One of the few points on which the philosophy of to-day is united is the knowledge that the only thing completely certain and undoubted for each one is the content of his own consciousness; and here the certainty is to be ascribed not to the content of consciousness in general, but only to the momentary content.

This momentary content we divide into two large groups, which we refer to the inner and outer world. If we call any kind of content of consciousness an experience, then we ascribe to the outer world such experiences as arise without the activity of our will and cannot be called forth by its activity alone. Such experiences never arise without the activity of certain parts of our body, which we call sense organs. In other words, the outer world is that which reaches our consciousness through the senses.

On the other hand, we ascribe to our inner world all experiences which arise without the immediate assistance of a sense organ. Here, first of all, belong all experiences which we call remembering and thinking. An exact and complete differentiation of the two territories is not intended here, for our purpose does not demand that this task be undertaken. For this purpose the general orientation in which every one recognizes familiar facts of his consciousness is sufficient.

Each experience has the characteristic of uniqueness. None of us doubts that the expression of the poet "Everything is only repeated in life" is really just the opposite of the truth, and that in fact nothing is repeated in life. But to express such a judgment we must be in position to compare different experiences with each other, and this possibility rests upon a fundamental phenomenon of our consciousness, memory. Memory alone enables us to put various experiences in relation to each other, so that the question as to their likeness or difference can be asked.

We find the simpler relations here in the inner experiences. A certain thought, such as twice two is four, I can bring up in my consciousness as often as I wish, and in addition to the content of the thought I experience the further consciousness that I have already had this thought before, that it is familiar to me.

A similar but somewhat more complex phenomenon appears in the experiences in which the outer world takes part. After I have eaten an apple, I can repeat the experience in two ways. First, as an inner experience, I can remember that I have eaten the apple and by an effort of my will I can re-create in myself, although with diminished strength and intensity, a part of the former experience—the part which belonged to my inner world. Another part, the sense impression which belonged to that experience, I cannot re-create by an effort of my will, but I must again eat an apple in order to have a similar experience of this sort. This is a complete repetition of the experience to which the external world also contributes. Such a repetition does not depend altogether on my own powers, for it is necessary that I have an apple, that is, that certain conditions which are independent of me and belong to the outer world be fulfilled.

Whether the outer world takes part in the repetition of an experience or not has no influence upon the possibility of the content of consciousness which we call memory. From this it follows that this content depends upon the inner experience alone, and that we remember an external event only by means of its inner constituents. The mere repetition of corresponding sense impressions is not sufficient for this, for we can see the same person repeatedly without recognizing him, if the inner accompanying phenomena were so insignificant, as a result of lack of interest, that their repetition does not produce the content of consciousness known as memory. If we see him quite frequently, the frequent repetition of the external impression finally causes the memory of the corresponding inner experience.

From this it results that for the "memory"-reaction a certain intensity of the inner experience is necessary. This threshold can be attained either at once or by continued repetition. The repetitions are the more effective the more rapidly they follow each other. From this we may conclude that the memory-value of an experience, or its capacity for calling forth the "memory"-reaction by repetition, decreases with the lapse of time. Further, we must consider the fact mentioned above, that an experience is never exactly repeated, and that therefore the "memory"-reaction occurs even where there is only resemblance or partial agreement in place of complete agreement. Here, too, there are different degrees; memory takes place more easily the more perfectly the two experiences agree, and vice versa.

If we look at these phenomena from the physiological side, we may say we have two kinds of apparatus or organs, one of which does not depend upon our will, whereas the other does. The former are the sense organs. The latter constitutes the organ of thought. Only the activities of the latter constitute our experiences or the content of our consciousness.

The activities of the former may call forth the corresponding processes of the latter, but this is not always necessary. Our sense organs can be influenced without our "noticing" it, that is, without the thinking apparatus being involved. An especially important reaction of the thinking apparatus is memory, that is, the consciousness that an experience which we have just had possesses more or less agreement with former experiences. With reference to the organ of thought, it is the expression of the general physiological fact that every process influences the organ in such a way that it has a different relation to the repetition of this process, from the first time, and moreover that the repetition is rendered easier. This influence decreases with time.

It is chiefly upon these phenomena that experience rests. Experience results from the fact that all events consist of a complete series of simultaneous and successive components. When a connection between some of those parts has become familiar to us by the repetition of similar occurrences (for instance, the succession of day and night), we do not feel such an occurrence as something completely new, but as something partially familiar, and the single parts or phases of it do not surprise us, but rather we anticipate their coming or expect them. From expectation to prediction is only a short step, and so experience enables us to prophesy the future from the past and present.

