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diff --git a/43791-0.txt b/43791-0.txt new file mode 100644 index 0000000..322b1ee --- /dev/null +++ b/43791-0.txt @@ -0,0 +1,5769 @@ +*** START OF THE PROJECT GUTENBERG EBOOK 43791 *** + +NATURAL PHILOSOPHY + + + BY + WILHELM OSTWALD + + TRANSLATED + BY + THOMAS SELTZER + + _With the author's special revision for the American edition_ + + + + NEW YORK + HENRY HOLT AND COMPANY + 1910 + + + + + COPYRIGHT, 1910, + BY + HENRY HOLT AND COMPANY + + _Published November_, 1910 + + THE QUINN & BODEN CO. PRESS + RAHWAY N. J. + + + + + The original of this book was published + as volume I in Reclam's BÜCHER DER + NATURWISSENSCHAFT. + + + + +PREFACE + + +The beginning of the twentieth century is marked by a sudden rise of +interest in philosophy. This is especially manifest in the vast growth +of philosophic literature. The present movement, it is noteworthy, is by +no means a revival proceeding from the academic philosophy traditionally +represented at the universities, but has rather the original character +of _natural philosophy_. It owes its origin to the fact that after the +specialization of the last half century, the synthetic factors of +science are again vigorously asserting themselves. The need finally to +consider all the numerous separate sciences from a general point of view +and to find the connection between one's own activity and the work of +mankind in its totality, must be regarded as the most prolific source of +the present philosophic movement, just as it was the source of the +natural philosophic endeavors a hundred years ago. + +But while that old natural philosophy soon ended in a boundless sea of +speculation, the present movement gives promise of permanent results, +because it is built upon an extremely broad basis of experience. The +laws of energy in the inorganic world and the laws of evolution in the +organic world furnish mental instruments for a conceptual elaboration +of the material provided by science, instruments capable not only of +unifying present knowledge, but also of evoking the knowledge of the +future. If it is not permissible to regard this unification as +exhaustive and sufficient for all time, yet there is still so much left +for us to do in working over the material we have on hand from the +general points of view just mentioned, that the need for systematizing +must be satisfied before we can turn our gaze upon things more remote. + +The present work is meant to serve as the first aid and guide in the +acquisition of these comprehensive notions of the external world and the +inner life. It is not meant to develop or uphold a "system of +philosophy." Through long experience as a teacher the writer has learned +that those are the best pupils who soon go their own way. However, it +_is_ meant to uphold a certain method, that is, the scientific (or, if +you will, the _natural_ scientific), which takes its problems, and +endeavors to solve its problems, from experience and for experience. If, +as a result, several points of view arise that differ from those of the +present day, and consequently demand a different attitude toward +important matters in the immediate future, this very fact affords proof +that our present natural philosophy does not lead away from life, but +aims to form a part of our life, and has a right to. + + + + +CONTENTS + + + PAGE + + INTRODUCTION 1 + + + PART I + + GENERAL THEORY OF KNOWLEDGE 11 + + 1. The Formation of Concepts 11 + + 2. Science 13 + + 3. The Aim of Science 13 + + 4. Concrete and Abstract 16 + + 5. The Subjective Part 17 + + 6. Empirical Concepts 18 + + 7. Simple and Complex Concepts 19 + + 8. The Conclusion 24 + + 9. The Natural Laws 28 + + 10. The Law of Causation 31 + + 11. The Purification of the Causal Relation 34 + + 12. Induction 38 + + 13. Deduction 40 + + 14. Ideal Cases 44 + + 15. The Determinateness of Things 47 + + 16. The Freedom of the Will 50 + + 17. The Classification of the Sciences 53 + + 18. The Applied Sciences 57 + + + PART II + + LOGIC, THE SCIENCE OF THE MANIFOLD, AND MATHEMATICS 61 + + 19. The Most General Concept 61 + + 20. Association 63 + + 21. The Group 65 + + 22. Negation 68 + + 23. Artificial and Natural Groups 69 + + 24. Arrangement of the Members 75 + + 25. Numbers 78 + + 26. Arithmetic, Algebra, and the Theory of Numbers 79 + + 27. Co-ordination 80 + + 28. Comparison 82 + + 29. Numbers 85 + + 30. Signs and Names 86 + + 31. The Written Language 89 + + 32. Pasigraphy and Sound Writing 92 + + 33. Sound Writing 96 + + 34. The Science of Language 97 + + 35. Continuity 101 + + 36. Measurement 107 + + 37. The Function 109 + + 38. The Application of the Functional Relation 112 + + 39. The Law of Continuity 113 + + 40. Time and Space 118 + + 41. Recapitulation 124 + + + PART III + + THE PHYSICAL SCIENCES 127 + + 42. General 127 + + 43. Mechanics 128 + + 44. Kinetic Energy 132 + + 45. Mass and Matter 136 + + 46. Energetic Mechanics 138 + + 47. The Mechanistic Theories 140 + + 48. Complementary Branches of Mechanics 144 + + 49. The Theory of Heat 147 + + 50. The Second Fundamental Principle 150 + + 51. Electricity and Magnetism 154 + + 52. Light 156 + + 53. Chemical Energy 159 + + + PART IV + + THE BIOLOGIC SCIENCES 163 + + 54. Life 163 + + 55. The Storehouse of Free Energy 168 + + 56. The Soul 171 + + 57. Feeling, Thinking, Acting 174 + + 58. Society 179 + + 59. Language and Intercourse 182 + + 60. Civilization 184 + + + INDEX 187 + + + + +INTRODUCTION + + +Natural science and natural philosophy are not two provinces mutually +exclusive of each other. They belong together. They are like two roads +leading to the same goal. This goal is the domination of nature by man, +which the various natural sciences reach by collecting all the +individual actual relations between the natural phenomena, placing them +in juxtaposition, and seeking to discover their interdependence, upon +the basis of which one phenomenon may be foretold from another with more +or less certainty. Natural philosophy accompanies these specialized +labors and generalizations with similar labors and generalizations, only +of a more universal nature. For instance, while the science of +electricity, as a branch of physics, deals with the relation of +electrical phenomena to one another and to phenomena in other branches +of physics, natural philosophy is not only concerned with the question +of the mutual connection of _all_ physical relations, but also endeavors +to include in the sphere of its study chemical, biological, +astronomical, in short, all the known phenomena. In other words, +_natural philosophy is the most general branch of natural science_. + +Here two questions are usually asked. First, how can we define the +boundary line between natural philosophy and the special sciences, +since, obviously, sharp lines of demarcation are out of the question? +Secondly, how can we investigate and teach natural philosophy, when it +is impossible for any one person to master all the sciences completely, +and so obtain a bird's-eye view of the general relations between all the +branches of knowledge? To the beginner especially, who must first learn +the various sciences, it seems quite hopeless to devote himself to a +study that presupposes a command of them. + +Since a discussion of the two questions will afford an excellent +preliminary survey of the work in hand, it will be well to consider them +in detail. In the first place, _the lack of complete and precise +boundary lines is a general characteristic of all natural things_, and +science is a natural thing. If, for instance, we try to differentiate +sharply between physics and chemistry, we are met with the same +difficulty. So also in biology if we try to settle beyond the shadow of +a doubt the line of separation between the animal and the vegetable +kingdoms. + +If, despite this well-known impossibility, we consider the division of +natural things into classes and orders as by no means useless and do not +discard it, but regard it as an important scientific work, this is +practical proof that such classification preserves its essential +usefulness, even if it does not attain ideal definiteness. For, this +imperfection notwithstanding, classification reaches its end, which is +a comprehensive view, and thus a mastery, of the manifoldness of +phenomena. For example, with the overwhelming majority of organic beings +there is no doubt whether they are animals or plants. Similarly, most +phenomena of inorganic nature can readily be designated as physical or +chemical. For all such cases, therefore, the existing classification is +good and useful. The few cases presenting difficulty may very well be +considered by themselves wherever they occur, and we need merely take +cognizance of them here. It follows from this, to be sure, that +classification will be all the _better fitted to its purpose the less +frequently_ such doubtful cases arise, and that we have an interest in +repeatedly testing existing classifications with a view to finding out +if they cannot be supplanted by more suitable ones. + +In these matters it is much the same as when we look upon the waves on +the surface of a large body of water. Our first glance tells us that a +number of waves are rolling there; and from a point giving us a +sufficiently wide outlook, we can count them and gauge their width. But +where is the line of division between one wave and the next? We +undoubtedly see one wave following another, yet it is impossible for us +to indicate precisely the end of one and the beginning of the next. Are +we then to deduce that it is superfluous or unfeasible to designate the +waves as different? By no means. On the contrary, in strictly +scientific work we will endeavor to find some suitable definition of the +boundary line between two consecutive waves. It may then be called an +arbitrary line, and in a degree arbitrary it will certainly be. But to +the investigator this does not matter. What concerns him is, if, with +the help of this definition, wave lengths can be unequivocally +determined, and if this is possible, he will use the definition as +suitable to the purposes of science, without dismissing from his mind +the idea that possibly some other definition may provide an even easier +or sharper determination. Such an one he would instantly prefer to the +old one. + +Thus we see that these questions of classification are not questions of +the so-called "essence" of the thing, _but pertain merely to purely +practical arrangements for an easier and more successful mastery of +scientific problems_. This is an extremely important point of view, much +more far-reaching than is apparent here at its first application. + +As to the second objection, I will admit its validity. But here, too, we +have a phenomenon appearing in all branches and forms of science. +Therefore we must familiarize ourselves with it in advance. Science was +created by man for man's purposes, and, consequently, like all human +achievements, possesses the indestructible quality of imperfection. But +the mere fact that a successful working science exists, with the help of +which human life has been fundamentally modified, signifies that _the +quality of incompleteness in human learning is no hindrance to its +efficiency_. For what science has once worked out always contains a +portion of truth, hence a portion of efficiency. The old corpuscular +theory of light, which now seems so childishly incomplete to us, was +adequate, none the less, for satisfactorily explaining the phenomena of +reflection and refraction, and the finest telescopes have been built +with its help. This is due to the _true elements_ in it, which taught us +correctly to calculate the direction of rays of light in reflection and +refraction. The rest was merely an arbitrary accessory which had to fall +when new, contradictory facts were discovered. These facts could not +have been taken into consideration when the theory was propounded, +because they were not yet known. But when the corpuscular theory of +light was replaced by the theory of waves of an elastic ether, geometric +optics at first remained quite unchanged, because the theory of straight +lines of rays could be deduced from the new views also, though not so +easily and smoothly. And geometric optics was then concerned with +nothing but these straight lines, in no wise with the question of their +propagation. It did not become clear until recently that this conception +of straight lines of rays is incomplete, though, it is true, it made a +first approach toward the presentation of actual phenomena. It fails +when it comes to characterize the behavior of a pencil of rays of large +aperture. The old idea of a straight line of rays was to be replaced by +a more complex concept with more varied characteristics, namely, the +wave-surface. The greater variety of this concept renders possible the +presentation of the greater variety of the optical phenomena just +mentioned. And from it proceed the very considerable advances that have +been made, since the new theory was propounded, in optical instruments, +especially the microscope and the photographic objective, for the +purposes of which pencils of rays of large aperture are required. The +astronomic objective with its small angle of aperture has not undergone +particularly important improvements. + +Experience in every province of science is the same as in this. Science +is not like a chain which snaps when only a single link proves to be +weak. It is like a tree, or, better still, like a forest, in which all +sorts of changes or ravages go on without causing the whole to pass out +of existence or cease to be active. The relations between the various +phenomena, once they become known, continue to exist as indestructible +components of all future science. It may come to pass, in fact, does +come to pass very frequently, that the form in which those relations +were first expressed prove to be imperfect, and that the relations +cannot be maintained quite generally. It turns out that they are +subjected to other influences which change them because they had been +unknown, and which could not have been taken into consideration at the +discovery and first formulation of these relations. But no matter what +changes science may undergo, a certain residue of that first knowledge +will remain and never be lost. In this sense, a truth that science has +once gained has life eternal, that is, it will exist as long as human +science exists. + +Applying this general notion to our case, we have the following. How far +and how generally at any given time the relations of the various +phenomena are summed up in fixed forms, that is, in natural laws, will +depend upon the stage attained by each of the special sciences. But +since science has been in existence it has yielded a certain number of +such general laws, and these, though they have been filed down a good +deal in form and expression, and have undergone many corrections as to +the limits of their application, nevertheless have preserved their +essence, since they began their existence in the brains of human +investigators. The net of the relations of phenomena grows ever wider +and more diversified, but its chief features persist. + +The same is true of an individual. No matter how limited the circle of +his knowledge, _it is a part of the great net, and therefore possesses +the quality by virtue of which the other parts readily join it as soon +as they reach the consciousness and knowledge of the individual_. The +man who thus enters the realm of science acquires advantages which may +be compared to those of a telephone in his residence. If he wishes to, +he may be connected with everybody else, though he will make extremely +limited use of his privilege, since he will try to reach only those with +whom he has personal relations. But once such relations have been +established, the possibility of telephone communication is +simultaneously and automatically established. Similarly, every bit of +knowledge that the individual appropriates will prove to be a regular +part of the central organization, the entire extent of which he can +never cover, though each individual part has been made accessible to +him, provided he wants to take cognizance of it. + +The mere beginner in learning, therefore, when receiving the most +elementary instruction in school, or from his parents, or even from his +personal experiences in his surroundings, is grasping one or more +threads of the mighty net, and can grope his way farther along it in +order to draw an increasing area of it into his life and the field of +his activity. _And this net has the valuable, even precious quality of +being the same that joins the greatest and most comprehensive intellects +in mankind to one another._ The truths a man has once grasped he need +never learn afresh so far as their _actual content_ is concerned, though +not infrequently--especially in newer sciences--he may have to see the +_form_ of their presentation and generalization change. For this reason +it is of such especial importance for each individual from the first to +perceive these unalterable facts and realize that they are unalterable +and learn to distinguish them from the alterable forms of their +presentation. It is in this very regard that the incompleteness of human +knowledge is most clearly revealed. Time and again in the history of +science form has been taken for content, and necessary changes of +form--a merely practical question--have been confused with revolutionary +modifications of the content. + +Thus, each presentation of a science has its natural philosophic +portion. In text-books, whether elementary or advanced, the chapter on +natural philosophy is found usually at the beginning of the book, +sometimes at the end, in the form of a "general introduction," or +"general summary." In the special works in which the latest advances of +science are made known by the investigators, the natural philosophic +portions are usually to be found in the form of theses, of principles, +which are not discussed, often not even explicitly stated, but upon the +acceptance of which depend all the special conclusions that are drawn, +in the case in hand, from the new facts or thoughts imparted. Whether at +the beginning or at the end of the book, these most general principles +do not quite occupy the place that befits them. If at the introduction +of the text-book, they are practically devoid of content, since the +facts they are meant to summarize are yet to be unfolded in the course +of the presentation. If at the end, they come too late, since they have +already been applied in numerous instances, though without reference to +their general nature. The best method is--and a good teacher always +employs this method, whether in the spoken or the written word--to let +the generalizations come whenever the individual facts imparted require +and justify them. + +Thus, all instruction in natural sciences is necessarily interspersed +with natural philosophy, good or bad, according to the clearheadedness +of the teacher. If we wish to obtain a perfect survey of a complex +structure, as, for instance, the confusion of streets in a large city, +we had better not try to know each street, but study a general plan, +from which we learn the comparative situation of the streets. So it is +well for us in studying a special science to look at our general plan, +if for no other reason than to keep from losing our way when it may +chance to lead through a quarter hitherto unknown. This is the purpose +of the present work. + + + + +PART I + +GENERAL THEORY OF KNOWLEDGE + + +=1. The Formation of Concepts.= To the human mind, as it slowly awakens +in every child, the world at first seems a chaos consisting of mere +individual experiences. The only connection between them is that they +follow each other consecutively. Of these experiences, all of which at +first are different from one another, certain parts come to be +distinguished by the fact that they are repeated more frequently, and +therefore receive a special character, that of _being familiar_. The +familiarity is due to our _recalling_ a former similar experience; in +other words, to our feeling that there is a relation between the present +experience and certain former experiences. The cause of this phenomenon, +which is at the basis of all mental life, is a quality common to all +living things, and manifesting itself in all their functions, while +appearing but rarely or accidentally in inorganic nature. It is the +quality by virtue of which _the oftener any process has taken place in a +living organism the more easily it is repeated_. Here is not yet the +place to show how almost all the characteristic qualities of living +beings, from the preservation of the species to the highest intellectual +accomplishments, are conditioned by this special peculiarity. Suffice it +to say that because of this quality all those processes which are +repeated frequently in any given living organism, assume spontaneously, +that is, from physiologic reasons, a character distinguishing them +essentially from those which appear only in isolated instances, or +sporadically. + +If a living being is equipped with consciousness and thought, like man, +then the conscious recollections of such uniform experiences form the +enduring or permanent part in the sum-total of his experiences. Each +time a complex event, like the change of seasons, for example, which we +know from experience repeats itself--each time a part of such an event +reaches our consciousness, we are prepared also for the other parts that +experience teaches are connected with it. This makes it possible for us +to foresee future events. What significance the foreseeing of future +events has for the preservation and the development of the individual as +well as the species can only be indicated here. To give one instance, it +is our ability to foretell the coming of winter with the impossibility +of obtaining food directly during the winter that causes us to refrain +from at once using up all the food we have and to preserve it for the +day of need. The ability to foretell, therefore, becomes the foundation +of the whole structure of economic life. + + +=2. Science.= The prophecy of future events based upon the knowledge of +the details of recurring events is called _science_ in its most general +sense. Here, as in most cases in which language became fixed long before +men had a clear knowledge of the things designated, the name of the +thing is easily associated with false ideas arising either from errors +that had been overcome or from other, still more accidental, causes. +Thus, the mere knowledge of _past_ events is also called science without +any thought of its use for prophesying future events. Yet a moment's +reflection teaches that mere knowledge of the past which is not meant +to, or cannot, serve as a basis for shaping the future is utterly +aimless knowledge, and must take its place with other aimless activities +called _play_. There are all sorts of plays requiring great acumen and +patient application, as for example the game of chess; and no one has +the right to prevent any individual from pursuing such games. But the +player for his part must not demand special regard for his activity. By +using his energies for his personal pleasure and not for a social +purpose, that is, for a general human purpose, he loses every claim to +the social encouragement of his activity, and must be content if only +his individual rights are respected; and that, too, only so long as the +social interests do not suffer by it. + + +=3. The Aim of Science.=These views are deliberately opposed to a very +widespread idea that science should be cultivated "for its own sake," +and not for the sake of the benefits it actually brings or may be made +to bring. We reply that there is nothing at all which is done merely +"for its own sake." Everything, without exception, is done for human +purposes. These purposes range from momentary personal satisfaction to +the most comprehensive social services involving disregard of one's own +person. But in all our actions we never get beyond the sphere of the +human. If, therefore, the phrase "for its own sake" means anything, it +means that science should be followed for the sake of the immediate +pleasure it affords, that is to say, as _play_ (as we have just +characterized it), and in the "for-its-own-sake" demand there is hidden +a misunderstood idealism, which, on closer inspection, resolves itself +into its very opposite, the degradation of science. + +The element of truth hidden in that misunderstood phrase is, that in a +higher state of culture it is found better to disregard the _immediate_ +technical application in the pursuit of science, and to aim only for the +greatest possible perfection and depth in the solution of its individual +problems. Whether this is the correct method of procedure and when it is +so, is solely a question of the general state of culture. In the early +stages of human civilization such a demand is utterly meaningless, and +all science is necessarily and naturally confined to immediate life. But +the wider and more complex human relations become, the wider and surer +must the ability to predict future events become. Then it is the +function of prophesying science to have answers ready for questions +which as yet have not become pressing, but which with further +development may sooner or later become so. + +In the net-like interlacing of the sciences, that is, of the various +fields of knowledge, described in the introduction, we must always +reckon with the fact that our anticipation of what kind of knowledge we +shall next need must always remain very incomplete. It is possible to +foresee future needs in general outline with more or less certainty, but +it is impossible to be prepared for particular individual cases which +lie on the _border line_ of such anticipation, and which may sometimes +become of the utmost importance and urgency. Therefore it is one of the +most important functions of science to achieve as _perfect_ an +elaboration as possible of _all_ the relations conceivable, and in this +practical necessity lies the foundation of the general or _theoretical_ +elaboration of science. + +=The Science of Concepts.= Here the question immediately arises: how can +we secure such perfection? The answer to this general preliminary +question of all the sciences belongs to the sphere of the first or the +most general of all the sciences, a knowledge of which is presupposed +for the pursuit of the other sciences. Since its foundation by the Greek +philosopher Aristotle it has borne the name of _logic_, which name, +etymologically speaking, hints suspiciously at the _word_, and the word, +as is known, steps in where ideas are wanting. Here, however, we have to +deal with the very science of ideas, to which language bears the +relation only of a means--and often an inadequate means--to an end. We +have already seen how, through the physiologic fact of _memory_, +experiences are found in our consciousness which are similar, that is, +partially coinciding with one another. These coinciding parts are those +concerning which we can make predictions, for the very reason that they +coincide in every single instance, and they alone, therefore, constitute +that part of our experience which bears results and hence has +significance. + + +=4. Concrete and Abstract.= Such coinciding or repeated parts of similar +experiences we call, as already stated, _concepts_. But here, too, +attention must immediately be drawn to a linguistic imperfection, which +consists in the fact that in such a group of coinciding experiences we +designate by the same name both the isolated experience or the object of +a special experience and the totality of _all_ the coinciding +experiences; in other words, all the similar experiences. Thus, _horse_ +means, on the one hand, quite a definite thing which for the moment +forms an object of our experience, and, on the other, the totality of +all possible similar objects which have been present in our former +experiences, and which we shall meet in our future experiences. It is +true that these two sorts of contents of consciousness bearing the same +name are distinguished also as _concrete_ and _abstract_, and there is +an inclination to attribute "reality" only to the first, while the +other, as "mere entities in thought," are relegated to a lesser degree +of reality. As a matter of fact, the difference, though important, is of +quite another kind. It is the difference between the _momentary +experience_, as opposed to the totality of the corresponding _memories_ +and _expectations_. Hence not so much a difference in _reality_ as in +_presence_. However, our observations have already made it apparent that +presence alone never yields knowledge. A necessary part of knowledge is +the memory of former similar experiences. For without such memory and +the corresponding comparison, it is quite impossible for us to get at +those things which agree and which, therefore, may be predicted; and we +should stand before every one of our experiences with the helplessness +of a new-born babe.[A] + +[A] Sometimes on suddenly awaking from a profound sleep a person finds +himself for the moment deprived of his personal stock of memories, +unable to recall where and in what circumstances he is. No one who has +experienced such a condition can ever forget the terrifying sense of +helplessness it brings. + + +=5. The Subjective Part.= We shall therefore have to recognize realities +in abstract ideas in so far as they must rest upon some experiences to +be at all intelligible to us. Since the formation of concepts depends +upon memories, and these may refer, according to the individual, to very +different parts of the same experience of different individuals, +concepts always possess an element dependent upon the individual, or a +_subjective_ element. This, however, does not consist in the _addition_ +by the individual of new parts not found in the experience, but, on the +contrary, in the different _choice_ out of what is found in the +experience. If every individual absorbed all parts of the experience, +the individual, or subjective, differences would disappear. And since +scientific experience endeavors to make the absorption of experiences as +complete as possible, it aims nearer and nearer to this ideal by seeking +to equalize the subjective deficiency of the individual memory through +the collocation of as many and as various memories as possible, thus +filling in the subjective gaps in experience as far as possible and +rendering them harmless. + + +=6. Empirical Concepts.= First and unconditionally those concepts +possess reality which always and without exception are based on +_experienced_ facts. But we can easily make manifold arbitrary +combinations of concepts from different experiences, since our memory +freely places them at our disposal, and from such a combination we can +form a new concept. Of course it is not necessary that our arbitrary +combination should also be found in our past or future experiences. On +the contrary, we may rather expect that there could be many more +arbitrary combinations not to be found in experience than combinations +later "confirmed" by experience. The former are purposeless because +unreal, the latter, on the contrary, are of the utmost consequence +because upon them is based the real aim of knowledge, prediction. The +former are those which have brought the very "reality" of the concepts +into ill repute, while the latter show that the formation and the mutual +reaction of the concepts practically constitute the entire content of +all science. It is of the greatest importance, therefore, to distinguish +between the two kinds of concept combinations, and the study of this +differentiation forms the content of that most general of all the +sciences which we have characterized as logic, or, better, the science +of concepts. + + +=7. Simple and Complex Concepts.= The formation of concepts consists, as +we have seen, in the selection of those parts of different but similar +experiences which coincide with one another and in the elimination of +those that are different in kind. The results of such a procedure may +vary greatly according to the number and the difference of the +experiences placed in relation with one another. If, for example, we +compare only a few experiences, and if, moreover, these experiences are +very similar to one another, then the resulting concepts will contain +very many parts that agree. But at the same time they will have the +peculiarity of not being applicable to other experiences, since these +are without some of the coinciding parts of that narrower circle. Thus, +for example, the concept which a rustic chained to the soil all his life +has of human work does not apply to the work of the city man. A concept +will embrace a larger number of individual cases in proportion as it +contains fewer different parts. And by systematically following out this +thought we arrive at the conclusion that the concepts that are simple +and have no different parts at all find the widest application or are +the most general. + +The elimination of the non-coinciding parts from the concept-forming +experience is called _abstraction_. Obviously abstraction must be +carried the farther the more numerous and the more varied the +experiences from which the concepts are abstracted, and the simplest +concepts are the most abstract. + +By looking back over the ground just traversed, the less abstract ideas +may also be regarded as the _more complex_ in contradistinction to the +simpler ones. Only we must guard against the error of literal +interpretation and not suppose that the less simple concepts have really +been compounded of the simpler ones. In point of origin they actually +existed first, since the experience contains the ensemble of all the +parts, those which have been retained as well as those which have been +eliminated. It is only later, by a characteristic mental operation, +after we have analyzed the more complex concept, that is, after we have +disclosed the simpler concepts existing in it, that we can compound it +again; in other words, execute its synthesis. + +These relations bear a striking resemblance to the relations known from +chemistry to exist between substances, namely, between elements and +compounds. From the chaos of all objects of experimentation (chemistry +purposely limits itself to ponderable bodies) the _pure_ substances are +sifted out--an operation corresponding to the formation of concepts. The +pure substances prove to be either _simple_ or _compound_, and the +compounds are so constituted that they can each be reduced to a limited +number of simple substances. The simple substances, or _elements_, +retain this quality of simplicity only until they are recalled; that is, +until it has been proved that they, too, can be resolved into still +simpler elements. The same is true of the simple concepts. They can +claim simplicity only until their complex nature is demonstrated. + +With all these similarities we must be extremely careful never to forget +the differences existing alongside the agreements. So hereafter we shall +make no further use of the chemical simile. It was brought into +requisition merely in order to acquaint the beginner the more readily +with the entire method of investigation by means of a more familiar +field of thought and study. It is quite certain, however, that side by +side with the given similarities there are also radical differences. +Moreover, the notion of simple and complex concepts or "ideas" had been +elaborated by John Locke long before chemistry reached its present state +of clearness concerning the concept of the elements. + +Nevertheless since then the relation has been completely reversed. While +the study of the chemical elements has in the meantime undergone great +development, so that not only have the elements of all the substances +coming under the observation of the chemist been discovered, but, +inversely, many compound substances have been constructed from their +elements, not even an approach to such a development is apparent in the +study of concepts. On the contrary, the whole matter has remained at +about the same point as that to which John Locke had brought it in the +second half of the seventeenth century. This is due above all to the +opinion of the most influential philosophers, that Aristotle's logic, or +science of concepts, is absolutely true as well as exhaustive and +complete, so that, at the utmost, what is left for later generations to +do is only to make a change in the form in which the matter is +presented. It is true that in more recent times the grave error of this +view is beginning to be recognized. We realize that Aristotle's logic +embraces but a very small part of the entire field, though in this part +he displays the greatest genius. But beyond this general recognition no +great step forward has been made. Not even a provisional table of the +elementary concepts has been propounded and applied since Locke. + +Hence in the following investigation we shall have to speak of the +elements or the simpler parts of a complex concept only in the sense +that these concept elements are simpler as compared with the complex +concepts, but not in the sense that the simplest or truly elementary +concepts have already been worked out. It must be left to later +investigators to find these, and it may be expected that the reduction +of some concepts until then considered elementary into still simpler +ones will take place chiefly in times of great intellectual progress. + +_Complex concepts_ can, in the first place, be formed from experience, +for in an empirical concept we meet with several conceptual component +parts which can be separated from one another by a process of +abstraction, but are always found together in the given experiences. For +example, the concept _horse_ has originated from a very frequent, +similarly repeated experience. On analysis it is found to contain a vast +number of other concepts, such as quadruped, vertebrate animal, +warm-blooded, hairiness, and so on. Horse, then, is obviously a _complex +empirical concept_. + +On the other hand, we can combine as many simple concepts as we please, +even if we did not find them combined in experience, for in reality +there is nothing to hinder us from uniting all the concepts provided by +memory into any combinations we please. In this way we obtain _complex +arbitrary concepts_. + +The task of science can now be even more sharply defined than before by +the fact that it _permits the construction of arbitrary concepts which +in circumstances to be foreseen become empirical concepts_. This is +another expression for _prediction_, which we recognized as the +characteristic of science. It goes deeper than the previous definition, +because here the means for its realization are given. + + +=8. The Conclusion.