Popular Science Monthly/Volume 68/March 1906/A Contribution to the Theory of Science




ONE of the few points on which the philosophy of to-day is agreed consists in the realization that the only thing that is absolutely certain and beyond doubt for every one is the content of his own consciousness; or rather not so much the content of consciousness in general as merely the content of any given moment.

This momentary content we divide into two great groups which we assign to the inner and to the outer world, respectively. If we call a single content of consciousness of any kind an experience, we refer such experiences to the external world as take place without the participation of our own will and which can not be produced by it alone. Such experiences come to us through the participation of certain parts of our body, the sense-organs; the external world, in other words, is that which reaches our consciousness through the senses.

Conversely, we refer to our internal world all the experiences which come to pass without the immediate help of our sense-apparatus. To this class belong all the experiences which we designate as 'remembering' or 'thinking.' An accurate and complete differentiation of both territories is at present not contemplated because it is not yet necessary. It is a problem which is not capable of being attacked and solved until later. For the present a general orientation in which every man may recognize the familiar facts of his own consciousness is sufficient.

Each and every experience has the property of being unique. None of us doubts that the words of the poet: 'Everything in life repeats itself,' strictly speaking, is the opposite of the truth, and that, as a matter of fact, nothing in life repeats itself. In order, however, to pronounce a judgment of this kind we must be in a position to compare different experiences with one another; and the possibility of doing so depends upon a fundamental phenomenon of our consciousness, 'memory.' By virtue of memory alone, are we able to bring different experiences into relation with one another, and thus make it possible to propound at all the question concerning their likeness or difference. The simpler relations are to be met with among internal experiences. Any given thought, such as twice two are four, I am able to produce in my consciousness as often as I choose; and, in addition to the content of the thought, I have the further consciousness that I have already had this thought, that it is familiar to me.

Similar, though somewhat more complicated, is the phenomenon in the case of experiences in which the external world had participated. After eating an apple, I may repeat this experience in two ways. I may repeat it as an inner experience, though with diminished intensity. Another part, the sensations that formed a piece of my experience, I am unable at will to reproduce in myself; but am compelled again to eat an apple in order to have a similar experience in this respect as well. This is a complete repetition that it is not necessarily in my power to produce because it is essential that I have an apple, i. e., that certain conditions belonging to the external world and independent of myself be fulfilled.

Whether in the repetition of an experience the outer world takes part or not has no influence upon the content of consciousness, called 'memory.' It follows, therefore, that the latter belongs wholly to internal experience, and that we remember an external event only through its inner constituents. The mere repetition of corresponding sense impressions is not sufficient. We may see the same individual repeatedly without recognizing him in case the accompanying inner phenomena have, through lack of interest, been so slight that their repetition does not produce this content of consciousness, 'memory.' If, however, we see him very often, the frequent repetition of the external impression finally produces the memory of the inner experience that goes with it.

Hence it follows that to call forth the reaction, 'memory,' a definite intensity of the internal experience is necessary. This threshold value may be attained either at one time by one strong impression, or by numerous repetitions of weak ones. The repetitions are the more effective, the more rapidly they follow upon one another. Hence we may conclude further that the memory value of an experience, or its power upon repetition to call forth the reaction, 'memory,' diminishes in the course of time.

Furthermore we must take into consideration the above-mentioned fact that a completely accurate reproduction of an experience never takes place. The reaction, 'memory,' must therefore be called forth when in place of complete correspondence there is merely similarity or partial correspondence. Here, too, there are gradations. Memory appears the more easily, the more completely the two experiences correspond; and vice versa.

If we look at these relations from a physiological point of view, we are able to say: We possess two kinds of contrivances or organs, of which one is independent of, and the other dependent upon, our will. The former are the organs of sense, the latter is the organ of thought. Only activity of the latter forms our experiences or the content of our consciousness. The participation of the former may call forth corresponding processes in the latter, though this is not always necessary. Our sensory apparatus may be influenced without our 'noticing' it, i. e., without the participation of the apparatus of thought. A particularly important reaction of the thought apparatus is 'memory,' i. e., the consciousness that an experience just taking place corresponds more or less with former experiences. It is the peculiar expression by the apparatus of thought of the general physiological law that every process influences an organ in such a way that it reacts to a repetition of this process in a way different from its reaction the first time, namely, by facilitating repetition. This effect decreases in time.

Upon these conditions experience in the abstract is based and is the result of the fact that experiences are composed of a whole series of simultaneous and successive components. If, as the result of the repetition of similar experiences (for instance the sequence of day and night), we have become familiar with the interdependence of certain experiences, we no longer perceive such an experience as an entirely new one, but rather as in part familiar, so that its separate parts or phases no longer astonish us. We anticipate or expect them. From expectation to prediction is but a very short step. Thus experience enables us to prophesy the future from the past and the present.