Now this is also the road to science: for science is nothing but systematized experience, that is, experience reduced to its simplest and clearest forms. Its purposes to predict from a part of a phenomenon which is known another part which is not yet known. Here it may be a question of spatial as well as of temporal phenomena. Thus the scientific zoölogist knows how to "determine," that is, to tell, from the skull of an animal, the nature of the other parts of the animal to which the skull belongs; likewise the astronomer is able to indicate the future, situation of a planet from a few observations of its present situation; and the more exact the first observations were, the more distant the future for which he can predict. All such scientific predictions are limited, therefore, with reference to their number and their accuracy. If the skull shown to the zoölogist is that of a chicken, then he will probably be able to indicate the general characteristics of chickens, and also perhaps whether the chicken had a top-knot or not; but not its color, and only uncertainly its age and its size. Both facts, the possibility of prediction and its limitation in content and amount, are an expression for the two fundamental facts, that among our experiences there is similarity, but not complete agreement.

The foregoing considerations deserve to be discussed and extended in several directions. First, the objection will be made that a chicken or a planet is not an experience; we call them rather by the most general name of thing. But our knowledge of the chicken begins with the experiencing of certain visual impressions, to which are added, perhaps, certain impressions of hearing and touch. The sight impressions (to discuss these first) by no means completely agree. We see the chicken large or small, according to the distance; and according to its position and movement its outline is very different. As we have seen, however, these differences are continually grading into one other and do not reach beyond certain limits; we neglect to observe them and rest contented with the fact that certain other peculiarities (legs, wings, eyes, bill, comb, etc.) remain and do not change. The constant properties we group together as a thing, and the changing ones we call the states of this thing. Among the changing properties, we distinguish further those which depend upon us (for example, the distance) and those upon which we have no immediate influence (for instance, the position or motion): the first is called the subjective changeable part of our experience, while the second is called the objective mutability of the thing.

This omission of both the subjectively and objectively changeable portion of the experience in connection with the retention of the constant portion and the gathering together of the latter into a unity is one of the most important operations which we perform with our experiences. We call it the process of abstraction, and its product, the permanent unity, we call a concept. Plainly this procedure contains arbitrary as well as necessary factors. Arbitrary or accidental is the circumstance that quite different phases of a given experience come to consciousness according to our attention, the amount of practice we have had, indeed according to our whole intellectual nature. We may overlook constant factors and attend to changeable ones. The objective factors, however, become necessary as soon as we have noticed them; after we have seen that the chicken is black, it is not in our power to see it red. Accordingly, in general, our knowledge of that which agrees must be less than it actually could be, since we have not been able to observe every agreement, and our concept is always poorer in constituents at any given time than it might be. To seek out such elements of concepts as have been overlooked, and to prove that they are necessary factors of the corresponding experiences, is one of the never-ending tasks of science. The other case, namely, that elements have been received in the concept which do not prove to be constant, also happens, and leads to another task. One can then leave that element out of the concept, if further experiences show that the other elements are found in them, or one can form a new concept which contains the former elements, leaving out those that have been recognized as unessential. For a long time the white color belonged to the concept swan. When the Dutch black swans became known, it was possible either to drop the element white from the concept swan (as actually happened), or to make a new concept for the bird which is similar to the swan but black. Which choice is made in a given case is largely arbitrary, and is determined by considerations of expediency.

Into the formation of concepts, therefore, two factors are operative, an objective empirical factor, and a subjective or purposive factor. The fitness of a concept is seen in relation to its purpose, which we shall now consider.

The purpose of a concept is its use for prediction. The old logic set up the syllogism as the type of thought-activity, and its simplest example is the well-known

In general, the scheme runs

One can say that this method of reasoning is in regular use even to this day. It must be added, however, that this use is of a quite different nature from that of the ancients. Whereas formerly the setting up of the first proposition or the major premise was considered the most important thing, and the establishment of the second proposition or minor premise was thought to be a rather trifling matter, now the relation is reversed. The major premise contains the description of a concept, the minor makes the assertion that a certain thing belongs under this concept. What right exists for such an assertion? The most palpable reply would be, since all the elements of the concept M (including B) are found in C, C belongs under the concept M. Such a conclusion would indeed be binding, but at the same time quite worthless, for it only repeats the minor premise. Actually the method of reasoning is essentially different, for the minor premise is not obtained by showing that all the elements of the concept M are found in C, but only some of them. The conclusion is not necessary, but only probable, and the whole process of reasoning runs: Certain elements are frequently found together, therefore they are united in the concept M. Certain of these elements are recognized in the thing C, therefore probably the other elements of the concept M will be found in C.

The old logic, also, was familiar with this kind of conclusion. It was branded, however, as the worst of all, by the name of incomplete induction, since the absolute certainty demanded of the syllogism did not belong to its results. One must admit, however, that the whole of modern science makes use of no other form of reasoning than incomplete induction, for it alone admits of a prediction, that is, an indication of relations which have not been immediately observed.

