If by experience we have found a proposition of the content, If A is, then B is also, the two concepts A and B generally consist of several elements which we will designate as a, a´, a´´, a´´´, etc., and as b, b´, b´´, b´´´. Now the question arises, whether or not all these elements are essential for the relation in question. It is quite possible, in fact, even highly probable, that at first only a special instance of the existing phenomena was found, that is, that theconcept A, which has been found to be connected with the concept B, contains other determining parts which are not at all requisite to the appearance of B.
The general method of convincing oneself of this is by eliminating one by one the component parts of the concept A, namely, a, a´, a´´, etc., and then seeing whether B still appears. It is not always easy to carry out this process of elimination. Our greater or less ability to conduct such investigations depends upon whether we deal with things that are merely the objects of ourobservation, and which we ourselves have not the power to change (as, for example, astronomical phenomena), or with things which are the objects of ourexperimentation, and which we can influence. In the latter case one or another factor is usually found which can be eliminated without the disappearance of B, and then we must proceed in such a way as to form a corresponding new concept A´ from the factors recognized as necessary (which new concept will be more general than the former A), and to express the given proposition in the improved form: If A´ is, then B is also.
Quite similar is the case with the other member of this relation. It often happens that when a, or a´´, a´´´ is found, somewhat different things appear, which do not fit the concept as first constructed. Then we must multiply the experiences as much as possible in order to determine what constant elements are found in the concept B, and to form from theseconstant elements the corresponding concept B´. The improved proposition will then read: if A´ is, then B´ is also.
This entire process may be called the purification of the causal relation. By this term we express the general fact that in first forming such a regular connection, the proper concepts are very seldom brought into relation with one another at once. The cause of it is that at first we make use ofexistingconcepts which had been formed for quite a different purpose. It must therefore be regarded as a special piece of good fortune if these old concepts should at once prove suited to the new purpose. Furthermore, the existing concepts are as a rule so vaguely characterized by their names, which we must employ to express the new relation, that for this reason also it is often necessary to determine empirically in what way the concept is to be definitely established.
The various sciences are constantly occupied with this work of the mutual adaptation of the concepts that enter into a causal relation. By way of example, we may take the "self-understood" proposition which we use when we call out to a careless child when it sticks its finger into the flame of a candle, "Fire burns!" We discover that there are self-luminous bodies which produce no increase of temperature, and therefore no sensation of pain. We discover that there are processes of combustion that develop no light, but heat enough to burn one'sfingers. And, finally, the scientific investigation of this proposition arrives at the general expression that, as a rule, chemical processes are accompanied by the development of heat, but that, conversely, such processes may also be accompanied by the absorption of heat. In this way that casual sentence which we call out to the child develops into the extensive science of thermo-chemistry when it is subjected to the continuous purification of the causal relation, which is the general task of science.
It remains to be added that in this process of adapting concepts it is necessary also sometimes to follow the opposite course. This is the case whenexceptionsare noticed in a relation as expressed for the time being; when, therefore, the proposition if A is present, then B is present also, is in a great many instances valid, but occasionally fails. This is an indication that in the concept A an element is still lacking. This element, however, is present in the instances that tally, but absent in the negative cases, and its absence is not noticed because it is not contained in A. Then it is necessary to seek this part, and after it has been found, to embody it in the concept A, which thus passes into the new concept A´.
This case is the obverse of the former one. Here the more suitable concept proves to be less general than the concept accepted temporarily, while in the first case the improved concept is more general. Hence we formulate the rule: exceptions to thetemporary rule require a limitation, while an unforeseen freedom requires an extension, of the accepted concept.
The form of conclusion previously discussed,because it has been so, I expect it will continue to be so in the future, is the form through which each science has arisen and has won its real content, that is, its value for the judgment of the future. It is calledinference by induction, and the sciences in which it is preponderatingly applied are calledinductive sciences. They are also called experiential or empirical sciences. At the basis of this nomenclature is the notion that there are other sciences, the deductive or rational sciences, in which a reverse logical procedure is applied, whereby from general principles admitted to be valid in advance, according to an absolutely sure logical process, conclusions of like absolute validity are drawn. At the present time people are beginning to recognize the fact that the deductive sciences must give up these claims one by one, and that they already have given them up to a certain extent; partly because on closer study they prove to be inductive sciences, and partly because they must forego the title and rank of a science altogether. The latter alternative applies especially to those provinces of knowledge which have not been used in prophesying the future or cannot be so used.
To return to the inductive method—it is to be noted thatAristotle, who was the first to describe it,proposed two kinds of induction, thecompleteand theincomplete. The first has this form: sinceallthings of a certain kind are so, eachindividual thingis so. While the incomplete induction merely says: sincemanythings of a certain kind are so,presumablyall things of this kind are so. One instantly perceives that the two conclusions are essentially different. The first lays claim to afford an absolutely certain result. But it rests upon the assumption thatallthe things of the kind in question are known and have been tested as to their behavior. This hypothesis is generally impossible of fulfilment, since we can never prove that there are not more things of the same kind other than those known to us or tested by us. Moreover, the conclusion issuperfluous, as it merely repeats knowledge that we have already directly acquired, since we have testedallthe things of the one kind, hence the special thing to which the predication refers.
On the other hand, theincompleteinduction affirms something that has not yet been tested, and therefore involves as a condition anextensionof our knowledge, sometimes an extremely important extension. To be sure, it must give up the claim to unqualified or absolute validity, but, to compensate, it acquires the irreplaceable advantage of lending itself to practical application. Indeed, in accordance with the scientific practice justified by experience, described onp. 29, the scientific inductive conclusionassumes the form: because it hasoncebeen found to be so, it willalwaysbe so. From this appears the significance of this method for the enlargement of science, which, without it, would have had to proceed at an incomparably slower pace.