= First let us consider the scientific import of the +complex empirical concepts. It consists in the fact that they accustom +us to the coexistence of the corresponding elements of a concept. So +that when, in a new experience, we meet with some of these elements +together, we immediately suppose that we shall find in the same +experience the other elements also which have not yet been ascertained. +Such a supposition is called a _conclusion_. A conclusion always exceeds +the present experience by predicting an expected experience. Therefore, +the form of a conclusion is the universal form of scientific +predication. + +A conclusion must contain at least two concepts, the one which is +experienced, and the one which, on the basis of this experience, is +expected. Every complex empirical concept makes such a conclusion +possible after it has been separated into simpler concepts. And the +simplest case is naturally the one in which there are only _two_ parts, +or in which only two parts are taken into consideration. + +To what extent such a conclusion is valid, that is to say, to what +extent the experience produces the anticipated concept, obviously +depends upon the reply to a very definite fundamental question. If in +experience the union of the two parts of the concept occurs +_invariably_, so that one part of the concept is never experienced +unless the other part is also experienced, then there is the _greatest_ +probability that the expected experience will also have the same +character, and that the conclusion will prove valid or true. To be sure, +there is no way of making certain that the coincident occurrence of the +two concepts, which experience has shown to be _without exception_ +hitherto, will continue to be so also in the future. For our only means +of penetrating into the future consists in applying that conclusion from +previous experiences to future experiences, and it can therefore by no +means claim absolute validity. There are, however, different _degrees of +certainty_, or, rather, _probability_, attaching to such a conclusion. +In experiences that occur but rarely the probability is that so far we +have experienced only certain combinations of simple concepts, while +others, though occurring, have not yet entered within the limited circle +of our experience. In such a case a conclusion of the kind mentioned +above may be right, but there is also some probability of its being +false. On the other hand, in experiences which happen extremely +frequently and in the most diverse circumstances, and in which we always +find the constant and unexceptional combination, the probability is very +strong that we shall find the combination in future experiences also, +and the probability of the conclusion approaches practical certainty. Of +course, we can never quite exclude the possibility that new relations +never as yet experienced might enter, by which the conclusion which +hitherto has always been true would now become false, whether because +the expectation entertained prove invalid in single instances or in all +cases. + +It follows from this that in general, our conclusions will have the +greater probability the more generally and the oftener the corresponding +experiences have occurred and are occurring. Such concepts as are found +consistently in many experiences otherwise different are called +_general_ concepts, and therefore the probability of the conclusions +described will be the greater the more _general_ the concepts to which +they refer. This obtains to such a degree that we feel that certain very +general conclusions must be true always and without exception, and it is +"unthinkable" to us that they can ever in any circumstances prove not +valid. Such a statement, however, is never anything else than a hidden +appeal to experience. For the mere putting of the question, whether the +conclusion can also be false, demonstrates that the opposite of what has +proved to be the experience so far can be conceived, and the assertion +of its "unthinkability" only signifies that such an experience cannot be +evoked in the mind by the _memory_ for the very reason that, as has been +premised, there are no such memories because the experiences did not +exist. But since, on the other hand, there is no hindrance to thinking +any combinations of concepts at will, we have not the least difficulty, +as everybody knows, in thinking any sort of "nonsense" whatsoever. Only +it is impossible to reproduce such combinations from memory. + +The scientific conclusion, therefore, first takes the form: if A is, +then B is also. Here A and B represent the two simple concepts which are +known from experience to be found together in the more complex concept +C. The word "is" signifies here some empirical reality corresponding to +the concepts. The conclusion may therefore also be expressed, somewhat +more circumstantially and more precisely, in this form: if A is +experienced, the experience of B is also expected. The evoking of this +expectation, which implies its justification, is due to the recollection +of the coincidence of the two concepts in former experiences, and the +probability depends, in the manner described above, upon the number of +valid cases. Here it must be observed that even individual cases in +which our expectations have been deceived do not for the most part lead +us to regard the conclusion as generally untrue, that is, to abandon the +expectation of B from A. For we know that our experience is always +_incomplete_, that in certain circumstances we fail to notice existing +factors, and that, therefore, our failure to find that relation valid +which, on other occasions, has been found to be valid, may be attributed +to subjective causes. In case, however, of the repeated occurrence of +such disappointments, we will look elsewhere for relations between these +and other elements of experience, in order that thereafter we may +foresee such cases also and include them in our anticipations. + + +=9. The Natural Laws.= The facts just described have very frequently +found expression in the doctrine of the _laws of nature_, in connection +with which we have often, as in the man-made social or political laws, +conceived of a lawmaker, who, for some reasons, or perhaps arbitrarily, +has ordained that things should be as they are and not otherwise. But +the intellectual history of the origin of the laws of nature shows that +here the process is quite a different one. The laws of nature do not +decree what shall happen, but _inform us what has happened and what is +wont to happen_. The knowledge of these laws, therefore, makes it +possible for us, as I have emphasized again and again, to foresee the +future in a certain degree and, in some measure, also to determine it. +We determine the future by constructing those relations in which the +desired results appear. If we cannot do so either because of ignorance +or because of inaccessibility to the required relations, then we have no +prospect of fashioning the future according to our desires. The wider +our knowledge of the natural laws, that is, of the actual behavior of +things, the more likely and more numerous the possibilities for +fashioning the future according to our desires. In this way science can +be conceived of as the study of how to become happy. For he is happy +whose desires are fulfilled. + +In this conception the natural laws indicate what simpler concepts are +found in complex concepts. The complex concept _water_ contains the +simpler ones _liquid_, a certain _density_, _transparency_, +_colorlessness_,[B] and many others. The sentences, water is a liquid, +water has a density of one, water is transparent, water is colorless, +or, pale blue, etc., are so many natural laws. + +[B] More precisely, a very pale blue. + +Now what predictions do those natural laws enable us to make? + +They enable us to predict that when we have recognized a given body as +water by virtue of the above properties, we are justified in expecting +to find in the same body all the other known properties of water. And so +far experience has invariably confirmed such expectations. + +Furthermore, we may expect that if in a given specimen of water we +discover a relation which up to that time was unknown, we shall find +this relation also in all the other specimens of water even though they +were not tested for that particular relation. It is obvious how +enormously this facilitates the progress of science. For it is only +necessary to determine this new relation in some one case accessible to +the investigator to enable us to predict the same relation in all the +other cases without subjecting them to a new test. As a matter of fact, +this is the general method that science pursues. It is this that makes +it possible for science to make regular and generally valid progress +through the labors of the most various investigators who work +independently of one another, and often know nothing of one another. + +Of course, it must not be forgotten that such conclusions are always +obtained in accordance with the following formula: _things have been so +until now, therefore we expect that they will be so in the future_. In +every such case, therefore, there is the possibility of error. Thus far, +whenever an expectation was not realized, it was almost always possible +to find an "explanation" for the error. Either the inclusion of the +special case in the general concept proved to be inadmissible because +some of its other characteristics were absent, or the accepted +characterization of the concept required an improvement (limitation or +extension). In other words, one way or another, there was a discrepancy +between the concept and the experience, and, as a rule, sooner or later +it becomes possible for us to arrive at a better adjustment between +them. + +This general truth has often been interpreted to mean that in the end +such an adjustment must of necessity always be possible to reach, +without exception; in other words, that absolutely every part of an +experience can be demonstrated as conditioned by natural law. Evidently +such an assertion far exceeds the demonstrable. And even the usual +conclusion cannot be applied here, that because it has happened so in +the past it will happen so in the future also. For the part of our +experiences that we can grasp by natural laws is infinitesimally small +in comparison with that in which our knowledge still fails us entirely. +I will mention only the uncertainty in predicting the weather for only +one day ahead. Moreover, when we consider that until now only the +_easiest_ problems had been solved, and naturally so, because they were +most accessible to the means at hand, then we can readily see that +experience offers no basis whatever for such a conclusion. We must not +say, therefore, that because we have been able so far to explain all +experiences by natural laws it will be so in the future likewise. For we +are far from being able to explain all experiences. In fact, it is only +a very small part that we have begun to investigate. We are as little +justified in saying that we have explained all the problems of our +experience that have been subjected to scientific investigation. We have +by no means explained all of them. Every science, even mathematics, +teems with unsolved problems. So we must resign ourselves to the present +status of human knowledge and ability, and may at best express the +_hope_ founded upon previous experience, that we shall be able to solve +more and more of the incalculable number of problems of our experience +without indulging in any illusions as to the perfection of this work. + + +=10. The Law of Causation.= By reason of its frequency and importance +the mental process above described has been subjected to the most +diverse investigations, and that most general form of the scientific +conclusion (which we apply in ordinary life even much more frequently +than in science) has been raised, under the name of the law of +causation, to a principle anteceding all experience and to the very +condition making experience possible. Of this so much is true, that +through the peculiar physiological organization of man, _memory in the +most general sense_--the easier execution of such processes as have +already repeatedly taken place in the organism, as against entirely new +kinds of processes--the formation of concepts (of the recurring parts in +the constantly changing variety of processes), is especially stimulated +and facilitated. By it the recurring parts of experience step into the +foreground, and on account of their paramount practical importance for +the security of life, it may well be said in the sense of the theory of +evolution and adaptation, that the entire structure and mode of life of +the organism, especially of the human organism, nay, perhaps life +itself, is indissolubly bound up with that foresight and, therefore, +with the law of causation also. Of course, there is nothing in the way +of calling such a relation an _a priori_ relation, if it is so desired. +As far as the individual is concerned it no doubt antedates all his +experience, since the entire organization which he inherits from his +parents had already been formed under such an influence. But that there +can be forms or existence _without_ such an attribute is shown by the +whole world of the _inorganic_, in which, as far as our knowledge goes, +there is no evidence of either memory or foresight, but only of an +immediate passive participation in the processes of the world around +them.[C] + +[C] It cannot be objected that inorganic nature also is known to be +subject to the law of causation. The causal mode of regarding inorganic +phenomena is a distinctly human one, and nothing justifies the assertion +that the same phenomena cannot be viewed in an entirely different +manner. + +Further, the circumstance that the causal relation is brought about by +the peculiar manner in which we react upon our experiences, has +sometimes been expressed in this way--the relation of cause and effect +does not exist in nature at all, but has been introduced by men. The +element of truth in this is, that a quite differently organized being, +it is to be supposed, would be able to, or would have to, arrange its +experiences according to quite different mutual relations. But since we +have no experience of such a being, we have no possibility of forming a +valid opinion of its behavior. On the other hand, we must recognize that +it is possible, at least formally, to conceive also of kinds of +experiences with no coinciding parts, or a world in which there are no +experiences at all with coinciding parts. In such, therefore, prediction +is impossible. Such a world will not call forth, even in a being endowed +with memory, a conception and generalization of the various experiences +in the shape of natural laws. Consequently we must recognize that in +addition to the _subjective_ factor in the formation of our knowledge of +the world, or that factor which is dependent upon our physico-psychical +structure, there is also the _objective_ character of the world with +which we must decidedly reckon, or that character which is independent +of us; and that in so far the natural laws contain also objective parts. +To represent the relation clearly to our minds by a figure, we may +compare the world to a heap of gravel and man to a pair of sieves, one +coarser than the other. As gravel passes through the double sieve +pebbles of apparently equal size accumulate between the sieves, the +larger ones being excluded by the first sieve and the smaller ones +allowed to pass by the second. It would be an error to assert that all +the gravel consisted of such pebbles of equal size. But it would be +equally false to assert that it was the sieves that _made_ the pebbles +equal. + + +=11. The Purification of the Causal Relation.= If by experience we have +found a proposition of the content, If A is, then B is also, the two +concepts A and B generally consist of several elements which we will +designate as a, a´, a´´, a´´´, etc., and as b, b´, b´´, b´´´. Now the +question arises, whether or not all these elements are essential for the +relation in question. It is quite possible, in fact, even highly +probable, that at first only a special instance of the existing +phenomena was found, that is, that the concept A, which has been found +to be connected with the concept B, contains other determining parts +which are not at all requisite to the appearance of B. + +The general method of convincing oneself of this is by eliminating one +by one the component parts of the concept A, namely, a, a´, a´´, etc., +and then seeing whether B still appears. It is not always easy to carry +out this process of elimination. Our greater or less ability to conduct +such investigations depends upon whether we deal with things that are +merely the objects of our _observation_, and which we ourselves have not +the power to change (as, for example, astronomical phenomena), or with +things which are the objects of our _experimentation_, and which we can +influence. In the latter case one or another factor is usually found +which can be eliminated without the disappearance of B, and then we must +proceed in such a way as to form a corresponding new concept A´ from the +factors recognized as necessary (which new concept will be more general +than the former A), and to express the given proposition in the improved +form: If A´ is, then B is also. + +Quite similar is the case with the other member of this relation. It +often happens that when a, or a´´, a´´´ is found, somewhat different +things appear, which do not fit the concept as first constructed. Then +we must multiply the experiences as much as possible in order to +determine what constant elements are found in the concept B, and to form +from these constant elements the corresponding concept B´. The improved +proposition will then read: if A´ is, then B´ is also. + +This entire process may be called the purification of the causal +relation. By this term we express the general fact that in first forming +such a regular connection, the proper concepts are very seldom brought +into relation with one another at once. The cause of it is that at first +we make use of _existing_ concepts which had been formed for quite a +different purpose. It must therefore be regarded as a special piece of +good fortune if these old concepts should at once prove suited to the +new purpose. Furthermore, the existing concepts are as a rule so vaguely +characterized by their names, which we must employ to express the new +relation, that for this reason also it is often necessary to determine +empirically in what way the concept is to be definitely established. + +The various sciences are constantly occupied with this work of the +mutual adaptation of the concepts that enter into a causal relation. By +way of example, we may take the "self-understood" proposition which we +use when we call out to a careless child when it sticks its finger into +the flame of a candle, "Fire burns!" We discover that there are +self-luminous bodies which produce no increase of temperature, and +therefore no sensation of pain. We discover that there are processes of +combustion that develop no light, but heat enough to burn one's +fingers. And, finally, the scientific investigation of this proposition +arrives at the general expression that, as a rule, chemical processes +are accompanied by the development of heat, but that, conversely, such +processes may also be accompanied by the absorption of heat. In this way +that casual sentence which we call out to the child develops into the +extensive science of thermo-chemistry when it is subjected to the +continuous purification of the causal relation, which is the general +task of science. + +It remains to be added that in this process of adapting concepts it is +necessary also sometimes to follow the opposite course. This is the case +when _exceptions_ are noticed in a relation as expressed for the time +being; when, therefore, the proposition if A is present, then B is +present also, is in a great many instances valid, but occasionally +fails. This is an indication that in the concept A an element is still +lacking. This element, however, is present in the instances that tally, +but absent in the negative cases, and its absence is not noticed because +it is not contained in A. Then it is necessary to seek this part, and +after it has been found, to embody it in the concept A, which thus +passes into the new concept A´. + +This case is the obverse of the former one. Here the more suitable +concept proves to be less general than the concept accepted temporarily, +while in the first case the improved concept is more general. Hence we +formulate the rule: exceptions to the temporary rule require a +limitation, while an unforeseen freedom requires an extension, of the +accepted concept. + + +=12. Induction.= The form of conclusion previously discussed, _because +it has been so, I expect it will continue to be so in the future_, is +the form through which each science has arisen and has won its real +content, that is, its value for the judgment of the future. It is called +_inference by induction_, and the sciences in which it is +preponderatingly applied are called _inductive sciences_. They are also +called experiential or empirical sciences. At the basis of this +nomenclature is the notion that there are other sciences, the deductive +or rational sciences, in which a reverse logical procedure is applied, +whereby from general principles admitted to be valid in advance, +according to an absolutely sure logical process, conclusions of like +absolute validity are drawn. At the present time people are beginning to +recognize the fact that the deductive sciences must give up these claims +one by one, and that they already have given them up to a certain +extent; partly because on closer study they prove to be inductive +sciences, and partly because they must forego the title and rank of a +science altogether. The latter alternative applies especially to those +provinces of knowledge which have not been used in prophesying the +future or cannot be so used. + +To return to the inductive method--it is to be noted that _Aristotle_, +who was the first to describe it, proposed two kinds of induction, the +_complete_ and the _incomplete_. The first has this form: since _all_ +things of a certain kind are so, each _individual thing_ is so. While +the incomplete induction merely says: since _many_ things of a certain +kind are so, _presumably_ all things of this kind are so. One instantly +perceives that the two conclusions are essentially different. The first +lays claim to afford an absolutely certain result. But it rests upon the +assumption that _all_ the things of the kind in question are known and +have been tested as to their behavior. This hypothesis is generally +impossible of fulfilment, since we can never prove that there are not +more things of the same kind other than those known to us or tested by +us. Moreover, the conclusion is _superfluous_, as it merely repeats +knowledge that we have already directly acquired, since we have tested +_all_ the things of the one kind, hence the special thing to which the +predication refers. + +On the other hand, the _incomplete_ induction affirms something that has +not yet been tested, and therefore involves as a condition an +_extension_ of our knowledge, sometimes an extremely important +extension. To be sure, it must give up the claim to unqualified or +absolute validity, but, to compensate, it acquires the irreplaceable +advantage of lending itself to practical application. Indeed, in +accordance with the scientific practice justified by experience, +described on p. 29, the scientific inductive conclusion assumes the +form: because it has _once_ been found to be so, it will _always_ be so. +From this appears the significance of this method for the enlargement of +science, which, without it, would have had to proceed at an incomparably +slower pace. + + +=13. Deduction.= In addition to the inductive method, science has (p. +38) another method, which, in a sense, should be the reverse of the +inductive and is claimed to provide absolutely correct results. It is +called the _deductive_ method, and it is described as the method that +leads from premises of general validity by means of logical methods of +general validity to results of general validity. + +As a matter of fact, there is no science that does or could work in such +a way. In the first place, we ask in vain, how can we arrive at such +general, or absolutely valid, premises, since all knowledge is of +empiric origin and is therefore equipped with the possibility of error +as ineradicable evidence of this origin. In the next place, we cannot +see how from principles at hand conclusions can be drawn the content of +which exceeds that of these principles (and of the other means +employed). In the third place, the absolute correctness of such results +is doubtful from the fact that blunders in the process of reasoning +cannot be excluded even where the premises and methods are absolutely +correct. In practice it has actually come to pass that in the so-called +deductive sciences doubts and contradictions on the part of the various +investigators of the same question are by no means excluded. To wit, +the discussion that has been carried on for centuries, and is not yet +ended, over Euclid's parallel theorem in geometry. + +If we ask whether, in the sense of the observations we have just made of +the formation of scientific principles, there is anything at all like +deduction, we can find a procedure which bears a certain resemblance +with that impossible procedure and which, as a matter of fact, is +frequently and to very good purpose applied in science. It consists in +the fact that general principles which have been acquired through the +ordinary incomplete induction are _applied to special instances which, +at the proposition of the principle, had not been taken into +consideration_, and whose connection with the general concept had not +become directly evident. Through such application of general principles +to cases that have not been regarded before, specific natural laws are +obtained which had not been foreseen either, but which, according to the +probability of the thesis and the correctness of the application are +also probably correct. However, the investigator, bearing in mind the +factor of uncertainty in these ratiocinations feels in each such +instance the need for testing the results by experience, and he does not +consider the _deduction_ complete until he had found _confirmation_ in +experience. + +Deduction, therefore, actually consists in the searching out of +particular instances of a principle established by induction and in its +confirmation by experience. This conduces to the growth of science, not +in breadth, but in profundity. I again resort to the comparison I have +frequently made of science with a very complex network. At first glance +we cannot obtain a complete picture of all the meshes. So, at the first +proposition of a natural law an immediate survey of the entire range of +the possible experiences to which it may apply is inachievable. It is a +regular, important, and necessary part of all scientific work to learn +the extent of this range and investigate the specific forms which the +law assumes in the remoter instances. Now, if an especially gifted and +far-seeing investigator has succeeded in stating in advance an +especially general formulation of an inductive law, it is everywhere +confirmed in the course of the trial applications, and the impression +easily arises that confirmation is superfluous, since it results simply +in what had already been "deduced." In point of fact, however, the +reverse is not infrequently the case, that the principle is _not_ +confirmed, and conditions quite different from those anticipated are +found. Such discoveries, then, as a rule, constitute the starting-point +of important and far-reaching modifications of the original formulation +of the law in question. + +As we see, deduction is a necessary complement of, in fact, a part of, +the inductive process. The history of the origin of a natural law is in +general as follows. The investigator notices certain agreements in +individual instances under his observation. He assumes that these +agreements are general, and propounds a temporary natural law +corresponding to them. Then he proceeds by further experimentation to +test the law in order to see whether he can find full confirmation of it +by a number of other instances. If not, he tries other formulations of +the law applicable to the contradictory instances, or exclusive of them, +as not allied. Through such a process of adjustment he finally arrives +at a principle that possesses a certain range of validity. He informs +other scientists of the principle. These in their turn are impelled to +test other instances known to them to which the principle can be +applied. Any doubts or contradictions arising from this again impel the +author of the principle to carry out whatever readjustments may have +become necessary. Upon the scientific imagination of the discoverer +depends the range of instances sufficing for the formulation of the +general inductive principle. It also frequently depends upon conscious +operations of the mind dubbed "scientific instinct." But as soon as the +principle has been propounded, even if only in the consciousness of the +discoverer, the deductive part of the work begins, and the consequent +test of the proposition has the most essential influence on the value of +the result. + +It is immediately evident that this _deductive_ part is of all the more +weight, the more _general_ the concepts in question are. If, in +addition, the inductive laws posited soon prove to be of a comparatively +high degree of perfection, we obtain the impression described above, +that an unlimited number of independent results can be deduced from a +premise. _Kant_ was keenly alive to the peculiarity of such a view, +which had been widely spread pre-eminently by _Euclid's_ presentation of +geometry, and he gave expression to his opinion of it in the famous +question: _How are a priori judgments possible?_ We have seen that it is +not always a question of _a priori_ judgments, but also of inductive +conclusions applied and tested according to deductive methods. + + +=14. Ideal Cases.= Each experience may generally be considered under an +indefinite number of various concepts, all of which may be abstracted +from that experience by corresponding observations. Accordingly an +indefinite number of natural laws would be required for prophesying that +experience in all its parts. Likewise the indefinite number of premises +must be known through the application of which those natural laws +acquire a certain content. Thus it seems as if it were altogether +impossible to apply natural laws for the determination of a single +experience to come, and in a certain sense this is true (p. 30). For +example, when a child is born, we are quite incapable of foretelling the +peculiar events that will occur in its life. Beyond the statement that +it will live a while and then die, we can make only the broadest +assertions qualified by numerous "ifs" and "buts." + +If, in spite of this, we arrange a very great part of our life and +activity according to the prophecies we make in regard to numerous +details in life, basing them upon natural laws, the question arises, how +we get over the difficulty, or, rather, the impossibility just referred +to. + +The answer is, that we repeatedly so find or can form our experiences +that certain natural relations _preponderatingly_ determine the +experience, while the other parts that remain undetermined fall into the +background. _The prophecy will cover so considerable a part of the +experience that we can forego previous knowledge of the rest._ We can +foretell enough to render a practical construction of life possible, and +increasing experience, whether the personal experience of the individual +or the general experience of science, constantly enlarges this +controllable part of future experiences. + +The procedure of science is similar to that of practical life, though +freer. Whenever an investigator seeks to test a natural law of the form: +if A is so, then B is so, he endeavors to choose or formulate the +experiences in such a way that the fewest possible extraneous elements +are present, and that those that are unavoidable should exert the least +possible influence upon the relation in question. He never succeeds +completely. In order, nevertheless, to reach a conclusion as to the form +the relation will take without extraneous influences, the following +general method is applied. + +A series of instances are investigated which are so adjusted that the +influence of the extraneous elements grows less and less. Then the +relation investigated approaches a limit which is never quite reached, +but to which it draws nearer and nearer, the less the influence of the +extraneous elements. And the conclusion is drawn that if it were +possible to exclude the extraneous elements entirely, the limit of the +relation would be reached. + +A case in which none of the extraneous elements of experience operate is +called an _ideal case_, and the inference from a series of values +leading to the limit-value is an _extrapolation_. _Such extrapolations +to the ideal case_ are a quite natural procedure in science, and a very +large part of natural laws, especially all quantitative laws, that is, +such as express a relation between measurable values, have precise +validity only in ideal cases. + +We here confront the fact that many natural laws, and among them the +most important, are expressed as, and taken to be, conditions _which +never occur in reality_. This seemingly absurd procedure is, as a matter +of fact, the best fitted for scientific purposes, since ideal cases are +to be distinguished by this, _that with them the natural laws take on +the simplest forms_. This is the result of the fact that in ideal cases +we intentionally and arbitrarily overlook every complication of the +determining factors, and in describing ideal cases we describe the +simplest conceivable form of the class of experiences in question. The +real cases are then constructed from the ideal cases by representing +them as the sum of all the elements that have an influence on the +experience or the result. Just as we can represent the unlimited +multitude of finite numbers by the figures up to ten, so we can +represent an unlimited quantity of complicated events by a finite number +of natural laws, and so reach a highly serviceable approximation to +reality. + +Thus geometry deals with absolutely straight lines, absolutely flat +surfaces, and perfect spheres, though such have never been observed, and +the results of geometry come the closer to truth, the more nearly the +real lines, surfaces, and spheres correspond to the ideal demands. +Similarly, in physics, there are no ideal gases or mirrors, or in +chemistry ideally pure substances, though the expressed simple laws in +these sciences are valid for only such bodies. The non-ideal bodies of +these sciences, which reality presents in various degrees of +approximation, correspond the more closely to these laws, the slighter +the deviation of the real from the ideal. And the same method is applied +in the so-called mental sciences, psychology and sociology, in which the +"normal eye" and a "state with an entirely closed door" are examples of +such idealized limit-concepts. + + +=15. The Determinateness of Things.