How does science get along with the defective certainty of this process of reasoning? The answer is, that the probability of the conclusion can run through all degrees from mere conjecture to the maximum probability, which is practically indistinguishable from certainty. The probability is the greater the more frequently an incomplete induction of this kind has proven correct in later experience. Accordingly we have at our command a number of expressions which in their simplest and most general form have the appearance: If an element A is met within a thing, then the element B is also found in it (in spatial or temporal relationship).

If the relation is temporal, this general statement is known by some such name as the law of causality. If it is spatial, one talks of the idea (in the Platonic sense), or the type of the thing, of substance, etc.

From the considerations here presented we get an easy answer to many questions which are frequently discussed in very different senses. First, the question concerning the general validity of the law of causality. All attempts to prove such a validity have failed, and there has remained only the indication that without this law we should feel an unbearable uncertainty in reference to the world. From this, however, we see very plainly that here it is merely a question of expediency. From the continuous flux of our experiences we hunt out those groups which can always be found again, in order to be able to conclude that if the element A is given, the element B will be present. We do not find this relationship as "given," but we put it into our experiences, in that we consider the parts which correspond to the relationship as belonging together.

The very same thing may be said of spatial complexes. Such factors as are always, or at any rate often, found together are taken by us as "belonging together," and out of them a concept is formed which embraces these factors. A question as to the why has here, as with the temporal complexes, no definite meaning. There are countless things that happen together once to which we pay no attention because they happen only once or but seldom. The knowledge of the fact that such a single concurrence exists amounts to nothing, since from the presence of one factor it does not lead to a conclusion as to the presence of another, and therefore does not make possible prediction. Of all the possible, and even actual combinations, only those interest us which are repeated, and this arbitrary but expedient selection produces the impression that the world consists only of combinations that can be repeated; that, in other words, the law of causality or of the type is a general one. However general or limited application those laws have, is more a question of our skill in finding the constant combinations among those that are present than a question of objective natural fact.

Thus we see the development and pursuit of all sciences going on in such a way that on the one hand more and more constant combinations are discovered, and on the other hand more inclusive relations of this kind are found out, by means of which elements are united with each other which before no one had even tried to bring together. So sciences are increasing both in the sense of an increasing complication and in an increasing unification.

If we consider from this standpoint the development and procedure of the various sciences, we find a rational division of the sum total of science in the question as to the scope and multiplicity of the combinations or groups treated of in them. These two properties are in a certain sense antithetical. The simpler a complex is, that is, the fewer elements brought together in it, the more frequently it is met with, and vice versa. One can therefore arrange all the sciences in such a way that one begins with the least multiplicity and the greatest scope, and ends with the greatest multiplicity and the least scope. The first science will be the most general, and will therefore contain the most general and therefore the most barren concepts; the last will contain the most specific and therefore the richest.

What are these limiting concepts? The most general is the concept of thing, that is, any piece of experience, seized arbitrarily from the flux of our experiences, which can be repeated. The most specific and richest is the concept of human intercourse. Between the science of things and the science of human intercourse, all the other sciences are found arranged in regular gradation. If one follows out the scheme the following outline results:

This table is arbitrary in so far as the grades assumed can be increased or diminished according to need. For example, mechanics and physics could be taken together; or between physics and chemistry, physical chemistry could be inserted. Likewise between physiology and psychology, anthropology might find a place; or the first five sciences might be united under mathematics. How one makes these divisions is entirely a practical question, which will be answered at any time in accordance with the purposes of division; and dispute concerning the matter is almost useless.

I should like, however, to call attention to the three great groups of mathematics, energetics, and biology (in the wider sense). They represent the decisive regulative thought which humanity has evolved, contributed up to this time, toward the scientific mastery of its experiences. Arrangement is the fundamental thought of mathematics. From mechanics to chemistry the concept of energy is the most important; and for the last three sciences it is the concept of life. Mathematics, energetics, and biology, therefore, embrace the totality of the sciences.

Before we enter upon the closer consideration of these sciences, it will be well to anticipate another objection which can be raised on the basis of the following fact. Besides the sciences named (and those which lie between them) there are many others, as geology, history, medicine, philology, which we find difficulty in arranging in the above scheme, which must, however, be taken into consideration in some way or other. They are often characterized by the fact that they stand in relation with several of the sciences named, but even more by the following circumstance. Their task is not, as is true of the pure sciences above named, the discovery of general relationships, but they relate rather to existing complex objects whose origin, scope, extent, etc., in short, whose temporal and spatial relationships they have to discover or to "explain." For this purpose they make use of relations which are placed at their disposal by the first-named pure sciences. These sciences, therefore, had better be called applied sciences. However, in this connection we should not think only or even chiefly of technical applications; rather the expression is used to indicate that the reciprocal relations of the parts of an object are to be called to mind by the application of the general rules found in pure science.

While in such a task the abstraction process of pure science is not applicable (for the omission of certain parts and the concentration upon others which is characteristic of these is excluded by the nature of the task), yet in a given case usually the necessity of bringing in various pure sciences for the purpose of explanation is evident.