In addition to the inductive method, science has (p. 38) another method, which, in a sense, should be the reverse of the inductive and is claimed to provide absolutely correct results. It is called thedeductivemethod, and it is described as the method that leads from premises of general validity by means of logical methods of general validity to results of general validity.
As a matter of fact, there is no science that does or could work in such a way. In the first place, we ask in vain, how can we arrive at such general, or absolutely valid, premises, since all knowledge is of empiric origin and is therefore equipped with the possibility of error as ineradicable evidence of this origin. In the next place, we cannot see how from principles at hand conclusions can be drawn the content of which exceeds that of these principles (and of the other means employed). In the third place, the absolute correctness of such results is doubtful from the fact that blunders in the process of reasoning cannot be excluded even where the premises and methods are absolutely correct. In practice it has actually come to pass that in the so-called deductive sciences doubts and contradictions on the part of the various investigatorsof the same question are by no means excluded. To wit, the discussion that has been carried on for centuries, and is not yet ended, over Euclid's parallel theorem in geometry.
If we ask whether, in the sense of the observations we have just made of the formation of scientific principles, there is anything at all like deduction, we can find a procedure which bears a certain resemblance with that impossible procedure and which, as a matter of fact, is frequently and to very good purpose applied in science. It consists in the fact that general principles which have been acquired through the ordinary incomplete induction areapplied to special instances which, at the proposition of the principle, had not been taken into consideration, and whose connection with the general concept had not become directly evident. Through such application of general principles to cases that have not been regarded before, specific natural laws are obtained which had not been foreseen either, but which, according to the probability of the thesis and the correctness of the application are also probably correct. However, the investigator, bearing in mind the factor of uncertainty in these ratiocinations feels in each such instance the need for testing the results by experience, and he does not consider thedeductioncomplete until he had foundconfirmationin experience.
Deduction, therefore, actually consists in the searching out of particular instances of a principleestablished by induction and in its confirmation by experience. This conduces to the growth of science, not in breadth, but in profundity. I again resort to the comparison I have frequently made of science with a very complex network. At first glance we cannot obtain a complete picture of all the meshes. So, at the first proposition of a natural law an immediate survey of the entire range of the possible experiences to which it may apply is inachievable. It is a regular, important, and necessary part of all scientific work to learn the extent of this range and investigate the specific forms which the law assumes in the remoter instances. Now, if an especially gifted and far-seeing investigator has succeeded in stating in advance an especially general formulation of an inductive law, it is everywhere confirmed in the course of the trial applications, and the impression easily arises that confirmation is superfluous, since it results simply in what had already been "deduced." In point of fact, however, the reverse is not infrequently the case, that the principle isnotconfirmed, and conditions quite different from those anticipated are found. Such discoveries, then, as a rule, constitute the starting-point of important and far-reaching modifications of the original formulation of the law in question.
As we see, deduction is a necessary complement of, in fact, a part of, the inductive process. The history of the origin of a natural law is in generalas follows. The investigator notices certain agreements in individual instances under his observation. He assumes that these agreements are general, and propounds a temporary natural law corresponding to them. Then he proceeds by further experimentation to test the law in order to see whether he can find full confirmation of it by a number of other instances. If not, he tries other formulations of the law applicable to the contradictory instances, or exclusive of them, as not allied. Through such a process of adjustment he finally arrives at a principle that possesses a certain range of validity. He informs other scientists of the principle. These in their turn are impelled to test other instances known to them to which the principle can be applied. Any doubts or contradictions arising from this again impel the author of the principle to carry out whatever readjustments may have become necessary. Upon the scientific imagination of the discoverer depends the range of instances sufficing for the formulation of the general inductive principle. It also frequently depends upon conscious operations of the mind dubbed "scientific instinct." But as soon as the principle has been propounded, even if only in the consciousness of the discoverer, the deductive part of the work begins, and the consequent test of the proposition has the most essential influence on the value of the result.
It is immediately evident that thisdeductivepart is of all the more weight, the moregeneralthe conceptsin question are. If, in addition, the inductive laws posited soon prove to be of a comparatively high degree of perfection, we obtain the impression described above, that an unlimited number of independent results can be deduced from a premise.Kantwas keenly alive to the peculiarity of such a view, which had been widely spread pre-eminently byEuclid'spresentation of geometry, and he gave expression to his opinion of it in the famous question:How are a priori judgments possible?We have seen that it is not always a question ofa priorijudgments, but also of inductive conclusions applied and tested according to deductive methods.
Each experience may generally be considered under an indefinite number of various concepts, all of which may be abstracted from that experience by corresponding observations. Accordingly an indefinite number of natural laws would be required for prophesying that experience in all its parts. Likewise the indefinite number of premises must be known through the application of which those natural laws acquire a certain content. Thus it seems as if it were altogether impossible to apply natural laws for the determination of a single experience to come, and in a certain sense this is true (p. 30). For example, when a child is born, we are quite incapable of foretelling the peculiar events that will occur in its life. Beyond the statement that it will live a while and then die, we can make onlythe broadest assertions qualified by numerous "ifs" and "buts."
If, in spite of this, we arrange a very great part of our life and activity according to the prophecies we make in regard to numerous details in life, basing them upon natural laws, the question arises, how we get over the difficulty, or, rather, the impossibility just referred to.
The answer is, that we repeatedly so find or can form our experiences that certain natural relationspreponderatinglydetermine the experience, while the other parts that remain undetermined fall into the background.The prophecy will cover so considerable a part of the experience that we can forego previous knowledge of the rest.We can foretell enough to render a practical construction of life possible, and increasing experience, whether the personal experience of the individual or the general experience of science, constantly enlarges this controllable part of future experiences.
The procedure of science is similar to that of practical life, though freer. Whenever an investigator seeks to test a natural law of the form: if A is so, then B is so, he endeavors to choose or formulate the experiences in such a way that the fewest possible extraneous elements are present, and that those that are unavoidable should exert the least possible influence upon the relation in question. He never succeeds completely. In order, nevertheless, to reach a conclusion as to the form the relationwill take without extraneous influences, the following general method is applied.