= A very widespread view and a very +grave one, because of its erroneous results, is _that by the natural +laws things are unequivocally and unalterably determined down to the +very minutest detail_. This is called _determinism_, and is regarded as +an inevitable consequence of every natural scientific generalization. +But an accurate investigation of actual relations produces something +rather different. + +The most general formulation of the natural law: _if A is experienced, +then we expect B_, necessarily refers in the first place only to certain +_parts_ of the thing experienced. For perfect similarity in two +experiences is excluded by the mere fact that we ourselves change +unceasingly and one-sidedly. Consequently, no matter how accurate the +repetition of a former experience may be, our very participation in it, +an element bound to enter, causes it to be different. Therefore we deal +with only a _partial_ repetition of any experience, and the common part +is all the smaller a fraction of the entire experience, the more +_general_ the concept corresponding to this part. But the most general +and most important natural laws apply to such very general ideas, and +accordingly they determine only a small part of the whole result. Other +parts are determined by other laws, but we can never point out an +experience that has been determined completely and unequivocally by +natural laws known to us. For example, we know that when we throw a +stone, it will describe an approximate parabolic curve in falling to the +ground. But if we should attempt to determine its course accurately, we +should have to take into consideration the resistance of the air, the +rotatory motion of the stone upon being thrown, the movement of the +earth, and numerous other circumstances, the exact determination of +which is a matter beyond the power of all sciences. Nothing but an +_approximate_ determination of the stone's course is possible, and every +step forward toward accuracy and absoluteness would require scientific +advances which it would probably take centuries to accomplish. + +Science, therefore, can by no means determine the exact linear course +that the stone will take in its fall. It can merely establish a certain +broader path within which the stone's movement will remain. And the path +is the wider the smaller the progress science has made in the branch in +question. The same conditions prevail in the case of every other +prediction based upon natural laws. Natural laws merely provide a +certain frame within which the thing will remain. But which of the +infinitely numerous possibilities within this frame will become reality +can never be absolutely determined by human powers. + +The belief that it is possible has been evoked merely by a far-reaching +method of abstraction on the part of science. By assuming in place of +the stone "a non-extended point of mass" and by disregarding all the +other factors which in some way (whether known or unknown) exercise an +influence on the stone's movement, we can effect an apparently perfect +solution of the problem. But the solution is not valid for real +experience, merely for an ideal case, which bears only a more or less +profound similarity to the real. It is only such an ideal world, that +is, a world arbitrarily removed from its actual complexity, that has the +quality of absolute determinateness which we are wont to ascribe to the +real world. + +We might point to the method of abstraction generally adopted in science +and to the extrapolation to ideal cases which has just been explained, +and regard the assertion of the absolute determinateness of events in +the world as a justified extrapolation to the ideal case. In other +words, we might say that we know all the natural laws and how to apply +them perfectly to the individual instances. In controversion of this it +must be said that the ulterior justification of such ideal extrapolation +is not yet feasible. The justification lies in the demonstration that +the real cases approximate the ideal the more closely the more we +actualize our presumptions. But in this case this is not feasible, +since, for the greater part of our experiences, we do not even know the +approximate or ideal natural laws by the help of which we can construct +such ideal cases. For instance, the whole province of organic life is at +present essentially like an unknown land, in which there are only a few +widely separated paths ending in _culs-de-sac_. + + +=16. The Freedom of the Will.= This relation explains why, on the one +hand, we assume a far-reaching determinateness for many things, that is, +for all those accessible to scientific treatment and regulation, and +why, on the other hand, we have the consciousness of acting _freely_, +that is, of being able to control future events according to the +relations they bear to our wishes. Essentially there is no objection to +be found to a fundamental determinism which explains that this feeling +of freedom is only a different way of saying _that a part of the causal +chain lies within our consciousness_, and that we feel these processes +(in themselves determined) as if we ourselves determined their course. +Nor can we prove this idea to be false, that, since the number of +factors which influence each experience is indefinitely great and their +nature indefinitely complex, each event would appear to be determined in +the eyes of an all-comprehensive intellect. But to our finite minds an +undetermined residue necessarily remains in each experience, and to that +extent the world must always remain in part practically undetermined to +human beings. Thus, both views, that the world is not completely +determined, and that it really is, though we can never recognize that it +is, lead practically to the same result: _that we can and must assume in +our practical attitude to the world that it is only partially +determined_. + +But if two different lines of thought in the whole world of experience +everywhere lead to the same result, they cannot be materially, but +merely formally or superficially, different. For those things are alike +which cannot be distinguished. There is no other definition of +alikeness. Thus, if we see that the age-long dispute between these two +views always breaks out afresh without seeming to be able to reach an +end, this is readily understood, from what has been said, since the very +same essential arguments which can be adduced of _one_ view can be used +as a prop for the _other_ view, because in their essential results the +two are the same. I have discussed this matter because it presents a +very telling example of a method to be applied in all the sciences when +dealing with the solution of old and ever recurrent moot questions. Each +time we encounter such problems, we must ask ourselves: what would be +the difference empirically if the one or the other view were correct? In +other words, we first assume the one to be correct, and develop the +consequences accordingly. Then we assume the second to be correct and +develop the consequences accordingly. If in the two cases the +consequences differ in a certain definite point, we at least have the +possibility of ascertaining the false view by investigating in favor of +which case experience decides on this point. However, we may not +conclude that by this the other view has been proved to be entirely +correct. It likewise may be false, only with the peculiar quality that +in the case in question it leads to the correct conclusions. That such +a thing is possible, every one knows who has attentively observed his +own experiences. How often we act correctly in actual practice, though +we have started out on false premises! The explanation of this +possibility resides in the highly composite nature of each experience +and each assumption. It is quite possible--and, in fact, it is the +general rule--that a certain view contains true elements, but _along +with them false elements also_. In applications of the view where the +true elements are the decisive factors, true results are obtained, +despite the errors present. Likewise, false results will be achieved +where the false elements are decisive, despite the true results that can +be had, or have been had, elsewhere, by means of the true elements. +Hence, in case of the "confirmation," we can only conclude that that +portion of the view essential for the instance in question is correct. + +One readily perceives that these observations find application in all +provinces of science and life. There are no absolutely correct +assertions, and even the falsest may in some respect be true. There are +only greater and lesser probabilities, and every advance made by the +human intellect tends to increase the degree of probability of +experiential relations, or natural laws. + + +=17. The Classification of the Sciences.= From the preceding +observations the means may be drawn for outlining a complete table of +the sciences. However, we must not regard it complete in the sense that +it gives every possible ramification and turn of each science, but that +it sets up a frame inside of which at given points each science finds +its place, so that, in the course of progressive enlargement, the frame +need not be exceeded. + +The basic thought upon which this classification rests is that of graded +abstraction. We have seen (p. 19) that a concept is all the more +general, that is, is applicable to all the more experiences, the fewer +parts or elementary concepts it contains. So we shall begin the system +of the sciences with the most general concepts, that is, the elementary +concepts (or with what for the time being we shall have to consider +elementary concepts), and, in grading the concept complexes according to +their increasing diversity, set up a corresponding graded series of +sciences. One thing more is to be noted here, that this graded series, +on account of the very large number of new concepts entering, must +produce a correspondingly great number of diverse sciences. For +practical reasons groups of such grades have been combined temporarily. +Thereby a rougher classification, though one easier to obtain a survey +of, has been made. The most suitable and lasting scheme of this sort was +originated by the French philosopher, _Auguste Comte_, since whom it has +undergone a few changes. + +Below is the table of the sciences, which I shall then proceed to +explain: + + I. _Formal Sciences._ Main concept: order + Logic, or the science of the Manifold + Mathematics, or the science of Quantity + Geometry, or the science of Space + Phoronomy, or the science of Motion + + II. _Physical Sciences._ Main concept: energy + Mechanics + Physics + Chemistry + + III. _Biological Sciences._ Main concept: life + Physiology + Psychology + Sociology + +As is evident, we first have to deal with the three great groups of the +formal, the physical, and the biological sciences. The formal sciences +treat of characteristics belonging to all experiences, characteristics, +consequently, that enter into every known phase of life, and so affect +science in the broadest sense. In order immediately to overcome a +widespread error, I emphasize the fact that these sciences are to be +considered just as experiential or empirical as the sciences of the +other two groups, as to which there is no doubt that they are empirical. +But because the concepts dealt with by the first group are so extremely +wide, and the experiences corresponding to them, therefore, are the most +general of all experiences, we easily forget that we are dealing with +experiences at all; and our very firmly rooted consciousness of the +unqualified similarity of these experiences causes them to seem native +qualities of the mind, or _a priori_ judgments. Nevertheless, +mathematics has been proved to be an empirical science by the fact that +in certain of its branches (the theory of numbers) laws are known which +have been found empirically and the "deductive" proof of which we have +as yet not succeeded in obtaining. The most general concept expressed +and operative in these sciences is the concept of order, of _conjugacy_ +or _function_, the content and significance of which will become clear +later in a more thorough study of the special sciences. + +In the second group, the physical sciences, the arbitrariness of the +classification becomes very apparent, since these sciences are among the +best known. We are perfectly justified in regarding mechanics as a part +of physics; and in our day physical chemistry, which in the last twenty +years suddenly developed into an extended and important special science, +thrust itself between physics and chemistry. + +The most general concept of the physical sciences is that of _energy_, +which does not appear in the formal sciences. To be sure it is not a +fundamental concept. On the contrary, its characteristic is undoubtedly +that of compositeness, or, rather, complexity. + +The third group comprehends all the relations of living beings. Their +most general concept, accordingly, is that of _life_. By physiology is +understood the entire science dealing with non-psychic life phenomena. +It therefore embraces what is called, in the present often chance +arrangement of scientific activities, botany, zoology, and physiology of +the plants, animals, and man. Psychology is the science of mental +phenomena. As such, it is not limited to man, even though for many +reasons he claims by far the preponderating part of it for himself. +Sociology is the science which deals with the peculiarities of the human +race. It may therefore be called anthropology, but in a far wider sense +than the word is now applied. + + +=18. The Applied Sciences.= It will be remarked that the grouping of the +table gives no place at all in its scheme to certain branches of +learning taught in the universities and equally good technical +institutions. We look in vain not only for theology and jurisprudence, +but also for astronomy, medicine, etc. + +The explanation and justification of this is, that for purposes of +systematization we must distinguish between _pure_ and _applied_ +sciences. By virtue of their strictly conceptual exclusiveness the pure +sciences constitute a regular hierarchy or graded series, so that all +the concepts that have been used and dealt with in the preceding +sciences are repeated in the following sciences, while certain +characteristic new concepts enter in addition. Thus logic, the science +of the manifold, exercises its dominion over all the other sciences, +while the specific concepts of physics and chemistry have nothing to do +with it, though they are of importance to all the biologic sciences. +Through this graded addition of new (naturally empiric) concepts, the +construction of the pure sciences proceeds in strict regularity, and +their problems arise exclusively from the application of new concepts to +all the earlier ones. In other words, their problems do not reach them +accidentally from without, but result from the action and reaction of +their concepts upon one another. + +At the same time there are problems that each day sets before us without +regard to system. These come from our endeavor to improve life and avert +evil. In the problems of life we are confronted by the whole variety of +possible concepts, and under the day's immediate compulsion we cannot +wait, if we are sowing crops or helping a sick man, until physiology and +all the other appropriate sciences have solved all the problems of plant +growth and the changes of the human body and human energy. When other +signs fail, we use the position of the stars for finding our way on the +high seas. In this manner we turn the teaching of the stars, or +astronomy, into an applied science, in which at first mechanics alone +seemed to have a part. Later physics took a share in it, then optics +took a particularly prominent share, and in recent times not only did +chemistry find its way into astronomy, but the specifically biologic +concept of evolution was applied in astronomy with success. + +Thus, side by side with the pure sciences are the applied, which are to +be distinguished from the pure sciences by the fact that they do not +unfold their problems systematically, but are assigned them by the +external circumstances of man's life. The pure sciences, therefore, +almost always have a larger or smaller share in the tasks of the applied +sciences. For instance, in building a bridge or railroad, physical +problems have to be taken into consideration as well as sociologic +problems (problems of trade), and a good physician should be a +psychologist as well as a chemist. + +But since all the individual questions arising in the applied sciences +may be considered essentially as problems of one or other pure science, +they need not be explicitly enumerated along with the pure sciences, +especially since their development is greatly dependent upon temporary +conditions and is therefore incapable of simple systematization. + + + + +PART II + +LOGIC, THE SCIENCE OF THE MANIFOLD, AND MATHEMATICS + + +=19. The Most General Concept.= If we try to conceive the whole +structure of science according to the principle of the increasing +complexity of concepts, the first question which confronts us is, What +concept is the _most general_ of all possible concepts, so general that +it enters into every concept formation and acts as a decisive factor? In +order to find this concept let us go back to the psycho-physical basis +of concept formation, namely, _memory_, and let us investigate what is +the general characteristic determining memory. We soon perceive that if +a being were to lead an absolutely uniform existence, _no_ memories +could be evoked. There would be nothing by which the past could be +distinguished from the present, hence nothing by which to compare them. +So the "primal phenomenon" of conscious thought is the realization of a +_difference_, a difference between memory and the present, or, to put +the same idea still more generally, between two memories. + +Our experiences, therefore, are divided into two parts, distinguished +from each other. In order to predicate something of a perfectly general +nature concerning those parts, without regard to their particular +content, we must, in accordance with the means employed in human +intercourse, designate them by a _name_. Now in all human languages +there is a great deal of arbitrariness and indefiniteness in the +relations between the concepts and the names applied to them, which +render all accurate work in the study of concepts extremely difficult. +It is necessary, therefore, to state definitely in each particular +instance with what conceptual content a given name is to be connected. +Every experience in so far as it is differentiated from other +experiences we shall call simply an _experience_ without making a +distinction between a so-called inner or outer experience. + +Many of the experiences remain isolated, because they are not repeated +in a similar form, and so do not remain in our memory. They depart from +our psychic life once for all and leave no further consequences or +associations. But some experiences recur with greater or less +uniformity, and become permanent parts of psychic life. Their duration +is by no means unlimited. For even memories fade and disappear. However, +they extend through a considerable part of life, and that suffices to +give them their character. + +The aggregate of similar experiences, hence of experiences conceptually +generalized, we shall call _things_. _A thing, therefore, is an +experience which has been repeated_, and is "recognized" by us. That +is, it is felt as repeated and conceptually comprehended. In other +words, all experiences of which we have formed concepts are things, and +_the concept of thing itself is the most general concept_, since, +according to its definition, it includes all possible concepts. Its +"essence," or determining characteristic, lies in the possibility of +differentiating any one thing from another. Things we do not +differentiate we call _the same_, or _identical_. Here we shall leave +undecided the question whether this lack of differentiation occurs +because we _cannot_, or because we _would not_, differentiate. All +experiences generalized into one concept are therefore felt or regarded +as the same in reference to this concept. Now, since concepts arise +unconsciously as well as consciously, the first is a case of identities +which had been directly felt as such. On the other hand, in the second +case, the process is that of consciously disregarding or abstracting the +existing differences in order to form a concept into which these do not +enter. This last process is applied in the highest degree possible in +obtaining the concept _thing_. + + +=20. Association.= The experience of the _connection_ or _relation_ +between various things is also derived from the nature of our +experiences in the most general sense. When we recall a thing A, another +thing B comes to our mind, the memory of which is called forth by A, and +_vice versa_. The cause of this invariably lies in some experiences in +which A and B occur together. In fact, A and B must have occurred +together a number of times. Otherwise they would have disappeared from +memory. In other words, it is the fact of the _complex concept_ which +appears in such connections between various things. Two things, A and B, +which are connected with each other in such a way, are said to be +associated. Association in the most general sense means nothing more +than that when we think of B we also have A in our consciousness, and +_vice versa_. However, we can at will make the association more +definite, so that quite definite thoughts or actions will be connected +with the association of B. These thoughts and actions are then the same +for all the individual cases occurring under the concept A and B. + +If we associate with the thing B another thing C, we obtain a relation +of the same nature as that obtained by the association of A and B. But +at the same time a new relation arises which was not directly sought, +namely, the association of A to C. If A recalls B, and B recalls C, A +must inevitably recall C also. This psychologic law of nature is +productive of numberless special results. For we can apply it directly +to still another case, the association of a fourth thing D to the thing +C, whereby new relations are necessarily established also between A and +D as well as between B and D. By positing the _one_ relation C : D there +arise two new relations not immediately given, namely, A : D and B : D. +The reason the other relations arise is because C was not taken free +from all relations, but had already attached to it the relations to A +and B. These relations of C, therefore, brought A and B into the new +relation with D. + +By this simplest and most general example we recognize the type of the +deductive process (p. 41), namely, the discovery of relations which, it +is true, have already been established by the accepted premises, but +which do not directly appear in undertaking the corresponding +operations. In the present case, to be sure, the deduction is so +apparent that the recognition of the relations in question offers not +the slightest difficulty. But we can easily imagine more complicated +cases in which it is much more difficult to find the actually existing +relations, and so in certain circumstances we may search for them a long +time in vain. + + +=21. The Group.= The aggregate of all individual things occurring in a +definite concept, or the common characteristics of which make up this +concept, is called a group. Such a group may consist of a limited or +finite number of members, or may be unlimited, according to the nature +of the concepts that characterize it. Thus, all the integers form an +unlimited or infinite group, while the integers between ten and one +hundred (or the two-digit numbers) form a limited or finite group. + +From the definition of the group concept follows the so-called classic +_process of argumentation_ of the syllogism. Its form is: _Group A is +distinguished by the characteristic of B_. _The thing C belongs to group +A. Therefore C has the characteristic of B._ The prominent part ascribed +by _Aristotle_ and his successors to this process is based upon the +_certainty_ which its results possess. Nevertheless, it has been pointed +out, especially by _Kant_, that judgments or conclusions of such a +nature (which he called analytic) have no significance at all for the +progress of science, since they express only what is already known. For +in order to enable us to say that the thing C belongs to group A, we +must already have recognized or proved the presence of the group +characteristic B in C, and in that case the conclusion only repeats what +is already contained in the second or minor premise. + +This is evident in the classic example: All men are mortal. Caius is a +man. Therefore Caius is mortal. For if Caius's mortality were not known +(here we are not concerned how this knowledge was obtained), we should +have no right to call him a man. + +At the same time the character of the really scientific conclusion based +upon the incomplete induction becomes clear. It proceeds according to +the following form. The attributes of the group A are the +characteristics of a, b, c, d. We find in the thing C the +characteristics a, b, c. Therefore we presume that the characteristic d +will also be found in C. The ground for this presumption is that we +have learned by experience that the characteristics mentioned have +always been found together. It is for this reason, and for this reason +only, that we may assume from the presence of a, b, c the presence of d. +In the case of an arbitrary combination, in which it is possible to +combine other characteristics, the conclusion is unfounded. But if, on +the other hand, the formation of the concept A with the characteristics +of a, b, c, d has been caused by repeated and habitual experience, then +the conclusion is well founded; that is, it is probable. + +As a matter of fact, however, that classic example which is supposed to +prove the absolute certainty of the regular syllogism turns out to be a +hidden inductive conclusion of the incomplete kind. The premise, Caius +is a man, is based on the attributes a, b, c (for example, erect +bearing, figure, language), while the attribute d (mortality) cannot be +brought under observation so long as Caius remains alive. In the sense +of the classic logic, therefore, we are not justified in the minor +premise, Caius is a man, while Caius is alive. The utter futility of the +syllogism is apparent, since, according to it, it is only of dead men +that we can assert that they are mortal. + +From these observations it becomes further apparent that logic, whether +it is the superfluous classic logic or modern effective inductive logic, +is nothing but a part of the group theory, or science of manifoldness, +which appears as the first, because it is the most general member of +the mathematical sciences (this word taken in its widest significance). +But according to the hierarchic system in harmony with which the scheme +of all the sciences had been consciously projected, we cannot expect +anything else than that those sciences which are needful for the pursuit +of all other sciences (and logic has always been regarded as such an +indispensable science, or, at least, art) should be found collected and +classified in the first science. + + +=22. Negation.= When the characteristics a, b, c, d of a group have been +determined, then the aggregate of all things existing can be divided +into two parts, namely, the things which belong to the group A and those +which do not belong to it. This second aggregate may then be regarded as +a group by itself. If we call this group "not-A," it follows from the +definition of this group that the two groups, A and not-A, together form +the aggregate of all things. + +This is the meaning and the significance of the linguistic form of +_negation_. It excludes the thing negated from any group given in a +proposition, and this relegates it to the second or complementary group. + +The characteristic of such a group is the common absence of the +characteristics of the positive group. We must note here that the +absence of even _one_ of the characteristics a, b, c, d excludes the +incorporation of the thing into the group A, while the mere absence of +this characteristic suffices to include it in the group not-A. We can +therefore by no means predicate of group not-A that each one of its +members must lack _all_ the characteristics a, b, c, d. We can only say +that each of its members lacks at least one of the characteristics, but +that one or some may be present, and several or all may be absent. From +this follows a certain asymmetry of the two groups, which we must bear +in mind. + +The consideration of this subject is especially important in the +treatment of negation in the conclusions of formal logic. As we shall +make no special use of formal logic, we need not enter into it in +detail. + + +=23. Artificial and Natural Groups.= The combination of the +characteristics which are to serve for the definition of a group is at +first purely arbitrary. Thus, when we have chosen such an arbitrary +combination, a, b, c, d, we can eliminate one of the characteristics, +as, for example, c, and form a group with the characteristics a, b, d. +Such a group, which is _poorer in characteristics_, will, in general, be +_richer in members_, for to it belong, in the first place, all the +things with the characteristics a, b, c, d, of which the first group +consisted, and in addition all the things which, though not possessing +c, possess a, b, and d. + +If we call such groups related as contain common characteristics, though +containing them in different members and combinations, so that the +definition of the one group can be derived from the other by the +elimination or incorporation of individual characteristics, then we can +postulate the general thesis _that in related groups those must be +richer in members which are poorer in characteristics, and inversely_. +This is the precise statement of the proposition of the less definite +thesis stated above. + +For the purposes of systematization we have assumed that we can +arbitrarily eliminate one or another characteristic of a group. In +experience, however, this often proves inadmissible. As a rule we find +that the things which lack one of the characteristics of a group will +also lack a number of other characteristics; in other words, that the +characteristics are not all independent of one another, but that a +certain number of them go together, so that they are present in a thing +either in common or not at all. + +This case, however, can be referred to the general one first described, +by treating the characteristics belonging together as being _one_ +characteristic, so that the group is defined solely by the independent +characteristics. Then, according to the definition, we can, without +losing our connection with experience, carry out that formal +manifoldness of all possible related groups which yields what is called +a _classification_ of the corresponding things. + +If for the determination of a group a definite number of independent +characteristics is taken, say, a, b, c, d, and e, then we have at first +the narrowest or poorest group abcde. By the elimination of one +characteristic we obtain the five groups, bcde, acde, abde, abce, and +abcd. If we omit one other characteristic we get ten different groups +abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde. Likewise, there are +ten groups with two characteristics each, and finally five groups with +one characteristic each. All these groups are related. There is a +science, the Theory of Combinations, which gives the rules by which, in +given elements or characteristics, the kind and number of the possible +groups can be found. The theory of combinations enables us to obtain a +complete table and survey of all possible complex concepts which can be +formed from given simple ones (whether they be really elementary +concepts, or only relatively so). When in any field of science the +fundamental concepts have been combined in this manner, a complete +survey can be had of all the possible parts of this science by means of +the theory of combinations. + +In order to present this process vividly to our minds, let us take as an +example the science of the chemical combination of substances which form +an important part of chemistry. There are about eighty elements in +chemistry, and this science has to treat of + + a) each of the eighty elements by itself + b) all substances containing two elements and no more + c) all substances containing three elements + d, e, f, etc.) the substances containing four, five, and six, etc., + elements, + +until finally we reach a group (not existing in experience) embracing +substances formed of _all_ the elements. That there is no such substance +in the present scope of human knowledge has, of course, no significance +for the structure of the scheme. What is significant is the fact that +the scheme really embraces and arranges all possible substances in such +a way that we cannot conceive of any case in which a newly discovered +substance cannot after examination immediately be classed with one of +the existing groups. + +To cite an example from another science. Physics, it will be recalled, +may be considered to be the science of the different kinds of energy. +This science, accordingly, is divided first into the study of the +properties of each energy, and then into the study of the relations of +two energies, of three energies, of four energies, etc. Here, too, we +may say that in the end there can be no physical phenomenon which cannot +be placed in one of the groups so obtained. + +Of course, neither in chemistry nor in physics does this mean that each +_new_ case will fall within the scheme obtained by the exhaustive +combination of elementary concepts (whether chemical elements or kinds +of energy) _known_ at the time. It is quite possible that a new thing +under investigation contains a _new_ elementary concept, so that on +account of it the scheme must be enlarged through the embodiment of +this new element. But simultaneously a corresponding number of new +groups appear in the scheme, and the investigator's attention is +directed to the fact that he still has a reasonable prospect, in +favorable circumstances, of discovering these new things also. Thus +combinatory schematization serves not only to bring the existing content +of science into such order that each single thing has its assigned +place, but the groups which have thereby been found to be vacant, to +which as yet nothing of experience corresponds, also point to the places +in which science can be completed by new discoveries. + +From the above presentation it is apparent how from the two concepts +"thing" and "association" alone a great manifoldness of various and +regular forms can be developed. They are purely empirical relations, for +the fact that several things can be combined in the graded series +described above according to a fixed rule does not follow merely from +the two concepts, but must be _experienced_. But, on the other hand, +both concepts are so general that the experiences obtained in some cases +can be applied to all possible experiences and may serve the purpose of +classifying and making a general survey of them. + +The above statements, however, have by no means exhausted the +possibilities. For it has been tacitly assumed that in the combination +of several things the _sequence_ according to which this combination +takes place should not condition a difference of the result. This is +true of a number of things, but not of all. In order, therefore, to +exhaust the possibilities the theory of combinations must be extended +also to cases in which the sequence is to be taken account of, so that +the form ab is regarded as different from ba. + +We will not undertake to work out the results of this assumption. It is +obvious that the manifoldness of the various cases is much greater than +if we neglect the sequence. On this point we have one more observation +to make, that further causes for diversity exist. It is true that a +chemical combination is not influenced by the sequence in which its +elements enter the combination, but there do occur with the same +elements differences in their _quantitative relations_, and thereby a +new complexity is introduced into the system, so that two or more +similar elements can form different combinations according to the +difference in the quantitative relations. Still, even with this, the +actual manifoldness is not exhausted, for from the same elements and +with the same quantitative relations there can arise different +substances called _isomeric_, which, for all their similarity, possess +different energy contents. But the first scheme is not demolished, nor +does it become impracticable because of this increase of manifoldness. +What simply happens is that _several_ different things instead of one +appear in the same group of the original scheme, the systematic +classification of which necessitates a further schematization by the use +of other characteristics. + + +=24. Arrangement of the Members.