If we have two groups A and B, and if we co-ordinate their members severally, three cases may arise. Either group A is exhausted while there are members remaining in B, or B is exhausted before A, or, finally, both groups allow of a mutual co-ordination ofalltheir members. In the first case A is called, in the broader sense of the word,smallerthan B, in the second B is called smaller than A, in the third the two groups are said to be ofequal magnitude. The expression, "B is greater than A," is equivalent to the expression, "A is smaller than B," and inversely.
It is to be noted that the relations mentioned above are true, whether the members are considered as individually different from one another or whether the difference of the members is disregarded, and they are treated as alike. This comes from the fact that every definite co-ordination of a group can be translated into every other possible co-ordination by exchanging two members at a time inpairs. Since in this process one member is each time substituted for another, and a gap therefore can never occur in its place, the group in the new arrangement can be co-ordinated with the other group as successfully as in the old arrangement. At the same time we learn from this that in every co-ordination of a group with itself, independently of the arrangement of its members, it must prove equal to itself.
By carrying out the co-ordination proof is further supplied of the following propositions:
{greater than}If group A isequal togroup Bsmaller than{greater than}and group B isequal togroup Csmaller than{greater than}then group A isequal togroup Csmaller than
From this it follows that any collection of finite groups whatsoever, of which no one is equal to the other, can always be so arranged that the series should begin with the smallest and end with the greatest, and that a larger should always follow a smaller.This order would be unequivocal, that is,there is only one series of the given groups which has this peculiarity. As we shall soon see, the series of integers is the purest type of a series so arranged.
In comparing two infinitely large groups by co-ordination, it may be said on the one hand that never will one group be exhausted while the other still contains members. Accordingly, it is possible to designate two unlimited or infinite groups (or as many such groups as we please) asequalto each other. On the other hand, the statement that in both groups each member of the one is co-ordinated with a member of the other has no definite meaning on account of the infinitely large number of members.The definition of equality is therefore not completely fulfilled, and we must not loosely apply a principle valid for finite groups to infinite groups. This consideration, which may assume very different forms according to circumstances, explains the "paradoxes of the infinite," that is, the contradictions which arise when concepts of a definite content are applied to cases possessing in part a different content. If we wish to attempt such an application, we must in each instance make a special investigation as to the manner in which the relations on their part change by the change of those contents (or premises). As a general rule we must expect that the former relations will not remain valid in these circumstances without any change at all.
In the course of these observations we have learned how co-ordination can be used for obtaining a number of fundamental and multifariously applied principles. From this alone the great importance of co-ordination is evident, and later we shall see that its significance is even more far-reaching.The entire methodology of all the sciences is based upon the most manifold and many-sided application of the process of co-ordination, and we shall have occasion to make use of it repeatedly. Its significance may be briefly characterized by stating that it is the most general means of bringing connection into the aggregate of our experiences.
The group of integral numbers, because of its fundamental simplicity and regularity, is by far the best basis of co-ordination. For while arithmetic and the theory of numbers give us a most thorough acquaintance with the peculiarities of this group, we secure by the process of co-ordination the right to presuppose these peculiarities and the possibility of finding them again in every other group which we have co-ordinated with the numerical group. The carrying out of such co-ordination is calledcounting, and from the premises made it followsthat we can count all things in so far as we disregard their differences.
We count when we co-ordinate in turn one member of a group after another with the members of the number series that succeed one another untilthe group to be counted is exhausted. The last number required for the co-ordination is called thesumof the members of the counted group. Since the number series continues indefinitely, every given group can be counted.
Numerals have been co-ordinated withnamesas well as withsigns. The former are different in the different languages, the latter are international, that is, they have the same form in all languages. From this proceeds the remarkable fact that the written numbers are understood by all educated men, while the spoken numbers are intelligible only within the various languages.
The purpose of counting is extremely manifold. Its most frequent and most important application lies in the fact that the amount affords ameasure for the effectiveness or the valueof the corresponding group, both increasing and decreasing simultaneously. A further number serves as a basis for divisions and arrangements of all kinds to be carried out within the group, whereby liberal use is made of the principle that everything that can be effected in the given number group can also be effected in the co-ordinated counted group.
The co-ordination of names and signs with numbers calls for a few general remarks on co-ordination of this nature.
The possibility of carrying out the formal operations effected in one of the groups upon the co-ordinated group itself facilitates to an extraordinaryextent the practical shaping of the reality for definite purposes. If by counting we have ascertained that a group of people numbers sixty, we can infer without actually executing the steps that it is possible to form these men in six rows of ten, or in five rows of twelve, or in four rows of fifteen, but that we cannot obtain complete rows if we try to arrange them in sevens or elevens. These and numberless other peculiarities we can learn of the group of men from its amount, that is, from its co-ordination with the numerical group of sixty. In co-ordination, therefore, we have a means of acquainting ourselves with facts without having to deal directly with the corresponding realities.
It is clear that men will very soon notice and avail themselves of so enormous an advantage for the mastery and shaping of life. Thus, we see the process of co-ordination in general use among the most primitive men. Even the higher animals know how to utilize co-ordination consciously. When the dog learns to answer to his name, when the horse responds to the "Whoa" and the "Gee" of his driver there is in each case a co-ordination of a definite action or series of actions, that is, of a concept with a sign, or, in other words, of a concept with a member of another group; and in this there need not be the least similarity between the things co-ordinated with each other. The only requirement is that on the one hand the co-ordinated sign should be easily and definitely expressed and beto the point, and that, on the other hand, it should be easily "understood," that is,comprehendedby the senses and unmistakablydifferentiatedfrom other signs co-ordinated with other things.