Astronomy is one of these applied sciences. Primarily it rests upon mechanics, and in its instrumental portion, upon optics; in its present development on the spectroscopic side, however, it borrows considerably of chemistry. In like manner history is applied sociology and psychology. Medicine makes use of all the sciences before mentioned, up to psychology, etc.

It is important to get clearly in mind the nature of these sciences, since, on account of their compound nature, they resist arrangement amongst the pure sciences, while, on account of their practical significance, they still demand consideration. The latter fact gives them also a sort of arbitrary or accidental character, since their development is largely conditioned by the special needs of the time. Their number, speaking in general, is very large, since each pure science may be turned into an applied science in various ways; and since in addition we have combinations of two, three, or more sciences. Moreover, the method of procedure in the applied sciences is fundamentally different from that in the pure sciences. In the first it is a question of the greatest possible analysis of a single given complex into its scientifically comprehensible parts; while pure science, on the other hand, considers many complexes together in order to separate out from them their common element, but expressly disclaims the complete analysis of a single complex.

In scientific work, as it appears in practice, pure and applied science are by no means sharply separated. On the one hand the auxiliaries of investigations, such as apparatus, books, etc., demand of the pure investigator knowledge and application in applied science; and, on the other hand, the applied scientist is frequently unable to accomplish his task unless he himself becomes for the time being a pure investigator and ascertains or discovers the missing general relationships which he needs for his task. A separation and differentiation of the two forms of science was necessary, however, since the method and the aim of each present essential differences.

In order to consider the method of procedure of pure science more carefully, let us turn back to the table on pages 339, 340, and attend to the single sciences separately. The theory of arrangement was mentioned first, although this place is usually assigned to mathematics. However, mathematics has to do with the concepts of number and magnitude as fundamentals, while the theory of arrangement does not make use of these. Here the fundamental concept is rather the thing or object of which nothing more is demanded or considered than that it is a fragment of our experience which can be isolated and will remain so. It must not be an arbitrary combination; such a thing would have only momentary duration, and the task of science, to learn the unknown from the given, could not find application. Rather must this element have such a nature that it can be characterized and recognized again, that is, it must already have a conceptual nature. Therefore only parts of our experience which can be repeated (which alone can be objects of science) can be characterized as things or objects. But in saying this we have said all that was demanded of them. In other respects they may be just as different as is conceivable.

If the question is asked, What can be said scientifically about indefinite things of this sort? it is especially the relations of arrangement and association which yield an answer. If we call any definite combination of such things a group, we can arrange such a group in different ways, that is, we can determine for each thing the relation in which it is to stand to the neighboring thing. From every such arrangement result not only the relationships indicated, but a great number of new ones, and it appears that when the first relationships are given the others always follow in like manner. This, however, is the type of the scientific proposition or natural law (page 335). From the presence of certain relations of arrangement we can deduce the presence of others which we have not yet demonstrated.

To illustrate this fact by an example, let us think of the things arranged in a simple row, while we choose one thing as a first member and associate another with it as following it; with the latter another is associated, etc. Thereby the position of each thing in the row is determined only in relation to the immediately preceding thing. Nevertheless, the position of every member in the whole row, and therefore its relation to every other member, is determined by this. This is seen in a number of special laws. If we differentiate former and latter members we can formulate the proposition, among others, if B is a later member with reference to A, and C with reference to B, then C is also a later member with reference to A.

The correctness and validity of this proposition seems to us beyond all doubt. But this is only a result of the fact that we are able to demonstrate it very easily in countless single cases, and have so demonstrated it. We know only cases which correspond to the proposition, and have never experienced a contradictory case. To call such a proposition, however, a necessity of thinking, does not appear to me correct. For the expression necessity of thinking can only rest upon the fact that every time the proposition is thought, that is, every time one remembers its demonstration, its confirmation always arises. But every sort of false proposition is also thinkable. An undeniable proof of this is the fact that so much which is false is actually thought. But to base the proof for the correctness of a proposition upon the impossibility of thinking its opposite is an impossible undertaking, because every sort of nonsense can be thought: where the proof was thought to have been given, there has always been a confusion of thought and intuition, proof or inspection.

With this one proposition of course the theory of order is not exhausted, for here it is not a question of the development of this theory, but of an example of the nature of the problems of science. Of the further questions we shall briefly discuss the problem of association.

If we have two groups A and B given, one can associate with every member of A one of B; that is, we determine that certain operations which can be carried on with the members of A are also to be carried on with those of B. Now we can begin by simply carrying out the association, member for member. Then we shall have one of three results: A will be exhausted while there are still members of B left, or B will be exhausted first, or finally A and B will be exhausted at the same time. In the first case we call A poorer than B; in the second B poorer than A; in the third both quantities are alike.