= Since we have started from the +proposition that all members of a group are different from one another, +we have perfect liberty to arrange them. The most obvious arrangement +according to which some _one_ definite member is followed by a _single_ +other member and so forth (as, for example, the arrangement of the +letters of the alphabet) is by no means the only mode of arrangement, +though it is the simplest. Besides this _linear_ arrangement, there is +also, for instance, the one in which two new members follow +simultaneously upon each previous one, or the members may be disposed +like a number of balls heaped up in a pyramid. However, we shall not +have much occasion to occupy ourselves with these complex types of +arrangement, and can therefore limit our considerations at first to the +simplest, that is, to the linear arrangement. + +This simplest of all possible forms expresses itself in the fact _that +the immediately experienced things of our consciousness are arranged in +this way_. In point of fact, the contents of our consciousness proceed +in linear order, one single new member always attaching itself to an +existing member. This law, however, is not strictly and invariably +adhered to. It sometimes happens that our consciousness continues for a +while to pursue the direction of thought it has once taken, although a +branching off had already taken place at a former point, at which a new +chain of thought had begun. Nevertheless, one of these chains usually +breaks off very soon, and the linear character of the inner experience +is immediately restored. Of certain specially powerful intellects it is +recorded that they could keep up several lines of thought for a +considerable length of time--Julius Cæsar, for instance. + +The biologic peculiarity here mentioned of the linear juxtaposition of +the contents of our consciousness has led to the concept of _time_, +which has been appropriately called a _form of inner life_. That all our +experiences succeed each other in time is equivalent to saying that our +thought processes represent a group in linear arrangement. As appears +from the above observations, this is by no means an absolute form, +unalterable for all times. On the contrary, a few highly developed +individuals have already begun to emancipate themselves from it. But the +existing form is so firmly fixed through heredity and habit that it +still seems impracticable for most men to imagine the succession of the +inner experiences in a different way than by a line or by _one +dimension_. Since, on the other hand, we have all learned to feel space +as _tri-dimensional_, although optically it appears to possess only two +dimensions (we see length and breadth, and only infer thickness from +secondary characteristics), we come to recognize that the linear form by +which we represent the succession of our experiences is a matter of +adaptation, and that because the change has been extremely slight in the +course of centuries it produces the impression of being unalterable.[D] + +[D] Mathematicians who busy themselves a great deal with the formal +theory of four-dimensional space, seem to acquire a capacity for +imagining this form as easily as the three-dimensional form with which +we are all familiar. Therefore, despite the oft-repeated statements to +the contrary, it is not impossible to imagine four-dimensional space. +Only, we must not attempt to represent to ourselves four-dimensional +space in three-dimensional space, especially not without a knowledge of +its properties. + +These discussions lead to a further difference that can exist in groups +of linear arrangement. While in the first example we chose, the +alphabet, the sequence was quite _arbitrary_, since any other sequence +is just as possible, the same cannot be said of experiences into which +the element of time enters. These are not arbitrary, but are arranged by +special circumstances depending upon the aggregate of things which +co-operate in the given experiences. + +While, therefore, a group with free members, that is, members not +determined in their arrangement by special circumstances, can be brought +into linear order in very different ways, there are groups in which only +one of those orders actually occurs. We see at once that in free groups +the number of different orders possible is the greater, the greater the +group itself. The theory of combinations teaches how to calculate these +numbers which play a very important rôle in the various provinces of +mathematics. The naturally ordered groups always represent a single +instance out of these possibilities, the source of which always lies +outside the group concept, that is, it proceeds from the things +themselves which are united into a group. + + +=25. Numbers.= An especially important group in the linear order is that +of the _integral numbers_. Its origin is as follows: + +First we abstract the difference of the things found in the group, that +is, we determine, although they are different, to disregard their +differences. Then we begin with some member of the group and form it +into a group by itself. It does not matter which member is chosen, since +all are regarded as equivalent. Then another member is added, and the +group thus obtained is again characterized as a special type. Then one +more member is added, and the corresponding type formed, and so on. +Experience teaches that never has a hindrance arisen to the formation of +new types of this kind by the addition of a single member at a time, so +that the operation of this peculiar group formation may be regarded as +_unlimited_ or _infinite_. + +The groups or types thus obtained are called the _integral numbers_. +From the description of the process it follows that every number has two +neighbors, the one the number from which it arose by the addition of a +member, and the other the number which arose from it by the addition of +a member. In the case of the number one with which the series begins, +this characteristic is present in a peculiar form, the preceding group +being _group zero_, that is, a group without content. This number in +consequence reveals certain peculiarities into which we cannot enter +here. + +Now, according to a previous observation (p. 64), not only does the +order bring every number into relation with the preceding one, but since +this last for its part already possesses a great number of relations to +all preceding, these relations exert their influence also upon the new +relation. This fact gives rise to extraordinarily manifold relations +between the various numbers and to manifold laws governing these +relations. The elucidation of them forms the subject of an extensive +science. + + +=26. Arithmetic, Algebra, and the Theory of Numbers.= From this regular +form of the number series numerous special characteristics can be +established. The investigations leading to the discovery of these +characteristics are purely scientific, that is, they have no special +technical aim. But they have the uncommonly great practical significance +that they provide for all possible arrangements and divisions of +numbered things, and so have instruments at hand ready for application +to each special case as it arises. I have already pointed out that in +this lies the positive importance of the theoretical sciences. For +_practical_ reasons the study of them must be as _general_ as possible. +This science is called _arithmetic_. + +Arithmetic undergoes an important generalization if the individual +numbers in a calculation are disregarded and _abstract signs_ standing +for any number at all are used in their place. At first glance this +seems superfluous, since in every real numerical calculation the numbers +must be reintroduced. The advantage lies in this, that in calculations +of the same form, the required steps are formally disposed of once for +all, so that the numerical values need be introduced only at the +conclusion and need not be calculated at each step. Moreover, the +general laws of numerical combination appear much more clearly if the +signs are kept, since the result is immediately seen to be composed of +the participating members. Thus, _algebra_, that is, calculation with +abstract or general quantities, has developed as an extensive and +important field of general mathematics. + +By the theory of numbers we understand the most general part of +arithmetic which treats of the properties of the "numerical bodies" +formed in some regular way. + + +=27. Co-ordination.= So far our discussion has confined itself to the +_individual_ groups and to the properties which each one of them +exhibits _by itself_. We shall now investigate the relations which exist +_between two or more groups_, both with regard to their several members +and to their aggregate. + +If at first we have two groups the members of which are all +differentiated from one another, then any one member of the one group +can be co-ordinated with any one member of the other group. This means +that we determine that the same should be done with every member of the +second group as is done with the corresponding member of the first +group. That such a rule may be carried out we must be able to do with +the members of all the groups whatever we do with the members of one +group. In other words, no properties peculiar to individual members may +be utilized, but only the properties that each member possesses as a +member of a group. As we have seen, these are the properties of +_association_. + +First, the co-ordination is _mutual_, that is, it is immaterial to which +of the two groups the processes are applied. The relation of the two +groups is reciprocal or symmetrical. + +Further, the process of co-ordination can be extended to a third and a +fourth group and so on, with the result that what has been done in one +of the co-ordinated groups must happen in all. If hereby the third group +is co-ordinated with the second, the effects are quite the same as if it +were co-ordinated directly with the first instead of indirectly through +the second. And the same is true for the fourth and the fifth groups, +etc. Thus, co-ordination can be extended to any number of groups we +please, and each single group proves to be co-ordinated with every +other. + +Finally, a group can be co-ordinated with itself, each of its members +corresponding to a certain definite other member. It is not impossible +that individual members should correspond to themselves, in which case +the group has _double members_, or _double points_. The limit-case is +_identity_, in which every member corresponds to _itself_. This last +case cannot supply any special knowledge in itself, but may be applied +profitably to throw light on those observations for which it represents +the extreme possibility. + + +=28. Comparison.= If we have two groups A and B, and if we co-ordinate +their members severally, three cases may arise. Either group A is +exhausted while there are members remaining in B, or B is exhausted +before A, or, finally, both groups allow of a mutual co-ordination of +_all_ their members. In the first case A is called, in the broader sense +of the word, _smaller_ than B, in the second B is called smaller than A, +in the third the two groups are said to be of _equal magnitude_. The +expression, "B is greater than A," is equivalent to the expression, "A +is smaller than B," and inversely. + +It is to be noted that the relations mentioned above are true, whether +the members are considered as individually different from one another or +whether the difference of the members is disregarded, and they are +treated as alike. This comes from the fact that every definite +co-ordination of a group can be translated into every other possible +co-ordination by exchanging two members at a time in pairs. Since in +this process one member is each time substituted for another, and a gap +therefore can never occur in its place, the group in the new arrangement +can be co-ordinated with the other group as successfully as in the old +arrangement. At the same time we learn from this that in every +co-ordination of a group with itself, independently of the arrangement +of its members, it must prove equal to itself. + +By carrying out the co-ordination proof is further supplied of the +following propositions: + + { greater than } + If group A is { equal to } group B + { smaller than } + + { greater than } + and group B is { equal to } group C + { smaller than } + + { greater than } + then group A is { equal to } group C + { smaller than } + +From this it follows that any collection of finite groups whatsoever, of +which no one is equal to the other, can always be so arranged that the +series should begin with the smallest and end with the greatest, and +that a larger should always follow a smaller. _This order would be +unequivocal_, that is, there is only one series of the given groups +which has this peculiarity. As we shall soon see, the series of integers +is the purest type of a series so arranged. + +In comparing two infinitely large groups by co-ordination, it may be +said on the one hand that never will one group be exhausted while the +other still contains members. Accordingly, it is possible to designate +two unlimited or infinite groups (or as many such groups as we please) +as _equal_ to each other. On the other hand, the statement that in both +groups each member of the one is co-ordinated with a member of the other +has no definite meaning on account of the infinitely large number of +members. _The definition of equality is therefore not completely +fulfilled_, and we must not loosely apply a principle valid for finite +groups to infinite groups. This consideration, which may assume very +different forms according to circumstances, explains the "paradoxes of +the infinite," that is, the contradictions which arise when concepts of +a definite content are applied to cases possessing in part a different +content. If we wish to attempt such an application, we must in each +instance make a special investigation as to the manner in which the +relations on their part change by the change of those contents (or +premises). As a general rule we must expect that the former relations +will not remain valid in these circumstances without any change at all. + +In the course of these observations we have learned how co-ordination +can be used for obtaining a number of fundamental and multifariously +applied principles. From this alone the great importance of +co-ordination is evident, and later we shall see that its significance +is even more far-reaching. _The entire methodology of all the sciences +is based upon the most manifold and many-sided application of the +process of co-ordination_, and we shall have occasion to make use of it +repeatedly. Its significance may be briefly characterized by stating +that it is the most general means of bringing connection into the +aggregate of our experiences. + + +=29. Counting.= The group of integral numbers, because of its +fundamental simplicity and regularity, is by far the best basis of +co-ordination. For while arithmetic and the theory of numbers give us a +most thorough acquaintance with the peculiarities of this group, we +secure by the process of co-ordination the right to presuppose these +peculiarities and the possibility of finding them again in every other +group which we have co-ordinated with the numerical group. The carrying +out of such co-ordination is called _counting_, and from the premises +made it follows _that we can count all things in so far as we disregard +their differences_. + +We count when we co-ordinate in turn one member of a group after another +with the members of the number series that succeed one another until +the group to be counted is exhausted. The last number required for the +co-ordination is called the _sum_ of the members of the counted group. +Since the number series continues indefinitely, every given group can be +counted. + +Numerals have been co-ordinated with _names_ as well as with _signs_. +The former are different in the different languages, the latter are +international, that is, they have the same form in all languages. From +this proceeds the remarkable fact that the written numbers are +understood by all educated men, while the spoken numbers are +intelligible only within the various languages. + +The purpose of counting is extremely manifold. Its most frequent and +most important application lies in the fact that the amount affords a +_measure for the effectiveness or the value_ of the corresponding group, +both increasing and decreasing simultaneously. A further number serves +as a basis for divisions and arrangements of all kinds to be carried out +within the group, whereby liberal use is made of the principle that +everything that can be effected in the given number group can also be +effected in the co-ordinated counted group. + + +=30. Signs and Names.= The co-ordination of names and signs with numbers +calls for a few general remarks on co-ordination of this nature. + +The possibility of carrying out the formal operations effected in one of +the groups upon the co-ordinated group itself facilitates to an +extraordinary extent the practical shaping of the reality for definite +purposes. If by counting we have ascertained that a group of people +numbers sixty, we can infer without actually executing the steps that it +is possible to form these men in six rows of ten, or in five rows of +twelve, or in four rows of fifteen, but that we cannot obtain complete +rows if we try to arrange them in sevens or elevens. These and +numberless other peculiarities we can learn of the group of men from its +amount, that is, from its co-ordination with the numerical group of +sixty. In co-ordination, therefore, we have a means of acquainting +ourselves with facts without having to deal directly with the +corresponding realities. + +It is clear that men will very soon notice and avail themselves of so +enormous an advantage for the mastery and shaping of life. Thus, we see +the process of co-ordination in general use among the most primitive +men. Even the higher animals know how to utilize co-ordination +consciously. When the dog learns to answer to his name, when the horse +responds to the "Whoa" and the "Gee" of his driver there is in each case +a co-ordination of a definite action or series of actions, that is, of a +concept with a sign, or, in other words, of a concept with a member of +another group; and in this there need not be the least similarity +between the things co-ordinated with each other. The only requirement is +that on the one hand the co-ordinated sign should be easily and +definitely expressed and be to the point, and that, on the other hand, +it should be easily "understood," that is, _comprehended_ by the senses +and unmistakably _differentiated_ from other signs co-ordinated with +other things. + +Thus, we find that the most frequent concepts of co-ordinated sound +signs form the beginnings of _language_ in the narrower sense. It is +very difficult to ascertain for what reasons the particular forms of +sound signs have been chosen, nor is it a matter of great importance. In +the course of time the original causes have disappeared from our +consciousness and the present connection is purely external. This is +evident from the enormous difference of languages in which hundreds of +different signs are employed for the same concept. + +Now it would be quite possible to solve the problem of co-ordinating +with each group of concepts a corresponding group of sounds, so that +each concept should have its own sound, or, in other words, that the +_co-ordination should be unambiguous_. It would not by any means be +beyond human power to accomplish this, if it were not for the fact that +the concepts themselves are still in so chaotic a state as they are at +present. We have seen that the attempts of Leibnitz and Locke to draw up +a system of concepts, if only in broad outline, have undergone no +further development since. Even the most regulated concepts as well as +the familiar concepts of daily life are in ceaseless flux, while the +co-ordinated signs are comparatively more stable. But they, too, +undergo a slow change, as the history of languages shows, and in +accordance with quite different laws from those which govern the change +of concepts. The consequence is that in language the co-ordination of +concepts and words is far from being unambiguous. The science of +language designates the presence of several names for the same concept +and of several concepts for the same name by the words synonym and +homonym. These forms, which have arisen accidentally, signify so many +_fundamental defects_ of language, since they destroy the _principle of +unambiguity_ upon which language is based. In consequence of the false +conception of its nature we have until now positively shrunk from +consciously developing language in such a way that it should more and +more approach the ideal of unambiguity. Such an ideal is in fact +scarcely known, much less recognized. + + +=31. The Written Language.= Sound signs, to be sure, possess the +advantage of being produced easily and without any apparatus, and of +being communicable over a not inconsiderable distance. But they suffer +under the disadvantage of transitoriness. They suffice for the purpose +of temporary understanding and are constantly being used for that. If, +on the other hand, it is necessary to make communications over greater +distances or longer periods of time, sound signs must be replaced by +more permanent forms. + +For this we turn to another sense, the sense of sight. Since optic +signs can travel much greater distances than sound signs without +becoming indistinguishable, we first have the optical telegraphs, which +find application, though rather limited application, in very varying +forms, the most efficient being the heliotrope. The other sort of optic +signs is much more generally used. These are objectively put on +appropriate solid bodies, and last and are understood as long as the +object in question lasts. Such signs form the _written language_ in the +widest sense, and here, too, it is a question of co-ordinating signs and +concepts. + +What I have said concerning the very imperfect state of our present +concept system is true also of these two groups. On the other hand, the +written signs are not subject to such great change as the sound signs, +because the sound signs must be produced anew each time, whereas the +written signs inscribed on the right material may survive hundreds, even +thousands of years. Hence it is that the written languages are, upon the +whole, much better developed than the spoken languages. In fact, there +are isolated instances in which it may be said that the ideal has +well-nigh been reached. + +As we have already pointed out, such a case is furnished by the _written +signs_ of numbers. By a systematic manipulation of the ten signs 0 1 2 3 +4 5 6 7 8 9 it is not only possible to co-ordinate a written sign with +any number whatsoever, but this co-ordination is strictly unambiguous, +that is, each number can be written in only one way, and each numerical +sign has only one numerical significance. This has been obtained in the +following manner: + +First, a special sign is co-ordinated to each of the group of numbers +from zero to nine. The same signs are co-ordinated with the next group, +ten to nineteen, containing as many numbers as the first. To distinguish +the second from the first group, the sign one is used as a prefix. The +third group is marked by the prefixed sign two, and so on, until we +reach group nine. The following group, in accordance with the principle +adopted, has as its prefix the sign ten, which contains two digits. All +the succeeding numbers are indicated accordingly. From this the +following result is assured: First, no number in its sequence escapes +designation; second, never is an aggregate sign used for two or more +different numbers. Both these circumstances suffice to secure +unambiguity of co-ordination. + +It is known that the system of rotation just described is by no means +the only possible one. But of all systems hitherto tried it is the +simplest and most logical, so that it has never had a serious rival, and +the clumsy notation with which the Greeks and Romans had to plague +themselves in their day was immediately crowded out, never to return +again upon the introduction of the Indo-Arabic notation, which has made +its way in the same form among all the civilized nations and constitutes +a uniform part of all their written languages. + +The comparison of the spoken and the written languages offers a very +illuminating proof of the much greater imperfection of the language of +_words_. The number 18654 is expressed in the English language by +eighteen thousand six hundred and fifty-four, that is, the second figure +is named first, then the first, the third, the fourth, and the fifth. In +addition, four different designations are used to indicate the place of +the figures, -teen, -thousand, -hundred, and -ty. A more aimless +confusion can scarcely be conceived. It would be much clearer to name +the figures simply in their sequence, as one-eight-six-five-four. +Besides, this would be unambiguous. If we should desire to indicate the +_place value_ in advance, we could do so in some conventional way, for +example, by stating the number of digits in advance. This, however, +would be superfluous, and ordinarily should be omitted.[E] + +[E] The usual designation of the larger groups, ten, hundred, thousand, +million, billion, etc., is also quite irrational. If it is our object to +secure expressions for place values in as few words as possible, we find +that the numbers of the form 10^{2n}, in which n is a whole number, must +receive their own names, that is, 10, 100, 10,000, 100,000,000 etc. In +this way the problem of designating as many numbers as possible by as +few words as possible is solved. + + +=32. Pasigraphy and Sound Writing.= There are two possibilities for +co-ordination between concepts and written signs. Either the +co-ordination is _direct_, so that it is only a matter of providing +every concept with a corresponding sign, or it is indirect, the signs +serving only the purpose of expressing the _language sound_. In the +latter case the written language is based entirely upon the sound +language, and the only problem, comparatively easy to solve, is to +construct _an unambiguous co-ordination between sound and sign_. The +Chinese script follows the direct process, but all the scripts of the +European-American civilized peoples are based on the indirect process. + +This, it is true, is the case only in ordinary, non-scientific language, +while for science the European nations also have to a large extent built +up a direct concept writing. One example of this we have seen in the +number signs. Musical notation furnishes another instance, though by far +not so perfect. The use of the different keys destroys the unambiguous +connection between the pitch and the note sign, and the signatures +placed at the beginning of a whole staff have the defect of removing the +sign from the place where it is applied. Despite this imperfection +musical notation is quite international, and every one who understands +European music also understands its signs.[F] + +[F] It is not difficult to perfect musical notation with a view to +unambiguity, a thing which would greatly facilitate the study of music. + +Fundamentally we need not hesitate to recognize in _concept writing_ or +_pasigraphy_ a more complete solution of the problem of sign +arrangement. Even the very incomplete Chinese pasigraphy renders +possible written intercourse, especially for mercantile purposes, +between the various East-Asiatic peoples who speak some dozens of +different languages. But each language community translates the common +signs into its own words, just as we do in the case of the number signs. +But in order that such a system of representation should be complete it +must fulfil a whole series of conditions for which scarcely a remote +possibility is to be discerned at present. + +At first the concepts could simply be taken as found in the words and +grammatical forms of the various languages, and each one provided with +an arbitrary sign. Such approximately is the Chinese system. But a +system of that sort entails an extreme burdening of the memory, which +results both from the great number of words and from the necessity of +keeping the signs within certain bounds of simplicity. If we consider +that the complex concepts are formed according to laws, to a large +extent still unknown, from a relatively small number of _elementary_ +concepts, we may attempt to build up the signs of the complex concepts +by the combination of those of the elementary concepts according to +corresponding rules. Then it would only be necessary to learn the signs +for the elementary concepts and the rules of combination in order for us +to be able to represent all the possible concepts. This would provide +even for the natural enlargement of the concept world, since every new +elementary concept would receive its sign and would then serve as the +basis from which to deduce all the complex concepts dependent upon it. +In fact, even should a concept hitherto regarded as elementary prove to +be complex, it would not be difficult to declare that its sign, like the +name of an extinct race, is dead, and after the lapse of sufficient time +to use it for other purposes. + +The numerical signs offer an excellent example for the elucidation of +this subject, and at the same time serve as a proof that in limited +provinces the ideal has already been attained. Another very instructive +example is furnished by the chemical formulas, which, though they use +the letters of the European languages, do not associate with them sound +concepts, but chemical concepts. Since the chemical concepts are +co-ordinated with certain letters, it is possible, in the first place, +to denote the composition of all combinations qualitatively by the +combination of the corresponding letters. But since quantitative +composition proceeds according to definite relations which are +determined by a variety of specific numbers peculiar to each element and +called its combining weight, we need only add to the sign of the element +the concept of the combining weight in order to represent in the second +place the quantitative composition. Further, the multiples mentioned can +also be given. Since, moreover, there are various substances which, +despite equal composition, possess different properties, the attempt +has been made to express this new manifoldness by the position of the +element signs on the paper, and in more recent times also by space +representation. And here, too, rules have been worked out in which the +scheme affords a close approach to experience. This example shows how, +by the constant increase of the complexity of a concept (here the +chemical composition), ever greater and more manifold demands are made +upon the co-ordinated scheme. The form of expression first chosen is not +always adequate to keep pace with the progress of science. In this case +it must be radically changed and formed anew to meet the new demands. + + +=33. Sound Writing.= In point of unambiguity of co-ordination _phonetic +writing_ is far more imperfect than concept writing. It is obvious that +in phonetic writing all the faults already present in the co-ordination +between concept and sound are transferred to the written language. To +these are added the defects as regards unambiguity occurring in +co-ordination between sound and sign from which no language is free. In +some languages, in fact, notably in English, these defects amount to a +crying calamity. The principle of unambiguity would require that there +should never be a doubt as to the way in which a spoken word is written, +and as little doubt as to the way in which a written word is spoken. It +needs no proof to show how often the principle is violated in every +language. In the German language the same sound is represented by f, v, +and ph; in the English by f and ph. And in both German and English quite +different sounds are associated with c, g, s, and other letters. _The +fact that orthographic mistakes can be made in the writing of any +language is direct proof of its imperfection_, and the oftener this +possibility occurs the more imperfect is the language in this respect. +We know that the spelling reforms begun in Germany more than ten years +ago and recently in America and England, have for their object +unambiguity in the co-ordination between sign and sound. Still it must +be admitted that this tendency has not always been pursued +undeviatingly. A few innovations, in fact, undoubtedly represent a step +backward. + + +=34. The Science of Language.