A series of instances are investigated which are so adjusted that the influence of the extraneous elements grows less and less. Then the relation investigated approaches a limit which is never quite reached, but to which it draws nearer and nearer, the less the influence of the extraneous elements. And the conclusion is drawn that if it were possible to exclude the extraneous elements entirely, the limit of the relation would be reached.
A case in which none of the extraneous elements of experience operate is called anideal case, and the inference from a series of values leading to the limit-value is anextrapolation.Such extrapolations to the ideal caseare a quite natural procedure in science, and a very large part of natural laws, especially all quantitative laws, that is, such as express a relation between measurable values, have precise validity only in ideal cases.
We here confront the fact that many natural laws, and among them the most important, are expressed as, and taken to be, conditionswhich never occur in reality. This seemingly absurd procedure is, as a matter of fact, the best fitted for scientific purposes, since ideal cases are to be distinguished by this,that with them the natural laws take on the simplest forms. This is the result of the fact that in ideal cases we intentionally and arbitrarily overlook every complication of the determining factors,and in describing ideal cases we describe the simplest conceivable form of the class of experiences in question. The real cases are then constructed from the ideal cases by representing them as the sum of all the elements that have an influence on the experience or the result. Just as we can represent the unlimited multitude of finite numbers by the figures up to ten, so we can represent an unlimited quantity of complicated events by a finite number of natural laws, and so reach a highly serviceable approximation to reality.
Thus geometry deals with absolutely straight lines, absolutely flat surfaces, and perfect spheres, though such have never been observed, and the results of geometry come the closer to truth, the more nearly the real lines, surfaces, and spheres correspond to the ideal demands. Similarly, in physics, there are no ideal gases or mirrors, or in chemistry ideally pure substances, though the expressed simple laws in these sciences are valid for only such bodies. The non-ideal bodies of these sciences, which reality presents in various degrees of approximation, correspond the more closely to these laws, the slighter the deviation of the real from the ideal. And the same method is applied in the so-called mental sciences, psychology and sociology, in which the "normal eye" and a "state with an entirely closed door" are examples of such idealized limit-concepts.
A very widespread view and a very grave one, because ofits erroneous results, isthat by the natural laws things are unequivocally and unalterably determined down to the very minutest detail. This is calleddeterminism, and is regarded as an inevitable consequence of every natural scientific generalization. But an accurate investigation of actual relations produces something rather different.
The most general formulation of the natural law:if A is experienced, then we expect B, necessarily refers in the first place only to certainpartsof the thing experienced. For perfect similarity in two experiences is excluded by the mere fact that we ourselves change unceasingly and one-sidedly. Consequently, no matter how accurate the repetition of a former experience may be, our very participation in it, an element bound to enter, causes it to be different. Therefore we deal with only apartialrepetition of any experience, and the common part is all the smaller a fraction of the entire experience, the moregeneralthe concept corresponding to this part. But the most general and most important natural laws apply to such very general ideas, and accordingly they determine only a small part of the whole result. Other parts are determined by other laws, but we can never point out an experience that has been determined completely and unequivocally by natural laws known to us. For example, we know that when we throw a stone, it will describe an approximate parabolic curve in falling to the ground. But if we should attempt to determine itscourse accurately, we should have to take into consideration the resistance of the air, the rotatory motion of the stone upon being thrown, the movement of the earth, and numerous other circumstances, the exact determination of which is a matter beyond the power of all sciences. Nothing but anapproximatedetermination of the stone's course is possible, and every step forward toward accuracy and absoluteness would require scientific advances which it would probably take centuries to accomplish.
Science, therefore, can by no means determine the exact linear course that the stone will take in its fall. It can merely establish a certain broader path within which the stone's movement will remain. And the path is the wider the smaller the progress science has made in the branch in question. The same conditions prevail in the case of every other prediction based upon natural laws. Natural laws merely provide a certain frame within which the thing will remain. But which of the infinitely numerous possibilities within this frame will become reality can never be absolutely determined by human powers.
The belief that it is possible has been evoked merely by a far-reaching method of abstraction on the part of science. By assuming in place of the stone "a non-extended point of mass" and by disregarding all the other factors which in some way (whether known or unknown) exercise an influenceon the stone's movement, we can effect an apparently perfect solution of the problem. But the solution is not valid for real experience, merely for an ideal case, which bears only a more or less profound similarity to the real. It is only such an ideal world, that is, a world arbitrarily removed from its actual complexity, that has the quality of absolute determinateness which we are wont to ascribe to the real world.
We might point to the method of abstraction generally adopted in science and to the extrapolation to ideal cases which has just been explained, and regard the assertion of the absolute determinateness of events in the world as a justified extrapolation to the ideal case. In other words, we might say that we know all the natural laws and how to apply them perfectly to the individual instances. In controversion of this it must be said that the ulterior justification of such ideal extrapolation is not yet feasible. The justification lies in the demonstration that the real cases approximate the ideal the more closely the more we actualize our presumptions. But in this case this is not feasible, since, for the greater part of our experiences, we do not even know the approximate or ideal natural laws by the help of which we can construct such ideal cases. For instance, the whole province of organic life is at present essentially like an unknown land, in which there are only a few widely separated paths ending inculs-de-sac.