Here for the first time we come upon the scientific concept of equality, which calls for discussion. There can be no question of a complete identity of the two groups which have been denominated equal, for we have made the assumption that the members of both groups can be of any nature whatever. They can then be as different as possible, considered singly, but they are alike as groups. However I may arrange the members of A, I can make a similar arrangement of the members of B, since every member of A has one of B associated with it; and with reference to the property of arrangement there is no difference to be observed between A and B. If, however, A is poorer or richer than B, this possibility ceases, for then one of the groups has members to which none of the members in the other group corresponds; so that the operations carried out with these members cannot be carried out with those of the other group.

Equality in the scientific sense, therefore, means equivalence, or the possibility of substitution in quite definite operations or for quite definite relations. Beyond this the things which are called like may show any differences whatever. The general scientific process of abstraction is again easily seen in this special case.

On the basis of the definitions just given, we can establish further propositions. If group A equals B, and B equals C, then A also equals C. The proof of this is that we can relate every member of A to a corresponding member of B and by hypothesis no member will be left. Then C is arranged with reference to B, and here also no member is left. By this process every member of A, through the connecting link of a member of B, is associated with a member of C, and this association is preserved even if we cut out the group B. Therefore A and C are equal. The same process of reasoning can be carried out for any number of groups.

Likewise it can be demonstrated that if A is poorer than B and B poorer than C, then A is also poorer than C. For in the association of B with A some members of B are left over by hypothesis, and likewise some members of C are left over if one associates C with B. Therefore in the association of C with A, not only those members are left over which could not be associated with B, but also those members of C which extend beyond B. This proposition can be extended to any number of groups, and permits the arrangement of a number of different groups in a simple series by beginning with the poorest and choosing each following so that it is richer than the preceding but poorer than the following. From the proposition just established, it follows that every group is so arranged with reference to all other groups that it is richer than all the preceding and poorer than all the following.^[3]

In this derivation of scientific proposition or laws of the simplest kinds, the process of derivation and the nature of the result becomes particularly clear. We arrive at such a proposition by performing an operation and expressing the result of it. This expression enables us to avoid the repetition of the operation in the future, since in accordance with the law we can indicate the result immediately. Thus an abbreviation and therefore, a facilitation of the problem is attained which is the more considerable the larger the number of operations saved.

If we have a number of equal groups, we know by the process of association that all of the operations with reference to arrangement which we can perform with one of them can be performed with all the others. It is sufficient, therefore, to determine the properties of arrangement of one of these groups in order to know forthwith the properties of all the others. This is an extremely important proposition, which is continually employed for the most various purposes. All speaking, writing, and reading rests upon the association of thoughts with sounds and symbols, and by arranging the signs in accordance with our thoughts we bring it to pass that our hearers or readers think like thoughts in like order. In a similar fashion we make use of various systems of formulæ in the different sciences, especially in the simpler sciences; and these formulæ we correlate with phenomena and use in place of the phenomena themselves, and can therefore derive from them certain characteristics of phenomena without being compelled to use the latter. The force of this process appears very strikingly in astronomy where, by the use of definite formulæ associated with the different heavenly bodies, we can foretell the future positions of these bodies with a high degree of approximation.

From the theory of order we come to the theory of number or arithmetic by the systematic arrangement or development of an operation just indicated (page 343). We can arrange any number of groups in such a way that a richer always follows a poorer. But the complex obtained in this manner is always accidental with reference to the number and the richness of its members. A regular and complete structure of all possible groups is evidently obtained only if we start from a group of one member or from a simple thing, and by the addition of one member at a time make further groups out of those that we have. Thus we obtain different groups arranged according to an increasing richness, and since we have advanced one member at a time, that is, made the smallest step which is possible, we are certain that we have left out no possible group which is poorer than the richest to which the operation has been carried.

This whole process is familiar; it gives the series of the positive whole numbers, that is, the cardinal numbers. It is to be noted that the concept of quantity has not yet been considered; what we have gained is the concept of number. The single things or members in this number are quite arbitrary, and especially they do not need to be alike in any manner. Every number forms a group-type, and arithmetic or the science of numbers has the task of investigating the properties of these different types with reference to their division and combination. If this is done in general form, without attention to the special amount of the number, the corresponding science is called algebra. On the other hand, by the application of formal rules of formation, the number system has had one extension after another beyond the territory of its original validity. Thus counting backward led to zero and to the negative numbers; the inversion of involution to the imaginary numbers. For the group-type of the positive whole numbers is the simplest but by no means the only possible one, and for the purpose of representing other manifolds than those which are met with in experience, these new types have proved themselves very useful.