Thus, we find that the most frequent concepts of co-ordinated sound signs form the beginnings oflanguagein the narrower sense. It is very difficult to ascertain for what reasons the particular forms of sound signs have been chosen, nor is it a matter of great importance. In the course of time the original causes have disappeared from our consciousness and the present connection is purely external. This is evident from the enormous difference of languages in which hundreds of different signs are employed for the same concept.
Now it would be quite possible to solve the problem of co-ordinating with each group of concepts a corresponding group of sounds, so that each concept should have its own sound, or, in other words, that theco-ordination should be unambiguous. It would not by any means be beyond human power to accomplish this, if it were not for the fact that the concepts themselves are still in so chaotic a state as they are at present. We have seen that the attempts of Leibnitz and Locke to draw up a system of concepts, if only in broad outline, have undergone no further development since. Even the most regulated concepts as well as the familiar concepts of daily life are in ceaseless flux, while the co-ordinated signs are comparatively more stable. But they,too, undergo a slow change, as the history of languages shows, and in accordance with quite different laws from those which govern the change of concepts. The consequence is that in language the co-ordination of concepts and words is far from being unambiguous. The science of language designates the presence of several names for the same concept and of several concepts for the same name by the words synonym and homonym. These forms, which have arisen accidentally, signify so manyfundamental defectsof language, since they destroy theprinciple of unambiguityupon which language is based. In consequence of the false conception of its nature we have until now positively shrunk from consciously developing language in such a way that it should more and more approach the ideal of unambiguity. Such an ideal is in fact scarcely known, much less recognized.
Sound signs, to be sure, possess the advantage of being produced easily and without any apparatus, and of being communicable over a not inconsiderable distance. But they suffer under the disadvantage of transitoriness. They suffice for the purpose of temporary understanding and are constantly being used for that. If, on the other hand, it is necessary to make communications over greater distances or longer periods of time, sound signs must be replaced by more permanent forms.
For this we turn to another sense, the sense ofsight. Since optic signs can travel much greater distances than sound signs without becoming indistinguishable, we first have the optical telegraphs, which find application, though rather limited application, in very varying forms, the most efficient being the heliotrope. The other sort of optic signs is much more generally used. These are objectively put on appropriate solid bodies, and last and are understood as long as the object in question lasts. Such signs form thewritten languagein the widest sense, and here, too, it is a question of co-ordinating signs and concepts.
What I have said concerning the very imperfect state of our present concept system is true also of these two groups. On the other hand, the written signs are not subject to such great change as the sound signs, because the sound signs must be produced anew each time, whereas the written signs inscribed on the right material may survive hundreds, even thousands of years. Hence it is that the written languages are, upon the whole, much better developed than the spoken languages. In fact, there are isolated instances in which it may be said that the ideal has well-nigh been reached.
As we have already pointed out, such a case is furnished by thewritten signsof numbers. By a systematic manipulation of the ten signs 0 1 2 3 4 5 6 7 8 9 it is not only possible to co-ordinate a written sign with any number whatsoever, but this co-ordination is strictly unambiguous, that is, eachnumber can be written in only one way, and each numerical sign has only one numerical significance. This has been obtained in the following manner:
First, a special sign is co-ordinated to each of the group of numbers from zero to nine. The same signs are co-ordinated with the next group, ten to nineteen, containing as many numbers as the first. To distinguish the second from the first group, the sign one is used as a prefix. The third group is marked by the prefixed sign two, and so on, until we reach group nine. The following group, in accordance with the principle adopted, has as its prefix the sign ten, which contains two digits. All the succeeding numbers are indicated accordingly. From this the following result is assured: First, no number in its sequence escapes designation; second, never is an aggregate sign used for two or more different numbers. Both these circumstances suffice to secure unambiguity of co-ordination.
It is known that the system of rotation just described is by no means the only possible one. But of all systems hitherto tried it is the simplest and most logical, so that it has never had a serious rival, and the clumsy notation with which the Greeks and Romans had to plague themselves in their day was immediately crowded out, never to return again upon the introduction of the Indo-Arabic notation, which has made its way in the same form among all the civilized nations and constitutes a uniform part of all their written languages.
The comparison of the spoken and the written languages offers a very illuminating proof of the much greater imperfection of the language ofwords. The number 18654 is expressed in the English language by eighteen thousand six hundred and fifty-four, that is, the second figure is named first, then the first, the third, the fourth, and the fifth. In addition, four different designations are used to indicate the place of the figures,-teen,-thousand,-hundred,and-ty.A more aimless confusion can scarcely be conceived. It would be much clearer to name the figures simply in their sequence, as one-eight-six-five-four. Besides, this would be unambiguous. If we should desire to indicate theplace valuein advance, we could do so in some conventional way, for example, by stating the number of digits in advance. This, however, would be superfluous, and ordinarily should be omitted.[E]
There are two possibilities for co-ordination between concepts and written signs. Either the co-ordination isdirect, so that it is only a matter of providing every concept with a corresponding sign, or it is indirect,the signs serving only the purpose of expressing thelanguage sound. In the latter case the written language is based entirely upon the sound language, and the only problem, comparatively easy to solve, is to constructan unambiguous co-ordination between sound and sign. The Chinese script follows the direct process, but all the scripts of the European-American civilized peoples are based on the indirect process.