= A comparison of our investigations--which +we cannot present in detail but only indicate--with the science of +language or philology as taught in the universities and in a great +number of books, reveals a great difference between them. This academic +philology makes a most exhaustive study of relations, which from the +point of view of the purpose of language are of no consequence whatever, +such as most of the rules and usages of grammar. A study of this sort +must naturally confine itself to a mere determination of whether certain +individuals or groups of individuals have or have not conformed to these +rules. Even the chief subject of modern comparative philology, the study +of the relations of the word forms to one another and their changes in +the course of history, both within the language communities and when +transferred to other localities, appear to be quite useless from the +point of view of the theory of co-ordination. For it is indeed of little +moment to us to learn by what process of change, as a rule utterly +superficial, a certain word has come to be co-ordinated with a concept +entirely different from the one with which it had been previously +co-ordinated. Of incomparably greater importance would be investigations +concerning the gradual change of the concepts themselves, although by no +means as important as the real study of concepts. To be sure, such +investigations are much more difficult than the study of word forms set +down in writing. + +Nevertheless, on account of a historical process, which it would lead us +too far afield to discuss, an idea of such word investigations has been +formed which is wholly disproportionate to their importance. And if we +ask ourselves what part such labors have taken in the progress of human +civilization, we are at a loss for an answer. Students of the _science_ +of language make a sharp distinction between it and the _knowledge_ of +language, which is regarded as incomparably lower. But while a knowledge +of language is important in at least one respect, in that it presents to +us the cultural material set down in other languages, or makes them +accessible in translation to those who do not know foreign languages, +philology is of no service in this respect at all, and the pursuit of it +will seem as inconceivably futile to future science as the +scholasticism of the middle ages seems to us now. + +The unwarranted importance attached to the historical study of language +forms is paralleled by the equally unwarranted importance ascribed to +grammatical and orthographic correctness in the use of language. This +perverse pedantry has been carried to such lengths that it is considered +almost dishonorable for any one to violate the usual forms of his mother +tongue, or even of a foreign language, like the French. We forget that +neither Shakespeare nor Luther nor Goethe spoke or wrote a "correct" +English or German, and we forget that it cannot be the object of a true +cultivation of language to _preserve_ as accurately as possible existing +linguistic usage, with its imperfections, amounting at times to +absurdities. Its real object lies rather in the appropriate +_development_ and _improvement_ of the language. We have already +mentioned the fact that in one department, orthography, the true +conception of the nature of language and of its development is gradually +beginning to assert itself. Among most nations efforts are being made to +improve orthography with a view to unambiguity, and when once sufficient +clearness is had as to the object aimed for in spelling, there will be +no special difficulty in finding the required means to attain it. + +But in all the other departments of language we are still almost wholly +without a conception of the genuine needs. Though the example of the +English language proves that we can entirely dispense with the manifold +co-ordinations in the same sentence as appearing in the special plural +forms of the adjective, verb, pronoun, etc., yet the idea of consciously +applying to other languages the natural process of improvement +unconsciously evolved in the English language seems not to have occurred +even to the boldest language reformers. So strongly are we all under the +domination of the "schoolmaster" ideal, that is to say, the ideal of +preserving every linguistic absurdity and impracticability simply +because it is "good usage." + +A twofold advantage will have been attained by the introduction of a +_universal auxiliary language_ (p. 183). Recently the efforts in that +direction have made considerable progress. In the first place it will +provide a general means of communication in all matters of common human +interest, especially the sciences. This will mean a saving of energy +scarcely to be estimated. In the second place, the superstitious awe of +language and our treatment of it will give way to a more appropriate +evaluation of its technical aim. And when by the help of the artificial +auxiliary language, we shall be able to convince ourselves daily how +much simpler and completer such a language can be made than are the +"natural" languages, then the need will irresistibly assert itself to +have these languages also participate in its advantages. The +consequences of such progress to human intellectual work in general +would be extraordinarily great. For it may be asserted that philosophy, +the most general of all the sciences, has hitherto made such extremely +limited progress only _because it was compelled to make use of the +medium of general language_. This is made obvious by the fact that the +science most closely related to it, mathematics, has made the greatest +progress of all, but that this progress began only after it had procured +both in the Indo-Arabic numerals and in the algebraic signs a language +which actually realizes very approximately the ideal of unambiguous +co-ordination between concept and sign. + + +=35. Continuity.= Up to this point our discussions have been based on +the general concept of the _thing_, that is, of the individual +experience differentiated from other experiences. Here the fact of +_being different_, which, as a general experience, led to the +corresponding elementary concept, appeared in the foreground in +accordance with its generality. But in addition to it there is another +general fact of experience, which has led to just as general a concept. +It is the concept of _continuity_. + +When, for example, we watch the diminution of light in our room as it +grows dark in the evening, we can by no means say that we find it darker +at the present moment than a moment before. We require a perceptibly +long time to be able to say with certainty that it is now darker than +before, and throughout the whole time _we have never felt the increase_ +of darkness from moment to moment, although theoretically we are +absolutely convinced that this is the correct conception of the process. + +This peculiar experience, our failure to perceive individual parts of a +change, the reality of which we realize when the difference reaches a +certain degree, is very general, and, like memory, is based upon a +fundamental physiological fact. It has already been noted by _Herbart_, +but its significance was first recognized by _Fechner_, and has since +then become generally known in physiology and psychology under the name +of _threshold_. _Next to memory the threshold determines the fundamental +lines of our psychic life._ + +The threshold therefore means that whatever state we are in _a certain +finite amount of difference or change must be stepped over_ before we +can perceive the difference or change. This peculiarity appears in all +our states or experiences. We have already given an example for the +phenomena of light and darkness. The same is true of differences in +color and of our judgments as to tone pitch and tone strength. Even the +transition from feeling well to feeling ill is usually imperceptible, +and it is only when the change occurs in a very brief time that we +become conscious of it. + +The physical causes of these psychic phenomena need be indicated only in +brief. In all our experiences an existing chemico-physical state in our +sense organs and in the central organ undergoes a change. Now +experiments with physical apparatus have shown that such a process +always requires a finite, though sometimes a very small, quantity of +work, or, generally speaking, energy, before it can be brought about at +all. Even the finest scale, sensitive to a millionth of a gram, remains +stationary when only a tenth of a millionth is placed upon it, although +we can _see_ a body of such minute weight under the microscope. In the +same way it requires a definite expenditure of energy in order to bring +the sense organs, or the central organ, into action, and all stimuli +less than this limit or threshold produce no experience of their +presence. + +By this the difficult concept of continuity is evoked in our experience. +The transition from the light of day to the darkness of evening proceeds +_continuously_, that is, at no point of the whole transition do we +notice that the state just passed is different from the present one, +while the difference over a wider extent of the experience is +unmistakable. If we wish to bring vividly to our minds the contradiction +to other habits of thought which this involves, we need only to +represent to ourselves the following instance. I will compare the thing +A at a certain time with the thing B, which is so constructed that +though objectively different from A, the difference has not yet reached +the threshold. From experience, therefore, I must take A to be equal to +B. Then I compare B with a thing C, which is objectively different from +B in the same way as A is from B, though here, too, the difference is +still within the threshold, though very near it. I shall also have to +take B as equal to C. But now if I compare A directly with C, the sum of +the two differences oversteps the threshold value, and I find that A is +different from C. This, then, is a contradiction of the fundamental +principle that if A = B and B = C, A = C. This principle is valid for +_counted_ things, which, in consequence, are discontinuous, but not for +continuous things susceptible by our senses. If in spite of this it is +applied to continuous things or _magnitudes_ in the narrower sense, we +must bear in mind that it is just as much a case of an _extrapolation to +the non-existing ideal instance_ (p. 46) as in the case of the other +general principles, which, though they are derived from experience, +nevertheless, for practical purposes, transcend experience in their use. + +The examples cited above prove also that these relations are by no means +confined to the judgments we derive on the basis of immediate +sensations. When by means of the scale we compare three weights, the +differences of which lie within the limit of its sensitiveness but +approach closely to it, we can arrive in a purely empirical and +objective way also at the contradiction A = B, B = C, but A [Not=] C. In +weight and measurement, therefore, we hold fast to the principle that +the relations cited have no claim to validity outside the limit of their +possible errors. Accordingly, though the non-equation of A [Not=] C can +be observed, the difference of both values cannot be greater than at +utmost the sum of the two threshold values. + +These considerations also give us a means of appraising the oft-repeated +statement that in contradistinction to the physical laws the +mathematical laws are absolutely accurate. The mathematical laws do not +refer to real things, but to imaginary ideal limit cases. Consequently +they cannot be tested by experience at all, and the demands science +makes on them lie in quite a different sphere. Their nature must be such +that _experience should approximate them infinitely_, if certain +definite well-known postulates are to be more and more fulfilled, and +that the various abstractions and idealizations should be so chosen as +not to contradict one another. Such contradictions have by no means +always been avoided. But we must not regard them as inherent in the +inner organization of our mind, as Kant did. These contradictions spring +from careless handling of the concept technique, by which postulates +elsewhere rejected are treated as valid. We have already come across an +instance of such relations in the application of the concept of equality +to unlimited groups (p. 84). + +We must be guided by the same rules of precaution in answering the +question whether the things felt as continuous--for example, space and +time--are "truly" continuous, or whether in the last analysis they must +not be conceived of as discontinuous. The various sense organs, and +still more, the various physical apparatus with which we examine given +states, are of very varying degrees of "sensibility," that is, the +threshold for distinguishing the differences may be of very different +magnitudes. Therefore, a thing which is discontinuous for a sensitive +apparatus will behave as if it were continuous with a less sensitive +apparatus. Accordingly, we shall find so many the more things continuous +the less sharply developed our ability is to differentiate. + +While this circumstance makes it possible that we should regard +discontinuous things as continuous, time relations in certain +circumstances produce the opposite effect. Even if in a process the +change is continuous but very rapid, and the new state remains unchanged +for a certain time, we easily conceive of this sequence as +discontinuous. We cannot resist this view of the process when the change +occurs in a shorter time than the threshold time of our mind for each +step in the process. But since this threshold changes with our general +condition, one and the same process can appear to us both continuous and +discontinuous according to circumstances. Here, therefore, we have a +cause through the operation of which, with advancing knowledge, more and +more things will become recognized as _continuous_. + +Now if we turn to _experience_, we find, as the sum total of our +knowledge, that for the sake of expediency we approach everything with +the presumption that it is _continuous_. This aggregate experience +finds its expression in such sayings as "Nature makes no jumps," and +similar proverbial generalizations. But we must emphasize the fact once +more that in deciding matters in this way we deal solely with questions +of expediency, not with questions of the nature of our mental capacity. + + +=36. Measurement.= Measuring is in a certain way the opposite of +counting. While, in counting, the things are regarded in advance as +_individual_, and the group, therefore, is a body compounded of +discontinuous elements, measuring, on the other hand, consists in +_co-ordinating numbers with continuous things_, that is, in applying to +continuous things a concept formed upon the hypothesis of discontinuity. + +It lies in the nature of such a problem that the difficulty of +adaptation must crop out somewhere in the course of its attempted +solution. This is actually shown by the fact that measurement proves to +be an unconcluded and inconcludable operation. If, in spite of this, +measurement may and must justly be denoted as one of the most important +advances in human thought, it follows that those fundamental +difficulties can practically be rendered harmless. + +Let us picture to ourselves some process of measurement--for example, +the determination of the length of a strip of paper. We place a rule +divided into millimeters (or some other unit) on the strip, and then we +determine the unit-mark at which the strip ends. It turns out that the +strip does not end exactly at a unit-mark, but _between_ two +unit-marks. And even if the rule is provided with divisions ten or a +hundred times finer, the case remains the same. In most cases a +microscopic examination will show that the end of the strip does not +coincide with a division. All that can be said, therefore, is that the +length must lie _between n and n + 1 units_, and even if a definite +number is given, the scientifically trained person will supplement this +number by the sign ± _f_, in which _f_ denotes the possible errors, that +is, the limit within which the given number may be false. + +We see at once how the characteristic concept of threshold, which has +led to the conception of the continuous, immediately asserts itself when +in connection with discontinuous numbers. The adaptation of the +threshold to numbers can be carried as far as it is possible to reduce +the threshold, but the latter can never be made to disappear entirely. + +The significance of measurement therefore lies in the fact that it +applies the operation of counting with all its advantages (see p. 85) to +_continuous_ things, which as such do not at first lend themselves to +enumeration. By the application of the unit measure a discontinuity is +at first artificially established through dividing the thing into +pieces, each piece equal to the unit, or imagining it to be so divided. +Then we count the pieces. When a quantity of liquid is _measured_ with a +liter this general process is carried out physically. In all other less +direct methods of measurement the physical process is substituted by an +easier process equally good. Thus, in the example of the strip of paper +we need not cut it up into pieces a millimeter in length. The divided +rule is available for comparing the length of any number of millimeters +that happen to come under consideration, and we need only read off from +the figures on the rule the quantity of millimeters equal to the length +of the strip, in order to infer that the strip can be cut up into an +equal number of pieces each a millimeter in length. + +After it has been made possible to count continuous things in this way, +the numeration of them can then be subjected to all the mathematical +operations first developed only for discrete, directly countable things. +When we reflect that our knowledge of things has given them to us +_preponderatingly as continuous_, we at once see what an important step +forward has been made through the invention of measurement in the +intellectual domination of our experience. + + +=37. The Function.= The concept of continuity makes possible the +development of another concept of greater universality, which can be +characterized as an extension of the concept of causation (p. 31). The +latter is an expression of the experience, if A is, B is also, in which +A is understood to be a definite thing at first conceived of as +immutable. Now it may happen that A is not immutable, but represents a +concept with continuously changing characteristics. Then, as a rule, B +will also be of that nature, so that _every special value or state of B +corresponds to every special value or state of A_. + +Here, in place of the reciprocal relation of two definite things, we +have the reciprocal relation of two more or less extended groups of +similar things. If these things are continuous, as is assumed here (and +which is extremely often the case), both groups or series, even though +they are finite, contain an endless quantity of individual cases. Such a +relation between two variable things is called a function. Although this +concept is used chiefly for the reciprocal relation of _continuous_ +things, there is nothing to hinder its application to discrete things, +and accordingly we distinguish between continuous and discontinuous +functions. + +The intellectual progress involved in the conception of the reciprocal +relation of entire _series_ or groups to one another, as distinguished +from the conception of the relations between _individual_ things, is of +the utmost importance and in the most expressive manner characterizes +the difference between modern scientific thought and ancient thought. +Ancient geometry, for example, knew only the cases of the acute, right, +and obtuse angled triangle, and treated them separately, while the +modern geometrician represents the side of the triangle as starting from +the angle zero and traversing the entire field of possible angles. +Accordingly, unlike his colleague of old, he does not ask for the +particular principles bearing upon these particular cases, but he asks +in what continuous relation do the sides and angles stand to one +another, and he lets the particular cases develop from out of one +another. In this way he attains a much profounder and more effectual +insight into the whole of the existing relations. + +It is in mathematics in especial that the introduction of the concept of +continuity and of the function concept arising from it has exercised an +extraordinarily deep influence. The so-called _Higher Analysis_, or +_Infinitesimal Analysis_, was the first result of this radical advance, +and the _Theory of Functions_, in the most general sense, was the later +result. This progress rests on the fact that the magnitudes appearing in +the mathematical formulas were no longer regarded as certain definite +values (or values to be arbitrarily determined), but as _variable_, that +is, values which may range through all possible quantities. If we +represent the relation between two things by the formula B = f(A), +expressed in spoken language by B _is a function_ of A, then in the old +conception A and B are each individual things, while in the modern +conception A and B represent an inexhaustible series of possibilities +embracing every conceivable individual case that may be co-ordinated +with a corresponding case. + +Herein lies the essential advantage of the concept of continuity. It is +true that it also introduces into calculation the above-mentioned +contradictions which crop up in the ever-recurring discussions +concerning the infinitely great and the infinitely small. The system +introduced by Leibnitz of calculating with _differentials_, that is, +with infinitely small quantities, which in most relations, however, +still preserve the character of finite quantities from which they are +considered to have been derived, has proved to be as fruitful of +practical results as it is difficult of intellectual mastery. We can +best conceive of these differentials as the expression of the law of the +threshold, which law gave rise to, or made possible, the relation +between the continuous and the discrete. + + +=38. The Application of the Functional Relation.= I have already shown +(p. 34) how the first formulation of a causal relation which experience +yields can be purified and elaborated by the multiplication of the +experience. The method described was based upon the fact that the +necessary and adequate factors of the result were obtained by +eliminating successively from the "cause" the various factors of which +its concept was or could be compounded, and by concluding from the +result, that is, the presence or absence of the "effect," as to the +necessity or superfluity of each factor. + +Obviously the application of this process presupposes the possibility of +eliminating each factor in turn. Very often it is not possible, and then +in place of the inadequate method of the individual case the _method of +the continuous functional relation_ steps in with its infinitely +greater effectiveness. If in most cases we cannot _eliminate_ the +factors one by one, there are very few instances in which it is not +possible to _change_ them, or to observe the result in the automatically +changed values of the factors. But then we have the principle that for +the causal relation _all such factors are essential the change of which +involves a change of the result_. + +It is clear that this signifies a generalization of the former and more +limited method. For the elimination of the factor means that its value +is reduced to zero. But now it is no longer necessary to go to this +extreme limit; it suffices merely to influence in some way the factor to +be investigated. + +It is true that here the difference in the result cannot be expressed +with a "yes" or a "no," as before. It can only be said that it has +changed _partly_, more or less. From this it can be seen that the +application of this process requires more refined methods of +observation, especially for measuring, that is, for determining values +or magnitudes. On the other hand, we must recognize how much deeper we +can penetrate into the knowledge of things by the application of the +measuring process. Each advance in precision of measurement signifies +the discovery of a new stratum of scientific truth previously +inaccessible. + + +=39. The Law of Continuity.= From the fact that natural phenomena in +general proceed continuously we can deduce a number of important and +generally applicable conclusions which are constantly used for the +development of science. + +When a relation of two continuously varying values of the form A = f(B) +is conjectured, we convince ourselves of its truth by observing for +different values of A the corresponding values of B, or reversely. If we +find that changes in the one correspond to changes in the other, the +existence of such a relation is proved, at first only for the observed +values, though we never hesitate to conclude that for the values of A +lying between the observed values, but themselves not yet observed, the +corresponding values of B will also lie between the observed values. For +example, if the temperature at a given place has been observed at +intervals of two hours, we assume without hesitancy that in the hours +between when no observations were made, the values lie between the +observed values. If we indicate the time in the usual manner by +horizontal lines and the temperature for the general periods of time by +longitudinal lines, the law of continuity asserts that all these +temperature points lie in a steady line, so that when a number of points +lying sufficiently near one another is known, the points between can be +derived from the steady line which may be drawn through the known +points. This very commonly applied process will yield the more accurate +results the nearer the known points are to one another, and the simpler +the line. + +The application of the law of continuity or steadiness, therefore, +means no less than that it is possible, from a finite, frequently not +even a very large, number of individual results, to obtain the means of +predicting the result for an infinitely large number of unexamined +cases. The instrument derived from this law, therefore, is an eminently +_scientific_ one. + +The value of this instrument is still greater if it succeeds in +expressing the relation A = f(B) in strict mathematical form. First, the +result of the determination of a number of individual values of that +function is represented as a table of co-ordinated values. By the +graphic process above described, or by its equivalent, the mathematical +process of interpolation, this table is so extended that it also +supplies all the intermediate values. But this is still a case of a +mechanical co-ordination of the corresponding values. Often we succeed, +especially in the relation of simple or pure concepts, in finding a +general mathematical rule by which the magnitude A can be derived from +the magnitude B, and reversely. This is the only instance in which we +speak of a natural law in the quantitative sense. + +Thus, for example, we can observe what volume a given quantity of air +occupies when successively subjected to different pressures. If we +arrange all these values together in a table, we can also calculate +the volume for all the intermediate pressures. But on close inspection +of the corresponding numbers of pressure and volume we notice that +they are in inverse ratio, or that when multiplied by one another +their products will be the same. If we denote the space by v and the +pressure by p, this fact assumes the mathematical form p·v = K, in +which K is a definite number depending upon the quantity of air, the +unit of pressure, etc., but remaining unchanged in an experimental +series in which these factors stay the same. The general functional +equation A = f(B) becomes the definite p = K/v. And this formula +enables us to determine by a simple calculation the volume for any +degree of pressure, provided the value of K has been once ascertained +by experiment. + +At first we have a right to such a calculation only within the province +in which the experiments have been made, and the simple mathematical +expression of the natural law has for the time being no further +significance than that of a specially convenient rule for interpolation. +But such a form immediately evokes a question which demands an +experimental answer. How far can the form be extended? That there must +be a limit is to be directly inferred from the consideration of the +formula itself. For if we let p = 0, then v = infinity, both of which +lie beyond the field of possible experience. + +Similar considerations obtain in all such mathematically formulated +natural laws, and each time, therefore, we must ask what the _range of +validity_ of such an expression is, and answer the question by +experiment. + +While in this discussion the mathematically formulated natural law seems +to have the nature only of a convenient formula of interpolation, we are +nevertheless in the habit of regarding the discovery of such a formula +as a great intellectual accomplishment, which so impresses us that we +frequently call it by the name of the discoverer. Now, wherein lies the +more significant value of such formulations? + +It lies in the fact that simple formulas are discovered only _when the +conceptual analysis of the phenomenon has advanced far enough_. The very +simplicity of the formula shows that the concept formation which is at +the basis of it is especially serviceable. In Ptolemy's theory of the +motion of the planets the means for calculating their positions in +advance was given just as in the theory of Copernicus. But Ptolemy's +theory was based on the assumption that the earth stands still, and that +the sun and the other planets move. The assumption that the sun stands +still and that the earth and the other planets move greatly facilitates +the calculation of the position of the planets. In this lay the primary +value of the advance made by Copernicus. It was not until much later +that it was found that a number of other actual relations could be +represented much more fittingly by means of the same hypothesis, and +thus the Copernican theory has come to be generally recognized and +applied. + +The significance of the law of continuity and its field of application +have by no means been exhausted by what has been said above. But later +we shall have a number of occasions to point out its application in +special instances, and so cause its use to become a steady mental habit +with the beginner in scientific research. + + +=40. Time and Space.= Time and space are two very general concepts, +though without doubt not elementary concepts. For besides the elementary +concept of continuity which both contain, time has the further character +of being one-seried or one-dimensional, of not admitting of the +possibility of return to a past point of time (absence of double points) +and of absolute onesidedness, that is, of the fundamental difference +between before and after. This last quality is the very one not found in +the space concept, which is in every sense symmetrical. On the other +hand, owing to the three dimensions it has a _three_fold manifoldness. + +That despite this radical distinction in the properties of space and +time all of our experiences can be expressed or represented within the +concepts of space and time, is very clear proof that experience is much +more limited than the formal manifoldness of the conceivable. In this +sense space and time can be conceived as natural laws which may be +applied to all our experiences. Here at the same time the +subjective-human element of the natural law becomes very clear. + +The properties of time are of so simple and obvious a nature that there +is no special science of time. What we need to know about it appears as +part of physics, especially of mechanics. Nevertheless time plays an +essential rôle in _phoronomy_, a subject which we shall consider +presently. In phoronomy, however, time appears only in its simplest form +as a one-seried continuous manifoldness. + +As for space, the presence of the three dimensions conditions a great +manifoldness of possible relations, and hence the existence of a very +extensive science of bodies in space, of _geometry_. Geometry is divided +into various parts depending upon whether or not the concept of +measurement enters. When dealing with purely spacial relations apart +from the concept of measurement it is called geometry of position. In +order to introduce the element of measurement a certain hypothesis is +necessary which is undemonstrable, and therefore appears to be arbitrary +and can be justified only because it is the simplest of all possible +hypotheses. This hypothesis takes for granted that a rigid body can be +moved in all directions in space without changing in measure. Or, to +state the inverse of this hypothesis, in space those parts are called +equal which a rigid body occupies, no matter how it is moved about. + +We are not conscious of the extreme arbitrariness of this assumption +simply because we have become accustomed to it in school. But if we +reflect that in daily experience the space occupied by a rigid body, say +a stick, seems to the eye to undergo radical changes as it shifts its +position in space and that we can maintain that hypothesis only by +declaring these changes to be "apparent," we recognize the arbitrariness +which really resides in that assumption. We could represent all the +relations just as well if we were to assume that those changes are real, +and that they are successively undone when we restore the stick to its +former relation to our eye. But though such a conception is +fundamentally practicable in so far as it deals merely with the space +picture of the stick, we nevertheless find that it would lead to such +extreme complications with regard to other relations (for example, the +fact that the weight of the stick is not affected by the change of the +optic picture) that we do better if we adhere to the usual assumption +that the optical changes are merely apparent. + +In this connection we learn what an enormous influence the various parts +of experience exert upon one another in the development of science. In +every special generalization of experiences, that is, in every +individual scientific theory, our aim is not only to generalize this +special group of experiences in themselves, but at the same time to join +such other experiences to them as expedience demands. If the effect of +this necessity is on the one hand to render the elaboration of an +appropriate theory more difficult, it has on the other hand the great +advantage of affording a choice among several theories of apparently +like value, and thus making possible a more precise notion of the +reality. For example, for the understanding of the mutual movements of +the sun and the earth it is the same whether we assume that the sun +moves about the earth or the earth about the sun. It is not until we try +to represent theoretically the position of the other planets that we see +the economic advantage of the second conception, and facts like +Foucault's experiment with a pendulum can be represented only according +to this second conception in our present state of knowledge. + +Likewise, the assumption on which scientific geometry goes, that space +has the same properties in all directions, conflicts with immediate +experience. In immediate experience we make a sharp distinction between +below and above, although we are prepared to admit the "homogeneity" of +space in the horizontal direction. This is due, as physics teaches, to +the fact that we are placed in a field of gravitation which acts only +from above downward and which permits free horizontal turnings, although +it imparts a characteristic difference to the third direction. Since +considerations of another kind enable us to place ourselves in a +position in which we ignore this field of gravitation in the +investigation of space, geometry abstracts this element and disregards +the corresponding manifoldness. In the theory of the gravitation +potential, on the other hand, this very manifoldness is made the subject +of scientific investigation. + +The common application of the concepts of space and time results in the +concept of _motion_, the science of which is called phoronomics. In +order to make this new variable subject to measurement we must arrive at +an agreement or convention as to the way in which to measure time. For +since past time can never be reproduced we actually experience only +unextended moments, and have no means of recognizing or defining the +equality of two periods of time by placing them side by side, as we can +in the case of spacial magnitudes. We help ourselves by saying _that in +uninfluenced motions equal periods of time must correspond to the equal +changes in space_. We regard the rotation of the earth on its axis and +its revolution about the sun as such uninfluenced motions. The two +depend upon dissimilar conditions, and the empirical fact that the +relation of the two motions, or the relation between the day and the +year, remains practically the same, sustains that assumption, and at the +same time shows the expediency of the given definition of time. + +_Analytic geometry_, the application of algebra to geometric relations, +occupies a noteworthy position, from the point of view of method, in the +science of space. It yields geometric results by means of calculation, +that is, by the application of the _algebraic_ material of symbols we +can obtain data concerning unknown _spacial_ relations. An explanation +is necessary of how by a method apparently so extraneous such results as +these can be attained. + +The answer lies again in the general principle of co-ordination, which +in this very case receives a particularly cogent illustration. Three +algebraic signs, x, y, and z, are co-ordinated with the three variable +dimensions of space. First, the same independent and constant +variability is ascribed to these signs, and, further, the same mutual +relations are assumed to subsist between them as actually exist between +the three-spacial dimensions. In other words, precisely the same kind of +manifoldness is imparted to these algebraic signs as the spacial +dimensions possess to which they are co-ordinated, and we may therefore +expect that all the conclusions arising from these assumptions will find +their corresponding parts in the spacial manifoldness. Accordingly, a +co-ordinated spacial relation corresponds to every change of those +algebraic formulas resulting from calculation, and if such changes lead +to an algebraically simple form, then the spacial form corresponding to +it must show an analogous simplicity. Here, therefore, we have a case +such as was described under simpler conditions on p. 86 of operations +undertaken with one group and repeated correspondingly in the +co-ordinated group. And it is only the great difference in the things +of which in this case the two groups are composed--spacial relations on +the one side and algebraic signs on the other--that creates the +impression of astonishment which was felt very strongly at the invention +of this method, and which is still felt by students with talent for +mathematics when they first become acquainted with analytical geometry. + + +=41. Recapitulation.