This relation explains why, on the one hand, we assume a far-reaching determinateness for many things, that is, for all those accessible to scientific treatment and regulation, and why, on the other hand, we have the consciousness of actingfreely, that is, of being able to control future events according to the relations they bear to our wishes. Essentially there is no objection to be found to a fundamental determinism which explains that this feeling of freedom is only a different way of sayingthat a part of the causal chain lies within our consciousness, and that we feel these processes (in themselves determined) as if we ourselves determined their course. Nor can we prove this idea to be false, that, since the number of factors which influence each experience is indefinitely great and their nature indefinitely complex, each event would appear to be determined in the eyes of an all-comprehensive intellect. But to our finite minds an undetermined residue necessarily remains in each experience, and to that extent the world must always remain in part practically undetermined to human beings. Thus, both views, that the world is not completely determined, and that it really is, though we can never recognize that it is, lead practically to the same result:that we can and must assume in our practical attitude to the world that it is only partially determined.
But if two different lines of thought in the wholeworld of experience everywhere lead to the same result, they cannot be materially, but merely formally or superficially, different. For those things are alike which cannot be distinguished. There is no other definition of alikeness. Thus, if we see that the age-long dispute between these two views always breaks out afresh without seeming to be able to reach an end, this is readily understood, from what has been said, since the very same essential arguments which can be adduced ofoneview can be used as a prop for theotherview, because in their essential results the two are the same. I have discussed this matter because it presents a very telling example of a method to be applied in all the sciences when dealing with the solution of old and ever recurrent moot questions. Each time we encounter such problems, we must ask ourselves: what would be the difference empirically if the one or the other view were correct? In other words, we first assume the one to be correct, and develop the consequences accordingly. Then we assume the second to be correct and develop the consequences accordingly. If in the two cases the consequences differ in a certain definite point, we at least have the possibility of ascertaining the false view by investigating in favor of which case experience decides on this point. However, we may not conclude that by this the other view has been proved to be entirely correct. It likewise may be false, only with the peculiar quality that in the case in questionit leads to the correct conclusions. That such a thing is possible, every one knows who has attentively observed his own experiences. How often we act correctly in actual practice, though we have started out on false premises! The explanation of this possibility resides in the highly composite nature of each experience and each assumption. It is quite possible—and, in fact, it is the general rule—that a certain view contains true elements, butalong with them false elements also. In applications of the view where the true elements are the decisive factors, true results are obtained, despite the errors present. Likewise, false results will be achieved where the false elements are decisive, despite the true results that can be had, or have been had, elsewhere, by means of the true elements. Hence, in case of the "confirmation," we can only conclude that that portion of the view essential for the instance in question is correct.
One readily perceives that these observations find application in all provinces of science and life. There are no absolutely correct assertions, and even the falsest may in some respect be true. There are only greater and lesser probabilities, and every advance made by the human intellect tends to increase the degree of probability of experiential relations, or natural laws.
From the preceding observations the means may be drawn for outlining a complete table of the sciences. However,we must not regard it complete in the sense that it gives every possible ramification and turn of each science, but that it sets up a frame inside of which at given points each science finds its place, so that, in the course of progressive enlargement, the frame need not be exceeded.
The basic thought upon which this classification rests is that of graded abstraction. We have seen (p. 19) that a concept is all the more general, that is, is applicable to all the more experiences, the fewer parts or elementary concepts it contains. So we shall begin the system of the sciences with the most general concepts, that is, the elementary concepts (or with what for the time being we shall have to consider elementary concepts), and, in grading the concept complexes according to their increasing diversity, set up a corresponding graded series of sciences. One thing more is to be noted here, that this graded series, on account of the very large number of new concepts entering, must produce a correspondingly great number of diverse sciences. For practical reasons groups of such grades have been combined temporarily. Thereby a rougher classification, though one easier to obtain a survey of, has been made. The most suitable and lasting scheme of this sort was originated by the French philosopher,Auguste Comte, since whom it has undergone a few changes.
Below is the table of the sciences, which I shall then proceed to explain:
I.Formal Sciences.Main concept: orderLogic, or the science of the ManifoldMathematics, or the science of QuantityGeometry, or the science of SpacePhoronomy, or the science of MotionII.Physical Sciences.Main concept: energyMechanicsPhysicsChemistryIII.Biological Sciences.Main concept: lifePhysiologyPsychologySociology
As is evident, we first have to deal with the three great groups of the formal, the physical, and the biological sciences. The formal sciences treat of characteristics belonging to all experiences, characteristics, consequently, that enter into every known phase of life, and so affect science in the broadest sense. In order immediately to overcome a widespread error, I emphasize the fact that these sciences are to be considered just as experiential or empirical as the sciences of the other two groups, as to which there is no doubt that they are empirical. But because the concepts dealt with by the first group are so extremely wide, and the experiences corresponding to them, therefore, are the most general of all experiences, we easily forget that we are dealing with experiences at all; and our very firmly rooted consciousness of the unqualified similarity of these experiencescauses them to seem native qualities of the mind, ora priorijudgments. Nevertheless, mathematics has been proved to be an empirical science by the fact that in certain of its branches (the theory of numbers) laws are known which have been found empirically and the "deductive" proof of which we have as yet not succeeded in obtaining. The most general concept expressed and operative in these sciences is the concept of order, ofconjugacyorfunction, the content and significance of which will become clear later in a more thorough study of the special sciences.
In the second group, the physical sciences, the arbitrariness of the classification becomes very apparent, since these sciences are among the best known. We are perfectly justified in regarding mechanics as a part of physics; and in our day physical chemistry, which in the last twenty years suddenly developed into an extended and important special science, thrust itself between physics and chemistry.
The most general concept of the physical sciences is that ofenergy, which does not appear in the formal sciences. To be sure it is not a fundamental concept. On the contrary, its characteristic is undoubtedly that of compositeness, or, rather, complexity.
The third group comprehends all the relations of living beings. Their most general concept, accordingly, is that oflife. By physiology is understood the entire science dealing with non-psychic lifephenomena. It therefore embraces what is called, in the present often chance arrangement of scientific activities, botany, zoology, and physiology of the plants, animals, and man. Psychology is the science of mental phenomena. As such, it is not limited to man, even though for many reasons he claims by far the preponderating part of it for himself. Sociology is the science which deals with the peculiarities of the human race. It may therefore be called anthropology, but in a far wider sense than the word is now applied.