This is the path to science which is nothing other than experience systematized, that is to say, reduced to simple and comprehensible terms. Its aim is to predict from the known part of a phenomenon the part that has remained unknown. It matters not whether we have to deal with phenomena of space or of time. Thus from a skull the scientific zoologist is able to determine the animal; that is to say, he is able to state the nature of all the other parts of the animal to which the skull belonged. In the same way an astronomer on the basis of a few observations of the position of a planet, is able to foretell its future position. He is able to do so for a future the more remote the more accurate his original observations. All such scientific predictions are limited in regard to their content and exactness. If the skull presented to the zoologist be that of a fowl, he is able to state the characteristics of fowls in general, perhaps too whether this particular fowl possessed a comb or not. He will be unable to tell its color and only within wide limits its age or its size. Both facts, the possibility of prediction and its limitations as regards content and extent, are expressions of the fundamental fact that our experiences may be similar, but are never completely identical.

These preliminary considerations need to be explained and enlarged upon in various directions. One may object to calling a fowl or a planet an experience. We call them by the most universal name— things. However, our knowledge of fowls always begins with the experience of certain sensations of sight with which sensations of hearing and of touch may be associated. The sensations of sight, to which we will limit ourselves for the present, are by no means completely identical. According to its distance from us, we see the fowl as large or small; with changes in its position and its movements its contour is very different. As, however, we observe that these differences pass continuously into one another without exceeding certain limits, we overlook them and confine ourselves to certain other peculiarities (legs, wings, eyes, beak, comb, etc.) which remain constant and do not change. The constant properties we gather together as a 'thing': the changing ones we call the states of this thing. Among the changing ones we distinguish those which are dependent upon ourselves (e. g., distance) from those upon which we ourselves have no immediate influence (e. g., position and movements). The former we call the subjectively variable part of our experience, whereas we term the latter the objective variability of the thing.

To ignore the subjectively and objectively variable portion of our experiences, while we retain their constant parts, and to combine the latter into a single unity, is one of the most important operations which we base on our experiences. We term this procedure abstraction, and its product, the constant unit, a concept. Obviously this procedure contains arbitrary as well as essential parts. Quite arbitrary, or rather accidental, is the fact that according to the state of our attention, our training, nay even our whole intellectual make-up, quite different parts of any given experience reach our consciousness. We may overlook constant components and notice changing ones. But all components become of necessity objective as soon as we have noticed them. After once seeing the fowl black, it is no longer in our power to see it red. It follows that in general our knowledge of corresponding characteristics is less extensive than it might be, inasmuch as we have never noticed all that correspond. Our concept is, therefore, poorer at any given moment in components than it might be. To search for these hitherto overlooked components of a concept and to prove them a constant part of the corresponding experience is one of the never-ceasing labors of science.

The other possibility, viz., that certain components which do not prove to be constant have been incorporated into a concept, also occurs and leads to another problem. These questionable components may, on the one hand, be eliminated from the concept if further experiences show that the remaining ones are contained in them; or, on the other hand, a new concept may be formed by including the constant components and eliminating the inconstant ones. Thus for a long time the white color was a part of the concept 'Swan.' When the black swans of Australia were discovered it was necessary either to eliminate the component 'white' from the concept 'swan' (as was actually done) or else to create a new concept for the bird which resembled a swan, but was black. Which course is decided upon is to a great extent arbitrary, and determined by considerations of fitness.

We have then two factors participating in the formation of a concept, one objective, the result of experience; and one subjective, dictated by fitness. The fitness of a concept is adaptation to the purpose to which it is to be put, and it is this case that we must now consider.

The purpose of a concept is its application in prediction. Ancient logic established the syllogism as the type of the process of thinking, the simplest example of which is the familiar:

All men are mortal;
Cajus is a man;
Therefore, Cajus is mortal.

The universal formula,is:

To the concept M belongs the component B;
C is included in the concept M;
Therefore the component B is found in C.

It may be said that this method of reasoning has remained in use up to the present day. We must, to be sure, add that its application is of a sort quite different from its ancient one. While formerly propounding the major premise was considered the more important, and the propounding of the minor premises was regarded as an almost self-evident and easy matter, to-day the relations have been reversed. The major premise contains the description of a concept, the minor makes the assertion that a certain thing must be classed under this concept. What justification is there for such an assertion?

The most evident answer is that because all the components of the concept M (including B) are to be found in C, C must be classed under the concept M. A conclusion of this kind would certainly be true, but at the same time quite valueless, for it only repeats the assertion of the minor premise.

As a matter of fact, our method of drawing conclusions is essentially different, for the minor premise is not obtained by proving all the constituents of the concept M to be present in C; but only some of them. The conclusion is, therefore, not binding, but merely probable. The whole method of drawing a conclusion is as follows: Certain constituents frequently occur together. They are combined to form the concept M. In the thing C some of the constituents may be recognized. Therefore, presumably the other components of the concept M occur in C.

Ancient logic was also familiar with this method of reasoning; but it was branded with the name of incomplete induction as the worst of all because it lacks absolute and unconditional certainty. One must admit, however, that all contemporary science makes use of no form of reasoning other than incomplete induction. It alone permits of prediction, that is to say, the determination of relations which have not yet been directly observed.