= Before we proceed to consider the fundamentals of +other sciences, it is well to make a general résumé of the field so far +traversed. Since the later sciences, as we have already observed, make +use of the entire apparatus of the earlier sciences, the mastery of them +must be assured in order to render their special application possible. + +This does not mean that one must have complete command of the entire +range of those earlier sciences in order to pursue a later one. Mere +human limitations would prevent the fulfilment of such a demand. As a +matter of fact, successful work can be done in one of the later sciences +even if only the most general features of the earlier ones have been +clearly grasped. Nevertheless, the rapidity and certainty of the results +are very considerably increased by a more thorough knowledge of the +earlier sciences, and the investigator, accordingly, should seek a +middle road between the danger of insufficient preparation for his +special science and the danger of never getting to it from sheer +preparation. In any circumstances he must be prepared always, even +though it be in later age, to acquire those fundamental aids so soon as +he feels the need of them for carrying out any special work. It is +generally acceded that without logic the adequate pursuit of science is +impossible. Nevertheless, the opinion is widely current, even among men +of science, that everybody has command of the needful logic without +having studied it. No more than a man can learn of himself to use the +calculus, even if he may have discovered unaided some of its elementary +principles, can he acquire certainty and readiness in the use of the +logical rules generally necessary, unless he has made the necessary +studies. It is true that the scientific works of the great pioneers and +leaders in the special sciences furnish practical examples of such +logical activity. But complete freedom and security are acquired only on +the basis of conscious knowledge. + +We have now seen how, from the physiological construction of our mental +apparatus, the process of concept formation and the experience of +concept connections are the basis of the whole of mental life. The laws +of the mutual interaction of the most general or elementary concepts +operated in the formation of the concepts, _thing_, _group_, +_co-ordination_. Here were found the fundamentals of logic or the +science of concepts. A special process of abstraction yielded the +concept of _number_, and with it the corresponding field of +_mathematics_, arithmetic, algebra, and the theory of numbers. + +By means of the second fundamental fact of physiology, the _threshold_, +another elementary fact was explained, that of _continuity_. The +co-ordination of individual things under the influence of this concept +was expanded into the _co-ordination of continuous phenomena-series_, +and yielded the correspondingly more general concept of the _function_. +From the application of the number concept to continuous things, the +idea of _measurement_ resulted. In mathematics the concept of continuity +led to higher _analysis_ and the _theory of functions_. Finally, the +concept of continuity proved to be an inexhaustible aid for the +extension of scientific knowledge and for the formulation of natural +laws in mathematical form. + + + + +PART III + +THE PHYSICAL SCIENCES + + +=42. General.= In the formal sciences we began the specialization of the +object from the most general concept of thing conceivable, possessing no +other characteristic attribute than its capability of being +distinguished from other things; and we carried the specialization so +far that we could follow in its movements an object definite as to time +and space. This object, to be sure, was defined only in that it occupied +a definite space, and accordingly had a definite form. As a matter of +fact, the spacial thing of geometry and phoronomy reveals no further +attributes. + +It is here that the physical sciences enter into their dominion one +after the other, and fill the empty space of the geometric thing with +definite attributes. These are the secondary qualities of Locke, of +which he assumed that they do not belong so much to the bodies +themselves as that they merely appear to us so on account of the nature +of our human sense organs. Now that our knowledge concerning the nature +of those properties as well as the structure of our sense organs is +much more thorough, we have more definite ideas also of the subjective +part of the corresponding experiences, and in a large measure are able +to separate it from the objective part. + +All properties which physical bodies in contradistinction to geometric +bodies possess can be traced back to a fundamental concept, which, in +conjunction with the concepts explained in the former chapter, serves to +characterize and distinguish the physical structure. For example, the +fact that we can distinguish cubes of equal size but of different +material, different temperature, and different luminosity, can be traced +back always and entirely to the different kinds of energy acting in the +geometric space in question. The concept of energy, therefore, plays +approximately the same rôle in the physical sciences as the concept of +thing in the formal sciences, and the essentials of this new field of +science are the comprehensive knowledge and development of this concept. +Because of its great importance it has long been known and applied in +individual forms. But the systematization of the physical sciences +relative to energy is a matter of only recent date. + + +=43. Mechanics.= Recently many scientists have taken exception to the +traditional division of mechanics into _statics_, or the science of +equilibrium, and _dynamics_, or the science of motion, because it does +not correspond to the essence of the thing, equilibrium being only the +limit-case of motion. However, the classic presentations of this +science are based on that division, so that it must express an essential +difference. This difference we can clearly recognize through the +application of the concept of energy to mechanics. We then learn that +statics is the science of work, or the energy of position, and that +dynamics is the science of living force, or of the energy of motion. + +By _work_ in the mechanical sense we mean the expenditure of force +required for the locomotion of physical bodies. While a cube of lead is +geometrically equal to a cube of glass, we experience a great difference +between them when we lift them from the floor to a table. We call the +cube of lead heavier than the glass cube, and we find it requires more +work to raise the former than the latter. For psychologic reasons this +judgment becomes especially clear when the work required to lift the +lead cube marks the limit of our physical capacity. + +Work depends not only upon the difference described above, but also upon +the distance through which it is exerted. It increases in proportion as +the distance increases. In mechanics work is proportional both to the +distance and to that peculiar property which in the given example we +call _weight_. But a more general concept has been formed for that +property in the mechanical sense, called _force_, of which weight +constitutes but a special instance. Whenever there is a resistance +combined with a change of place we speak of a force, _and the product +of the force and the distance we call work_. + +The cause of this kind of concept formation is the following: There are +a great number of different machines, all of them possessing the +peculiarity that work can be put into them at a definite place and taken +out at another place. Now, centuries of experience have shown that it is +impossible to obtain more work from such mechanical machines than has +been put into them. As a matter of fact, the work obtained is always +less than the work put in, and the two approach equality as the machine +approaches perfection. It is to such ideal machines, therefore, that +_the law of the conservation of work_ applies. This law states that, +though a given quantity of work may be changed in the most manifold ways +as to direction, force, etc., it is impossible to change its _quantity_. + +The reason we can judge of this fact with such certainty is because for +many centuries a number of the ablest mechanicians have sought for a +solution of the problem of perpetual motion, that is, for the +construction of a machine from which more work can be gotten than is put +into it. All such attempts have failed. But the positive result secured +from these apparently futile efforts is the law of the conservation of +work. The greatness and importance of this result will become apparent +in the further course of our study. + +Here for the first time we meet with a law expressing the +_quantitative_ conservation of a thing, which may none the less undergo +the most varied qualitative changes. With the knowledge of this fact we +involuntarily combine the notion that it is the "same" thing that passes +through all these transformations, and that it only changes its outward +form without being changed in its essence. Such ideas, it is true, are +widespread, but they have a very doubtful side to them, since they +correspond to no distinct concept. If we want to call the quantitative +magnitude of the product of the force and distance the "essence" of +work, and the determination of the force and the distance according to +magnitude and direction, which come under consideration for each special +value, as its "form," then, of course, there is no objection to be made +to mere nomenclature. But we must bear in mind that the difference +obtaining here lies exclusively in the fact that the amount of work +measured quantitatively remains unchanged, while its factors undergo +simultaneous and opposite changes. + +This discovery, that there is a magnitude which can be quantitatively +determined, and which, as experience shows, remains unchanged, however +much its factors may change, invariably results not only in a very +simple and clear formulation of the corresponding natural law, but also +corresponds to the general tendency of the human mind to work out +conceptually "the permanent in change." If, in accordance with the +word-sense, we denote everything which persists under changing +conditions by the name of _substance, we encounter in work the first +substance_ of which we have attained knowledge in our scientific +journeys. In the history of the evolution of human thought this +substance has been preceded by others, especially by the weight and mass +of ponderable bodies (which are also subject to a law of conservation), +so that at present we are inclined to connect with the word substance a +tacit secondary sense of ponderability. But this is a remnant of the +still very widely spread mechanistic theory of the universe, which, +though it has almost finished its rôle in physics, will presumably +continue to persist for a long time to come in the popularly scientific +consciousness in accordance with the laws of collective thought. + + +=44. Kinetic Energy.= The law of the conservation of work is by no means +true of all cases in which work is expended or converted, but, as has +been said, only of _ideal_ machines, that is, of such cases which do not +exist in reality. But while in imperfect machines there is at least an +approximation to this law, there are besides countless normal cases in +which we cannot even speak of an approximation. When, for example, a +stone falls to the ground from a certain height, a certain quantity of +work is expended, which is equal to that by means of which the stone can +be raised again to its original height. This quantity of work apparently +disappears entirely when the stone remains lying on the ground. We +shall discuss this case later. Or the falling of the stone can be so +guided that it can lift itself again. This happens, for instance, when, +by fastening the stone to a thread, it is forced to move in a curved +path, or to perform pendular oscillations. In that case, it is true, the +stone will fall to the lowest point which the thread permits, and so +will there have lost its work without having done any other work in the +meantime. But it has entered a condition by virtue of which it raises +itself again, so that (as before, only in the ideal limit-case) it +reaches its former height, and so has lost no work. For this moment, +too, then, the law of the conservation of work obtains. But in the +meantime new relations have arisen. + +What distinguishes the stone moving like a pendulum from the stone which +simply falls is, that at its lowest point it has not remained lying +still, but possesses a certain velocity. By means of this it lifts +itself again, and after it has reached its former height, it has lost +its velocity. _Therefore, there is a reciprocal relation between the +work which it loses and the velocity which it gains_, and the question +may therefore be put, How can this relation be represented +mathematically? Experience teaches that in every such case a function of +the velocity and of another property of the body, called _mass_, can be +established in such a way that this function, called the _kinetic +energy_ of the body, increases precisely as much as the amount of work +the body has expended, and _vice versa_. The sum of the kinetic energy +of the body and of the _work_ is therefore _constant_, and the clearest +mode of conceiving of this relation is by assuming _that work can be +transformed into kinetic energy and vice versa_ in such a way that given +amounts of the two magnitudes are equal or equivalent to one another. +Naturally, this is only an abbreviated way of expressing the actual +relations, for it might just as well be assumed that the work really +disappears and the kinetic energy really originates anew, and that the +disappearance of the one substance only happens regularly to coincide +with the origin of the other. But it is this regular conjunction of +phenomena that constitutes the sole ground of every _causal_ relation, +and in such a sense we are justified _in regarding the disappearing work +as the cause of the kinetic energy that arises_, and to designate this +relation summarily as a transformation. + +By the inclusion of cases in which work is converted into kinetic energy +the law of the conservation of work therefore becomes _the law of the +conservation of the sum of work and kinetic energy_. We are thereby +compelled to extend the concept of substance, which at first contains +only work, to the sum of both magnitudes, and to introduce a new name +for this enlarged concept. + +It will soon appear that all cases of imperfect machines, in which work +disappears without giving rise to an equivalent amount of kinetic +energy, can, with a corresponding enlargement of the concept, be +likewise included in the law of conservation. For experience has shown +that in such cases something else arises, heat, light, or electric +force, etc. This generalized concept, which embraces all natural +processes and permits the sum of all corresponding values to be +expressed by a law of conservation, we call _energy_. The law in +question, therefore, is: + +_In all processes the sum of the existing energies remains unchanged._ + +The principle of the conservation of work in perfect machines proves to +be an ideal special instance of this general law. A perfect machine is +one in which work changes into nothing but _work_ of another kind, and +not into a different kind of energy. Then each side of the equation +which expresses the general law of energy, namely, + +Energy that has disappeared = energy that has arisen, + +contains only the magnitude of the work, and expresses the law of the +conservation of work. If, on the other hand, as in the case of the +pendulum, the work increasingly changes part by part into kinetic +energy, and _vice versa_, the equation during the first period is: + +Work that has disappeared = kinetic energy that has arisen, + +and during the second period in which the pendulum rises again, + +Kinetic energy that has disappeared = work that has arisen. + +Thus, while work can be called a substance only in a limited sense, +since its conservation is limited only to perfect machines, we may call +energy a substance unqualifiedly, since in every instance of which we +know the principle has been maintained _that a quantity of any energy +never disappears unless an equivalent quantity of another energy +arises_. Accordingly, this law of the conservation of energy must be +taken as a fundamental law of the physical sciences. But not only do all +the phenomena of physics, including chemistry, occur within the limits +of the law of conservation, but until the contrary is proved the law of +conservation must also be regarded as operative in all the later +sciences, that is, in all the activities of organisms, so that all the +phenomena of life must also take place within the limits of the law of +conservation. This corresponds to the general fact, which I have +emphasized a number of times, that all the laws of a former science find +application in all the following sciences, since the latter can only +contain concepts which by specialization, that is, by the addition of +further characteristics, have sprung from the concepts of the former or +more general sciences. + + +=45. Mass and Matter.= It has been noted above that kinetic energy +depends upon another magnitude beside velocity. A conception of its +nature can be obtained when we try to put different bodies in motion. +In doing so the muscles of the arm perform certain quantities of work, +and we feel whether the quantities are greater or smaller. In this way +we obtain a clear consciousness of the fact that different bodies +require quite different quantities of work for the same velocity. The +property which comes into play here is called _mass_, and mass is +proportional to the work which the various bodies require to attain the +same velocity. Since the work and the velocity can be measured very +accurately by appropriate means, mass also lends itself to a +correspondingly accurate measurement. + +All known ponderable bodies have mass. That means there is a regular +connection between the property which makes a body tend to the earth +with a certain definite force (called weight) and the property by virtue +of which a body assumes certain velocities under the influences of +motive causes. We can readily conceive that it is possible for us to +learn only of such bodies as are heavy, that is, bodies which are _held_ +by the earth, since the others, if they exist at all, would naturally +have left the earth long ago. That all these bodies also have mass is to +be explained in a similar way. For a body of mass zero would at each +impulse assume infinitely great velocity, and could therefore never be +the object of our observation. Consequently, by reason of the physical +conditions obtaining on the earth's surface, the bodies known to us must +combine both properties, mass and weight. + +The name given to this concept of the combined presence of mass and +weight in space is _matter_. Experience shows that there is a law of +_conservation_ for these magnitudes also, according to which _whatever +changes we may produce in bodies possessing weight and mass, no change +will occur in the sum of their weight and mass_. According to the +nomenclature previously introduced we must therefore call weight and +mass substances, since they remain the same as to quantity, no matter +what changes they may undergo. However, it is usual to apply the name +substance to the concept of matter composed of mass and weight. In fact, +scientists often go so far as to limit the name to this single instance +of the various laws of conservation, and to take substance to mean +exclusively the combination of mass and weight. This is connected with +the conception which we are about to discuss, that all natural phenomena +can ultimately be conceived as the motion of matter. Through the greater +part of the nineteenth century this conception, called _scientific +materialism_, was accepted almost without opposition. At present it is +being more and more recognized that it was only an unproved assumption, +which the development of science daily proves to be more untenable. + + +=46. Energetic Mechanics.= In the light of our previous observations the +branch of science traditionally known as mechanics appears as the +science of work and of kinetic energy. Furthermore, statics is shown to +be the science of work, while dynamics, besides treating of kinetic +energy in itself, also treats of the phenomena of the change of work +into kinetic energy, and _vice versa_. We shall find the same relation +again later, only in more manifold forms. Every branch of physics proves +to be the science of a special kind of energy, and to the knowledge of +each kind of energy must be added the knowledge of the relations by +which it changes to the other forms of energy and _vice versa_. It is +true that in the traditional division of physics this system has not +been strictly carried out, since an additional and very influential +motive for classification has been the regard paid to the various human +sense organs. + +Nevertheless this ground does not lie in the field of physics, but in +that of physiology, and must, therefore, be abandoned in the interest of +strict systematization. + +Of the physical sciences mechanics was the first to develop in the +course of historical evolution. A number of factors contributed to this +end--the wide distribution of mechanical phenomena, their significance +to human life, and the comparative simplicity of the principles of +mechanics, which made it possible to discover them at an early date. +Most to be noted is, that of all departments of physics mechanics is the +first which lent itself to comprehensive _mathematical_ treatment. It is +true that the mathematical treatment of mechanics was possible only +after idealizing assumptions had been made--perfect machines and the +like--so that the results of this mathematical treatment not +infrequently had very little to do with reality. The mistake of losing +sight of the physical problem and of making mechanics a chapter of +mathematics has not always been avoided, and it is only in most recent +times that the consciousness has again arisen that the classical +mechanics, in arbitrarily limiting itself to extreme idealized cases, +sometimes runs the risk of losing sight of the aim of science. + + +=47. The Mechanistic Theories.= Because the evolution of mechanics +antedates that of the other branches of physics, mechanics has largely +served as a model for the formal organization of the other physical +sciences, just as geometry, which has been handed down to us from +antiquity in the very elaborate form of Euclid, has largely been used as +a model for scientific work in general. Such methods of analogy prove to +be extremely useful at first because they serve as a guide to indicate +when and where new sciences, in which all possibilities are open, can be +got hold of. But later on such analogies are apt to be harmful. For each +new science soon requires new methods, by reason of the peculiar +manifoldness which it has to deal with, and the finding and the +introduction of these new methods are easily delayed, and, as a matter +of fact, often have been delayed, because scientists could not free +themselves soon enough from the old analogy. + +By its being based upon memory the human mind is so constructed that it +cannot assimilate something entirely new. The new must in some way be +connected with the known in order that it may be organically embodied in +the aggregate of concepts. Therefore, it is the first involuntary +impulse of our mind, in the presence of new experiences or thoughts, to +look about for such points at which a linking of the unknown to the +known seems possible. In the case of mechanics this necessity for +finding connecting links has acted in such a way that the attempt has +been made, and is still being made, to conceive and represent all +physical phenomena as mechanical. + +The impulse to this was first given by the extraordinary successes which +mechanics has attained in the generalization and prediction of the +_motions of the heavenly bodies_. The names of Copernicus, Kepler, and +Newton mark the individual steps in the mechanization of astronomy. The +cause of this lies in the fact that the heavenly bodies actually +approximate very closely the ideal of the purely mechanical form with +which classical mechanics deals. These successes encourage the attempt +to apply these mental instruments that were productive of such rich +results to all other natural phenomena. An old theory, according to +which all physical things are composed of the most minute solid +particles of matter called _atoms_, supported these tendencies and +invited the attempt to regard the little world of atoms as subject to +the same laws as had been found to apply so successfully to the great +world of the stars. + +Thus we see how this mechanistic hypothesis, the assumption that all +natural phenomena can be reduced to mechanical phenomena, comes as if it +were a self-understood matter, and with its claim to be a profound +interpretation of nature it scarcely permits the question as to its +justification to be raised at all. And the effects here have been the +same as I described above in cases in which inferences from analogy are +accepted too extensively or too credulously. While it is true, no doubt, +that the mechanical hypothesis at first was fruitful of results in +special research, because it facilitated the putting of the +question--for example, we need think only of the atomic hypothesis in +chemistry--later, the efforts to find further hypothetic help for the +inadequacies of the hypothesis that gradually came to light, have not +infrequently led scientific research to pseudo-problems, that is, to +questions which are questions only in hypothesis, but to which no actual +reality can be shown to correspond. Such problems, therefore, are by +their very nature _insoluble_, and constitute an inexhaustible source of +differences of scientific opinion. + +The most flagrant of the injurious consequences of the mechanistic +hypothesis appear in the scientific treatment of the mental phenomena. +Ready as scientists were to represent all other life phenomena, such as +digestion, assimilation, and even generation and propagation, as the +consequence of an extremely complicated play of certain atoms, their +courage never went so far as to apply this principle to mental life and +to consider that by mechanics the last word had been said on the +subject. + +It is because of this hesitancy to bring mental phenomena under the same +mechanistic principle as all the other phenomena that the philosophical +systems had to search for some other means to connect the mental world +with the mechanical, and the efforts of the philosophers to bring about +this end have been most varied. Of the various doctrines that have come +down to us, that of the _pre-established harmony_ proposed by Leibnitz +is in the ascendant in our day, and is now called the theory of the +_psycho-physical parallelism_. According to this theory it is assumed +that the mental world exists alongside, and quite independent of, the +mechanical, but that the things have been so prearranged that mental +processes take place simultaneously with certain mechanical processes +(according to some, with all mechanical processes) in such a way that, +although the two series do not influence each other in the least, they +always correspond to each other precisely. How such a relation has come +about and how it is maintained remains unsaid, or is left to future +explanation. + +We need only think of the content of this hypothesis with an unbiased +mind to lose all relish for it at once. In fact, it has no other _raison +d'être_ than the presumption that the mental and the mechanical world +are opposed to each other. As soon as we abandon the thesis that the +non-mental world is exclusively mechanical, we acquire the possibility +again of finding for the theory of mental phenomena a constant and +regular connection with the theories of all other phenomena, especially +with the phenomena of life. Therefore it will be found most expedient in +every respect, instead of rendering scientific research one-sided and +almost blind to nonconforming facts by preconceived hypotheses, such as +the mechanistic hypothesis, to seek, as hitherto, from step to step, the +new elements of manifoldness which must be taken account of in the +progressive upbuilding of science and to limit ourselves faithfully to +them in the formation of general ideas. + + +=48. Complementary Branches of Mechanics.= The field of pure or +classical mechanics is limited to the above two kinds of energy, work +and kinetic energy, though these do not exhaust the manifoldness of the +mechanical energies. Accordingly, other branches of mechanics dealing +with the corresponding phenomena are added to the classical mechanics +described above. + +If by mechanical energies we understand all energies in which _changes +of space are connected with changes of energy_, there are as many +different forms as there are spacial concepts that seem applicable. +_Form_, _Volume_, and _Surface_ of bodies in space are especially +recognizable as the field of action for energy, which shows different +properties or manifoldnesses according to each of these relations. + +The _energy of form_ is manifested in bodies (solid or rigid bodies) +that maintain a definite shape because every change of shape is +connected with work or with the expenditure of some other energy. If the +changes are small, the bodies are of such a nature that they return to +their former condition of their own accord after the force exerted upon +them has ceased to act. This property is called _elasticity_. However, +the theory of elasticity, which has been extensively and rationally +developed, is regarded as belonging rather to mathematical physics in +general than to mechanics in particular. In greater changes of shape the +energy of form, or elastic energy, passes into other forms, and the body +does not return to its former shape after the force has been removed. + +Other bodies have no energy of form (or only in an infinitesimally +slight degree), so that they allow of changes of form without the +expenditure of work, but their volume can be changed only by work. These +are divided into two classes. First, the _liquids_, which have a +definite volume (corresponding to the definite shape of solids), the +changes of which in _every_ sense, both compression and expansion, +require work. Secondly, the _gases_ with volume energy in only one sense +of the word, in which only the compression of volume requires work, +while in expansion a certain amount of work is thrown off. Such bodies +can exist only so long as the expenditure of their volume energy by +spontaneous expansion is prevented by the presence of a counter energy, +as, for example, the elasticity of the walls of a vessel. This tendency +is called _pressure_. + +Finally, there are energy qualities at the surfaces between various +kinds of bodies which come into play at the change of these surfaces. +They always lie in such a direction that the enlargement of the surfaces +requires work, and hence, by reason of the law of conservation of +energy, cannot proceed by itself. In cases where there has been an +inverse kind of energy present, that is, one which diminishes with +increasing surface, it also has been active as a rule, thus bringing +about the disappearance of the existing boundaries. + +Since the seat of this kind of energy is in the surfaces (or +superficies), it is called _surface-energy_. The phenomena depending +upon it manifest themselves most clearly at the surface boundaries +between _liquids_ and _gases_. They are called _capillary phenomena_. +This strange name, derived from the word _capilla_, hair, has its origin +in the fact that because of surface-energy liquids rise in tubes which +they wet, and the narrower the tube the higher they rise. If the lumen +of the tube is as fine as a _hair_, a considerable rise can be observed. +This is the entire connection between the name and the thing. + +The mechanics of liquids is called _hydromechanics_, that of gases, +_aeromechanics_, after the most familiar liquid, water, and the most +familiar gas, air. The study of surface-energy under the name of the +capillary theory forms part of theoretical physics. While formerly this +branch, too, was regarded as a working part, or, rather, as a playing +part, of mathematical problems, in more recent times extensive +experimental research has made its entry in this province also, and has +demonstrated the necessity of passing from the former abstractions or +idealizations, which were carried altogether too far, to a better and +profounder regard for the actually existing complexities. + + +=49. The Theory of Heat.