This, it is true, is the case only in ordinary, non-scientific language, while for science the European nations also have to a large extent built up a direct concept writing. One example of this we have seen in the number signs. Musical notation furnishes another instance, though by far not so perfect. The use of the different keys destroys the unambiguous connection between the pitch and the note sign, and the signatures placed at the beginning of a whole staff have the defect of removing the sign from the place where it is applied. Despite this imperfection musical notation is quite international, and every one who understands European music also understands its signs.[F]
Fundamentally we need not hesitate to recognize inconcept writingorpasigraphya more complete solution of the problem of sign arrangement. Even the very incomplete Chinese pasigraphy renderspossible written intercourse, especially for mercantile purposes, between the various East-Asiatic peoples who speak some dozens of different languages. But each language community translates the common signs into its own words, just as we do in the case of the number signs. But in order that such a system of representation should be complete it must fulfil a whole series of conditions for which scarcely a remote possibility is to be discerned at present.
At first the concepts could simply be taken as found in the words and grammatical forms of the various languages, and each one provided with an arbitrary sign. Such approximately is the Chinese system. But a system of that sort entails an extreme burdening of the memory, which results both from the great number of words and from the necessity of keeping the signs within certain bounds of simplicity. If we consider that the complex concepts are formed according to laws, to a large extent still unknown, from a relatively small number ofelementaryconcepts, we may attempt to build up the signs of the complex concepts by the combination of those of the elementary concepts according to corresponding rules. Then it would only be necessary to learn the signs for the elementary concepts and the rules of combination in order for us to be able to represent all the possible concepts. This would provide even for the natural enlargement of the concept world, since every newelementary concept would receive its sign and would then serve as the basis from which to deduce all the complex concepts dependent upon it. In fact, even should a concept hitherto regarded as elementary prove to be complex, it would not be difficult to declare that its sign, like the name of an extinct race, is dead, and after the lapse of sufficient time to use it for other purposes.
The numerical signs offer an excellent example for the elucidation of this subject, and at the same time serve as a proof that in limited provinces the ideal has already been attained. Another very instructive example is furnished by the chemical formulas, which, though they use the letters of the European languages, do not associate with them sound concepts, but chemical concepts. Since the chemical concepts are co-ordinated with certain letters, it is possible, in the first place, to denote the composition of all combinations qualitatively by the combination of the corresponding letters. But since quantitative composition proceeds according to definite relations which are determined by a variety of specific numbers peculiar to each element and called its combining weight, we need only add to the sign of the element the concept of the combining weight in order to represent in the second place the quantitative composition. Further, the multiples mentioned can also be given. Since, moreover, there are various substances which, despite equal composition, possess different properties, theattempt has been made to express this new manifoldness by the position of the element signs on the paper, and in more recent times also by space representation. And here, too, rules have been worked out in which the scheme affords a close approach to experience. This example shows how, by the constant increase of the complexity of a concept (here the chemical composition), ever greater and more manifold demands are made upon the co-ordinated scheme. The form of expression first chosen is not always adequate to keep pace with the progress of science. In this case it must be radically changed and formed anew to meet the new demands.
In point of unambiguity of co-ordinationphonetic writingis far more imperfect than concept writing. It is obvious that in phonetic writing all the faults already present in the co-ordination between concept and sound are transferred to the written language. To these are added the defects as regards unambiguity occurring in co-ordination between sound and sign from which no language is free. In some languages, in fact, notably in English, these defects amount to a crying calamity. The principle of unambiguity would require that there should never be a doubt as to the way in which a spoken word is written, and as little doubt as to the way in which a written word is spoken. It needs no proof to show how often the principle is violated in every language. In the German language the same sound is represented by f, v,and ph; in the English by f and ph. And in both German and English quite different sounds are associated with c, g, s, and other letters.The fact that orthographic mistakes can be made in the writing of any language is direct proof of its imperfection, and the oftener this possibility occurs the more imperfect is the language in this respect. We know that the spelling reforms begun in Germany more than ten years ago and recently in America and England, have for their object unambiguity in the co-ordination between sign and sound. Still it must be admitted that this tendency has not always been pursued undeviatingly. A few innovations, in fact, undoubtedly represent a step backward.
A comparison of our investigations—which we cannot present in detail but only indicate—with the science of language or philology as taught in the universities and in a great number of books, reveals a great difference between them. This academic philology makes a most exhaustive study of relations, which from the point of view of the purpose of language are of no consequence whatever, such as most of the rules and usages of grammar. A study of this sort must naturally confine itself to a mere determination of whether certain individuals or groups of individuals have or have not conformed to these rules. Even the chief subject of modern comparative philology, the study of the relations of the word forms to one another and their changes in the course of history,both within the language communities and when transferred to other localities, appear to be quite useless from the point of view of the theory of co-ordination. For it is indeed of little moment to us to learn by what process of change, as a rule utterly superficial, a certain word has come to be co-ordinated with a concept entirely different from the one with which it had been previously co-ordinated. Of incomparably greater importance would be investigations concerning the gradual change of the concepts themselves, although by no means as important as the real study of concepts. To be sure, such investigations are much more difficult than the study of word forms set down in writing.
Nevertheless, on account of a historical process, which it would lead us too far afield to discuss, an idea of such word investigations has been formed which is wholly disproportionate to their importance. And if we ask ourselves what part such labors have taken in the progress of human civilization, we are at a loss for an answer. Students of thescienceof language make a sharp distinction between it and theknowledgeof language, which is regarded as incomparably lower. But while a knowledge of language is important in at least one respect, in that it presents to us the cultural material set down in other languages, or makes them accessible in translation to those who do not know foreign languages, philology is of no service in this respect at all, and the pursuit of it will seemas inconceivably futile to future science as the scholasticism of the middle ages seems to us now.