But how does science get along with this lack of certainty in its method of drawing conclusions? The reply to this question is that the probability of the conclusions may run the gamut of all possible gradations from mere supposition to the maximum of probability which is no longer to be distinguished, practically, from certainty. The probability is greater the more frequently any given incomplete induction of this kind has been found consistent with subsequent experience. Thus we have at our disposal a number of propositions which in their simplest and most general shape take the form: If the component A is to be found in a given thing, the component B is also to be found in it (in relation to either time or space).

If the relation is one of time, we term this general proposition the law of causality. If it is one of space we speak of the idea (in the Platonic sense) or type of the thing, of substance, etc.

These considerations yield answers to many questions which have been repeatedly propounded in various forms. We have first the question of the universal validity of the law of causality. All attempts to establish this kind of validity have failed, and only the fact remains that without this law we should feel an unendurable uncertainty as to the world. Hence it follows that we have to deal in this matter with a question of fitness. From the constant stream of our experience we select relations which we encounter again and again in order that whenever the component A is given us we may conclude that the component B is also to be found. Hence we do not find these correlations occurring as 'given,' but we ourselves bring them into our experiences, by ourselves regarding the components which show such a connection as belonging together.

We may make quite the same statements in regard to space relations. The components which are always, or at any rate frequently, encountered together we interpret as forming a unit; and we shape from them a concept which includes these components. As in the case of time relation, there is no sense in propounding the question why. There are thousands of correlations to which we pay no attention because they are unique or rare. Knowledge of such a unique correlation leads to nothing, because it does not enable us to infer the presence of one component from the presence of another and therefore does not render prediction possible. Of all possible and actual combinations only those interest us which are repeated. This arbitrary though fit selection creates the impression that these are the only recurring combinations, or that, in other words, the law of casuality or of type rules unqualifiedly. How universal or how limited is the application of these laws is therefore rather a question of our skill in selecting the constant combinations from among all those occurring than a question of objective natural phenomena.

Thus we observe the development and practise of all sciences progressing in this manner by the discovery, on the one hand, of ever more numerous individual constant combinations, and, on the other, of more and more universal laws by means of which components are brought into relation with one another, which formerly no one had attempted to bring together. Thus sciences grow in that they become complex and at the same time unified.

If now we consider the development and course of the various sciences from this point of view we shall attain a rational classification of all science by an inquiry into the extent and complexity of the combinations or complexes which they treat. Both characteristics are in a sense antagonistic. The simpler a complex is, that is, the fewer the components united in it, the more frequently will it occur; and vice versa. It will therefore be possible to classify all the sciences in this fashion by beginning with the minimum of complexity and the maximum of extent and ending with the maximum of complexity and the minimum of extent. The first science will include the most general and therefore the poorest and most meager concepts; the last the most specialized and therefore the richest.

What then are these limiting concepts? The most universal is the thing, any fraction of experience arbitrarily selected from the stream of our experiences and capable of repetition. The most specialized and the richest is the concept of human society. Between the study of things and the study of human society all the remaining sciences may be interpolated in an orderly series. The attempt to follow out this scheme leads to the following table:

1. Science of manifoldness or assemblages, Logic.
2. Science of numbers or arithmetic, Mathematics.
3. Science of time,
4. Science of space or geometry,
5. Mechanics, Energetics.
6. Physics,
7. Chemistry,
8. Physiology, Biology.
9. Psychology,
10. Sociology,

This table contains an arbitrary element inasmuch as the steps assumed in it may be multiplied. Thus mechanics and physics might be combined or physical chemistry interpolated between physics and chemistry. Similarly anthropology might be placed between physiology and psychology while the first four sciences might be combined as mathematics. How this subdivision is to be carried out is purely a practical question which will be answered differently by different ages and concerning which it is useless to quarrel.

I would like, however, to call attention to the three great divisions: Mathematics, energetics and biology (in the broader sense). They represent the three guiding thoughts which up to the present mankind has brought forth for the purpose of mastering scientifically its experiences. Order is the fundamental thought of mathematics, energy the guiding concept from mechanics to chemistry; for the last three sciences it is life. Mathematics, energetics and biology therefore embrace the whole body of the sciences, while logic precedes them all.

Before entering upon a more detailed consideration of these sciences, it is well to anticipate an objection which may be raised on the basis of the following fact. There are in addition to the previously mentioned sciences (as well as the intermediate ones) many others such as geology, history, medicine, philology, which present difficulties if a place is sought for them in our scheme, and which, nevertheless, demand consideration. It is often characteristic of them that they bear relation to several of the sciences which have been enumerated; and still more characteristic of them is it that they do not search for universal relations as do the pure sciences, but rather treat existing complex objects in order to 'explain' or discover their origin, extent, distribution, in a word their time and space relations. To accomplish this object they make use of the relations which the pure sciences have put at their disposal. It is best to designate these sciences as applied sciences. This term is not meant to imply either exclusively or even chiefly technical application. It is merely intended to express the fact that here interrelations of the parts of a ready-made object are rendered intelligible by the application of the general laws which have been discovered by pure science.

