Chapter VI: Part II: Logic, the Science of the Manifold, and Mathematics (1)

=19. The Most General Concept.= 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 the _most general_ of 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, _no_ memories 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 a _difference_, 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 two parts, 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 a _name_. 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 an _experience_ without 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 call _things_. _A thing, therefore, is an experience which has 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, and _the 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 call _the same_, or _identical_. Here we shall leave undecided the question whether this lack of differentiation occurs because we _cannot_, or because we _would 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 concept _thing_.

=20. Association.= The experience of the _connection_ or _relation_ between 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, and _vice versa_. The cause of this invariably lies in some experiences in which 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 the _complex concept_ which 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, and _vice 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 the _one_ relation C : D there arise two new relations not immediately given, namely, A : D and B : 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.

=21. The Group.= 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 classic _process of argumentation_ of the syllogism. 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 by _Aristotle_ and his successors to this process is based upon the _certainty_ which its results possess. Nevertheless, it has been pointed out, especially by _Kant_, 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 we have 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 is the 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.

=22. Negation.= 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 of _negation_. 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 even _one_ of the characteristics a, b, c, d excludes the incorporation of the thing into the group A, while the mere absence 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 lack _all_ the 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.

=23. Artificial and Natural Groups.= 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 is _poorer in characteristics_, will, in general, be _richer 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 of the one group can be derived from the other by the elimination or incorporation of individual characteristics, then we can postulate the general thesis _that 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 being _one_ characteristic, 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 a _classification_ of 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 narrowest or 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 itself
b) all substances containing two elements and no more
c) all substances containing three elements
d, 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 of _all_ the 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 each _new_ case will fall within the scheme obtained by the exhaustive combination of elementary concepts (whether chemical elements or kinds of energy) _known_ at the time. It is quite possible that a new thing under investigation contains a _new_ elementary concept, so that on account of 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 be _experienced_. 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 the _sequence_ according to which this combination takes 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 their _quantitative 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 called _isomeric_, 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 that _several_ different things instead of one appear in the same group of the original scheme, the systematic classification of which necessitates a further schematization by the use of other characteristics.

=24. Arrangement of the Members.= 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 some _one_ definite member is followed by a _single_ other 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 this _linear_ arrangement, 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 fact _that 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 already taken 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 of _time_, which has been appropriately called a _form 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 by _one dimension_. Since, on the other hand, we have all learned to feel space as _tri-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 matter of adaptation, and that because the change has been extremely slight in the course of centuries it produces the impression of being unalterable.[D]

[D] Mathematicians who busy themselves a great deal with the formal theory of four-dimensional space, seem to acquire a capacity for imagining this form as easily as the three-dimensional form with which we are all familiar. Therefore, despite the oft-repeated statements to the contrary, it is not impossible to imagine four-dimensional space. Only, we must not attempt to represent to ourselves four-dimensional space in three-dimensional space, especially not without a knowledge of its properties.

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 quite _arbitrary_, 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 of mathematics. 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.

=25. Numbers.= An especially important group in the linear order is that of the _integral 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 as _unlimited_ or _infinite_.

The groups or types thus obtained are called the _integral 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 series begins, this characteristic is present in a peculiar form, the preceding group being _group 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.

=26. Arithmetic, Algebra, and the Theory of Numbers.= 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. For _practical_ reasons the study of them must be as _general_ as possible. This science is called _arithmetic_.

Arithmetic undergoes an important generalization if the individual numbers in a calculation are disregarded and _abstract signs_ standing 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.

=27. Co-ordination.= So far our discussion has confined itself to the _individual_ groups and to the properties which each one of them exhibits _by itself_. We shall now investigate the relations which exist _between 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, then any 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 of _association_.

First, the co-ordination is _mutual_, 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 has _double members_, or _double points_. The limit-case is _identity_, in which every member corresponds to _itself_. 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.

=28. Comparison.= 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 of _all_ their members. In the first case A is called, in the broader sense of the word, _smaller_ than B, in the second B is called smaller than A, in the third the two groups are said to be of _equal 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 in pairs. 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 is { equal to } group B
{ smaller than }

{ greater than }
and group B is { equal to } group C
{ smaller than }

{ greater than }
then group A is { equal to } group C
{ smaller 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) as _equal_ to 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.

=29. Counting.= 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 called _counting_, and from the premises made it follows _that 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 until the group to be counted is exhausted. The last number required for the co-ordination is called the _sum_ of the members of the counted group. Since the number series continues indefinitely, every given group can be counted.

Numerals have been co-ordinated with _names_ as well as with _signs_. 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 a _measure for the effectiveness or the value_ of 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.

=30. Signs and Names.= 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 extraordinary extent 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 be to the point, and that, on the other hand, it should be easily "understood," that is, _comprehended_ by the senses and unmistakably _differentiated_ from other signs co-ordinated with other things.

Thus, we find that the most frequent concepts of co-ordinated sound signs form the beginnings of _language_ in 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 the _co-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 many _fundamental defects_ of language, since they destroy the _principle of unambiguity_ upon 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.

=31. The Written Language.= 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 of sight. 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 the _written language_ in 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 the _written signs_ of 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, each number 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 of _words_. 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 the _place value_ in 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]

[E] The usual designation of the larger groups, ten, hundred, thousand, million, billion, etc., is also quite irrational. If it is our object to secure expressions for place values in as few words as possible, we find that the numbers of the form 10^{2n}, in which n is a whole number, must receive their own names, that is, 10, 100, 10,000, 100,000,000 etc. In this way the problem of designating as many numbers as possible by as few words as possible is solved.

=32. Pasigraphy and Sound Writing.= There are two possibilities for co-ordination between concepts and written signs. Either the co-ordination is _direct_, 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 the _language 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 construct _an 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]

[F] It is not difficult to perfect musical notation with a view to unambiguity, a thing which would greatly facilitate the study of music.

Fundamentally we need not hesitate to recognize in _concept writing_ or _pasigraphy_ a more complete solution of the problem of sign arrangement. Even the very incomplete Chinese pasigraphy renders possible 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 of _elementary_ concepts, 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 new elementary 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, the attempt 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.

=33. Sound Writing.= In point of unambiguity of co-ordination _phonetic writing_ is 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.