The unwarranted importance attached to the historical study of language forms is paralleled by the equally unwarranted importance ascribed to grammatical and orthographic correctness in the use of language. This perverse pedantry has been carried to such lengths that it is considered almost dishonorable for any one to violate the usual forms of his mother tongue, or even of a foreign language, like the French. We forget that neither Shakespeare nor Luther nor Goethe spoke or wrote a "correct" English or German, and we forget that it cannot be the object of a true cultivation of language topreserveas accurately as possible existing linguistic usage, with its imperfections, amounting at times to absurdities. Its real object lies rather in the appropriatedevelopmentandimprovementof the language. We have already mentioned the fact that in one department, orthography, the true conception of the nature of language and of its development is gradually beginning to assert itself. Among most nations efforts are being made to improve orthography with a view to unambiguity, and when once sufficient clearness is had as to the object aimed for in spelling, there will be no special difficulty in finding the required means to attain it.
But in all the other departments of language we are still almost wholly without a conception of the genuine needs. Though the example of the Englishlanguage proves that we can entirely dispense with the manifold co-ordinations in the same sentence as appearing in the special plural forms of the adjective, verb, pronoun, etc., yet the idea of consciously applying to other languages the natural process of improvement unconsciously evolved in the English language seems not to have occurred even to the boldest language reformers. So strongly are we all under the domination of the "schoolmaster" ideal, that is to say, the ideal of preserving every linguistic absurdity and impracticability simply because it is "good usage."
A twofold advantage will have been attained by the introduction of auniversal auxiliary language(183). Recently the efforts in that direction have made considerable progress. In the first place it will provide a general means of communication in all matters of common human interest, especially the sciences. This will mean a saving of energy scarcely to be estimated. In the second place, the superstitious awe of language and our treatment of it will give way to a more appropriate evaluation of its technical aim. And when by the help of the artificial auxiliary language, we shall be able to convince ourselves daily how much simpler and completer such a language can be made than are the "natural" languages, then the need will irresistibly assert itself to have these languages also participate in its advantages. The consequences of such progress to human intellectual work in general would be extraordinarilygreat. For it may be asserted that philosophy, the most general of all the sciences, has hitherto made such extremely limited progress onlybecause it was compelled to make use of the medium of general language. This is made obvious by the fact that the science most closely related to it, mathematics, has made the greatest progress of all, but that this progress began only after it had procured both in the Indo-Arabic numerals and in the algebraic signs a language which actually realizes very approximately the ideal of unambiguous co-ordination between concept and sign.
Up to this point our discussions have been based on the general concept of thething, that is, of the individual experience differentiated from other experiences. Here the fact ofbeing different, which, as a general experience, led to the corresponding elementary concept, appeared in the foreground in accordance with its generality. But in addition to it there is another general fact of experience, which has led to just as general a concept. It is the concept ofcontinuity.
When, for example, we watch the diminution of light in our room as it grows dark in the evening, we can by no means say that we find it darker at the present moment than a moment before. We require a perceptibly long time to be able to say with certainty that it is now darker than before, and throughout the whole timewe have never felt the increaseof darkness from moment to moment, althoughtheoretically we are absolutely convinced that this is the correct conception of the process.
This peculiar experience, our failure to perceive individual parts of a change, the reality of which we realize when the difference reaches a certain degree, is very general, and, like memory, is based upon a fundamental physiological fact. It has already been noted byHerbart, but its significance was first recognized byFechner, and has since then become generally known in physiology and psychology under the name ofthreshold.Next to memory the threshold determines the fundamental lines of our psychic life.
The threshold therefore means that whatever state we are ina certain finite amount of difference or change must be stepped overbefore we can perceive the difference or change. This peculiarity appears in all our states or experiences. We have already given an example for the phenomena of light and darkness. The same is true of differences in color and of our judgments as to tone pitch and tone strength. Even the transition from feeling well to feeling ill is usually imperceptible, and it is only when the change occurs in a very brief time that we become conscious of it.
The physical causes of these psychic phenomena need be indicated only in brief. In all our experiences an existing chemico-physical state in our sense organs and in the central organ undergoes a change. Now experiments with physical apparatushave shown that such a process always requires a finite, though sometimes a very small, quantity of work, or, generally speaking, energy, before it can be brought about at all. Even the finest scale, sensitive to a millionth of a gram, remains stationary when only a tenth of a millionth is placed upon it, although we canseea body of such minute weight under the microscope. In the same way it requires a definite expenditure of energy in order to bring the sense organs, or the central organ, into action, and all stimuli less than this limit or threshold produce no experience of their presence.
By this the difficult concept of continuity is evoked in our experience. The transition from the light of day to the darkness of evening proceedscontinuously, that is, at no point of the whole transition do we notice that the state just passed is different from the present one, while the difference over a wider extent of the experience is unmistakable. If we wish to bring vividly to our minds the contradiction to other habits of thought which this involves, we need only to represent to ourselves the following instance. I will compare the thing A at a certain time with the thing B, which is so constructed that though objectively different from A, the difference has not yet reached the threshold. From experience, therefore, I must take A to be equal to B. Then I compare B with a thing C, which is objectively different from B in the same way as A is from B, though here, too, the difference is stillwithin the threshold, though very near it. I shall also have to take B as equal to C. But now if I compare A directly with C, the sum of the two differences oversteps the threshold value, and I find that A is different from C. This, then, is a contradiction of the fundamental principle that ifA = BandB = C,A = C.This principle is valid forcountedthings, which, in consequence, are discontinuous, but not for continuous things susceptible by our senses. If in spite of this it is applied to continuous things ormagnitudesin the narrower sense, we must bear in mind that it is just as much a case of anextrapolation to the non-existing ideal instance(p. 46) as in the case of the other general principles, which, though they are derived from experience, nevertheless, for practical purposes, transcend experience in their use.