In a problem of this nature it is usually necessary to make use of several different pure sciences simultaneously for an explanation, because the abstract method of the pure sciences is not permissible here. To omit certain parts and to limit oneself to certain others is from the very nature of the problem out of the question. Astronomy is an applied science of this sort. It is based immediately upon mechanics; in its instrumental part upon optics; while in its contemporaneous spectroscopic development it borrows much from chemistry. Thus history is applied sociology and psychology; medicine makes use of all preceding sciences up to psychology, etc.

It is important to realize the nature of these applied sciences since their composite nature renders it impossible to classify them among the pure sciences, though, because of their practical importance, they demand consideration. The latter characteristic gives them to a certain extent an arbitrary or accidental character since their development depends upon the particular requirements of the times. Their number, broadly speaking, is very great because every pure science may be changed to an applied one in many ways and may be combined for this purpose with one or more other sciences. Furthermore, the method of applied science is fundamentally different from that of pure science inasmuch as the former seeks to analyze any given complex into its scientifically manageable parts, whereas conversely the latter considers many complexes in order to extract from them their common feature and explicitly refrains from the complete analysis of each individual complex.

In scientific work, as carried out in practise, pure and applied science are by no means always to be sharply separated. On the one hand, the means of research, apparatus, books, etc., demand the knowledge and the practise of applied science even by the 'pure' investigator. On the other hand, the 'applied' investigator is often able to solve his problem only by becoming temporarily a 'pure' investigator and himself ferreting out or discovering the universal relations which he needs for the solution of his problem. The separation and differentiation of these two kinds of science was, however, necessary, because each employs quite different methods and pursues essentially different ends.

In order that we may attain a clear understanding of the method of pure science, we will turn to the table on page 225 and consider the individual sciences separately. The first place is ordinarily given to mathematics as to the science of quantity. However, mathematics deals with number and size as its fundamental concepts, while the science of assemblages does not as yet use them. Moreover, in the latter, the fundamental concept is the thing or object of which no more is required than merely that it be a fraction of our experience capable of being isolated and remaining so. It may not be any indiscriminate fraction, for such a one could have but a momentary duration; and the aim of science, to discover the unknown from what is given, could not be accomplished with it. This part of experience must rather be of such a nature that it may be distinguished and recognized, that is to say, it must already be of the nature of a concept. Only those parts of our experience which are capable of repetition (for these alone can form the subject-matter of science) can be called things or objects. This statement, however, includes everything that is required of them. Otherwise they may differ as much as is conceivable.

If it be asked what scientific statements it is possible to make concerning such uncertain things, one will find that the relations of order and classification are the ones to yield results. If we designate any limited combination of things of this nature a group, we may arrange a group in different ways, that is, we may determine for each thing the relation in which it is to stand toward neighboring things. Such an arrangement produces not merely the relations prescribed above, but in addition a large number of new ones; and it becomes plain that, given the first relations, the others may be observed at once. This gives us the type of a law of nature: the possibility of inferring from the presence of a definite classification-relation the presence of others which we have not yet tested.

To illustrate by an example: Let us imagine the things arranged in a simple series formed by choosing one thing for the first member; placing another next to this one; then a new one next to the latter; another next to the last, etc. The position of each thing in the series is determined in relation to the immediately preceding one. Nevertheless the position of every member of the entire series is determined; and thus its relation to every other member. This fact appears in a number of special laws. If we distinguish between preceding and succeeding members one of the laws we may observe is: If B is a succeeding member in relation to A, and if C is a succeeding member in relation to B, then C is also a succeeding member in relation to A.

The correctness and universal validity of this proposition seems to us beyond any possible doubt. It depends, however, merely upon the fact that we are able to test it with the greatest ease in innumerable individual instances and have so tested it. We know none other than instances agreeing with this proposition and none that contradict. Therefore, to designate such a statement as a necessity of thought seems to me misleading. Now the expression necessity of thought can only be based upon the fact that each time one thinks this proposition, that is, remembers having tested it, one always has in mind its confirmation. Any wrong proposition is, however, conceivable as the fact that so much that is wrong is actually thought indisputably shows. To base the proof of the truth of a proposition upon the inconceivability of its converse is an undertaking that can not be carried out, because it is possible to think any sort of nonsense. Whenever this proof was believed to have been demonstrated, thinking was always confused with considering, demonstrating or proving.

Of course the theory of groups is not exhausted with this single statement. We do not, however, care to develop this theory here, but rather to give an example of the nature of the problems of science. Of the other questions only the method of coordination will be briefly treated.

