all the faculties are concentrated and drawn toward one single point, while all ex- traneous impulses and all criticism are inhibited by the conscious- ness: credo quia absurdum est. Such a state of mind, essentially emotional and religious, con- stitutes a problem for the psychologist, but the philosopher who is dealing with the theory of knowledge has no motive for concerning himself with it. To explain how and why, together with the weakening of re- ligious faith, in modern times, the absolute has become the object of a research which tries to construct science out of it, would be indeed instructive, and not alone from the historical point of view. But such an explanation would need a much longer examination. It must suffice us to show, by means of the preceding con- siderations that the absolute, taken as an object of rational con- struction, is no longer absolute, and, through the very criticism itself, loses the value of supreme guide, dominating all the aims of the will, and all the motives of intellect and of sense. What then remains of this pretended metaphysic but a mere document of the human mind, a mind at once feeble and domi- neering? Icarus, poised for a flight through the heavens, fell headlong into the depths of the sea. Reason, discouraged, sinks into the depths of the unknowable. 16. SUBSTANCE AND APPEARANCE. We have seen how the wide-spread sophism which says that "the relative presupposes the absolute," rests upon a verbal illusion, that conceals a process of definition wholly void of sense. All the similar antinomies, of which classical philosophy is full, may be explained in an analogous way. They may be put into the form of an infinite process, or they may appear as a simple nega- tion ; but in the latter case the negation is purely formal, and serves in the same way to construct for us a statement void of meaning. The antinomies of this sort, being true up to a certain point, if we do not give a strict sense to the terms, gain just from this fact all their force. We speak, for example, of the substance of things in contra- distinction to their appearance. Carbon and diamond are of the same substance though quite different in appearance. On the con- 20 PROBLEMS OF SCIENCE. trary, under the similar appearance of a diamond and of rock crystal, are hidden quite different substances. In such cases an important distinction is established between the immediate sensations which we refer to any object whatever, and the totality of its relations to the surrounding world, of which we can partially acquire a mediate knowledge. And this observation goes back to Aristotle, for the philosopher remarked, for example, how the oar, plunged into water, looks as if broken, when it is really whole. But the original sense of that distinction underwent a change during the Middle Ages, when the idea of the absolute was superimposed upon the Hellenic civilization. Let us see what Kant has made of this distinction! Appear- ance, or as Kant expresses it, the phenomenon, is conceived as con- trasted with the essence, or noumenon, taking this last term in an ab- solute sense.^ Kant abstracts, in considering any body, from all its relations, which are perceptible to us, either in an immediate or in a mediate way ; what remains is the true essence of the body. Such a non-sense would be truly inexplicable, were it not joined with an anthropomorphic representation of the world. Here it is fancied that, if we could enter into a stone, we should experience sensations adapted to reveal to us its real essence. The confused state of mind which corresponds to the absolute mode of considering essence, recalls precisely the anthropomorphism through which we must probably have passed during our earliest infancy. But really it would not be strange if we should arrive at agnostic conclusions as to this so-called essence,^ so defined that no meaning at all remains in the word! 17. THE UNKNOWN. We do not wish thus to deny the feeling of a "great mystery of the universe," which reflections upon the idea of substance excite in our minds. Since we conceive that manifold relations bind all things together, we are led to discern behind these an unknown to be unveiled, and to represent to ourselves the impossibility of exhausting 'We here allude to the noumenon understood in a negative sense, as in Kant's discussion in section III of his "Transcendental Analytik." "The analysis of Kant leads precisely to the unknowability of the nou- menon, which he expresses by affirming the impossibility of understanding a noumenon in a positive sense. INTRODUCTION. 21 the objects that belong to this realm. But the affirmation of the unknowable does not adequately express this conception! In agreement with our own Ardigo in his criticism of this point in the doctrine of Spencer, we cannot admit that to a reality acces- sible to our knowledge, there is opposed a mysterious reality, which necessarily must escape all efforts of our thought. On the contrary the foregoing observations show a series of objects equally acces- sible to research ; but as that series appears to us to be without limit, we see that our desire for knowledge can never be completely satisfied. Truly this is a fortunate thing for humanity, before whom an endless progress is thus opened ! 18. THE DISTINCTION BETWEEN SUBJECTIVE AND OBJECTIVE ACCORDING TO KANT. Akin to the distinction between appearance and substance is that between subject and object, or between subjective and objective. We may now dispense with repeating that such a distinction, taken absolutely, is devoid of sense, in that it corresponds on the one hand with a transcendental view of the thing in itself (which is confounded with the substance), on the other hand again, with a transcendental view of the ego, considered as a substratum inde- pendent of the various individual attributes which are added to it in one person. It is always the same spirit of founding one's criti- cism on the basis of absolute distinctions, which furnishes the foun- dation for Kant's agnosticism. But only he who considers things from a special point of view can interpret Kant's philosophy as leading towards a skeptical re- sult that should agree with the postulates of the positive philosophy. This side of Kant's doctrine seems rather to be intended to reconcile certain practical requirements, in so far as "it allows precisely prac- tical data to fill, in the extension of knowledge, the place which speculative reason is seen to be incapable of filling."