The examples cited above prove also that these relations are by no means confined to the judgments we derive on the basis of immediate sensations. When by means of the scale we compare three weights, the differences of which lie within the limit of its sensitiveness but approach closely to it, we can arrive in a purely empirical and objective way also at the contradictionA = B,B = C,butA ≠ C.In weight and measurement, therefore, we hold fast to the principle that the relations cited have no claim to validity outside the limit of their possible errors. Accordingly, though the non-equation ofA ≠ Ccan be observed, the difference of both valuescannot be greater than at utmost the sum of the two threshold values.
These considerations also give us a means of appraising the oft-repeated statement that in contradistinction to the physical laws the mathematical laws are absolutely accurate. The mathematical laws do not refer to real things, but to imaginary ideal limit cases. Consequently they cannot be tested by experience at all, and the demands science makes on them lie in quite a different sphere. Their nature must be such thatexperience should approximate them infinitely, if certain definite well-known postulates are to be more and more fulfilled, and that the various abstractions and idealizations should be so chosen as not to contradict one another. Such contradictions have by no means always been avoided. But we must not regard them as inherent in the inner organization of our mind, as Kant did. These contradictions spring from careless handling of the concept technique, by which postulates elsewhere rejected are treated as valid. We have already come across an instance of such relations in the application of the concept of equality to unlimited groups (p. 84).
We must be guided by the same rules of precaution in answering the question whether the things felt as continuous—for example, space and time—are "truly" continuous, or whether in the last analysis they must not be conceived of as discontinuous. The various sense organs, and still more, the variousphysical apparatus with which we examine given states, are of very varying degrees of "sensibility," that is, the threshold for distinguishing the differences may be of very different magnitudes. Therefore, a thing which is discontinuous for a sensitive apparatus will behave as if it were continuous with a less sensitive apparatus. Accordingly, we shall find so many the more things continuous the less sharply developed our ability is to differentiate.
While this circumstance makes it possible that we should regard discontinuous things as continuous, time relations in certain circumstances produce the opposite effect. Even if in a process the change is continuous but very rapid, and the new state remains unchanged for a certain time, we easily conceive of this sequence as discontinuous. We cannot resist this view of the process when the change occurs in a shorter time than the threshold time of our mind for each step in the process. But since this threshold changes with our general condition, one and the same process can appear to us both continuous and discontinuous according to circumstances. Here, therefore, we have a cause through the operation of which, with advancing knowledge, more and more things will become recognized ascontinuous.
Now if we turn toexperience, we find, as the sum total of our knowledge, that for the sake of expediency we approach everything with the presumption that it iscontinuous. This aggregate experiencefinds its expression in such sayings as "Nature makes no jumps," and similar proverbial generalizations. But we must emphasize the fact once more that in deciding matters in this way we deal solely with questions of expediency, not with questions of the nature of our mental capacity.
Measuring is in a certain way the opposite of counting. While, in counting, the things are regarded in advance asindividual, and the group, therefore, is a body compounded of discontinuous elements, measuring, on the other hand, consists inco-ordinating numbers with continuous things, that is, in applying to continuous things a concept formed upon the hypothesis of discontinuity.
It lies in the nature of such a problem that the difficulty of adaptation must crop out somewhere in the course of its attempted solution. This is actually shown by the fact that measurement proves to be an unconcluded and inconcludable operation. If, in spite of this, measurement may and must justly be denoted as one of the most important advances in human thought, it follows that those fundamental difficulties can practically be rendered harmless.
Let us picture to ourselves some process of measurement—for example, the determination of the length of a strip of paper. We place a rule divided into millimeters (or some other unit) on the strip, and then we determine the unit-mark at which the strip ends. It turns out that the strip does not end exactly at a unit-mark, butbetweentwo unit-marks.And even if the rule is provided with divisions ten or a hundred times finer, the case remains the same. In most cases a microscopic examination will show that the end of the strip does not coincide with a division. All that can be said, therefore, is that the length must liebetween n andn + 1units, and even if a definite number is given, the scientifically trained person will supplement this number by the sign ±f, in whichfdenotes the possible errors, that is, the limit within which the given number may be false.
We see at once how the characteristic concept of threshold, which has led to the conception of the continuous, immediately asserts itself when in connection with discontinuous numbers. The adaptation of the threshold to numbers can be carried as far as it is possible to reduce the threshold, but the latter can never be made to disappear entirely.
The significance of measurement therefore lies in the fact that it applies the operation of counting with all its advantages (seep. 85) tocontinuousthings, which as such do not at first lend themselves to enumeration. By the application of the unit measure a discontinuity is at first artificially established through dividing the thing into pieces, each piece equal to the unit, or imagining it to be so divided. Then we count the pieces. When a quantity of liquid ismeasuredwith a liter this general process is carried out physically. In all other less direct methods of measurement the physical process is substitutedby an easier process equally good. Thus, in the example of the strip of paper we need not cut it up into pieces a millimeter in length. The divided rule is available for comparing the length of any number of millimeters that happen to come under consideration, and we need only read off from the figures on the rule the quantity of millimeters equal to the length of the strip, in order to infer that the strip can be cut up into an equal number of pieces each a millimeter in length.
After it has been made possible to count continuous things in this way, the numeration of them can then be subjected to all the mathematical operations first developed only for discrete, directly countable things. When we reflect that our knowledge of things has given them to uspreponderatingly as continuous, we at once see what an important step forward has been made through the invention of measurement in the intellectual domination of our experience.
The concept of continuity makes possible the development of another concept of greater universality, which can be characterized as an extension of the concept of causation (p. 31). The latter is an expression of the experience, if A is, B is also, in which A is understood to be a definite thing at first conceived of as immutable. Now it may happen that A is not immutable, but represents a concept with continuously changing characteristics. Then, as a rule, B will also be ofthat nature, so thatevery special value or state of B corresponds to every special value or state of A.