= The various forms of energies the aggregate of +which is comprehended in physics, have very different special +characters. A systematic investigation has not yet been made of the +characters of manifoldness by which, for example, work is distinguished +from heat, electrical energy from kinetic energy, etc., nor of what are +the essential properties peculiar to each individual energy. We feel +certain that differences do exist, for otherwise the energies could not +be distinguished, and we feel certain that these differences are very +important, for doubt seldom arises as to the kind of energy to which a +certain phenomenon is to be assigned. But just as we have no systematic +table of the elementary concepts, so we are still without a systematic +natural history of the forms of energy in which the peculiarities of +every species are characterized, and in which the entire material is so +arranged according to these characteristics that we can take a general +survey of it. + +As regards heat energy, its foremost and most striking characteristic is +its physiological effect. In our skin there are organs for the +perception of heat as well as of cold, that is, for temperatures above +and below the temperature of the skin. However, the temperature that +these organs can bear without injury to themselves is of a very small +range, beyond which physical apparatuses of all kinds must be used, such +as "thermometers." + +Heat is the simplest kind of energy from the point of view of +manifoldness. Every heat quantity is marked by a temperature, just as a +kinetic energy is marked by velocity. But while a velocity is determined +in space so that velocities of equal magnitude have in addition a +threefold infinite manifoldness in reference to direction, a temperature +is characterized completely and unambiguously by a simple number, the +degree of temperature. Two temperatures of equal degree can in no wise +be distinguished, since temperature possesses no other possible +manifoldness than degree. + +The same property is found in heat energy itself. In heat energy we +measure the quantity of energy itself and call it the _heat quantity_, +while in some of the other kinds of energy, only the factors into which +they can be divided are measured, and no habitual conception of the +energy itself is developed. A heat quantity is likewise fully indicated +by its measure number. + +That heat is an energy, that is, that it is developed in equal +quantities from other kinds of energy, and can change back again into +them, is a discovery which, despite its fundamental and general +character, was not made before the forties of the nineteenth century. As +often happens in cases of important scientific advances, the same idea +came simultaneously to a number of investigators. The first to grasp and +fully comprehend this idea was _Julius Robert Mayer_ of Heilbronn, who +published his results in 1842. Mayer not only showed that the imperfect +machines (p. 134), which limit the validity of the law of the +conservation of work, owe this peculiarity to the fact that they +transform a part of the work into _heat_, and that when we take account +of this part, the law of conservation holds perfectly good, but he also +calculated, with extraordinary acumen, the mechanical equivalent of heat +from the then existing data of physics. That is to say, he determined +how many units of heat (in the measure then in use) correspond to a unit +of work (in its specific measure) in the change from one to the other, +and back. And this fundamental knowledge of the existence of a +quantitatively unchangeable substance, arising from work, and capable of +being transformed into it, Mayer did not limit in its application merely +to heat. He was the first to construct a table, which he made as +complete as possible, of all the forms of energy then known, and to +assert and prove the possibility of their reciprocal change into each +other. + +In view of this relation of the quantitative equivalent of the various +forms of energy when transformed into one another, an attempt is being +made at present to measure them all with the _same unit_. That is, some +easily obtained quantity of energy is arbitrarily chosen as a unit and +it is determined that in every other form of energy the unit shall be +equal to the quantity obtained from that unit on its transformation into +the energy in question. For formal reasons the kinetic energy of a mass +of two grams which moves with the velocity of one centimeter in a second +has been chosen as the unit. It is called _erg_, an abbreviation of +energy. The amount is very small, and for technical reasons 10^{10} +times greater unit is used. To raise the temperature of a gram of water +one degree a quantity of energy equal to 41,830,000 ergs is required. + + +=50. The Second Fundamental Principle.= Another fundamental discovery +has been made in connection with the heat form of energy, which, like +the law of conservation, relates to all forms of energy, but has found +its first and most important application in heat. While the law of +conservation answers the question, how much of the new form of energy is +developed if a given quantity of energy changes, but gives no clue as to +when such a change occurs, this second law asserts the condition under +which such changes arise, and is therefore called the _second +fundamental principle_. + +The discovery of this law antedates _Mayer's_ discovery of the law of +conservation by about twenty years, and was made by a French military +engineer, _Sadi Carnot_, who died soon afterward without having lived to +see the recognition his great work obtained. _Carnot_ asked himself the +question, Upon what does the action of the steam engine, which had just +then come into use, depend? This led him first to the more general +question of the action of heat engines in general. He found that no heat +engine could work unless the heat dropped from a higher to a lower +temperature, just as no water wheel can work unless the water flows from +a higher to a lower level, and he determined the conditions which an +_ideal heat engine_ must fulfil, that is, a machine in which the +greatest possible value in work is obtained from heat. However, an ideal +machine of this nature can be constructed in very different ways, and +Carnot's discovery consists in the recognition of the fact _that the +quantity of work obtained from the heat unit does not at all depend upon +the peculiar construction of the ideal machine, but is determined solely +by the temperature between which the heat transition takes place_. This +follows from the following considerations: + +In the first place an ideal engine must be _reversible_, that is, it +must be capable of working both ways, changing heat into work and work +back into heat. Now, if we have two ideal engines between the same +temperatures, and if we assume that engine A produces more work from the +same quantity of heat than engine B, then let A move one way and let B +move the other way with the work obtained from A. Since B produces less +work from a given amount of heat, hence more heat from an equal amount +of work, there will in the end be more heat at the higher temperature +than was originally there. But experience teaches _that there is no +means in nature by which heat in the absence of concomitant change could +be caused to rise to a higher temperature_. Therefore an engine so +constructed as to produce this result is impossible, And B cannot be of +such a nature as to produce less work from the same quantity of heat +than A. + +The reverse is also impossible. For then we need merely couple the +engines in the reverse way in order to obtain the same effect. +Therefore, since B can do neither less nor more work than A, the two +must do the same amount of work--which was to be proved. + +It is obvious that this process of proof is similar to that by which the +law of conservation was established. Because the arbitrary creation of +energy from nothing is impossible there must be definite and immutable +relations of change between the forms of energy. Because energy at rest +does not spontaneously pass into conditions in which it can do work, +the efficiencies of the machines must have definite and unchangeable +values. If, for example, we could cause heat of its own accord to rise +to a higher temperature, we could also construct a perpetual motion +machine which would always yield work at no expense. But this perpetual +motion would not be one that creates work out of nothing, but one that +extracts it from energy at rest. A perpetual motion machine of this +nature, too, is, according to our experience, impossible, and this +impossibility forms the content of the second fundamental principle. + +On the face of it this apparently "self-evident" proposition does not +reveal how fruitful of results it is when applied to the discovery of +simple but not obvious relations. It can only be said here that the +deductions from this principle form the chief content of the extensive +science of thermodynamics, which deals with the changes of heat into +other forms of energy. We must only emphasize the fact that the +application of this law, as was already observed in stating it, is not +confined to the changes of heat alone. It is a law rather which finds +application in _all_ the forms of energy. For in every form of energy +there is a property which corresponds to temperature in heat, and upon +the equality or the inequality of which depends whether the energy in +question is at rest or ready for transformations. This property is +called the _intensity_ of the energy. In work, for instance, it is +_force_, in volume-energy it is _pressure_. If once the intensity in a +body is equal, its energy is at rest, and it never again moves of its +own accord. + +Another form in which to present these relations is to make a +distinction between _free_ energy and energy _at rest_. If we have a +heat quantity the temperature of which is higher than that of the +surrounding objects, it can be used to do work only until its +temperature has dropped to that of the surrounding objects. Although +energy in abundance is still present, there is no longer any energy +_capable of change_, or _free_ energy. Since differences of temperature, +like other differences of intensity, have a constant tendency to +diminish, the amount of free energy on earth is constantly decreasing, +and yet it is only this free energy that has value. For since all +phenomena depend upon change of energy, and change of energy is possible +only through free energy, _free energy is the condition of all +phenomena_. + + +=51. Electricity and Magnetism.= While the knowledge of heat energy goes +back to the most ancient periods of civilization, electrical and +magnetic energies are relatively young acquisitions. The highly +developed technical application of both with the rich harvests they have +yielded belongs exclusively to most recent times. + +Both these forms of energy, like those discussed above, are connected in +the main with ponderable "matter," but in a much slighter and less +regular measure. While it is not possible as yet to render any given +body free of heat (although lately the absolute zero point has been +considerably approximated), freedom from electrical and magnetic energy +is the normal condition of most bodies. This is connected with the +peculiarity that electrical and magnetic properties are decidedly +bi-symmetrical or _polar_. This property is not found in any other form +of energy, and can serve as the special scientific characteristic of +electricity and magnetism. This peculiarity shows itself in the concepts +of positive and negative magnetism, and positive and negative +electricity, and is due to the fact that two equal opposite quantities +of electricity or magnetism, when added together, do not produce double +their value, but nullify each other.[G] + +[G] For the sake of the layman it must be observed that those +"quantities" are not energy magnitudes but factors of the electrical and +magnetic energies. Energy itself in its various forms is an _exclusively +positive magnitude_, and the result of the additions of their various +amounts is always the sum, never the difference, of their numerical +values. By the negative sign is understood the energy _expended_ in +contradistinction to the energy _received_. It is therefore nothing more +than the indication of a mathematical operation. + +The fact that electrical and magnetic energies generally exist only in a +transitory state (with the notable exception of the magnetic condition +of the earth) is probably the cause of our not having developed a sense +organ for them, especially since their phenomena as they occur in nature +have only occasionally and in very rare instances (thunderstorms) an +influence upon us. On the other hand, the modern development of +electrotechnics is based upon that property of electrical energy by +virtue of which large quantities of it can be conducted along a thin +wire over great distances without any considerable loss, and at the +point desired can be easily changed into any other forms of energy. But +since the collection and conservation of large quantities of electrical +energy is hardly possible technically, the electrical apparatus must be +so constructed that the quantities each time required should be produced +at the moment they are used. The chief source of electricity is the +chemical energy of coal, which is first transformed into heat, then into +mechanical energy, and finally into electrical energy. This extremely +roundabout process is necessary because a method technically practicable +of transforming the chemical energy of coal directly into electrical +energy has not yet been invented. On the other hand, mechanical energy +can be easily and completely changed into electrical energy. Upon this +is based the exploitation of much "water power," the energy of which +could not be utilized but for the great capacity for change of the +electrical form. + + +=52. Light.= The case of light in our day seems to be similar to that of +sound, which, although it has its special sense organ in man, is yet no +particular form of energy, but has been found to be a combination of +mechanical energies in an oscillatory or mutually changing state. It +seems highly probable that light, too, is not a special form of energy, +but a peculiar oscillatory combination of electrical and magnetic +energies. It is true that the circle of proof is not yet quite closed, +but the gaps have become so small that the above conclusion may at any +rate be accepted as highly probable. + +However that may be, light is an energy which, according to the known +laws, travels through space with tremendous rapidity. We will call it +_radiant energy_, since the part optically visible, to which alone the +name light in its original sense belongs, represents an extremely small +portion of a vast field, the properties of which change quite +continuously from one end to the other. + +Radiant energy is characterized as an oscillatory or wave-like process. +So long as this fact was unknown (up to the beginning of the nineteenth +century) it was thought that light consisted of minute spherical +particles, which shot through space in a straight line with the +tremendous velocity mentioned above. Later, in order to "explain" its +wave nature, which in the meantime has come to be recognized, it was +assumed to be due to the elastic vibrations of an all-pervading thing +called _ether_, of which we know nothing else. This elastic undulatory +theory has been abandoned in our time in favor of an _electromagnetic_ +theory supported by quite considerable experiential grounds. Whether it +will be spared the fate that has overtaken the older theories (or rather +hypotheses) of light cannot as yet be predicted with any degree of +certainty. + +Radiant energy is of very marked importance in human relations. As light +it serves, with the aid of the corresponding receiving organs, the eyes, +as a more manifold means of intercommunication between our bodies and +the outer world than any other form of energy. The energy quantities +penetrating to us from the extreme limits of the world space mark the +outermost limits of which we have knowledge in any way whatsoever, and +finally the energy quantities radiating to us from the sun constitute +the supply of free energy at the expense of which all organic life on +earth is maintained. Even the chemical energy stored up in coal +represents nothing else than accumulations of former sun radiation, +which had been transformed by the plants into the permanent form of +chemical energy. + +Very recently other newly discovered forms of radiant energy have been +added to light. They are produced in manifold circumstances, and some +bodies emit them constantly. The scientific elaboration of these +extremely manifold and unusual phenomena has not yet been carried so far +that they can be reduced to a doubt-free system. But so much, it seems, +is already apparent, that they are presumably not purely new forms of +energy, but rather very composite phenomena which may yield one or more +new energies as component parts. But despite the peculiarity of these +new rays, nothing certain has as yet been proved against the law of +conservation itself. + + +=53. Chemical Energy.= Since chemical energy is only one of several +forms of energy, there seems to be no justification for allotting it to +a special science, since all the other forms of energy must be +incorporated in physics. + +But the actual existence of chemistry as a special science which has +already many subdivisions is justified in the first place by the +external fact that in practical life and in industry chemistry occupies +a very wide field comparable, if not superior, to that of the whole of +physics. In the next place, from the psychological point of view, it is +found that the chemist's methods of reasoning and working are so +different from those of the physicist that a division seems to be in +order for that reason also. Finally, there is in the nature of chemical +energy itself an important distinction which marks it off from the other +forms. + +While, for example, there is only one form of heat or of kinetic energy, +and in electricity there are only the two forms of polar opposites, +chemistry, even after the greatest theoretical reduction, possesses at +least about eighty forms. That is, it possesses as many forms as there +are _chemical elements_. The experiential law, that the elements cannot +be changed into one another,[H] also limits the corresponding changes +of the chemical energies into one another, and thus characterizes the +independence of these various forms. From this results a +disproportionately greater manifoldness of relations, which find their +expression in the many thousands of the individualized chemical +substances or combinations. + +[H] Lately changes of elements into one another have been observed in +individual instances, but in such peculiar circumstances that for the +present we need not consider these discoveries, which have only just +begun. + +This great manifoldness and the slight regularity hitherto found in +connection with the properties and reciprocal relations of the numerous +chemical elements renders modern chemistry more a descriptive than a +rational science. It was no more than twenty years ago that an earnest +and successful attempt was begun to apply the stricter methods of +physics to the investigation of chemical phenomena. These labors, so far +as they have gone, have yielded a great many far-reaching and +comprehensive principles. + +The significance of chemistry in human life is twofold. In the first +place the energy of the human body, just as that of all other living +organisms, depends chiefly upon the action of chemical energies in the +most manifold forms. Of all the physical sciences, therefore, chemistry +is the most important for biology, particularly for physiology. In the +second place, as I have emphasized a number of times, it possesses the +peculiar property which enables it to be _preserved_ for a long time +without passing into other forms and being dissipated. Furthermore, +energy in this form permits of the most powerful _concentration_. More +of chemical energy can be stored in a given space than of any other form +of energy. Both these properties, then, may be considered as the reason +why organic beings are constituted chiefly by means of chemical energy. +At any rate, it is due to these two peculiarities that chemical energy +serves as the primary source for almost all the energy used in industry. + +Further, the manifoldness of chemical energy is the cause of the +peculiar manner in which it is transformed into other forms. In the +other forms of energy the transformation can be effected by the body +itself. Nothing else is required. If a stone is thrown and it hits +against a wall, it loses its kinetic energy, the greater part of which +changes into heat. But in order to liberate the _chemical_ energy of, +say, coal, the coal _alone_ is not sufficient; _another_ chemical +substance is required, the oxygen of the air. The interaction of the two +substances produces a new substance, and it is only during this process +that a corresponding part of the chemical energy is liberated. There are +a few chemical processes also (allotropic and isomeric changes) in which +a single substance without the co-agency of another substance can give +off energy. But the quantity of energy thus obtained is infinitely +small as compared to that liberated by the interaction of two or more +substances. Because of the necessity of two or more substances to +co-operate in giving off chemical energy, the opportunity for the +transformation of chemical energy is less than for the transformation of +the other forms of energy, and this is the main reason why it can be +conserved so long and so easily. All that is necessary is to prevent +contact with another substance. This is a problem, it is true, which +from the point of view of strict theoretical rigor it is almost +impossible to solve. In practice, however, it can be easily solved for +periods of time long enough at least to require special means to enable +us to recognize that it is only a temporary and not a fundamental +solution. Scientifically expressed, the cause of this is that the +_diffusion_ of the various substances in one another can theoretically +never be completely eliminated, while on the other hand the velocity of +the diffusion over distances measured only by decimeters is extremely +low. + + + + +PART IV + +THE BIOLOGIC SCIENCES + + +=54. Life.= Among the bodies in our environment that are ponderable and +have mass the animate beings are so strikingly distinguished from the +inanimate that in most cases we have not the slightest doubt whether a +body belongs to the one kind or to the other, even if in some cases we +happen not to be familiar with its peculiar form. In the first place, +therefore, we must answer the question in a general way and tell what +the distinguishing peculiarities are that mark them off one from the +other. + +The first peculiarity is this, that living organisms are not _stable_ +but _stationary_ forms. This distinction is based upon the fact that a +stable form is at rest or unchangeable in all its parts, while a +stationary body, though it seems unchangeable in its form, internally +undergoes a constant change of its parts. Thus, a brass faucet is a +stable body, since it not only preserves its form and function +permanently, but consists at all times of the same material and shows +the same peculiarities, such as stains, defects in form, etc. It cannot +be said, it is true, that it will remain completely unchanged for all +time. Its metal suffers a gradual chemical and mechanical deterioration. +But this is not essential to the existence of the faucet, since the +deterioration varies greatly with circumstances, and if conditions are +ideal it can be reduced to zero. + +On the other hand, the jet of water flowing from the faucet is a +stationary body. In favorable circumstances it can assume a constant +form, so that at a hasty glance it might be taken for a stable glass +rod. On closer examination it will be found that the parts of water of +which it is formed are not the same at any given instant as the instant +before, each part that has flowed away being replaced by another just as +large following it. + +From this difference in the nature of the two bodies results a +difference in their behavior. If I make a mark on the faucet with a +file, the mark remains permanent. But even if I sever the entire water +jet with a knife, the cut is healed the next moment, because by reason +of the continuous flow of the water, the severed place is instantly +eliminated from the body. Owing to this nature peculiar to stationary +bodies, they have the capacity of _being healed_ or of _regeneration_. + +For a body to continue permanently in a stationary condition the +material of which it is composed must be permanently _supplied_. If we +turn off the faucet, the water jet immediately disappears or "dies." +Evidently, therefore, a stationary body can subsist by its own means +only if it has the property or capacity to provide itself continually +with the necessary material. This material consists in the main of +ponderable or chemical substances of definite physical and chemical +properties, and thus the _change of substance_, _metabolism_, appears as +a necessary property of the stationary body. In order, however, that +metabolism should take place we must have free _energy_, or energy +having the capacity to work, since it is only free energy that can cause +substances to change, just as every phenomenon in the world implies the +equalization of free energy. For a stationary body to exist +independently, therefore, it must have the property of being able +spontaneously to possess itself of the necessary substances and of free +energy. But since, as we have already said, the energy of organisms is +stored up and used in the main in the form of chemical energy, the two +tasks which a stationary body has to perform, that of meeting the need +for substances and for energy, are as a rule externally combined. In +organisms these two necessities combined are called _nutrition_, and +thus we recognize in the capacity for _self-acquisition of nutrition_ +another essential property of organisms. + +A third essential property of organisms is the capacity for +_reproduction_, for the bringing forth of similar beings. It is never +impossible that the balance between the receipts and expenditures of a +stationary body should, in consequence of some external causes, be +disturbed, even when under normal conditions it possesses the property +of self-nutrition. If the disturbance remains below a certain point, +then, as we have already stated, regeneration sets in. But the +disturbance may rise above that point, in which case the body ceases to +exist, or dies. Then a similar body will not arise unless the manifold +necessities that have led to the origin of the first will combine again +to produce the second. That such a thing is possible, that, in fact, it +often happens, is shown, for example, by the waves of the ocean, which +have a stationary character since, while they are composed of constantly +changing masses of water, their form remains unchanged. The waves are +destroyed in the breakers, but arise again and again through the action +of the wind upon the surface of the water. But the more complex such +bodies are the less easily they are formed, while once they have been +formed and have found the conditions of their existence, their +preservation is much easier. + +Beings, therefore, which have the capacity to form similar bodies out of +themselves regularly and at the right time can preserve their species +much more easily than those in which this property is absent. Death has +to a great extent lost its power over beings capable of reproduction. By +way of illustration let us take another stationary thing, a flame. A +flame is not an organism because it is not self-sustaining. Yet it +multiplies itself. And while a single little flame soon dies out, the +sea of flame of a burning forest, which started from a single small +flame, is well-nigh inextinguishable, and it cannot be fought in any +other way than by letting it die its natural death and burn to the end. + +Thus, while the fulfilment of the first two conditions, the stationary +change and the self-supply of food, could produce bodies, which would be +able to exist for a longer or shorter period, but which at some time +would have to give way to other bodies of different form and nature, the +capacity for reproduction creates the condition that forms of the _same +species_ continue to exist even after the existence of the individual +has ceased. + +These three properties constitute the essential characteristics of +animate things or organisms. + +That the organisms are all constructed upon the basis of chemical energy +is a fact of experience which may be understood to imply that the other +forms of energy are not capable of producing the above-mentioned +conditions. This is due to the properties of chemical energy to which I +have already called attention: its great concentration and, at the same +time, its capacity for prolonged preservation. That chemical energy is +the only form of energy suitable to life is obvious from the fact that +in airship navigation, for example, the kinetic energy required for +steering can be supplied only in the form of gasoline or hydrogen, that +is, in the form of chemical energy, because any of the other forms would +be much too heavy. The flight of a bee or the swimming of a dolphin +cannot be conceived of except as brought about through chemical energy. + +That this chemical energy is essentially that of _carbon_ has also been +established by experience, although it is not quite universal, for the +sulphur bacteria found their household upon the energy of sulphur. The +cause of the preference of carbon is again to be sought in its special +fitness for the purpose, due, on the one hand, to its wide distribution, +and, on the other hand, to the exceeding manifoldness of its +combinations. + +Finally, the construction of the organisms from a peculiar combination +of solid and liquid substances can be proved to be equally due to +technical relations. + +These three last-named peculiarities are therefore to be regarded as the +special characteristics of the organisms with which we are acquainted on +the surface of the earth in the conditions there prevailing. We need not +regard them conceptually as unchangeable or irreplaceable. But the first +three characteristics, namely, the stationary nature, self-supply of +nutrition, and reproduction, we may regard as the _essential +characteristics of organisms_. They constitute the frame within which +everything must be found which we should recognize as living in the +widest sense. + + +=55. The Storehouse of Free Energy.= If we ask whence the organisms +obtain the free energy which they require for the maintenance of their +stationary existence, the answer is that _solar radiation_ alone +furnishes this supply. Without this permanent supply the free energies +upon the earth, so far as our knowledge goes, would long ago have +reached a state of equilibrium, and the earth's bodies would be stable, +that is, dead and not stationary and living. + +It is comprehensible, therefore, that machines should have evolved in +the organism for _transforming the radiant energy of the sun into a +permanent form_, and, as we have already learned, chemical energy is +permanent, while radiant energy is an extremely transitory form of +energy, that is, it changes very readily. The very fact that, owing to +the change from day to night, the supply of radiant energy periodically +ceases, makes the storing-up of energy for the night necessary to the +existence of a form dependent upon it. Thus, we recognize in the +_photochemical_ processes, that is, in the transformation of radiant +energy into chemical energy, the foundation of life on earth. + +This work is done by the plants, which thus provide a store of free +energy not only for their own needs but also for all the other organisms +which possess themselves directly or indirectly of the plant-chemical +supplies in order to utilize them for their individual purposes. In this +manner nourishment in the widest sense is secured for all organisms, +being based upon the regular supply of free energy derived from the +sun. This also explains the great chemical similarity of all organisms, +which could not subsist if they were not so constructed as to be able to +utilize the chemical energy in the form in which it is provided by the +plants. + +Of the great stream of free energy poured out from the sun into cosmic +space the earth receives an extremely small portion (corresponding to +the bit of space it occupies in the heavenly sphere as seen from the +sun), and the plants collect and store up only a very small fraction of +this portion received by the earth. Measurements have shown that in most +favorable circumstances a plant leaf changes only about 1/50 of the +radiant energy it receives into chemical energy. If we consider that +only a small part of the surface of the earth is covered with plants and +that during the winter no energy from the sun is stored up at all, we +perceive what infinite possibilities for development there still are in +arresting and storing up free energy. The part stored up by the plants +flows from these into the countless streams, brooks, and veins of the +other organisms, to end finally as used-up energy, or energy at rest. +This energy is at rest, it is true, only in relation to the earth's +surface. We do not know whether the radiation from the earth, which at +present amounts to about as much as the radiation from the sun to the +earth, is in its turn somewhere utilized. + +While the free energy is poured out in such a stream in one direction, +the ponderable substances of which the organisms are made up _circulate_ +through plants and animals and back again. This is especially true of +_carbon_, which is freed from its combination with oxygen, that is, from +carbonic acid, by the sun energy transformed in the plants. While carbon +serves to build up the plant body and represents its supply of chemical +energy, the oxygen is returned to the air. These two substances are +again chemically combined in the various organisms and the quantities of +energy which were necessary for their decomposition are again available +for the manifold functions of life. The product of the chemical +combination, carbonic acid, returns to the air and is ready for renewed +decomposition in the plants. + +Thus, the entire mechanism of life can be compared to a water-wheel. The +free energy corresponds to the water, which must flow in one direction +through the wheel in order to provide it with the necessary amount of +work. The chemical elements of the organisms correspond to the wheel, +which constantly turns in a circle as it transfers the energy of the +falling water to the individual parts of the machine. + + +=56. The Soul.= Our observations so far have shown the organisms to be +extremely specialized individual instances of physico-chemical machines. +Now we have to take into consideration a property which seems markedly +to distinguish them from the lifeless machines, and which we have +already encountered in the very beginning of our treatise. + +It is the property which we there called _memory_, and which we will +define in a very general way as the quality by virtue of which the +repetition in organisms of a process which has taken place a number of +times is preferred to new processes, because it originates more easily +and proceeds more smoothly. It is readily apparent that by this property +the organisms are enabled to travel on the sea of physical possibilities +as if equipped with a keel, by which the voyage is made stable and the +keeping of the course is assured. + +If we ask whether this is exclusively a quality of organisms the +question cannot be answered affirmatively. Inanimate bodies also have +something like the quality of adaptation. An accurate clock attains its +valuable qualities only after it has been going for some time, and the +best violin is "raw" until it has been "broken in." An accumulator must +be "formed" before it can do its normal amount of work. All these +processes are due to the fact that the repetition of the same process +improves the action, that is, it facilitates or increases it. + +Adaptation or memory, then, is not limited to organisms. In inanimate +things, however, this property is comparatively rare. Memory, therefore, +is to be regarded as another property of organisms representing an +extreme specialization of the inorganic possibilities. This is an +important point of view for what follows. + +In the first place, this property of adaptation facilitates and assures +nourishment. If we take the fundamental idea developed by Darwin, that +that predominates in the world which by virtue of its properties endures +the longest time, then it is evident that a body which teleologically +preserves and elaborates its nourishment will live longer than a similar +body without this property. Moreover, by the general process of +adaptation, these "teleological" properties come to be more greatly +developed and more readily exercised in the body that lives longer, so +that its long life gives it another advantage over its rival. Thus we +can understand how this property of adaptation, which at first is to be +conceived of as a purely physico-chemical quality is found developed in +all organisms. + +In its most primitive forms the quality of adaptation gives rise to the +_phenomena of reaction_, or to _reflex_ actions, that is, to a series of +processes in the organism in response to the stimulus of an outward +energy. This response is made in furtherance of the life of the +organism. Reactions that serve a certain end, that is, teleological +reactions, can naturally be developed to such stimuli alone to which the +organism is frequently and regularly subjected. This is why adaptation +to unusual phenomena is generally lacking, and in relation to them the +organisms are often extremely unfit. The typical example of this is the +moth, which flies into the light and is burned. + +As the reactions become more fixed they develop into longer and more +complicated series, which then appear to us as _instinctive actions_. +But here, too, we find the characteristic unsuitability when unwonted +circumstances arise, even if the teleologic reactions to stimuli become +more manifold. + +Finally, there are the _conscious acts_ which appear to us to be the +highest degree of the series. It is with the teleologic regulation of +these conscious acts, including the very highest activities of mankind, +that this book deals. They are distinguished from instinctive action by +the fact that they no longer proceed in a single and definite series, +but are combined at need in the most manifold ways. But the fundamental +fact, namely, that actions are based upon the repetition of coinciding +experiences, at once appears here also, since the basis of the entire +conscious life of the soul, the formation of _concepts_, is made +possible only through _repetition_. Thus, we are justified in regarding +the various degrees of mental activity from the simplest reflex +manifestation to the highest mental act as a connected series of +increasingly manifold and purposive actions proceeding from the same +physico-chemical and physiological foundation. + + +=57. Feeling, Thinking, Acting.