Given two quantities A and B, one may assign to every member of A a member of B, that is, one determines that certain operations which are to be carried out with the members of A, shall also be carried out with those of B. We may begin by simply associating them member for member. Then one of three things will happen: either A will be exhausted while members of B remain, or B is first exhausted, or finally A and B are exhausted simultaneously. In the first case we say that A is poorer than B; in the second B is poorer than A; and in the third case that both quantities are equal.

We meet now for the first time the scientific concept of equality; and it is necessary that we enlarge upon it. Absolutely complete identity of both groups is obviously out of the question, inasmuch as we made the assumption that the members of both groups might be of any nature whatsoever. Regarded singly they may be as different as possible. They are, however, equal as groups. For, however I arrange the members of A, inasmuch as a member of B is assigned to every member of A, I am able to carry out every arrangement of A upon B as well. As regards the possibilities of arrangement there is no apparent difference between A and B. As soon, however, as A is either poorer or richer than B, this similarity disappears, for one of the two quantities possesses members to which no members of the other groups correspond. The operations that may be performed upon these members can not be carried out upon the second group.

Equality, in the scientific sense of the word, signifies, therefore, equivalence or the possibility of substitution as regards definite operations or relations. In all other respects the things that have been pronounced equal may differ in any way. It is easy in this special case to recognize the universal method of abstraction of science.

It is possible on the basis of these definitions to make further propositions. If the quantity A is equal to B and if B is equal to C, then A is also equal to C. This may be proved by first arranging A with reference to B. According to our presupposition no member remains. Thereupon C is arranged with reference to B with no member remaining. In this way every member of A is, through the intervention of B, assigned to a member of C. Moreover, this arrangement remains unchanged even after the removal of B, i. e., A and C are equal. The same method may be applied to any number of quantities.

It is possible to prove in a similar manner that, if A is poorer than B, and B is poorer than C, A must also be poorer than C. For in assigning the members of B to A, some members of B will, according to our assumption, remain, and the same will be true of C if we assign the members of C to those of B. Hence in assigning the members of C to those of A there are left not merely the members which can not be assigned to B, but also the members of C which have been assigned to such members of B as are supernumerary in respect to A. This proposition is applicable to all groups and renders it possible to arrange different groups in a series by beginning with the poorest and choosing each succeeding one so that it is richer than its predecessor though poorer than its successor. Through a proposition that has been already proved (p. 229) it follows that each group is thus also arranged with reference to all other groups in such a way that it is richer than all its predecessors as well as poorer than all its successors.[2]

In developing most simple propositions or laws, this method of their discovery and the nature of the results become most clear to us. We achieve such a proposition by carrying out an operation and giving expression to its results. This expression enables us thereafter to save ourselves the trouble of repeating the operation. We are able to give the result immediately in accordance with the law. Thus we shorten and facilitate the procedure more or less according to the number of operations avoided.

Given any number of equal groups, we recognize that by arranging them with relation to one another as above, we are able to carry out upon all of them each and every operation involving arrangement that we are able to carry out upon one of them. It is therefore sufficient to determine the characteristics of arrangement of any one of these groups in order to know those of all the others. This is a most important proposition which is constantly applied for manifold purposes. Thus talking, writing and reading are founded upon the coordinating of thoughts to sounds and signs; and by arranging the signs in accordance with our thoughts we cause our hearers or our readers to think the same thoughts in the same sequence. We manipulate many formulæ in a similar manner (especially in the simpler sciences), applying the results to phenomena, instead of dealing with the phenomena themselves; and we are able to deduce some properties of the latter without being compelled to work with the phenomena themselves. The force of this procedure is most striking in astronomy, where, by manipulating certain formulas which have been applied to certain celestial bodies, we are able to predict their future positions with a great degree of accuracy.

From the science of order we pass to the science of numbers or arithmetic by the systematic development of an operation that has just been indicated. We are able to arrange any given number of quantities in such a manner that the richer always succeeds the poorer. The system obtained in this fashion is, however, quite accidental as regards the number and richness of its members. Obviously we can only obtain an orderly structure of all possible groups by starting with a group having but one member, i. e., a simple thing, and forming new members of the series from old ones by adding a single member. By this process we at once obtain the different groups arranged according to their richness. Further, inasmuch as we advanced by a single member, that is, we have made the smallest step possible, we are certain to have omitted no possible group that is poorer than the richest to which we have advanced our operation.

This whole procedure is well known. It yields the entire series of positive numbers—the cardinal numbers. It is to be noted that the concept of magnitude does not appear as yet. What we have obtained is merely the concept of number. The individual members may be chosen quite arbitrarily. They need in no way be equal. Each number represents a quantity type; and it is the sphere of arithmetic to examine these different types in respect to subdivision and combination. If this be done without considering the amount of the number, we call the corresponding science algebra. On the other hand, the extension of formal rules beyond their original application has led to one development of numbers after the other. Thus counting backwards leads to zero and the negative numbers, the square root of the latter to the imaginary numbers. The quantity-type of all the positive numbers is, to be sure, the simplest, though by no means the only possible one. For the purpose of representing other arrangements such as occur among our experiences these new types have proved very useful.