^ Still we must admit that, by bringing to light some of his defects, we have not lessened the value of the revolution of the older metaphysics accom- plished by Kant. Broadly interpreting the spirit of Kant, we per- ceive in fact that the distinction between subjective and objective does not remain a sterile antinomy in the thought of the philosopher, but becomes for him the starting-point for a new conception of scien- tific reality, a conception which positivism has reached by another road. * Cf. Critique of Pure Reason, 2d ed., Preface. 22 PROBLEMS OF SCIENCE. We believe rather that to the more restricted view of positivism the philosophy of Kant may even add something scientific if we agree to keep only the spirit that appears in its best possessions, and if we advance from other foundations to a new construction. The teaching that "in consciousness a personal element (subjec- tive) and a real element (objective) differentiate themselves from one another, and that the latter, varying in different persons, is joined with certain general forms of human sensibility and intelli- gence" — ^this teaching is in fact full of positive significance, if we properly estimate the distinction, excluding all pretense of finding in it anything absolute. But we must therefore not only set aside the extremes of post- Kantian speculation, but likewise the very judgments of the master about the "a priori anticipation of forms of possible experience," (compare op. cit.) and about the "objective reality which is given to our knowledge a priori by the possibility of experience" (cf. op. cit.). For these judgments are in themselves ambiguous, and in their legitimate interpretation and application, retain traces of their orig- inal vice of understanding transcendentally the distinction between subjective and objective. And by their judgment in their recog- nizing implicitly a certain objectivity of the structural laws of mind in the constructions of geometry or mechanics, — the door is after all reopened to that metaphysics which Kant wished to have con- demned forever. Indeed apart from the development of philosophy in the past century, what was most vital in that critical philosophy has left the clearest traces in the realm of science, where it has had a most awakening effect. And upon exactly this basis the new criticism should arise, and throw its light upon the positive problems of knowledge ! 19. THE DISTINCTION BETWEEN SUBJECTIVE AND OBJECTIVE CONSIDERED POSITIVELY. Has the distinction between subjective and objective a positive content with regard to our knowledge? In what manner will it assert itself since we have seen the antinomy between subject and object arise, when taken transcendentally? Let us consider certain examples which seem adapted to lead us by induction to a suitable definition. Let us take a little box of cube-shaped blocks, such as in Froe- bel's system are given to children for toys. The bottom of the box INTRODUCTION. 23 is divided by two black lines into four squares, equal to the faces of our blocks. The child thus learns that "to cover the bottom of the box one needs four blocks." We all agree in granting that this assertion contains an objec- tive element. If the box were larger in comparison with the blocks, six or eight of them might be needed. Nevertheless some subjective element enters into the expression of this very knowledge. In the first place, there is the verbal form of the word "four," which is differently spoken by different people, and for which a Frenchman substitutes the sound "quatre," a German "vier" etc. In the second place there is the psychological form of the idea in question.^ One child imagines the "four" by ideally associating the blocks with the fingers of his hand with the thumb turned in. Another child asso- ciates them with four of the balls on his counting-frame, and a third child, who cannot yet count, has taught himself to shape with stones upon the blocks the arcs of a circle, each being a quadrant. Thus he succeeds in laying apart as many blocks as are needed to cover the bottom of the box, by placing them in turn, one beside the other, in such a way that the arcs form approximately a circle. The last child, like the others, has the objective knowledge in question, but he has gained it, not through the representation of number, but rather by means of the geometrical figure. We may say that these children have objectively the same knowledge, since they agree in the prevision required of them by providing equally well the blocks which will cover the bottom of the box, while this prevision would not be found to be verified in the case of another box, or of blocks of different sizes. We may say that the knowledge of the children is subjectively diverse, because they obtain their prevision in different ways, by means of different images. But if we carry our criticism a little further we perceive: 1. That such a prevision is not possible without some mental image, and therefore that a purely objective consciousness is im- possible. 2. That the subjective manner of representation has an in- fluence over the prevision itself, and over other analogous previsions, so that we must admit that it contains some objectivity. This comes out very clearly in the example just used. The 'A psychological difference is sometimes expressed by differences in the words themselves: as for example, the number ottanta (80) is represented by an Italian as 10X8, by a Frenchman as 20X4 (quatre-vingts) . 