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diff --git a/43791-8.txt b/43791-8.txt deleted file mode 100644 index 4900be9..0000000 --- a/43791-8.txt +++ /dev/null @@ -1,6167 +0,0 @@ -The Project Gutenberg EBook of Natural Philosophy, by Wilhelm Ostwald - -This eBook is for the use of anyone anywhere at no cost and with -almost no restrictions whatsoever. You may copy it, give it away or -re-use it under the terms of the Project Gutenberg License included -with this eBook or online at www.gutenberg.org/license - - -Title: Natural Philosophy - -Author: Wilhelm Ostwald - -Translator: Thomas Seltzer - -Release Date: September 22, 2013 [EBook #43791] - -Language: English - -Character set encoding: ISO-8859-1 - -*** START OF THIS PROJECT GUTENBERG EBOOK NATURAL PHILOSOPHY *** - - - - -Produced by Chris Curnow, John Welch and the Online -Distributed Proofreading Team at http://www.pgdp.net (This -file was produced from images generously made available -by The Internet Archive) - - - - - - - - - -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 Cĉsar, 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 ħ _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 - - - Cĉsar, 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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