At the same time the numerical series yields a most useful type of arrangement. From its very origin it is arranged in an orderly fashion and it is therefore employed for the purpose of arranging other quantities. Thus we are accustomed to apply the signs of the numerical series to any objects which we desire to use in a definite order, such as the pages of a book, the seats in a theater, as well as countless other groups. We, however, tacitly make the assumption that the arranged groups are to be used in the same sequence in which the natural numbers follow one another. These sequence-numbers represent no magnitudes nor do they represent the only type of arrangement possible. They are, however, the very simplest.

We do not reach the concept of magnitude until we reach the science of time and space. A science of time has not been developed separately. On the contrary, what there is to say about time usually appears for the first time in mechanics. However, it is possible for us to state the fundamental characteristics of time here, so that the want of a distinct science of time will not be felt.

The first and most important property of time (and also of space) is that it is continuous. In other words, any portion of time may be divided at any point. In the numerical series this is not the case; it may be divided only between numbers. The series one to ten has nine places of division, and only nine. A minute or a second, on the other hand, has an unlimited number of possible points of division. In other words, there is nothing in the passage of time preventing us at any desired moment from separating or distinguishing in thought the time that has passed from that which is to follow. Space is of the same nature, except that time is simple, space threefold.

Nevertheless, we are accustomed to describe both time and space by means of numbers whenever we measure them. If we examine into this procedure, for instance in the case of measuring length, we find that it consists in applying a length considered fixed, the measure, as often to the length to be measured as is necessary to cover it. The number of applications gives us the measure or magnitude of the length. It merely amounts to forcing an artificial discontinuity upon a continuous length by marking off arbitrarily chosen points, allowing us to refer it to the discontinuous numerical series.

The equality of the portions of distance set off by the measuring-rod is an essential part of the concept of measuring. We assume this condition fulfilled no matter how the measuring-rod be shifted. As we see, this is a more forced definition of equality than heretofore made, for it is actually quite impossible to substitute a given portion of a distance for another in order to become convinced that the validity of our definition is not impaired, that nothing is changed thereby. It is quite as impossible to prove that the measuring-rod in being shifted in space remains of the same length. We may only affirm that such distances as are determined in various places by means of the measuring-rod are declared or defined as equal. As a matter of fact, the measuring-rod in perspective looks smaller the further it is away from us.

This example demonstrates anew the great arbitrariness with which we shape science. It is conceivable that a geometry might be developed in which the distances are considered equal which subjectively appear to our eye to be so, and we should then be quite as able to develop a consistent system or science. A geometry of this kind would, however, be of too complicated a nature to be advantageous for any objective purpose (e. g., surveying). Therefore we endeavor to develop a science as free as possible from subjective factors. Historically the Ptolemaic astronomy and that of Copernicus present an illustration in point. The former was formed according to subjective appearances in its assumption that the stars revolved about the earth. It proved most complicated when confronted with the problem of expressing these motions mathematically. The latter gave up the subjective point of view of the observer who regarded himself as the center; and, by transferring the center of motion to the sun, produced: an enormous simplification.

A few more words are necessary at this point concerning the application of arithmetic and algebra in geometry. It is well known that under certain assumptions (coordinates) geometric figures may be expressed in algebraic formulæ so that it is possible to deduce the geometric properties of the figures from the calculatory properties of the formulas, and vice versa. We must inquire how such close and unambiguous relations can exist between things so diverse. The answer is that we have to deal in this instance with a particularly obvious case of association. The manifoldness of numbers is far greater than that of planes or space, for, whereas the latter are determined by but two or three independent measurements, any number of independent numerical series may be made to react upon one another. We therefore arbitrarily limit the manifoldness of the numbers to two or three independent series, and determine (by means of the cosinus law) their mutual relations so that a manifoldness corresponding exactly to that of the space arises to which we are able completely to refer it. We have then two manifoldnesses of identical character; and all properties of arrangement and size of the one are 'depicted' in the other. In this an extremely important scientific procedure is indicated which consists in giving to the experience-content of a given field a formal manifoldness to which we impart the same manifoldness-character as that possessed by the former. Every science thus develops a formula language of its own, perfect in proportion to the accuracy with which the manifoldness-character of the object has been recognized, and the fitness of the formulas selected. Whereas, in arithmetic and algebra this problem has been solved quite perfectly (though by no means absolutely so), chemical formulas, for instance, express only a relatively small part of the characteristics which they ought to express, while in biology and sociology we have hardly progressed beyond the very beginnings of the solution of this problem.

One of these universal manifoldnesses designed to express our experiences is speech. Inasmuch as it was developed in a primitive civilization it is by no means regular and complete enough to fulfil its purpose satisfactorily. On the contrary, it is quite as unsystematic as were the events in the history of the various peoples. The need to express the infinite variety of events in daily life has been filled by allowing word and concept to correspond only within a wide limit of variation. Therefore all research in the sciences which are forced to employ this means of expression (psychology and sociology or philosophy generally) is greatly impeded by the struggle with the indefiniteness and ambiguity of language. An improvement of these conditions is to be attained only by the introduction, as rapidly as the progress of the science warrants it, of symbols to which we refer the manifoldness which experience tells us is peculiar to the concept.