Here, in place of the reciprocal relation of two definite things, we have the reciprocal relation of two more or less extended groups of similar things. If these things are continuous, as is assumed here (and which is extremely often the case), both groups or series, even though they are finite, contain an endless quantity of individual cases. Such a relation between two variable things is called a function. Although this concept is used chiefly for the reciprocal relation ofcontinuousthings, there is nothing to hinder its application to discrete things, and accordingly we distinguish between continuous and discontinuous functions.
The intellectual progress involved in the conception of the reciprocal relation of entireseriesor groups to one another, as distinguished from the conception of the relations betweenindividualthings, is of the utmost importance and in the most expressive manner characterizes the difference between modern scientific thought and ancient thought. Ancient geometry, for example, knew only the cases of the acute, right, and obtuse angled triangle, and treated them separately, while the modern geometrician represents the side of the triangle as starting from the angle zero and traversing the entire field of possible angles. Accordingly, unlike his colleague of old, he does not ask for the particular principles bearing upon these particularcases, but he asks in what continuous relation do the sides and angles stand to one another, and he lets the particular cases develop from out of one another. In this way he attains a much profounder and more effectual insight into the whole of the existing relations.
It is in mathematics in especial that the introduction of the concept of continuity and of the function concept arising from it has exercised an extraordinarily deep influence. The so-calledHigher Analysis, orInfinitesimal Analysis, was the first result of this radical advance, and theTheory of Functions, in the most general sense, was the later result. This progress rests on the fact that the magnitudes appearing in the mathematical formulas were no longer regarded as certain definite values (or values to be arbitrarily determined), but asvariable, that is, values which may range through all possible quantities. If we represent the relation between two things by the formulaB = f(A),expressed in spoken language by Bis a functionof A, then in the old conception A and B are each individual things, while in the modern conception A and B represent an inexhaustible series of possibilities embracing every conceivable individual case that may be co-ordinated with a corresponding case.
Herein lies the essential advantage of the concept of continuity. It is true that it also introduces into calculation the above-mentioned contradictionswhich crop up in the ever-recurring discussions concerning the infinitely great and the infinitely small. The system introduced by Leibnitz of calculating withdifferentials, that is, with infinitely small quantities, which in most relations, however, still preserve the character of finite quantities from which they are considered to have been derived, has proved to be as fruitful of practical results as it is difficult of intellectual mastery. We can best conceive of these differentials as the expression of the law of the threshold, which law gave rise to, or made possible, the relation between the continuous and the discrete.
I have already shown (p. 34) how the first formulation of a causal relation which experience yields can be purified and elaborated by the multiplication of the experience. The method described was based upon the fact that the necessary and adequate factors of the result were obtained by eliminating successively from the "cause" the various factors of which its concept was or could be compounded, and by concluding from the result, that is, the presence or absence of the "effect," as to the necessity or superfluity of each factor.
Obviously the application of this process presupposes the possibility of eliminating each factor in turn. Very often it is not possible, and then in place of the inadequate method of the individual case themethod of the continuous functional relationsteps in with its infinitely greater effectiveness. If in most cases we cannoteliminatethe factors one by one, there are very few instances in which it is not possible tochangethem, or to observe the result in the automatically changed values of the factors. But then we have the principle that for the causal relationall such factors are essential the change of which involves a change of the result.
It is clear that this signifies a generalization of the former and more limited method. For the elimination of the factor means that its value is reduced to zero. But now it is no longer necessary to go to this extreme limit; it suffices merely to influence in some way the factor to be investigated.
It is true that here the difference in the result cannot be expressed with a "yes" or a "no," as before. It can only be said that it has changedpartly, more or less. From this it can be seen that the application of this process requires more refined methods of observation, especially for measuring, that is, for determining values or magnitudes. On the other hand, we must recognize how much deeper we can penetrate into the knowledge of things by the application of the measuring process. Each advance in precision of measurement signifies the discovery of a new stratum of scientific truth previously inaccessible.
From the fact that natural phenomena in general proceed continuously we can deduce a number of important and generallyapplicable conclusions which are constantly used for the development of science.
When a relation of two continuously varying values of the formA = f(B)is conjectured, we convince ourselves of its truth by observing for different values of A the corresponding values of B, or reversely. If we find that changes in the one correspond to changes in the other, the existence of such a relation is proved, at first only for the observed values, though we never hesitate to conclude that for the values of A lying between the observed values, but themselves not yet observed, the corresponding values of B will also lie between the observed values. For example, if the temperature at a given place has been observed at intervals of two hours, we assume without hesitancy that in the hours between when no observations were made, the values lie between the observed values. If we indicate the time in the usual manner by horizontal lines and the temperature for the general periods of time by longitudinal lines, the law of continuity asserts that all these temperature points lie in a steady line, so that when a number of points lying sufficiently near one another is known, the points between can be derived from the steady line which may be drawn through the known points. This very commonly applied process will yield the more accurate results the nearer the known points are to one another, and the simpler the line.
The application of the law of continuity or steadiness,therefore, means no less than that it is possible, from a finite, frequently not even a very large, number of individual results, to obtain the means of predicting the result for an infinitely large number of unexamined cases. The instrument derived from this law, therefore, is an eminentlyscientificone.