At the same time the number series gives us an extremely useful type of arrangement. In the process of arising it is already ordered, and we make use of it for the purpose of arranging other groups. Thus, we are accustomed to furnish the pages in a book, the seats in a theatre, and countless other groups which we wish to make use of in any kind of order with the signs of the number series, and thereby we make the tacit assumption that the use of that corresponding group shall take place in the same order as the natural numbers follow each other. The ordinal numbers arising therefrom do not represent quantities, nor do they represent the only possible type of arrangement, but they are again the simplest of all. We come to the concept of magnitude only in the theory of time and space. The theory of time has not been developed as a special science; on the contrary, what we have to say about time first appears in mechanics. Meantime we can present the fundamental concepts, which arise in this connection, with reference to such well-known characteristics of time that the lack of a special science of time is no disadvantage.

The first and most important characteristic of time (and of space, too) is that it is a continuous manifold; that is, every portion of time chosen can be divided at any place whatever. In the number series this is not the case; it can be divided only between the single numbers. The series one to ten has only nine places of division and no more. A minute, or a second, on the other hand, has an unlimited number of places of division. In other words, there is nothing in the lapse of any time which hinders us from separating or distinguishing in thought at any given instant the time which has elapsed till then from the following time. It is just the same with space, except that time is a simple manifold and space a threefold, continuous manifold.

Nevertheless, when we measure them, we are accustomed to indicate times and spaces with numbers. If we first examine, for example, the process of measuring a length, it consists in our applying to the distance to be measured a length conceived as unchangeable, the unit of measure, until we have passed over the distance. The number of these applications gives us the measure or magnitude of the distance. The result is that by the indication of arbitrarily chosen points upon the continuous distance, we place upon it an artificial discontinuity which enables us to associate it with the discontinuous number series.

A still further assumption, however, belongs to the concept of measuring, namely, that the parts of the distance cut off by the unit used as a measure be equal, and it is taken for granted that this requirement will be fulfilled to whatever place the unit of measure is shifted. As may be seen, this is a definition of equality carried further than the former, for one cannot actually replace a part of the distance by another in order to convince one's self that it has not changed. Just as little can one assert or prove that the unit of measure in changing its place in space remains of the same length; we can only say that such distances as are determined by the unit of measure in different places are declared or defined as equal. Actually, for our eye, the unit of measure becomes smaller in perspective the farther away from it we find ourselves.

From this example we see again the great contribution which arbitrariness or free choice has made to all our structure of science. We could develop a geometry in which distances which seem subjectively equal to our eye are called equal, and upon this assumption we would be able to develop a self-consistent system or science. Such a geometry, however, would have an extremely complex and impractical structure for objective purposes (as, for example, land measurement), and so we strive to develop a science as free as possible from subjective factors. Historically, we have before us a process of this sort in the astronomy of Ptolemy and that of Copernicus. The former corresponded to the subjective appearances in the assumption that all heavenly bodies revolved around the earth, but proved to be very complicated when confronted with the task of mastering these movements with figures. The latter gave up the subjective standpoint of the observer, who looked upon himself as the centre, and attained a tremendous simplification by placing the centre of revolution in the sun.

A few words are to be said here about the application of arithmetic and algebra to geometry. It is well known that under definite assumptions (coördinates), geometrical figures can be represented by means of algebraic formulæ, so that the geometrical properties of the figure can be deduced from the arithmetical properties of the formulæ, and vice versa. The question must be asked how such a close and univocal relationship is possible between things of such different nature. The answer is, that here is an especially clear case of association. The manifold of numbers is much greater than that of surface or space, for while the latter are determined by two or three independent measurements, one can have any number of independent number series working together. Therefore the manifold of numbers is arbitrarily limited to two or three independent series, and in so far determines their mutual relations (by means of the laws of cosine) that there results a manifold, corresponding to the spatial, which can be completely associated with the spatial manifold. Then we have two manifolds of the same manifold character, and all characteristics of arrangement and size of the one find their likeness in the other.

This again characterizes an extremely important scientific procedure which consists, namely, in constructing a formal manifold for the content of experience of a certain field, to which one attributes the same manifold character which the former possesses. Every science reaches by this means a sort of formal language of corresponding completeness, which depends upon how accurately the manifold character of the object is recognized and how judiciously the formulæ have been chosen. While in arithmetic and algebra this task has been performed fairly well (though by no means absolutely perfectly), the chemical formulæ, for instance, express only a relatively small part of the manifold to be represented; and in biology as far as sociology, scarcely the first attempts have been made in the accomplishment of this task.

Language especially serves as such a universal manifold to represent the manifolds of experience. As a result of its development from a time of less culture, it has by no means sufficient regularity and completeness to accomplish its purpose adequately and conveniently. Rather, it is just as unsystematic as the events in the lives of single peoples have been, and the necessity of expressing the endlessly different particulars of daily life has only allowed it to develop so that the correspondence between word and concept is kept rather indefinite and changeable, according to need within somewhat wide limits. Thus all work in those sciences which must make vital use of these means, as especially psychology and sociology, or philosophy in general, is made extremely difficult by the ceaseless struggle with the indefiniteness and ambiguity of language. An improvement of this condition can be effected only by introducing signs in place of words for the representation of concepts, as the progress of science allows it, and equipping these signs with the manifold which from experience belongs to the concept.