24 PROBLEMS OF SCIENCE. Fig. I. child who pictures to himself geometrically the fact we have stated, knows less than the others, in that he does not quickly see that his four marked blocks will cover the bottom of the box, even if they are distributed so as to form, with their four arcs, a figure dififering from the circle (v. Fig. 1). From another point of view the above geometrical represen- tation teaches more, that is, to recognize that the bottom of the box can be covered by ar- ranging the blocks in a given way and in a given order etc. From this example we can already see that the subjective element and the objective element are not two irreducible terms of knowl- edge, but they are rather two aspects of this knowledge resulting from its being compared with other knowl- edge in one same person or in different persons, in relation to one thing or to different things. We find the objective element wherever there is an agreement of previsions, in however diverse modes these may be obtained by one or more persons. We find the subjective element in the plurality of these possible modes. But to whatever degree these two elements become more and more distinct with the growth of knowledge, yet absolutely distinct they will never be. For the concepts of objective and of subjective arise all the time through abstraction, from a com- parison of cognitions; which cognitions will be always capable of extension. Consequently some objective element will always be contained in the subjective aspect of knowledge, and so also will there be some subjective element in its objective aspect. In any case the process of distinction may be followed much further than appears from the example just cited. Always using for our basis the comparison of cognitions, we may in fact rise above that objectivity, which appears whenever we compare different people's knowledge of one same thing. And if in a similar way we compare the kinds of knowledge relating to different objects, we may succeed in recognizing the subjectivity of representations which are yet common to all men. This is a new scientific stage of the distinction between sub- jective and objective, in which a new condition reveals itself , namely, that "the mode of representation leading in a given case to a veri- fied prevision, leads us into error with regard to other possible pre INTRODUCTION. 25 visions." This may be expressed by saying that to the subjective aspect of knowledge are joined the appearances of things which do not correspond with their reality. The distinction thus obtained then leads us to correct our knowledge as primarily gained by divid- ing it into two parts, one of which (the objective element) corre- sponds better to the larger complex of previsions, while the other (the subjective element) added to the former offers a prevision, relative to the single incongruous case or explains, as we say, the fallacious appearance. 20. SUBJECTIVE AND OBJECTIVE IN THE PROCESS OF MEAS- UREMENT. We may learn something as to this point by the analysis of a simple and instructive example. When we speak of the size or of the dimensions of an object, we affirm a complex cognition which includes the agreement of manifold previsions, in relation to possible experiences of touch, of sight etc. Now let us suppose as known the dimensions of an object placed before our eyes. Sight furnishes us a certain immediate knowledge of these dimensions, in which are mingled, nevertheless, various elements which we ought to subject to an analysis. First we must take account of the distance and of the position of the object, and correct the causes of error which arise from this element of relation between the subject and the object, thus changing in several ways the conditions of our observation. But even then there still remain, in our consciousness, some elements depending upon the structure of the visual organ, which, as physiological psychology teaches us, varies in different men, though in any case it obeys certain general laws of sensation. Thus, for example, two points of the object, which are separated by empty space, appear nearer than two other equidistant points which are joined by a tract of continuous matter. And in like manner equal lengths are differently judged according to their uniform or varied coloring etc. We have seen already, in the example cited, the influence of the subjective element which is inherent in sight, and how the sensations of touch enable us to correct that influence. But even without re sorting to the more exact processes of measurement, we can easily understand by means of touch how experiences may lead us to 26 PROBLEMS OF SCIENCE. recognize directly the organic causes of error, without leaving the domain of visual observation. It may suffice simply to notice the coloring of different parts of the object, to substitute empty space for such and such filled parts etc. And when once rules which may be applied to aerial perspective, taken in a broad sense, have been established we have the means of correcting our sight impressions, and hence of reaching a more adequate knowledge of the dimensions submitted to our judgment, so as to obtain previsions more in agreement with the various possible experiences of sight, in one person or in different persons. If now we drop all restrictions as to the use of one sense rather than another we shall find, without leaving the field of measurement, material for further interesting reflections. The determination of a measured length, constituting a com- parative judgment between an unknown length and another taken as the unit of measure, is effected by means of the senses, aided by suitable instruments. The instrument used secures a certain degree of precision, which may be regarded as an objective datum for the judgment. But beyond this we should take into consideration the manner in which the said instrument is used, the physical and psychological conditions of the experimenter, and a thousand other accessory causes, by which the results of experiments, requiring a certain degree of accuracy, vary from one time to another for the