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 a.s.sociation.
If we have two groups A and B given, one can a.s.sociate 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 a.s.sociation, 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 quant.i.ties 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 ident.i.ty of the two groups which have been denominated equal, for we have made the a.s.sumption 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 a.s.sociated 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 subst.i.tution 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 a.s.sociated with a member of C, and this a.s.sociation 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 a.s.sociation of B with A some members of B are left over by hypothesis, and likewise some members of C are left over if one a.s.sociates C with B. Therefore in the a.s.sociation of C with A, not only those members are left over which could not be a.s.sociated 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]
[Footnote 3: Equal groups cannot be distinguished here, and therefore represent only a group.]
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 repet.i.tion 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 a.s.sociation 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 a.s.sociation of thoughts with sounds and symbols, and by arranging the signs in accordance with our thoughts we bring it to pa.s.s that our hearers or readers think like thoughts in like order. In a similar fashion we make use of various systems of formulae in the different sciences, especially in the simpler sciences; and these formulae 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 formulae a.s.sociated 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 quant.i.ty 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 a.s.sumption 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 quant.i.ties, 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 s.p.a.ce. 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 s.p.a.ce, 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 s.p.a.ce, except that time is a simple manifold and s.p.a.ce a threefold, continuous manifold.
Nevertheless, when we measure them, we are accustomed to indicate times and s.p.a.ces 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 pa.s.sed 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 a.s.sociate it with the discontinuous number series.
A still further a.s.sumption, 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 a.s.sert or prove that the unit of measure in changing its place in s.p.a.ce 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 a.s.sumption 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 a.s.sumption 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 a.s.sumptions (coordinates), geometrical figures can be represented by means of algebraic formulae, so that the geometrical properties of the figure can be deduced from the arithmetical properties of the formulae, 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 a.s.sociation. The manifold of numbers is much greater than that of surface or s.p.a.ce, 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 a.s.sociated 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 formulae have been chosen. While in arithmetic and algebra this task has been performed fairly well (though by no means absolutely perfectly), the chemical formulae, 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, s.p.a.ce, 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 quant.i.ty, which is known to us most familiarly as mechanical work, on account of its qualitative transformability and quant.i.tative constancy, can be shown to be a const.i.tuent 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 quant.i.tative 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 quant.i.ty 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 a.s.sist 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 a.n.a.logy 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 const.i.tutes the chief task of psychology. To this science belong those fields which are generally allotted to philosophy, especially logic and epistemology, while aesthetics, 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 a.n.a.logy 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 a.s.surance 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.
THE CONTENT AND VALIDITY OF THE CAUSAL LAW
BY BENNO ERDMANN
(_Translated from the German by Professor Walter T. Marvin, Western Reserve University_)
[Benno Erdmann, Professor of Philosophy, University of Bonn, since 1898. b. October 5, 1851, Glogau in Schlesien, Germany. Ph.D.; Privy Councilor. Academical Lecturer, Berlin, 1876- ; Special Professor, Kiel, 1878-79; Regular Professor, _ibid._ 1879-84; _ibid._ Breslau, 1884-90; _ibid._ Halle, 1890-98. Member various scientific and learned societies. Author of _The Axioms of Geometry_; _Kant"s Criticism_; _Logic_; _Psychological Researches on Reading_ (together with Prof. Ramon Dodge); _The Psychology of the Child and the School_; _Historical Researches an Kant"s Prolegomena_, and many other works and papers in Philosophy.]
We have learned to regard the real, which we endeavor to apprehend scientifically in universally valid judgments, as a whole that is connected continuously in time and in s.p.a.ce and by causation, and that is accordingly continuously self-evolving. This continuity of connection has the following result, namely, every attempt to cla.s.sify the sum total of the sciences on the basis of the difference of their objects leads merely to representative types, that is, to species which glide into one another. We find no gaps by means of which we can separate sharply physics and chemistry, botany and zoology, political and economic history and the histories of art and religion, or, again, history, philology, and the study of the prehistoric.
As are the objects, so also are the methods of science. They are separable one from another only through a division into representative types; for the variety of these methods is dependent upon the variety of the objects of our knowledge, and is, at the same time, determined by the difference between the manifold forms of our thought, itself a part of the real, with its elements also gliding into One another.[4]
[Footnote 4: Cf. the author"s "Theorie der Typeneinteilungen," _Philosophische Monatshefte_, vol. x.x.x, Berlin, 1894.]
The threads which join the general methodology of scientific thought with neighboring fields of knowledge run in two main directions. In the one direction they make up a closely packed cable, whereas in the other their course diverges into all the dimensions of scientific thought.
That is to say, first, methodology has its roots in logic, in the narrower sense, namely, in the science of the elementary forms of our thought which enter into the make-up of all scientific methods.