An intermediate position in this respect is taken by the sciences which were indicated above as parts of energetics. In this realm there is added to the concepts order, number, size, space, and time, a new concept, that of energy, which finds application to every single phenomenon in this whole field, just as do those more general concepts. This is due to the fact that a certain quantity, which is known to us most familiarly as mechanical work, on account of its qualitative transformability and quantitative constancy, can be shown to be a constituent of every physical phenomenon, that is, every phenomenon which belongs to the field of mechanics, physics, and chemistry. In other words, one can perfectly characterize every physical event by indicating what amounts and kinds of energy have been present in it and into what energies they have been transformed. Accordingly, it is logical to designate the so-called physical phenomena as energetical.

That such a conception is possible is now generally admitted. On the other hand, its expediency is frequently questioned, and there is at present so much the more reason for this because a thorough presentation of the physical sciences in the energetical sense has not yet been made. If one applies to this question the criterion of the scientific system given above, the completeness of the correspondence between the representing manifold and that to be represented, there is no doubt that all previous systematizations in the form of hypotheses which have been tried in these sciences are defective in this respect. Formerly, for the purpose of representing experiences, manifolds whose character corresponded to the character of the manifold to be represented only in certain salient points without consideration of any rigid agreement, indeed, even without definite question as to such an agreement, have been employed.

The energetical conception admits of that definiteness of representation which the condition of science demands and renders possible. For each special manifold character of the field a special kind of energy presents itself: science has long distinguished mechanical, electric, thermal, chemical, etc., energies. All of these different kinds hold together by the law of transformation with the maintenance of the quantitative amount, and in so far are united. On the other hand, it has been possible to fix upon the corresponding energetical expression for every empirically discovered manifold. As a future system of united energetics, we have then a table of possible manifolds of which energy is capable. In this we must keep in mind the fact that, in accordance with the law of the conservation, energy is a necessarily positive quantity which also is furnished with the property of unlimited possibility of addition; therefore, every particular kind of energy must have this character.

The very small manifold which seems to lack this condition is much widened by the fact that every kind of energy can be separated into two factors, which are only subject to the limitation that their product, the energy, fulfills the conditions mentioned while they themselves are much freer. For example, one factor of a kind of enemy can become negative as well as positive; it is only necessary that at the same time the other factor should become negative, viz., positive.

Thus it seems possible to make a table of all possible forms of energy, by attributing all thinkable manifold characteristics to the factors of the energy and then combining them by pairs and cutting out those products which do not fulfill the above-mentioned conditions. For a number of years I have tried from time to time to carry out this programme, but I have not yet got far enough to justify publication of the results obtained.

If we turn to the biological sciences, in them the phenomenon of life appears to us as new. If we stick to the observed facts, keeping ourselves free from all hypotheses, we observe as the general characteristics of the phenomena of life the continuous stream of energy which courses through a relatively constant structure. Change of substance is only a part, although a very important part, of this stream. Especially in plants we can observe at first hand the great importance of energy in its most incorporeal form, the sun's rays. Along with this, self-preservation and development and reproduction, the begetting of offspring of like nature, are characteristic. All of these properties must be present in order that an organism may come into existence; they must also be present if the reflecting man is to be able by repeated experience to form a concept of any definite organism, whether of a lion or of a mushroom. Other organisms are met with which do not fulfill these conditions; on account of their rarity, however, they do not lead to a species concept, but are excluded from scientific consideration (except for special purposes) as deformities or monsters.

While organisms usually work with kinds of energy which we know well from the inorganic world, organs are found in the higher forms which without doubt cause or assist transfers of energy, but we cannot yet say definitely what particular kind of energy is active in them. These organs are called nerves, and their function is regularly that, after certain forms of energy have acted upon one end of them, they should act at the other end and release the energies stored up there which then act in their special manner. That energetical transformations also take place in the nerve during the process of nervous transmission can be looked upon as demonstrated. We shall thus be justified in speaking of a nerve energy, while leaving it undecided whether there is here an energy of a particular kind, or perhaps chemical energy, or finally a combination of several energies.

While these processes can be shown objectively by the stimulation of the nerve and its corresponding releasing reaction in the end apparatus (for instance, a muscle), we find in ourselves, connected with certain nervous processes, a phenomenon of a new sort which we call self-consciousness. From the agreement of our reactions with those of other people we conclude with scientific probability that they also have self-consciousness; and we are justified in making the same conclusion with regard to some higher animals. How far down something similar to this is present cannot be determined by the means at hand, since the analogy of organization and of behavior diminishes very quickly; but the line is probably not very long, in view of the great leap from man to animal. Moreover, there are many reasons for the view that the gray cortical substance in the brain, with its characteristic pyramidal cell, is the anatomical substratum of this kind of nervous activity.