It will be remarked that the grouping of the table gives no place at all in its scheme to certain branches of learning taught in the universities and equally good technical institutions. We look in vain not only for theology and jurisprudence, but also for astronomy, medicine, etc.
The explanation and justification of this is, that for purposes of systematization we must distinguish betweenpureandappliedsciences. By virtue of their strictly conceptual exclusiveness the pure sciences constitute a regular hierarchy or graded series, so that all the concepts that have been used and dealt with in the preceding sciences are repeated in the following sciences, while certain characteristic new concepts enter in addition. Thus logic, the science of the manifold, exercises its dominion over all the other sciences, while the specific concepts of physics and chemistry have nothing to do with it, thoughthey are of importance to all the biologic sciences. Through this graded addition of new (naturally empiric) concepts, the construction of the pure sciences proceeds in strict regularity, and their problems arise exclusively from the application of new concepts to all the earlier ones. In other words, their problems do not reach them accidentally from without, but result from the action and reaction of their concepts upon one another.
At the same time there are problems that each day sets before us without regard to system. These come from our endeavor to improve life and avert evil. In the problems of life we are confronted by the whole variety of possible concepts, and under the day's immediate compulsion we cannot wait, if we are sowing crops or helping a sick man, until physiology and all the other appropriate sciences have solved all the problems of plant growth and the changes of the human body and human energy. When other signs fail, we use the position of the stars for finding our way on the high seas. In this manner we turn the teaching of the stars, or astronomy, into an applied science, in which at first mechanics alone seemed to have a part. Later physics took a share in it, then optics took a particularly prominent share, and in recent times not only did chemistry find its way into astronomy, but the specifically biologic concept of evolution was applied in astronomy with success.
Thus, side by side with the pure sciences are theapplied, which are to be distinguished from the pure sciences by the fact that they do not unfold their problems systematically, but are assigned them by the external circumstances of man's life. The pure sciences, therefore, almost always have a larger or smaller share in the tasks of the applied sciences. For instance, in building a bridge or railroad, physical problems have to be taken into consideration as well as sociologic problems (problems of trade), and a good physician should be a psychologist as well as a chemist.
But since all the individual questions arising in the applied sciences may be considered essentially as problems of one or other pure science, they need not be explicitly enumerated along with the pure sciences, especially since their development is greatly dependent upon temporary conditions and is therefore incapable of simple systematization.
If we try to conceive the whole structure of science according to the principle of the increasing complexity of concepts, the first question which confronts us is, What concept is themost generalof all possible concepts, so general that it enters into every concept formation and acts as a decisive factor? In order to find this concept let us go back to the psycho-physical basis of concept formation, namely,memory, and let us investigate what is the general characteristic determining memory. We soon perceive that if a being were to lead an absolutely uniform existence,nomemories could be evoked. There would be nothing by which the past could be distinguished from the present, hence nothing by which to compare them. So the "primal phenomenon" of conscious thought is the realization of adifference, a difference between memory and the present, or, to put the same idea still more generally, between two memories.
Our experiences, therefore, are divided into twoparts, distinguished from each other. In order to predicate something of a perfectly general nature concerning those parts, without regard to their particular content, we must, in accordance with the means employed in human intercourse, designate them by aname. Now in all human languages there is a great deal of arbitrariness and indefiniteness in the relations between the concepts and the names applied to them, which render all accurate work in the study of concepts extremely difficult. It is necessary, therefore, to state definitely in each particular instance with what conceptual content a given name is to be connected. Every experience in so far as it is differentiated from other experiences we shall call simply anexperiencewithout making a distinction between a so-called inner or outer experience.
Many of the experiences remain isolated, because they are not repeated in a similar form, and so do not remain in our memory. They depart from our psychic life once for all and leave no further consequences or associations. But some experiences recur with greater or less uniformity, and become permanent parts of psychic life. Their duration is by no means unlimited. For even memories fade and disappear. However, they extend through a considerable part of life, and that suffices to give them their character.
The aggregate of similar experiences, hence of experiences conceptually generalized, we shall callthings.A thing, therefore, is an experience whichhas been repeated, and is "recognized" by us. That is, it is felt as repeated and conceptually comprehended. In other words, all experiences of which we have formed concepts are things, andthe concept of thing itself is the most general concept, since, according to its definition, it includes all possible concepts. Its "essence," or determining characteristic, lies in the possibility of differentiating any one thing from another. Things we do not differentiate we callthe same, oridentical. Here we shall leave undecided the question whether this lack of differentiation occurs because wecannot, or because wewould not, differentiate. All experiences generalized into one concept are therefore felt or regarded as the same in reference to this concept. Now, since concepts arise unconsciously as well as consciously, the first is a case of identities which had been directly felt as such. On the other hand, in the second case, the process is that of consciously disregarding or abstracting the existing differences in order to form a concept into which these do not enter. This last process is applied in the highest degree possible in obtaining the conceptthing.
The experience of theconnectionorrelationbetween various things is also derived from the nature of our experiences in the most general sense. When we recall a thing A, another thing B comes to our mind, the memory of which is called forth by A, andvice versa. The cause of this invariably lies in some experiences inwhich A and B occur together. In fact, A and B must have occurred together a number of times. Otherwise they would have disappeared from memory. In other words, it is the fact of thecomplex conceptwhich appears in such connections between various things. Two things, A and B, which are connected with each other in such a way, are said to be associated. Association in the most general sense means nothing more than that when we think of B we also have A in our consciousness, andvice versa. However, we can at will make the association more definite, so that quite definite thoughts or actions will be connected with the association of B. These thoughts and actions are then the same for all the individual cases occurring under the concept A and B.