= For good reasons it is generally +assumed that the organisms have not always been what they are now, but +have "developed" from previous simpler forms. It is undecided whether +originally there were one or several forms from which the present forms +sprang, nor is it known how life first made its appearance on earth. So +long as the various assumptions with regard to this question have not +led to decisive, actually demonstrable differences in the results, a +discussion of it is fruitless, and therefore unscientific. The usual +word evolution is non-purposive in so far as it signifies the appearance +of something already existing. Another conception is better according to +which the influence of _changed_ conditions of existence has yielded the +most important factor of change. + +The change that the organisms undergo is always in a definite direction. +More and more complex and manifold forms are evolved, and the evolution +of these forms is characterized by an ever greater specialization of the +functions of life, so that every specially developed organ comes to +perform but one function. It is true that by this means the organism is +better fitted to perform those functions, but at the same time it grows +more susceptible to injury, since its existence depends upon the proper +simultaneous activity of many different organs. Such an evolution, +therefore, can occur only when the general conditions of life have grown +steadier, so that the danger of disturbance becomes less. We are +accustomed to regard changes in this direction as higher developments, +and the progressive simplifications of the organization (as for example +in parasites) as backward steps. + +Since our opinion as to what constitutes a higher and a lower organism +is doubtless arbitrary, let us ask whether it is not possible to find an +_objective_ standard by which to measure the relative perfection of the +different organisms. The question must be answered in the affirmative +when we take into consideration the following. Since the quantity of +available free energy upon the earth is limited, the organism which +transforms the energy at its disposal more completely and with the least +loss into the forms of energy necessary for the function of life, must +be regarded as the more perfect organism. In fact, we observe that with +increasing complexity of the organisms there is for the most part also +an increasing improvement in that direction, and we can therefore speak +of some beings as more perfect than others. This view-point is +especially significant in the evaluation of _human_ progress, appearing, +as it does, as the general standard of all civilization. + +The perfection of the organism shows itself in relation to the outer +world in the development of the _sense organs_. While a single-celled +animal reacts almost exclusively to chemical, sometimes also to optical, +stimuli, and receives these with the entire surface of its body, special +parts of the body develop more and more toward perfection. These are the +parts that respond with special ease to the appropriate stimuli, that +is, react to them with an increasingly smaller expenditure of energy. +Then the points at which the stimuli are received are separated from +those in which the reaction occurs, and the two are connected by +_conducting paths_, the nerves, in which an energy process takes place. +Our present knowledge of this process still leaves much to be desired. +It is a process which moves with fairly great but by no means +extraordinary rapidity (about ten to thirty meters per second) along the +conducting paths. At the one end of this path it is caused by actions of +various kinds, chiefly that of the specific energy, for which the sense +organ is developed. At the other end it discharges specific effects. +There is no doubt that here we have in each instance a case of energy +transformation connected with a _discharge_, that is, with the action of +other energies which lie at the ends ready for change. Hence there is no +equivalence between the different kinds of energy, the discharging and +the discharged, mostly not even a proportional relation, although both +increase and decrease simultaneously. + +What the form of the energy is that is propagated in the nerves is +unknown. It can be either a special form which arises only under the +conditions here present (just as, for example, a galvanic stream +develops only under definite chemical and spacial conditions), or a +special combination of known energies, as in sound and probably in +light. Some day, it is likely, we shall have a more accurate knowledge +of the nerve process which will solve the question. + +When such a process is caused by some energy impulse from without, it +may produce various results. In the simplest case it discharges the +corresponding reaction, just as the leaves of the sensitive plant close +when they are touched. Or it may give rise to a series of processes +following one another like the instinctive actions. Or, finally, it may +bring about a series of inner processes which lead to an extreme +differentiation of slight differences of this stimulus and to a +corresponding graded reaction with the anticipation of success. We call +this conscious thinking, willing, and acting. + +Through the age-long effect of the blunder committed by Plato in making +a fundamental distinction between mental life and physical life, we +experience the utmost difficulty in habituating ourselves to the thought +of the regular connection between the simplest physiological and the +highest intellectual acts. Moreover, this contrast has been accentuated +by the mechanical hypothesis. If we abandon the mechanical hypothesis +and adhere to the summarization of experience free from all hypotheses, +as represented in the science of energy, this contrast disappears. For +even if we concede the impossibility of conceiving thought as +_mechanical_, there is no difficulty in conceiving of it as _energetic_, +especially since we know that mental work is connected with expenditure +of energy and exhaustion just as physical work is. However, the +elucidation of this subject lies almost entirely in the future since the +idea just developed has but only begun to influence scientific work in +this field. But judging from the results that have already been obtained +we may hope for a speedy development. + + +=58. Society.= The external circumstance that as an organism multiplies +the new being must come to life in the proximity of the older one, is in +itself cause for the formation of closed groups confined to certain +localities by animal organisms of the same species. But they become +scattered if the advantage of their living together is not such as to +outweigh the disadvantage of having a narrow field of competition for +the means of sustenance. Thus we see different plants and animals +behaving differently in this respect. While some species live in as +great isolation as possible, others form communities, even if there is +no mechanical tie to hold them together by a common integument. + +Since the second case is true of man in a highly marked degree, his +_social_ characteristics and needs form a large and important part of +his life. And since, further, the socialization of man makes continuous +headway with increasing civilization--we need but think of the +development of the former little groups and tribes into states and the +present very active internationalization of the most important affairs +of mankind, especially of the sciences--the social problems also +occupy an ever larger place in the organization of human life. + +What distinguishes man most essentially from the other animals, even the +most advanced, is his capacity for perfection, which in the lower animal +can be paralleled at best by its capacity for _self-preservation_. While +the organization of the animals within the short period of which we have +any historical knowledge has to all appearances remained essentially +unchanged, the world of mankind has changed in quite a remarkable way. +This change consists in an increasing subjection of the external world +to human purposes, and rests upon the increasing socialization of his +capacities. + +Memory and heredity (the latter being but an extension of memory to the +offspring, which is to be conceived of as a part of the older organism) +secures in the first place only the preservation of the stock and the +renewed development of the new individual in the average type. If a +specially favored individual succeeds in accomplishing greater +achievements, he may in favorable circumstances transmit this capacity +for higher attainments to his offspring. But such individuals gain an +advantage in the struggle for existence only if the other sides of their +activity do not suffer curtailment as a result. With the limited amount +of energy at the individual's disposal every extraordinary +accomplishment involves a corresponding _one-sidedness_, and as soon as +a certain measure is slightly overstepped, it will cause a reduction of +the other functions which will render the individual less fit in the +struggle for existence. But this is true only so long as an individual +must live _by himself_. As soon as he forms part of a social +organization which benefits by his particular activity, the organization +compensates for the personal disadvantages by its collective activity, +and a social community not only finds room for such special +developments, but it even encourages and promotes them. + +We have already seen that such manifestations occur within the organism +itself. Higher functions, depending upon the higher susceptibility of +the sense organs, can only be attained at the expense of the general +functions by the organ in question. We observe this fact in all socially +organized beings, like bees and ants, which display a high degree of +specialization in the functions of the individual subordinate groups; +the specialization often being carried so far that the individual groups +can no longer subsist by themselves alone. It is only the organization +as a whole that is capable of permanent existence. + +While the evolution of such superior functions involves a corresponding +differentiation, and therefore a _division_ and _separation_ of the +superior functions within the social structure, the necessity for +_communication_ and for _mutual support_ results in an _approximation_ +of the individuals and the groups. In every society, therefore, the +centrifugal and the centripetal forces work simultaneously in +co-operation and in opposition to one another. While the extreme +specialization on the one hand seems to make for the best individual +functioning, on the other hand it renders the entire collective +structure much more dependent, and therefore much more subject to +injury, as is shown by the example of the queen bee, whose departure +threatens the existence of the entire hive. Thus a medium degree of +differentiation will as a general rule produce the most permanent social +structure. + + +=59. Language and Intercourse.= The essential value of the social +organization resides in the fact that the work of the individual, in so +far as it is adapted to it, accrues to the benefit of the collective +whole. For this it is absolutely essential that the members of the +collectivity should be able to _have intercourse_ with one another in +order that every part of the general activity may be communicated to the +others. This intercourse is obtained through language in the most +general sense. + +We have already learned that the essence of language consists in the +co-ordination of concept to sign. The social application of language +demands that the signs co-ordinated to the concepts in use should be the +same for all the members of the social organization. Only in this way +can the members make themselves mutually understood. But intelligible +means of communication and division of labor impart to the social +knowledge that is set down in writing a kind of independent existence. +Many centuries ago the possibility ceased for one person to store in his +memory the entire stock of human knowledge. Nowadays we have men who are +versed only in single parts of separate sciences, and the aggregate +knowledge appears at first to be a unity existing only in thought. But +because this knowledge is set down in signs which endure far beyond the +life of the individual and at the appropriate moment can unfold its +entire power even after a long period of inactivity, it has acquired an +existence of a social character independent of the individual. For +although it survives the individual, it cannot survive the death of +human society. + +As the socialization of all mankind advances to ever greater unities, +the linguistic limitations sprung from former stages of evolution prove +to be a hindrance. The mother tongue, of course, forms the first and +most important entry for the individual to the common store of +knowledge. But in view of the linguistic limitation of which I have just +spoken the efforts in our day are carried on with renewed zeal to create +a _universal auxiliary language_ (p. 100) by means of which intercourse +should be made possible beyond the language boundaries. There have +already been gratifying results.[I] + +[I] At the present time "Ido" is the best. It is a highly practicable +artificial language, and its advocates have succeeded in organizing it +to insure its normal development. An older and still rather widespread +form called "Esperanto" has failed to organize itself so as to insure +its development and it must inevitably die out. + + +=60. Civilization.= Everything which serves the social progress of +mankind is appropriately called civilization or culture, and the +objective characteristic of progress consists in improved methods for +seizing and utilizing the raw energies of nature for human purposes. +Thus it was a cultural act when a primitive man discovered that he could +extend the radius of his muscle energy by taking a pole in his hand, and +it was another cultural act when a primitive man discovered that by +throwing a stone he could send his muscle energy a distance of many +meters to the desired point. The effect of the knife, the spear, the +arrow, and of all the other primitive implements can be called in each +case a purposive transformation of energy. And at the other end of the +scale of civilization the most abstract scientific discovery, by reason +of its generalization and simplification, signifies a corresponding +economy of energy for all the coming generations that may have anything +to do with the matter. Thus, in fact, the concept of progress as here +defined embraces the entire sweep of human endeavor for perfection, or +the entire field of culture, and at the same time it shows the great +scientific value of the concept of energy. + +If we consider further that, according to the second fundamental +principle, the free energy accessible to us can only decrease, but not +increase, while the number of men whose existence depends directly on +the consumption of a due amount of free energy is constantly on the +increase, then we at once see the objective necessity of the development +of civilization in that sense. His foresight puts man in a position to +act culturally. But if we examine our present social order from this +point of view, we realize with horror how barbarous it still is. Not +only do murder and war destroy cultural values without substituting +others in their place, not only do the countless conflicts which take +place between the different nations and political organizations act +anticulturally, but so do also the conflicts between the various social +classes of one nation, for they destroy quantities of free energy which +are thus withdrawn from the total of real cultural values. At present +mankind is in a state of development in which progress depends much less +upon the leadership of a few distinguished individuals than upon the +collective labor of all workers. Proof of this is that it is coming to +be more and more the fact that the great scientific discoveries are made +simultaneously by a number of independent investigators--an indication +that society creates in several places the individual conditions +requisite for such discoveries. Thus we are living at a time when men +are gradually approximating one another very closely in their natures, +and when the social organization therefore demands and strives for as +thorough an equalization as possible in the conditions of existence of +all men. + + + + +INDEX + + + Above and below, distinction between, 121 + + Abstract, concrete and, 16 ff. + + Abstraction, 20 + + Action, conscious, 174; + instinctive, 174 + + Adaptation, 172 ff. + + Aeromechanics, 147 + + Algebra, 80 + + Alikeness, definition of, 51 ff. + + Allotropic changes, 161 + + Analysis, infinitesimal, 111 + + Analytic geometry, 122 ff. + + Analytic judgments, 66 + + Anthropology, 57 + + Ants, specialization of, 181 + + Applied sciences, 57 ff. + + _A priori_ judgments, 44 + + Aristotle, 38, 66 + + Aristotle's logic, 22 + + Arithmetic, 79 ff. + + Assertions, never absolutely correct, 53 + + Association, 63 ff., 81 + + Astronomic objective, 6 + + Astronomy as an applied science, 58 + + Atomic hypothesis in chemistry, 142 + + Atoms, 141 + + + Bees, specialization of, 181 + + Biological sciences, 55; + life most general concept in, 56 + + Botany, 56 + + + Cæsar, Julius, 76 + + Capillary phenomena, 146 + + Capillary theory, 147 + + Carbon, its circulation through plants and animals, 171; + life based on the energy of, 168 + + Carbonic acid, 171 + + Carnot, Sadi, 151 + + Causal relation, purification of, 34 ff. + + Causation, the law of, 31 ff. + + Chemical combinations, 71 ff.; + quantitative relations in, 74 + + Chemical energy, 159 ff.; + capable of powerful concentration, 161; + different forms of, 159 + + Chemical formulas represent concepts not sounds, 95 + + Chemistry, 20, 47, 55; + significance of, 160 ff. + + Chinese script based on direct co-ordination, 93 + + Civilization, 184 ff. + + Classification, not definite, 2; + purpose of, 2-4 + + Classification of the sciences, 53 ff. + + Collective activity, 181 + + Combination, sequence in, 73 ff. + + Combinations, theory of, 71 + + Combinatory schematization, 73; + in chemistry, 71 ff.; + in physics, 72 + + Communication, 181 + + Community among plants and animals, 179 + + Comparison, 82 + + Comte, Auguste, 54 + + Concept, the most general, 61 ff. + + Concepts, arbitrary, 23; + complex, 23; + complex empirical, 23; + definition of, 16; + empirical, 18; + formation of, 19; + general, 26; + in ceaseless flux, 88; + science of, 15 ff., 122; + simple, 20; + simple and complex, 19 ff. + + Conclusion, the, 24 ff.; + analytic, 66; + scientific, 27, 30, 66 ff. + + Concrete and abstract, 16 ff. + + Conjugacy, most general concept in formal sciences, 56 + + Conscious action, 174 + + Conscious thinking, willing, and acting, 178 + + Conservation of energy, the law of the, 135 ff. + + Conservation of matter, 138 + + Conservation of the sum of work and kinetic energy, the law of the, 134 + + Conservation of work, the law of the, 130 + + Conservation, quantitative, 131 + + Continuity, 101 ff.; + the law of, 113 ff. + + Co-ordinated signs, change in, 88 ff. + + Co-ordination, 80 ff.; + a means of obtaining facts without dealing directly with the + corresponding realities, 87; + between concept and word not unambiguous, 89; + between concept and written sign, direct and indirect, 92 ff.; + identity the limit case in, 82; + integral numbers the best basis of, 85; + in use among primitive men and higher animals, 87; + its importance, 85; + methodology of the sciences based upon, 85; + of numbers with signs, 90 ff.; + possibility of unambiguous, 88 + + Copernican theory, 117 ff. + + Copernicus, 117, 141 + + Corpuscular theory of light, 5, 157 + + Counting, 85 ff.; + defined, 85; + purpose of, 86 + + Culture, see Civilization + + + Darwin, his fundamental theory, 173 + + Deduction, 40 ff.; + the process of, 41 ff. + + Deductive sciences, 38 + + Determinateness, absolute, only in ideal world, 50 + + Determinateness of things, the, 47 ff. + + Determinism, 48, 51 + + Differential Calculus, see Differentials + + Differentials, 112 + + Double numbers or double points in a group, 82 + + Dynamics, 128 ff.; + definition of, 139 + + + Elasticity, 145 + + Elastic undulatory theory of light, see Wave theory of light + + Electricity, principal source of, 156 + + Electricity and magnetism, 154 ff. + + Electromagnetic theory of light, 157 ff. + + Electrotechnics, 156 + + Empirical sciences, 38 + + Energetic mechanics, 138 ff. + + Energy, a substance, 136; + at rest, 154; + free, 154; + importance of concept of, 128; + in nerves, 177; + the most general concept in the physical sciences, 56; + of form, 145; + of volume, 145 + + Energy intensity, 153 + + Erg, definition of, 150 + + Esperanto, 183, note + + Euclid, 44, 140 + + European-American scripts based on indirect co-ordination, 93 + + Experience, incompleteness of, 27; + more limited than the conceivable, 118 + + Experiences, distinguished by _being familiar_, 31; + limited knowledge of, 31 + + Experiential sciences, see Empirical sciences + + Extrapolation, 46, 50, 104 + + + Familiarity due to recalling former similar experiences, 11 + + Fechner, 102 + + Feeling, thinking, acting, 174 ff. + + Force, 129 ff., 153 + + Formal sciences, 54; + are empirical sciences, 55; + order most general concept in, 56 + + Foucault's pendulum experiment, 121 + + Freedom of the will, 50 ff. + + Frequency of process facilitates repetition, 11 ff. + + Function, 109 ff.; + continuous and discontinuous, 110; + most general concept in formal sciences, 56 + + Functional relation, the application of the, 112 ff. + + Functions, the theory of, 111 + + Fundamental principle, the second, 150 ff. + + + Gases, 145 + + Generalization, suitable place for, in text-books, 9 ff. + + Geometry, 47, 54, 119, 127; + ancient and modern methods of, 110 ff. + + Goethe, 99 + + Good usage in language, 100 + + Grammatical correctness, importance attached to, 99 + + Grammatical rules, 97 + + Gravitation potential, the, 112 + + Group, the, 65 ff.; + double members or double points in, 82; + linear arrangement of members of, 75 ff. + + Groups, artificial and natural, 69 ff.; + closed, among animals, 179; + infinite, equality of, 84; + related, 69 ff.; + unequivocal order of, 83 + + + Heat, mechanical equivalent of, 149; + theory of, 147 ff. + + Heat energy, 148 ff. + + Heat engine, 151; + ideal, 151 ff. + + Heat quantity, 148 ff. + + Heliotrope, 90 + + Herbart, 102 + + Heredity, 180 + + Higher analysis, 111 + + Homonym, 89 + + Hydromechanics, 147 + + + Ideal cases, 44 ff. + + Ideal machines, 132 + + Identity, the limit case in co-ordination, 82 + + Ido, 183, note + + Imperfection, indestructible quality of science, 4 + + Incompleteness, no hindrance to efficiency of science, 5 + + Indestructibility of matter, see Conservation of matter + + Indo-Arabic notation, 91 + + Induction, 38; + the complete and the incomplete, 39 + + Inductive sciences, 38 + + Inference, by induction, 38; + from analogy, 140 + + Infinitesimal analysis, 111 + + Inorganic world, lack of memory and foresight in, 33 + + Insoluble problems, 142 + + Instinctive action, 174 + + Intercourse, language and, 182 ff. + + Isolation among plants and animals, 179 + + Isomeric, 74 + + Isomeric changes, 161 + + + Judgments, analytic, 66 + + + Kant, 44, 66, 105 + + Kepler, 141 + + Kinetic energy, 132; + and work, their sum constant, 133 ff.; + transformed into work and _vice versa_, 134 + + Knowledge, aim of, 19; + individual, compared to telephone, 7 ff.; + limited, 31; + possibility of error in, ineradicable, 40; + social character of, 183 + + + Language, beginnings of, 88; + defective in co-ordination, 96; + distinction between science and knowledge of, 98; + good usage in, 100; + and intercourse, 182 ff.; + needless inflections in, 99 ff.; + of words more imperfect than written language, 92; + purpose of its cultivation, 99; + the science of, 97 ff.; + unambiguity the ideal of, 89; + a universal auxiliary, 100; + written, 89 ff. + + Leibnitz, 88; + his doctrine of pre-established harmony, 143; + inventor of differentials, 112 + + Life, 163 ff.; + the most general concept in the biological sciences, 56 + + Light, 5, 156 ff. + + Liquids, 145 + + Locke, John, 21 ff., 88; + his elaboration of the notion of simple and complex "ideas," 21; + his secondary qualities, 127 + + Logic, 54, 67 ff.; + content of, 19; + definition of, 15 ff. + + Luther, 99 + + + Magnetism, electricity and, 154 ff. + + Man, compared to pair of sieves, 34; + his capacity for perfection, 180 + + Manifold, the science of the, 54 + + Mass, 132 ff., 136 ff.; + a substance, 138 + + Mathematical laws, accuracy of, 105 + + Mathematics, 54; + an empirical science, 55; + influence on, of concept of continuity, 111; + its progress after introduction of Indo-Arabic numerals and algebraic + signs, 101 + + Matter, definition of, 138 + + Mayer, Julius Robert, 149; + his discovery of the law of conservation, 151 + + Measurement, 107 + + Mechanical energies, 144 + + Mechanics, 55, 128 ff.; + complementary branches of, 144 ff.; + definition of, 138; + early development of, 139; + energetic, 138 ff.; + the first branch of physics treated mathematically, 139; + pure or classical, 144 + + Mechanistic hypothesis, the, as an interpretation of all + natural phenomena, 142; + especially injurious in study of mental phenomena, 142 + + Mechanistic theories, 140 ff. + + Mechanistic theory of the universe, 132 + + Mechanization of astronomy, 141 + + Memory, 16, 32, 180; + definition of, 172; + general characteristic of, 61; + lack of, in inorganic world, 53 + + Metabolism, 165 + + Methodology of the sciences based upon co-ordination, 85 + + Microscope, 6 + + Motion, the science of, 54, 122; + uninfluenced, 122 + + Musical notation, 93 + + + Names, arbitrariness of, 62; + signs and, 86 ff. + + Natural laws, 28 ff.; + definition of, 28; + their extent dependent upon stage of knowledge in each science, 7; + their usual origin, 42 ff.; + prediction from, only approximate, 48 + + Natural philosophy, definition of, 1; + importance of, in study of science, 10; + place of, in text-books, 9 ff. + + Negation, 68 ff. + + Nerves, 177 + + Nervous discharge, 177 + + Newton, Sir Isaac, 141 + + Number groups, unlimited, 78 + + Numbers, 78 ff.; + theory of, 80 + + Numerals, co-ordination of, with signs, 86 + + Numerical names different in different languages, 86 + + Numerical signs international, 86 + + Nutrition, 165 + + + Objective, astronomic, 6; + photographic, 6 + + Objective character of the world, 34 + + Optical telegraph, 90 + + Optics, geometric, 5 + + Optic signs, 90 + + Order, most general concept in formal sciences, 56 + + Organisms, standard for measuring relative perfection of, 176; + stationary forms, 163 + + Orthography, efforts to improve, 99; + English, defective in co-ordination, 96; + exaggerated importance of correctness in, 99; + mistakes in, 97; + reform of, 97 + + + Parabolic curve, 48 + + Paradoxes of the infinite, 84 + + Pasigraphy, 92 ff.; + Chinese system of, 94 + + Permanent in change, the, 131 + + Perpetual motion, 130 + + Perpetual motion machine, 153 + + Philology, 97 ff. + + Philosophy, limited progress in, 101 + + Phonetic writing, 33 ff. + + Phoronomy, 54, 119, 122, 127 + + Photochemical processes, foundation of terrestial life, 169 + + Photographic objective, 6 + + Physical sciences, 55 + + Physics, 47, 55; + each branch of, treats of a special kind of energy, 139 + the science of the different kinds of energy, 72; + + Physiology, 55 ff. + + Plato, his distinction between mental and physical life, 178 + + Polarity of electricity and magnetism, 155 + + Political organizations, conflicts between, 185 + + Prediction, 12 + + Pre-established harmony, 143 + + Pressure, 146, 154 + + Progress, depends on collective labor, 185; + economy of energy, 184; + evaluation of, 176 + + Pseudo-problems in science, 142 + + Psychology, 47, 55 ff. + + Psycho-physical parallelism, 143 + + Ptolemy's system, 117 + + Pure science, 57 + + + Quantity, the science of, see Mathematics, 54 + + + Radiant energy, 157; + its importance to man, 158 + + Rational sciences, see Deductive sciences + + Rays, straight lines of, 5 + + Reaction, teleological, 173 + + Reality, 16 ff. + + Reflection, 5 + + Reflex action, 173 + + Refraction, 5 + + Repetition, basis of conscious life, 174 + + Reproduction, 165 ff. + + Roman notation, 91 + + + Science, aim of, 13 ff.; + comparison of, to a network, 42; + comparison of, to a tree or forest, 6; + definition of, 13; + eternal truth of, 6 ff.; + "for its own sake," 13 ff.; + the facts of, unalterable, 8 ff.; + the function of, 23, 37; + importance of theoretical, 15; + its procedure, 45; + the study of happiness, 28 + + Sciences, the table of the, 54 ff. + + Scientific discoveries, independent simultaneous, 185 + + Scientific instinct, 43 + + Scientific materialism, 138 + + Scientific written language based on direct co-ordination, 93 + + Self-preservation, 180 + + Sense organs, 176 ff. + + Shakespeare, 99 + + Signs and names, 86 ff. + + Social characteristics, importance of, 179 ff. + + Social classes, conflicts between, 185 + + Socialization of human capacities, 180 + + Social order still barbarous, 185 + + Social organization, 180; + how best obtained, 182; + its tendency to equalize conditions, 185; + secures permanence among specialized individuals, 181 + + Social problems, 179 ff. + + Society, 179 ff.; + centrifugal and centripetal forces in, 181 ff.; + division of functions in, 181 + + Sociology, 47, 55, 57 + + Solar radiation, 169 + + Soul, the, 171 ff. + + Sound signs, advantage and disadvantage of, 89 ff. + + Sound writing, 33 ff., 92 ff. + + Space, four-dimensional, 77, note; + homogeneity of, in + horizontal direction, 121; + the science of, 54; + symmetrical and tri-dimensional, 118; + time and, 118 ff.; + tri-dimensional, 76 + + Specialization, one-sidedness of, 180 ff. + + Spelling reform, 97 + + Stable forms, 163 + + Statics, 128 ff.; + definition of, 138 ff. + + Stationary bodies, capable of regeneration, 164 + + Stationary forms, 163 + + Substance, 132 + + Surface-energy, 146 + + Syllogism, the, classic method of argumentation, 65 ff. + + Synonym, 89 + + + Table of the sciences, 54 ff. + + Telegraph, optical, 90 + + "Teleological" properties of organisms, 173 + + Teleological reaction, 173 + + Telescope, 5 + + Temperature, 148 + + Theoretical science, importance of, 15 + + Theory of functions, 111 + + Theory of numbers, 80 + + Thermo-chemistry, 37 + + Thermo-dynamics, 153 + + Thing, definition of, 62 ff. + + Thought conceived of as energetic, 178 + + Threshold, 102 + + Time, a form of inner life, 76; + measurement of, 122; + one-seried, or one-dimensional, 118; + and space, 118 ff. + + + Unambiguity, in language, 89; + of co-ordination of numbers to signs, 90 + + Universal auxiliary language, 100, 183 + + + Velocity, 133 + + Volume energy, 145 + + + War, 185 + + Wave surface, 6 + + Wave theory of light, 5, 157 + + Weight, 132, 137 ff.; + a substance, 138 + + Work, mechanical, 129; + product of the force and the distance, 130; + a substance in a limited sense, 136 + + Written language, 89 ff. + + Written signs, 90 + + + Zoology, 56 + + * * * * * + + +American Public Problems + +EDITED BY + +RALPH CURTIS RINGWALT + + +IMMIGRATION: And Its Effects Upon the United States + +By PRESCOTT F. 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Author of "Materials and Methods of Fiction." $1.50 +net; by mail, $1.60. + +CONTENTS: + +THE THEORY OF THE THEATRE.--What is a Play?--The Psychology of Theatre +Audiences.--The Actor and the Dramatist.--Stage Conventions in Modern +Times.--Economy of Attention in Theatrical Performances.--Emphasis in +the Drama.--The Four Leading Types of Drama: Tragedy and Melodrama; +Comedy and Farce.--The Modern Social Drama. + +OTHER PRINCIPLES OF DRAMATIC CRITICISM.--The Public and the +Dramatist.--Dramatic Art and the Theatre Business.--The Happy Endings in +the Theatre.--The Boundaries of Approbation.--Imitation and Suggestion +in the Drama.--Holding the Mirror up to Nature.--Blank Verse on the +Contemporary Stage.--Dramatic Literature and Theatric Journalism.--The +Intention of Performance.--The Quality of New Endeavor.--The Effect of +Plays upon the Public.--Pleasant and Unpleasant Plays.--Themes in the +Theatre.--The Function of Imagination. + + +DRAMATISTS OF TO-DAY + +ROSTAND, HAUPTMANN, SUDERMANN, PINERO, SHAW, PHILLIPS, MAETERLINCK + +By PROF. EDWARD EVERETT HALE, JR., of Union College. With gilt top, +$1.50 net. (By mail, $1.60.) + +An informal discussion of their principal plays and of the performances +of some of them. The volume opens with a paper "On Standards of +Criticism," and concludes with "Our Idea of Tragedy," and an appendix of +all the plays of each author, with dates of their first performance or +publication. + +_New York Evening Post:_ "It is not often nowadays that a theatrical +book can be met with so free from gush and mere eulogy, or so weighted +by common sense ... an excellent chronological appendix and full index +... uncommonly useful for reference." + +_Dial:_ "Noteworthy example of literary criticism in one of the most +interesting of literary fields.... Well worth reading a second time." + + +THE GERMAN DRAMA OF THE NINETEENTH CENTURY + +By GEORG WITKOWSKI. Translated by PROF. L. E. HORNING. 12mo. $1.00. + +Kleist, Grillparzer, Hebbel, Ludwig, Wildenbruch, Sudermann, Hauptmann, +and minor dramatists receive attention. + +_New York Times Review:_ "The translation of this brief, clear, and +logical account was an extremely happy idea. Nothing at the same time so +comprehensive and terse has appeared on the subject, and it is a subject +of increasing interest to the English-speaking public." + + + HENRY HOLT AND COMPANY + PUBLISHERS NEW YORK + + * * * * * + +Transcriber's Notes: + + Bold text is denoted by =equal signs=. The caret ^ indicates that the following character or [ + {expression} is superscripted. + + Mid-sentence capital letters are used by the Author to indicate the + beginning of a quote or question, which terminates at the end of the + sentence. + + Typographical errors corrected: + + p. 100: approprate changed to appropriate (... to a more appropriate + evaluation ...). + + p. 108: meassure changed to measure (By the application of the unit + measure ...). + + p. 184: correspondng changed to corresponding (... signifies a + corresponding economy ...). + + p. 191: A single period deleted from index. + + P. 188, 189: limit-case changed to limit case (2 occurrences), to + mirror text (3 occurrences). + + Alphabetical sequencing adjusted in index: + + P. 189: Two 'Energy' entries moved after Energetic mechanics. + + P. 191: Photographic objective moved below Photochemical processes. + + P. 191: Physics: The order of the sub-entries swapped. + + P. 192: Pure science moved down four places to end of "P" entries. + + P. 193: Two 'Teleological' entries moved after Telegraph, optical. + + + + + +End of the Project Gutenberg EBook of Natural Philosophy, by Wilhelm Ostwald + +*** END OF THE PROJECT GUTENBERG EBOOK 43791 *** |