The value of this instrument is still greater if it succeeds in expressing the relationA = f(B)in strict mathematical form. First, the result of the determination of a number of individual values of that function is represented as a table of co-ordinated values. By the graphic process above described, or by its equivalent, the mathematical process of interpolation, this table is so extended that it also supplies all the intermediate values. But this is still a case of a mechanical co-ordination of the corresponding values. Often we succeed, especially in the relation of simple or pure concepts, in finding a general mathematical rule by which the magnitude A can be derived from the magnitude B, and reversely. This is the only instance in which we speak of a natural law in the quantitative sense.
Thus, for example, we can observe what volume a given quantity of air occupies when successively subjected to different pressures. If we arrange all these values together in a table, we can also calculate the volume for all the intermediate pressures. But on close inspection of the correspondingnumbers of pressure and volume we notice that they are in inverse ratio, or that when multiplied by one another their products will be the same. If we denote the space by v and the pressure by p, this fact assumes the mathematical formp. v = K,in which K is a definite number depending upon the quantity of air, the unit of pressure, etc., but remaining unchanged in an experimental series in which these factors stay the same. The general functional equationA = f(B)becomes the definitep = K/v.And this formula enables us to determine by a simple calculation the volume for any degree of pressure, provided the value of K has been once ascertained by experiment.
At first we have a right to such a calculation only within the province in which the experiments have been made, and the simple mathematical expression of the natural law has for the time being no further significance than that of a specially convenient rule for interpolation. But such a form immediately evokes a question which demands an experimental answer. How far can the form be extended? That there must be a limit is to be directly inferred from the consideration of the formula itself. For if we letp = 0,thenv = infinity,both of which lie beyond the field of possible experience.
Similar considerations obtain in all such mathematically formulated natural laws, and each time,therefore, we must ask what therange of validityof such an expression is, and answer the question by experiment.
While in this discussion the mathematically formulated natural law seems to have the nature only of a convenient formula of interpolation, we are nevertheless in the habit of regarding the discovery of such a formula as a great intellectual accomplishment, which so impresses us that we frequently call it by the name of the discoverer. Now, wherein lies the more significant value of such formulations?
It lies in the fact that simple formulas are discovered onlywhen the conceptual analysis of the phenomenon has advanced far enough. The very simplicity of the formula shows that the concept formation which is at the basis of it is especially serviceable. In Ptolemy's theory of the motion of the planets the means for calculating their positions in advance was given just as in the theory of Copernicus. But Ptolemy's theory was based on the assumption that the earth stands still, and that the sun and the other planets move. The assumption that the sun stands still and that the earth and the other planets move greatly facilitates the calculation of the position of the planets. In this lay the primary value of the advance made by Copernicus. It was not until much later that it was found that a number of other actual relations could be represented much more fittingly by means of the same hypothesis,and thus the Copernican theory has come to be generally recognized and applied.
The significance of the law of continuity and its field of application have by no means been exhausted by what has been said above. But later we shall have a number of occasions to point out its application in special instances, and so cause its use to become a steady mental habit with the beginner in scientific research.
Time and space are two very general concepts, though without doubt not elementary concepts. For besides the elementary concept of continuity which both contain, time has the further character of being one-seried or one-dimensional, of not admitting of the possibility of return to a past point of time (absence of double points) and of absolute onesidedness, that is, of the fundamental difference between before and after. This last quality is the very one not found in the space concept, which is in every sense symmetrical. On the other hand, owing to the three dimensions it has athreefold manifoldness.
That despite this radical distinction in the properties of space and time all of our experiences can be expressed or represented within the concepts of space and time, is very clear proof that experience is much more limited than the formal manifoldness of the conceivable. In this sense space and time can be conceived as natural laws which may be applied to all our experiences. Here at the sametime the subjective-human element of the natural law becomes very clear.
The properties of time are of so simple and obvious a nature that there is no special science of time. What we need to know about it appears as part of physics, especially of mechanics. Nevertheless time plays an essential rôle inphoronomy, a subject which we shall consider presently. In phoronomy, however, time appears only in its simplest form as a one-seried continuous manifoldness.
As for space, the presence of the three dimensions conditions a great manifoldness of possible relations, and hence the existence of a very extensive science of bodies in space, ofgeometry. Geometry is divided into various parts depending upon whether or not the concept of measurement enters. When dealing with purely spacial relations apart from the concept of measurement it is called geometry of position. In order to introduce the element of measurement a certain hypothesis is necessary which is undemonstrable, and therefore appears to be arbitrary and can be justified only because it is the simplest of all possible hypotheses. This hypothesis takes for granted that a rigid body can be moved in all directions in space without changing in measure. Or, to state the inverse of this hypothesis, in space those parts are called equal which a rigid body occupies, no matter how it is moved about.
We are not conscious of the extreme arbitrarinessof this assumption simply because we have become accustomed to it in school. But if we reflect that in daily experience the space occupied by a rigid body, say a stick, seems to the eye to undergo radical changes as it shifts its position in space and that we can maintain that hypothesis only by declaring these changes to be "apparent," we recognize the arbitrariness which really resides in that assumption. We could represent all the relations just as well if we were to assume that those changes are real, and that they are successively undone when we restore the stick to its former relation to our eye. But though such a conception is fundamentally practicable in so far as it deals merely with the space picture of the stick, we nevertheless find that it would lead to such extreme complications with regard to other relations (for example, the fact that the weight of the stick is not affected by the change of the optic picture) that we do better if we adhere to the usual assumption that the optical changes are merely apparent.