The sciences which have been classed above as a part of energetics occupy an intermediate position. In addition to the concepts of order, number, magnitude, space and time we meet in this branch of knowledge with the new concept of energy, which is applied as universally to every phenomenon in this field as the other universal concepts. This is so because a certain magnitude, known to us immediately as mechanical work, may be proved to be a constituent of every physical phenomenon, i. e., of mechanics, physics and chemistry, by virtue of its qualitative transformability, and its quantitative immutability. In other words, it is possible to characterize every physical phenomenon completely by stating what quantities and kinds of energy are present and into what kinds of energy they are transformed. It is, therefore, more rational to term the so-called physical phenomena, energetic.

That such a conception is possible is now generally acknowledged, but its utility is usually doubted. These doubts are at present justified, inasmuch as a complete exposition of the physical sciences from the point of view of energetics has not been thoroughly carried out. If the above-mentioned criterion of a scientific system, viz., the conformity of the representing manifoldness to the one depicted, be applied to this question, we shall find unmistakably that all previous systematizations, which in the form of hypotheses have been attempted in these sciences, are faulty in this respect. Hitherto manifoldnesses have been used for the purpose of 'depicting' experiences the character of which corresponded to the depicted object only in a few main points. No attention was paid to the necessity of exact correspondence. There was no concise formulation or investigation of this side of the problem.

Now the energetic point of view permits as great a certainty in the method of depiction or expression as is necessary or possible for the state of the science at the time. For the manifoldness-character of each department there is a special form of energy. Thus science has long since distinguished between mechanical, electrical, thermal, chemical and other forms of energy. All these different varieties are related through the law of transformation with the conservation of energy. They are therefore organically connected. On the other hand, it has been possible to find the energetic expression for every manifoldness as yet discovered empirically. The future system of energetics in its entirety will therefore be a table of all the possible manifoldnesses of which energy is capable. It must be noted, however, that as a consequence of the law of the conservation of energy, energy is of necessity a positive magnitude which furthermore is without limit additive. Each special kind of energy must therefore also have this character.

The very slight degree of manifoldness which these conditions seem to leave us is increased very much by the fact that every form of energy may be resolved into two factors. The latter are subject to but a single limitation, viz., that their product, energy, fulfils the above conditions, while they themselves are far more free. Thus, one of the factors of a form of energy may become negative instead of positive, if the other factor also becomes negative.

Accordingly it would seem possible to construct a table of all the possible forms of energy by assigning to the factors of energy all possible manifoldness characteristics and combining them in pairs, with the subsequent elimination of the products which do not conform to the conditions stated on p. 234. By comparison with all the forms of energy known at the time, it would be possible to discover the forms that were still unknown and to outline their most important properties. Experience would merely have to discover their specific constants. For some years I have myself from time to time attempted to carry out this program; but hitherto I have not progressed far enough to justify the publication of the results already obtained.

If now we turn to the biological sciences, the new phenomenon we meet is life. If we limit ourselves to observable facts excluding all hypotheses, we shall recognize as the universal characteristic of all life-phenomena the stationary stream of energy which flows through a comparatively constant structure. Metabolism is merely a part, though a most important part, of this stream. Plants, particularly, demonstrate immediately the paramount importance of energy in its most immaterial form, the sunbeams. Self-maintenance and repair with the production of similar descendants are other essential characteristics. All these characteristics must be present in order that an organism may arise. Furthermore they must be present if the knowing individual is to be capable of forming, by repeated experiences, a concept of any given organism, say a lion or a mold. Other organisms not fulfilling these conditions may occur. Because they are unique, they do not lead to the concept of a species, but are excluded (except of course for special purposes) from scientific consideration as 'malformations' or 'monsters.'

Whereas organisms mostly deal with forms of energy which are familiar to us in the inorganic world, we find the higher forms possessed of organs which undoubtedly produce, or are active in, the transformation of energy, though we do not know which form of energy acts within them. These organs are termed nerves; and their functioning is regularly of such a nature that upon the application of a definite form of energy to one end, they call into action at the other end forms of energy there present and which there act in their own peculiarly characteristic way. That energetic changes do occur in the process of nerve-transmission may be regarded as settled. We are therefore justified in speaking of nerve energy, leaving the question open as to whether it be a special form of energy or merely chemical energy or lastly a combination of several forms of energy.

While these processes of nerve stimulation with corresponding reaction in the end-organ, a muscle, for example, may be observed objectively, we find within ourselves connected with this nervous process a new kind of phenomenon which we term self-consciousness. From the correspondence of our reactions with those of other people we conclude with scientific probability that they too are possessed of self-consciousness. We draw the same conclusion concerning a few of the higher animals. How far down the scale anything similar is present is not to be ascertained with the means at our command to-day, for the analogy between organization and action rapidly diminishes as we pass down the scale. Still in view of the very great gulf between man and the higher animals, this series is presumably not very long. Moreover, there are many reasons for regarding the gray cortical substance of the brain with its characteristic pyramidal cells as the anatomical substratum for this kind of nervous activity.