If we associate with the thing B another thing C, we obtain a relation of the same nature as that obtained by the association of A and B. But at the same time a new relation arises which was not directly sought, namely, the association of A to C. If A recalls B, and B recalls C, A must inevitably recall C also. This psychologic law of nature is productive of numberless special results. For we can apply it directly to still another case, the association of a fourth thing D to the thing C, whereby new relations are necessarily established also between A and D as well as between B and D. By positing theonerelationC : Dthere arise two new relations not immediately given, namely,A : DandB : D. The reason the other relations arise is because C was not taken free from all relations, but had already attached to it the relations to A and B. These relations of C, therefore, brought A and B into the new relation with D.
By this simplest and most general example we recognize the type of the deductive process (p. 41), namely, the discovery of relations which, it is true, have already been established by the accepted premises, but which do not directly appear in undertaking the corresponding operations. In the present case, to be sure, the deduction is so apparent that the recognition of the relations in question offers not the slightest difficulty. But we can easily imagine more complicated cases in which it is much more difficult to find the actually existing relations, and so in certain circumstances we may search for them a long time in vain.
The aggregate of all individual things occurring in a definite concept, or the common characteristics of which make up this concept, is called a group. Such a group may consist of a limited or finite number of members, or may be unlimited, according to the nature of the concepts that characterize it. Thus, all the integers form an unlimited or infinite group, while the integers between ten and one hundred (or the two-digit numbers) form a limited or finite group.
From the definition of the group concept follows the so-called classicprocess of argumentationof thesyllogism. Its form is:Group A is distinguished by the characteristic of B.The thing C belongs to group A. Therefore C has the characteristic of B.The prominent part ascribed byAristotleand his successors to this process is based upon thecertaintywhich its results possess. Nevertheless, it has been pointed out, especially byKant, that judgments or conclusions of such a nature (which he called analytic) have no significance at all for the progress of science, since they express only what is already known. For in order to enable us to say that the thing C belongs to group A, we must already have recognized or proved the presence of the group characteristic B in C, and in that case the conclusion only repeats what is already contained in the second or minor premise.
This is evident in the classic example: All men are mortal. Caius is a man. Therefore Caius is mortal. For if Caius's mortality were not known (here we are not concerned how this knowledge was obtained), we should have no right to call him a man.
At the same time the character of the really scientific conclusion based upon the incomplete induction becomes clear. It proceeds according to the following form. The attributes of the group A are the characteristics of a, b, c, d. We find in the thing C the characteristics a, b, c. Therefore we presume that the characteristic d will also be found in C. The ground for this presumption is that wehave learned by experience that the characteristics mentioned have always been found together. It is for this reason, and for this reason only, that we may assume from the presence of a, b, c the presence of d. In the case of an arbitrary combination, in which it is possible to combine other characteristics, the conclusion is unfounded. But if, on the other hand, the formation of the concept A with the characteristics of a, b, c, d has been caused by repeated and habitual experience, then the conclusion is well founded; that is, it is probable.
As a matter of fact, however, that classic example which is supposed to prove the absolute certainty of the regular syllogism turns out to be a hidden inductive conclusion of the incomplete kind. The premise, Caius is a man, is based on the attributes a, b, c (for example, erect bearing, figure, language), while the attribute d (mortality) cannot be brought under observation so long as Caius remains alive. In the sense of the classic logic, therefore, we are not justified in the minor premise, Caius is a man, while Caius is alive. The utter futility of the syllogism is apparent, since, according to it, it is only of dead men that we can assert that they are mortal.
From these observations it becomes further apparent that logic, whether it is the superfluous classic logic or modern effective inductive logic, is nothing but a part of the group theory, or science of manifoldness, which appears as the first, because it isthe most general member of the mathematical sciences (this word taken in its widest significance). But according to the hierarchic system in harmony with which the scheme of all the sciences had been consciously projected, we cannot expect anything else than that those sciences which are needful for the pursuit of all other sciences (and logic has always been regarded as such an indispensable science, or, at least, art) should be found collected and classified in the first science.
When the characteristics a, b, c, d of a group have been determined, then the aggregate of all things existing can be divided into two parts, namely, the things which belong to the group A and those which do not belong to it. This second aggregate may then be regarded as a group by itself. If we call this group "not-A," it follows from the definition of this group that the two groups, A and not-A, together form the aggregate of all things.
This is the meaning and the significance of the linguistic form ofnegation. It excludes the thing negated from any group given in a proposition, and this relegates it to the second or complementary group.
The characteristic of such a group is the common absence of the characteristics of the positive group. We must note here that the absence of evenoneof the characteristics a, b, c, d excludes the incorporation of the thing into the group A, while the mereabsence of this characteristic suffices to include it in the group not-A. We can therefore by no means predicate of group not-A that each one of its members must lackallthe characteristics a, b, c, d. We can only say that each of its members lacks at least one of the characteristics, but that one or some may be present, and several or all may be absent. From this follows a certain asymmetry of the two groups, which we must bear in mind.
The consideration of this subject is especially important in the treatment of negation in the conclusions of formal logic. As we shall make no special use of formal logic, we need not enter into it in detail.
The combination of the characteristics which are to serve for the definition of a group is at first purely arbitrary. Thus, when we have chosen such an arbitrary combination, a, b, c, d, we can eliminate one of the characteristics, as, for example, c, and form a group with the characteristics a, b, d. Such a group, which ispoorer in characteristics, will, in general, bericher in members, for to it belong, in the first place, all the things with the characteristics a, b, c, d, of which the first group consisted, and in addition all the things which, though not possessing c, possess a, b, and d.
If we call such groups related as contain common characteristics, though containing them in different members and combinations, so that the definition ofthe one group can be derived from the other by the elimination or incorporation of individual characteristics, then we can postulate the general thesisthat in related groups those must be richer in members which are poorer in characteristics, and inversely. This is the precise statement of the proposition of the less definite thesis stated above.