In this connection we learn what an enormous influence the various parts of experience exert upon one another in the development of science. In every special generalization of experiences, that is, in every individual scientific theory, our aim is not only to generalize this special group of experiences in themselves, but at the same time to join such other experiences to them as expedience demands. If the effect of this necessity is on the one hand to renderthe elaboration of an appropriate theory more difficult, it has on the other hand the great advantage of affording a choice among several theories of apparently like value, and thus making possible a more precise notion of the reality. For example, for the understanding of the mutual movements of the sun and the earth it is the same whether we assume that the sun moves about the earth or the earth about the sun. It is not until we try to represent theoretically the position of the other planets that we see the economic advantage of the second conception, and facts like Foucault's experiment with a pendulum can be represented only according to this second conception in our present state of knowledge.
Likewise, the assumption on which scientific geometry goes, that space has the same properties in all directions, conflicts with immediate experience. In immediate experience we make a sharp distinction between below and above, although we are prepared to admit the "homogeneity" of space in the horizontal direction. This is due, as physics teaches, to the fact that we are placed in a field of gravitation which acts only from above downward and which permits free horizontal turnings, although it imparts a characteristic difference to the third direction. Since considerations of another kind enable us to place ourselves in a position in which we ignore this field of gravitation in the investigation of space, geometry abstracts this element anddisregards the corresponding manifoldness. In the theory of the gravitation potential, on the other hand, this very manifoldness is made the subject of scientific investigation.
The common application of the concepts of space and time results in the concept ofmotion, the science of which is called phoronomics. In order to make this new variable subject to measurement we must arrive at an agreement or convention as to the way in which to measure time. For since past time can never be reproduced we actually experience only unextended moments, and have no means of recognizing or defining the equality of two periods of time by placing them side by side, as we can in the case of spacial magnitudes. We help ourselves by sayingthat in uninfluenced motions equal periods of time must correspond to the equal changes in space. We regard the rotation of the earth on its axis and its revolution about the sun as such uninfluenced motions. The two depend upon dissimilar conditions, and the empirical fact that the relation of the two motions, or the relation between the day and the year, remains practically the same, sustains that assumption, and at the same time shows the expediency of the given definition of time.
Analytic geometry, the application of algebra to geometric relations, occupies a noteworthy position, from the point of view of method, in the science of space. It yields geometric results by means of calculation,that is, by the application of thealgebraicmaterial of symbols we can obtain data concerning unknownspacialrelations. An explanation is necessary of how by a method apparently so extraneous such results as these can be attained.
The answer lies again in the general principle of co-ordination, which in this very case receives a particularly cogent illustration. Three algebraic signs, x, y, and z, are co-ordinated with the three variable dimensions of space. First, the same independent and constant variability is ascribed to these signs, and, further, the same mutual relations are assumed to subsist between them as actually exist between the three-spacial dimensions. In other words, precisely the same kind of manifoldness is imparted to these algebraic signs as the spacial dimensions possess to which they are co-ordinated, and we may therefore expect that all the conclusions arising from these assumptions will find their corresponding parts in the spacial manifoldness. Accordingly, a co-ordinated spacial relation corresponds to every change of those algebraic formulas resulting from calculation, and if such changes lead to an algebraically simple form, then the spacial form corresponding to it must show an analogous simplicity. Here, therefore, we have a case such as was described under simpler conditions onp. 86of operations undertaken with one group and repeated correspondingly in the co-ordinated group. And it is only the great differencein the things of which in this case the two groups are composed—spacial relations on the one side and algebraic signs on the other—that creates the impression of astonishment which was felt very strongly at the invention of this method, and which is still felt by students with talent for mathematics when they first become acquainted with analytical geometry.
Before we proceed to consider the fundamentals of other sciences, it is well to make a general résumé of the field so far traversed. Since the later sciences, as we have already observed, make use of the entire apparatus of the earlier sciences, the mastery of them must be assured in order to render their special application possible.
This does not mean that one must have complete command of the entire range of those earlier sciences in order to pursue a later one. Mere human limitations would prevent the fulfilment of such a demand. As a matter of fact, successful work can be done in one of the later sciences even if only the most general features of the earlier ones have been clearly grasped. Nevertheless, the rapidity and certainty of the results are very considerably increased by a more thorough knowledge of the earlier sciences, and the investigator, accordingly, should seek a middle road between the danger of insufficient preparation for his special science and the danger of never getting to it from sheer preparation. In any circumstances he must be prepared always,even though it be in later age, to acquire those fundamental aids so soon as he feels the need of them for carrying out any special work. It is generally acceded that without logic the adequate pursuit of science is impossible. Nevertheless, the opinion is widely current, even among men of science, that everybody has command of the needful logic without having studied it. No more than a man can learn of himself to use the calculus, even if he may have discovered unaided some of its elementary principles, can he acquire certainty and readiness in the use of the logical rules generally necessary, unless he has made the necessary studies. It is true that the scientific works of the great pioneers and leaders in the special sciences furnish practical examples of such logical activity. But complete freedom and security are acquired only on the basis of conscious knowledge.
We have now seen how, from the physiological construction of our mental apparatus, the process of concept formation and the experience of concept connections are the basis of the whole of mental life. The laws of the mutual interaction of the most general or elementary concepts operated in the formation of the concepts,thing,group,co-ordination. Here were found the fundamentals of logic or the science of concepts. A special process of abstraction yielded the concept ofnumber, and with it the corresponding field ofmathematics, arithmetic, algebra, and the theory of numbers.
By means of the second fundamental fact of physiology, thethreshold, another elementary fact was explained, that ofcontinuity. The co-ordination of individual things under the influence of this concept was expanded into theco-ordination of continuous phenomena-series, and yielded the correspondingly more general concept of thefunction. From the application of the number concept to continuous things, the idea ofmeasurementresulted. In mathematics the concept of continuity led to higheranalysisand thetheory of functions. Finally, the concept of continuity proved to be an inexhaustible aid for the extension of scientific knowledge and for the formulation of natural laws in mathematical form.