The study of the processes of self-consciousness is the subject-matter of psychology. Some departments usually considered a part of philosophy really belong to this science, viz., the theory of knowledge. Esthetics, and still more ethics, are, however, a part of social science.

The latter deals with beings in so far as they may be combined in groups with common functions. In place of an individual mind we have here a collective one. The latter, by virtue of the average struck between the variations of the individuals, presents simpler relations than the former. Thence we may deduce the problem of the historical sciences. The events of our world depend partly upon physical, partly upon psychological factors. Both show a one-sidedness in regard to time. Thus arises, on the one hand, a history of the sky and the earth; and, on the other, a history of the organisms up to man.

The problem of history is to fix past facts through the effects they have wrought. Where the latter are not present we are dependent for a conception of the facts upon that most uncertain procedure, analogy. We must observe, however, that an event which has left no trail has absolutely no interest for us. Our interest in an event is directly proportional to the extent of the change it has produced upon the present. The problem of history is, however, as little exhausted by determining past facts as is that of physics by ascertaining an isolated fact, such as determining the temperature of a given place at a given time. The individual facts serve rather to discover the general properties of the collective mind; and the much-discussed laws of history are laws of collective psychology. Just as physical and chemical laws are discovered in order that with their help we may predict future events (such as those produced in experiment or technology), so laws of history should render possible the control and the development of society and of politics. We observe that the great statesmen of all times assiduously studied history; and hence we may conclude that, despite the doubts expressed by many scholars, numerous laws actually exist in history

. If after this cursory survey we review the ground we have covered, we shall recognize the following general relations: In each case the development of a science consists in correlating concepts formed from definite abstractions derived from experience; and by this means we achieve in our minds a mastery over certain parts of our experiences. Such correlations are termed according to the degree of their universality and reliability, rules or laws. A law is the more important, the more definite its statement concerning the greatest possible number of things; and the more accurately it consequently permits of predicting the future. Every law is based upon incomplete induction and is therefore liable to modification by experience. Hence the development of science is of necessity twofold.

In the first place actual relations are examined to see whether or not new relations other than those already known may not be discovered, i. e., constant relations between individual peculiarities. This is the inductive method. And because the possibilities of experience are unlimited it must ever be an incomplete method.

In the second place, relations discovered by induction are applied to cases which have not yet been investigated. Cases resulting from the combination of several inductive laws are particularly liable to be studied. If the combination is correctly made and if the inductive laws are absolutely certain, the result has a claim to unconditional validity. This is the limiting case which all sciences strive to approach. It is almost attained by the simplest sciences, mathematics and certain parts of mechanics. This is termed the deductive method.

In the actual practise of every science both methods of investigation constantly alternate. The best method to discover new and significant inductions is to make a deduction even though its basis be insufficient, requiring subsequent proof from experience. Sometimes the investigator is not conscious of the separate steps of his deduction. In such cases scientific instinct is spoken of. On the other hand, great mathematicians have informed us that they used to find their general laws by induction, by trying and considering individual cases, and that their deductive derivation from other known laws is an independent operation which at times did not follow until much later. Even today there are a number of mathematical propositions which have not reached the second stage and which are therefore at present of a purely inductive and empirical character. The part that such laws play in the sciences rapidly increases as we pass up the series.

Another peculiarity which may be mentioned here is that in the series all preceding sciences assume the characteristics of applied sciences in respect to succeeding ones because they are essential to the course of the last without being themselves increased. They are merely helps to the latter.

If, in conclusion, we ask what influence investigations such as have just been sketched in outline can have upon the development of the future the following may be said: Whether a great and influential man of science develop and where, has hitherto been regarded as an event quite beyond control. All are agreed that such a one is one of the most precious treasures a nation (or indeed mankind) may possess. The conscious and regular training of such rarities had not been considered possible. While this is still true in the case of the quite exceptional genius, nevertheless countries of old civilization, at present notably Germany, exhibit an educational system at their universities which yields a regular harvest of young men of science, masters not merely of existing knowledge, but also of the technique of discovery. In this fashion the growth of science has been rendered sure and regular while its practise has been raised to a higher plane. These results have hitherto been attained by essentially empirical or even accidental means. It is the problem of the philosophy of science to regulate and systematize this activity in order that success may no longer depend solely upon individual talent, but may also be achieved by less original minds. Mastery of method, moreover, leads the exceptionally gifted individual to considerably higher achievements than he could attain without it.

  1. Address before the Section of Methodology, Congress of Arts and Science, St. Louis, translated by Dr. Carl. L. Alsberg.
  2. Equal groups can not be distinguished here; and represent merely a single quantity.