PART II

LOGIC, THE SCIENCE OF THE MANIFOLD, AND MATHEMATICS

=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.

=34. The Science of Language.= 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 the _science_

of language make a sharp distinction between it and the _knowledge_ of

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 seem as 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 to _preserve_ as accurately as possible existing

linguistic usage, with its imperfections, amounting at times to

absurdities. Its real object lies rather in the appropriate

_development_ and _improvement_ of 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

English language 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 a

_universal auxiliary language_ (p. 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 extraordinarily great. For it may be asserted that philosophy,

the most general of all the sciences, has hitherto made such extremely

limited progress only _because 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.

=35. Continuity.= Up to this point our discussions have been based on

the general concept of the _thing_, that is, of the individual

experience differentiated from other experiences. Here the fact of

_being 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 of _continuity_.

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 time _we have never felt the increase_

of darkness from moment to moment, although theoretically 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 by _Herbart_,

but its significance was first recognized by _Fechner_, and has since

then become generally known in physiology and psychology under the name

of _threshold_. _Next to memory the threshold determines the fundamental

lines of our psychic life._

The threshold therefore means that whatever state we are in _a certain

finite amount of difference or change must be stepped over_ before 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 apparatus have 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 can _see_ a 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 proceeds

_continuously_, 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

still within 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 if A = B and B = C, A = C. This principle is valid for

_counted_ things, 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 or _magnitudes_ in the narrower sense, we

must bear in mind that it is just as much a case of an _extrapolation 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 contradiction A = B, B = C, but A [Not=] 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 of A [Not=] C can

be observed, the difference of both values cannot 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

that _experience 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 various physical 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 as _continuous_.

Now if we turn to _experience_, we find, as the sum total of our

knowledge, that for the sake of expediency we approach everything with

the presumption that it is _continuous_. This aggregate experience

finds 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.

=36. Measurement.= Measuring is in a certain way the opposite of

counting. While, in counting, the things are regarded in advance as

_individual_, and the group, therefore, is a body compounded of

discontinuous elements, measuring, on the other hand, consists in

_co-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, but _between_ two

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 lie _between n and n + 1 units_, and even if a definite

number is given, the scientifically trained person will supplement this

number by the sign ± _f_, in which _f_ denotes 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 (see p. 85) to

_continuous_ things, 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 is _measured_ with a

liter this general process is carried out physically. In all other less

direct methods of measurement the physical process is substituted by 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 us

_preponderatingly 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.

=37. The Function.= 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 of that nature, so that _every 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 of _continuous_

things, 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 entire _series_ or groups to one another, as distinguished

from the conception of the relations between _individual_ things, 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 particular cases, 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-called _Higher Analysis_, or

_Infinitesimal Analysis_, was the first result of this radical advance,

and the _Theory 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 as _variable_, that

is, values which may range through all possible quantities. If we

represent the relation between two things by the formula B = f(A),

expressed in spoken language by B _is a function_ of 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

contradictions which crop up in the ever-recurring discussions

concerning the infinitely great and the infinitely small. The system

introduced by Leibnitz of calculating with _differentials_, 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.

=38. The Application of the Functional Relation.= 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 the _method of

the continuous functional relation_ steps in with its infinitely

greater effectiveness. If in most cases we cannot _eliminate_ the

factors one by one, there are very few instances in which it is not

possible to _change_ them, or to observe the result in the automatically

changed values of the factors. But then we have the principle that for

the causal relation _all 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

changed _partly_, 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.

=39. The Law of Continuity.= From the fact that natural phenomena in

general proceed continuously we can deduce a number of important and

generally applicable conclusions which are constantly used for the

development of science.

When a relation of two continuously varying values of the form A = 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 eminently

_scientific_ one.

The value of this instrument is still greater if it succeeds in

expressing the relation A = 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 corresponding numbers 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 form p·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

equation A = f(B) becomes the definite p = 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 let p = 0, then v = 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 the _range of

validity_ of 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 only _when 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.

=40. Time and Space.= 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 a _three_fold 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 same time 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 in _phoronomy_, 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, of _geometry_. 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 arbitrariness of 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 render the 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 and disregards

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 of _motion_, 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 saying _that 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 the _algebraic_ material of symbols we

can obtain data concerning unknown _spacial_ relations. 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 on p. 86 of operations

undertaken with one group and repeated correspondingly in the

co-ordinated group. And it is only the great difference in 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.

=41. Recapitulation.= 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 of _number_, and with it the corresponding field of

_mathematics_, arithmetic, algebra, and the theory of numbers.

By means of the second fundamental fact of physiology, the _threshold_,

another elementary fact was explained, that of _continuity_. The

co-ordination of individual things under the influence of this concept

was expanded into the _co-ordination of continuous phenomena-series_,

and yielded the correspondingly more general concept of the _function_.

From the application of the number concept to continuous things, the

idea of _measurement_ resulted. In mathematics the concept of continuity

led to higher _analysis_ and the _theory 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.