same person, and at all times together for different persons. Now, leaving out of account the systematic errors, and con- cerning ourselves only with the accidental errors, we meet with the singular and instructive fact that the average of the results obtained by the same experimenter tends to differ from the actual measure by a regular error, which goes by the name of personal equation. This fact may be perceived in two ways, either by using more perfect instruments, or by comparing the observations of different persons. But the personal equation once established is sufficient in itself to correct systematically the determination of measures, and we reach a greater degree of precision, which shows itself in a more exact agreement between the results of the same experimenter and of different experimenters. The observations recorded above throw light upon the positive content of the statement relating to the subjective element in con- sciousness. In fact, through the preceding example we see once more that INTRODUCTION. 27 there can be no question of giving to the relations of the subject and the object a strictly transcendental meaning. The distinction certainly loses none of its importance through its relativity. The existence of a datum due to the nature of the sense organs is clearly established, however true it is that the very knowledge itself which we accept as exact is mediately derived from the same senses whose errors we are bringing to light. But there is another aspect of the above considerations which it is important to bring out. In view of the law of error above referred to, one substitutes for the totality of many and varied observations, their arithmetical mean, which is supposed to be nearer to the exact measurement. Two questions arise. First of all we see that the idea of an extract measure forms part of a mental process which leads us to correct the error of our sense impressions. And the followers of Kant lay especial stress upon this essential point. What is an exact measure? Is it not the absolute in the world of measures? And are we not thus forced to grant a place to that same absolute, which we have already declared to be senseless? Let us examine the question dispassionately. If we were asked: "Can one conceive an ultimate term in the constantly approximating determination of a measure?" we should answer "No." In this realistic sense, the exact measure has no meaning. The hypothesis of a strictly immediate determination of a measure meets with various kinds of objections. What we know or suppose about the constitution of matter, and what we admit in regard to light (above all the idea of a wave-length), throws ob- stacles in the way of such an hypothesis. So that it is not difficult, for example, to assign to the least length visible with a microscope a theoretical limit, not very far from the practical limit actually reached. But all this makes little difference to us if we are using the hypothesis of an exact measure as the starting-point of a reasoning process. The worth of the hypothesis consists only in admitting that the facts inherent in the experimental determination of measure can be represented by a logically well defined concept (the concept of a number), in such a way that the reasoning processes founded upon such a concept will lead to previsions verified by experience. Now in this subordination of the data of sensation to such concepts, there is actually an element of psychological nature, the importance of 28 PROBLEMS OF SCIENCE. which for the theory of knowledge we shall have occasion to notice later. A few words further as to the postulate of the mean, which is less directly bound up with our subject. Lipmann cleverly said to Poincare that in such questions, re- lating to probability, the agreement of scientific men is unanimous, since mathematicians consider them to be decided by physical ex- periment, and the physicists believe that the mathematicians have settled the matter. Whereupon Poincare, reporting this remark, justly observes that the mathematicians are right in their view. For the principle in question cannot be established by mathematics. It is an empirical rule. The choice of the arithmetical mean to represent a series of observations, corresponds to the choice of that number for which the sum of the squares of the differences between the results of several partial observations is smallest. This choice is a priori arbitrary. But the principle upon which it depends is justified by the fuller agreement obtained by comparing various series of observa- tions. In sum, the physical notion of measure corresponds to an interval, which we try to make as small as we can, within which are enclosed the numbers furnished by the process of measurement. This interval is diminished if for the numbers obtained by single observations we substitute the averages of similar observations. Such is the value of the postulate of the mean upon which Gauss has built up his theory of errors. One takes the arithmetical mean, reducing to its minimum the sum of the squares of the differences pointed out before ; precisely because one seeks, as a result, to reduce the interval which separates, in either direction, the extreme num- bers obtained, from the number equidistant from them. 21. SUBJECTIVE AND OBJECTIVE IN SCIENTIFIC CONSTRUC- TION. In consequence of the preceding observations we shall be led to discuss the problem which deals with the positive definition of reality. We shall consider this problem more fully in Chap. II, by undertaking a critique of facts and theories. Let us here merely notice how very instructive the example of measure is as to the light it throws upon the scientific value of the distinction between subjective and objective in knowledge. For this distinction, if we lay aside any pretense of understanding it transcendentally, becomes the starting point of a method of pro- INTRODUCTION. 29 gressive correction for the extension of science. If such a view cannot satisfy those who, in whatsoever field, lii