For the purposes of systematization we have assumed that we can arbitrarily eliminate one or another characteristic of a group. In experience, however, this often proves inadmissible. As a rule we find that the things which lack one of the characteristics of a group will also lack a number of other characteristics; in other words, that the characteristics are not all independent of one another, but that a certain number of them go together, so that they are present in a thing either in common or not at all.
This case, however, can be referred to the general one first described, by treating the characteristics belonging together as beingonecharacteristic, so that the group is defined solely by the independent characteristics. Then, according to the definition, we can, without losing our connection with experience, carry out that formal manifoldness of all possible related groups which yields what is called aclassificationof the corresponding things.
If for the determination of a group a definite number of independent characteristics is taken, say, a, b, c, d, and e, then we have at first the narrowestor poorest group abcde. By the elimination of one characteristic we obtain the five groups, bcde, acde, abde, abce, and abcd. If we omit one other characteristic we get ten different groups abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde. Likewise, there are ten groups with two characteristics each, and finally five groups with one characteristic each. All these groups are related. There is a science, the Theory of Combinations, which gives the rules by which, in given elements or characteristics, the kind and number of the possible groups can be found. The theory of combinations enables us to obtain a complete table and survey of all possible complex concepts which can be formed from given simple ones (whether they be really elementary concepts, or only relatively so). When in any field of science the fundamental concepts have been combined in this manner, a complete survey can be had of all the possible parts of this science by means of the theory of combinations.
In order to present this process vividly to our minds, let us take as an example the science of the chemical combination of substances which form an important part of chemistry. There are about eighty elements in chemistry, and this science has to treat of
a) each of the eighty elements by itselfb) all substances containing two elements and no morec) all substances containing three elementsd, e, f, etc.) the substances containing four, five, and six, etc., elements,
until finally we reach a group (not existing in experience) embracing substances formed ofallthe elements. That there is no such substance in the present scope of human knowledge has, of course, no significance for the structure of the scheme. What is significant is the fact that the scheme really embraces and arranges all possible substances in such a way that we cannot conceive of any case in which a newly discovered substance cannot after examination immediately be classed with one of the existing groups.
To cite an example from another science. Physics, it will be recalled, may be considered to be the science of the different kinds of energy. This science, accordingly, is divided first into the study of the properties of each energy, and then into the study of the relations of two energies, of three energies, of four energies, etc. Here, too, we may say that in the end there can be no physical phenomenon which cannot be placed in one of the groups so obtained.
Of course, neither in chemistry nor in physics does this mean that eachnewcase will fall within the scheme obtained by the exhaustive combination of elementary concepts (whether chemical elements or kinds of energy)knownat the time. It is quite possible that a new thing under investigation contains anewelementary concept, so that on accountof it the scheme must be enlarged through the embodiment of this new element. But simultaneously a corresponding number of new groups appear in the scheme, and the investigator's attention is directed to the fact that he still has a reasonable prospect, in favorable circumstances, of discovering these new things also. Thus combinatory schematization serves not only to bring the existing content of science into such order that each single thing has its assigned place, but the groups which have thereby been found to be vacant, to which as yet nothing of experience corresponds, also point to the places in which science can be completed by new discoveries.
From the above presentation it is apparent how from the two concepts "thing" and "association" alone a great manifoldness of various and regular forms can be developed. They are purely empirical relations, for the fact that several things can be combined in the graded series described above according to a fixed rule does not follow merely from the two concepts, but must beexperienced. But, on the other hand, both concepts are so general that the experiences obtained in some cases can be applied to all possible experiences and may serve the purpose of classifying and making a general survey of them.
The above statements, however, have by no means exhausted the possibilities. For it has been tacitly assumed that in the combination of several things thesequenceaccording to which this combinationtakes place should not condition a difference of the result. This is true of a number of things, but not of all. In order, therefore, to exhaust the possibilities the theory of combinations must be extended also to cases in which the sequence is to be taken account of, so that the form ab is regarded as different from ba.
We will not undertake to work out the results of this assumption. It is obvious that the manifoldness of the various cases is much greater than if we neglect the sequence. On this point we have one more observation to make, that further causes for diversity exist. It is true that a chemical combination is not influenced by the sequence in which its elements enter the combination, but there do occur with the same elements differences in theirquantitative relations, and thereby a new complexity is introduced into the system, so that two or more similar elements can form different combinations according to the difference in the quantitative relations. Still, even with this, the actual manifoldness is not exhausted, for from the same elements and with the same quantitative relations there can arise different substances calledisomeric, which, for all their similarity, possess different energy contents. But the first scheme is not demolished, nor does it become impracticable because of this increase of manifoldness. What simply happens is thatseveraldifferent things instead of one appear in the same group of the original scheme, the systematicclassification of which necessitates a further schematization by the use of other characteristics.
Since we have started from the proposition that all members of a group are different from one another, we have perfect liberty to arrange them. The most obvious arrangement according to which someonedefinite member is followed by asingleother member and so forth (as, for example, the arrangement of the letters of the alphabet) is by no means the only mode of arrangement, though it is the simplest. Besides thislineararrangement, there is also, for instance, the one in which two new members follow simultaneously upon each previous one, or the members may be disposed like a number of balls heaped up in a pyramid. However, we shall not have much occasion to occupy ourselves with these complex types of arrangement, and can therefore limit our considerations at first to the simplest, that is, to the linear arrangement.
This simplest of all possible forms expresses itself in the factthat the immediately experienced things of our consciousness are arranged in this way. In point of fact, the contents of our consciousness proceed in linear order, one single new member always attaching itself to an existing member. This law, however, is not strictly and invariably adhered to. It sometimes happens that our consciousness continues for a while to pursue the direction of thought it has once taken, although a branching off had alreadytaken place at a former point, at which a new chain of thought had begun. Nevertheless, one of these chains usually breaks off very soon, and the linear character of the inner experience is immediately restored. Of certain specially powerful intellects it is recorded that they could keep up several lines of thought for a considerable length of time—Julius Cæsar, for instance.