The study of the processes of self-consciousness constitutes the chief task of psychology. To this science belong those fields which are generally allotted to philosophy, especially logic and epistemology, while æsthetics, and still more ethics, are to be reckoned with the social sciences.

The latter have to do with living beings in so far as they can be united in groups with common functions. Here in place of the individual mind appears a collective mind, which owing to the adjustment of the differences of the members of society shows simpler conditions than that. From this comes especially the task of the historical sciences. The happenings in the world accessible to us are conditioned partly by physical, partly by psychological factors, and both show a temporal mutability in one direction. Thus arises on the one hand a history of heaven and earth, on the other hand a history of organisms up to man.

All history has primarily the task of fixing past events through the effects which have remained from them. Where such are not accessible, only analogy is left, a very doubtful means for gaining a conception of those events. But it must be kept in mind that an event which has left no evident traces has no sort of interest for us, for our interest is directly proportional to the amount of change which that event has caused in what we have before us. The task of historical science is just as little exhausted, however, with the fixing of former events as, for instance, the task of physics with the establishment of a single fact, as the temperature of a given place at a given time. Rather the individual facts must serve to bring out the general characteristics of the collective mind, and the much discussed historical laws are laws of collective psychology. Just as physical and chemical laws are deduced in order with their help to predict the course of future physical events (to be called forth either experimentally or technically), so should the historical laws contribute to the formation and control of social and political development. We see that the great statesmen of all time have eagerly studied history for this purpose, and from that we derive the assurance that there are historical laws in spite of the objections of numerous scholars.

After this brief survey, if we look back over the road we have come, we observe the following general facts. In every case the development of a science consists in the formation of concepts by certain abstractions from experience, and setting of these concepts in relation with each other so that a systematical control of certain sides of our experience is made possible. These relations, according to their generality and reliability, are called rules or laws. A law is the more important the more it definitely expresses concerning the greatest possible number of things, and the more accurately, therefore, it enables us to predict the future. Every law rests upon an incomplete induction, and is therefore subject to modification by experience. From this there results a double process in the development of science.

First, the actual conditions are investigated to find out whether, besides those already known, new rules or laws, that is, constant relations between individual peculiarities, cannot be discovered between them. This is the inductive process, and the induction is always an incomplete one on account of the limitlessness of all possible experience.

Immediately the relationship found inductively is applied to cases which have not yet been investigated. Especially such cases are investigated as result from a combination of several inductive laws. If these are perfectly certain, and the combination is also properly made, the result has claim to unconditional validity. This is the limit which all sciences are striving to reach. It has almost been reached in the simpler sciences: in mathematics and in certain parts of mechanics. This is called the deductive process.

In the actual working of every science the two methods of investigation are continually changing. The best means of finding new successful inductions is in the making of a deduction on a very insufficient basis, perhaps, and subsequently testing it in experience. Sometimes the elements of his deductions do not come into the investigator's consciousness; in such cases we speak of scientific instinct. On the other hand we have much evidence from great mathematicians that they were accustomed to find their general laws by the method of induction, by trying and considering single cases; and that the deductive derivation from other known laws is an independent operation which sometimes does not succeed until much later. Indeed there is to-day a number of mathematical propositions which have not yet reached the second stage and therefore have at present a purely inductive empirical character. The proportion of such laws in science increases very quickly with the rise in the scale (page 339).

Another peculiarity which may be mentioned here is that in the scale all previous sciences have the character of applied sciences (page 341) with reference to those which follow, since they are everywhere necessary in the technique of the latter, yet do not serve to increase their own field but are merely auxiliaries to the latter.

If we ask finally what influence upon the shaping of the future such investigations as those which have been sketched in outline above can have, the following can be said. Up till now it has been considered a completely uncontrollable event whether and where a great and influential man of science has developed. It is obvious that such a man is among the most costly treasures which a people (and, indeed, humanity) can possess. The conscious and regular breeding of such rarities has not been considered possible. While this is still the case for the very exceptional genius, we see in the countries of the older civilization, especially in Germany at present, a system of education in vogue in the universities by which a regular harvest of young scientific men is gained who not only have a mastery of knowledge handed down, but also of the technique of discovery. Thereby the growth of science is made certain and regular, and its pursuit is raised to a higher plane. These results were formerly attained chiefly by empirically and oftentimes by accidental processes. It is a task of scientific theory to make this activity also regular and systematic, so that success is no more dependent solely upon a special capacity for the founding of a "school" but can also be attained by less original minds. By the mastery of methods the way to considerably higher performances than he could otherwise attain will be open for the exceptionally gifted.

[3] Equal groups cannot be distinguished here, and therefore represent only a group.

THE CONTENT AND VALIDITY OF THE CAUSAL LAW