The biologic peculiarity here mentioned of the linear juxtaposition of the contents of our consciousness has led to the concept oftime, which has been appropriately called aform of inner life. That all our experiences succeed each other in time is equivalent to saying that our thought processes represent a group in linear arrangement. As appears from the above observations, this is by no means an absolute form, unalterable for all times. On the contrary, a few highly developed individuals have already begun to emancipate themselves from it. But the existing form is so firmly fixed through heredity and habit that it still seems impracticable for most men to imagine the succession of the inner experiences in a different way than by a line or byone dimension. Since, on the other hand, we have all learned to feel space astri-dimensional, although optically it appears to possess only two dimensions (we see length and breadth, and only infer thickness from secondary characteristics), we come to recognize that the linear form by which we represent the succession of our experiences is a matterof adaptation, and that because the change has been extremely slight in the course of centuries it produces the impression of being unalterable.[D]
These discussions lead to a further difference that can exist in groups of linear arrangement. While in the first example we chose, the alphabet, the sequence was quitearbitrary, since any other sequence is just as possible, the same cannot be said of experiences into which the element of time enters. These are not arbitrary, but are arranged by special circumstances depending upon the aggregate of things which co-operate in the given experiences.
While, therefore, a group with free members, that is, members not determined in their arrangement by special circumstances, can be brought into linear order in very different ways, there are groups in which only one of those orders actually occurs. We see at once that in free groups the number of different orders possible is the greater, the greater the group itself. The theory of combinations teaches how to calculate these numbers which play a very important rôle in the various provinces ofmathematics. The naturally ordered groups always represent a single instance out of these possibilities, the source of which always lies outside the group concept, that is, it proceeds from the things themselves which are united into a group.
An especially important group in the linear order is that of theintegral numbers. Its origin is as follows:
First we abstract the difference of the things found in the group, that is, we determine, although they are different, to disregard their differences. Then we begin with some member of the group and form it into a group by itself. It does not matter which member is chosen, since all are regarded as equivalent. Then another member is added, and the group thus obtained is again characterized as a special type. Then one more member is added, and the corresponding type formed, and so on. Experience teaches that never has a hindrance arisen to the formation of new types of this kind by the addition of a single member at a time, so that the operation of this peculiar group formation may be regarded asunlimitedorinfinite.
The groups or types thus obtained are called theintegral numbers. From the description of the process it follows that every number has two neighbors, the one the number from which it arose by the addition of a member, and the other the number which arose from it by the addition of a member. In the case of the number one with which the seriesbegins, this characteristic is present in a peculiar form, the preceding group beinggroup zero, that is, a group without content. This number in consequence reveals certain peculiarities into which we cannot enter here.
Now, according to a previous observation (p. 64), not only does the order bring every number into relation with the preceding one, but since this last for its part already possesses a great number of relations to all preceding, these relations exert their influence also upon the new relation. This fact gives rise to extraordinarily manifold relations between the various numbers and to manifold laws governing these relations. The elucidation of them forms the subject of an extensive science.
From this regular form of the number series numerous special characteristics can be established. The investigations leading to the discovery of these characteristics are purely scientific, that is, they have no special technical aim. But they have the uncommonly great practical significance that they provide for all possible arrangements and divisions of numbered things, and so have instruments at hand ready for application to each special case as it arises. I have already pointed out that in this lies the positive importance of the theoretical sciences. Forpracticalreasons the study of them must be asgeneralas possible. This science is calledarithmetic.
Arithmetic undergoes an important generalization if the individual numbers in a calculation are disregarded andabstract signsstanding for any number at all are used in their place. At first glance this seems superfluous, since in every real numerical calculation the numbers must be reintroduced. The advantage lies in this, that in calculations of the same form, the required steps are formally disposed of once for all, so that the numerical values need be introduced only at the conclusion and need not be calculated at each step. Moreover, the general laws of numerical combination appear much more clearly if the signs are kept, since the result is immediately seen to be composed of the participating members. Thus,algebra, that is, calculation with abstract or general quantities, has developed as an extensive and important field of general mathematics.
By the theory of numbers we understand the most general part of arithmetic which treats of the properties of the "numerical bodies" formed in some regular way.
So far our discussion has confined itself to theindividualgroups and to the properties which each one of them exhibitsby itself. We shall now investigate the relations which existbetween two or more groups, both with regard to their several members and to their aggregate.
If at first we have two groups the members of which are all differentiated from one another, thenany one member of the one group can be co-ordinated with any one member of the other group. This means that we determine that the same should be done with every member of the second group as is done with the corresponding member of the first group. That such a rule may be carried out we must be able to do with the members of all the groups whatever we do with the members of one group. In other words, no properties peculiar to individual members may be utilized, but only the properties that each member possesses as a member of a group. As we have seen, these are the properties ofassociation.
First, the co-ordination ismutual, that is, it is immaterial to which of the two groups the processes are applied. The relation of the two groups is reciprocal or symmetrical.
Further, the process of co-ordination can be extended to a third and a fourth group and so on, with the result that what has been done in one of the co-ordinated groups must happen in all. If hereby the third group is co-ordinated with the second, the effects are quite the same as if it were co-ordinated directly with the first instead of indirectly through the second. And the same is true for the fourth and the fifth groups, etc. Thus, co-ordination can be extended to any number of groups we please, and each single group proves to be co-ordinated with every other.
Finally, a group can be co-ordinated with itself,each of its members corresponding to a certain definite other member. It is not impossible that individual members should correspond to themselves, in which case the group hasdouble members, ordouble points. The limit-case isidentity, in which every member corresponds toitself. This last case cannot supply any special knowledge in itself, but may be applied profitably to throw light on those observations for which it represents the extreme possibility.