Natural Philosophy

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Title: Natural Philosophy
Author: Ostwald, Wilhelm
Language: English
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*** Start of this LibraryBlog Digital Book "Natural Philosophy" ***

NATURAL PHILOSOPHY

BY
WILHELM OSTWALD

TRANSLATED
BY
THOMAS SELTZER

_With the author's special revision for the American edition_

NEW YORK
HENRY HOLT AND COMPANY
1910

_Published November_, 1910

THE QUINN & BODEN CO. PRESS
RAHWAY N. J.

The original of this book was published
as volume I in Reclam's BÜCHER DER
NATURWISSENSCHAFT.

PREFACE

The beginning of the twentieth century is marked by a sudden rise of
interest in philosophy. This is especially manifest in the vast growth
of philosophic literature. The present movement, it is noteworthy, is by
no means a revival proceeding from the academic philosophy traditionally
represented at the universities, but has rather the original character
of _natural philosophy_. It owes its origin to the fact that after the
specialization of the last half century, the synthetic factors of
science are again vigorously asserting themselves. The need finally to
consider all the numerous separate sciences from a general point of view
and to find the connection between one's own activity and the work of
mankind in its totality, must be regarded as the most prolific source of
the present philosophic movement, just as it was the source of the
natural philosophic endeavors a hundred years ago.

But while that old natural philosophy soon ended in a boundless sea of
speculation, the present movement gives promise of permanent results,
because it is built upon an extremely broad basis of experience. The
laws of energy in the inorganic world and the laws of evolution in the
organic world furnish mental instruments for a conceptual elaboration
of the material provided by science, instruments capable not only of
unifying present knowledge, but also of evoking the knowledge of the
future. If it is not permissible to regard this unification as
exhaustive and sufficient for all time, yet there is still so much left
for us to do in working over the material we have on hand from the
general points of view just mentioned, that the need for systematizing
must be satisfied before we can turn our gaze upon things more remote.

The present work is meant to serve as the first aid and guide in the
acquisition of these comprehensive notions of the external world and the
inner life. It is not meant to develop or uphold a "system of
philosophy." Through long experience as a teacher the writer has learned
that those are the best pupils who soon go their own way. However, it
_is_ meant to uphold a certain method, that is, the scientific (or, if
you will, the _natural_ scientific), which takes its problems, and
endeavors to solve its problems, from experience and for experience. If,
as a result, several points of view arise that differ from those of the
present day, and consequently demand a different attitude toward
important matters in the immediate future, this very fact affords proof
that our present natural philosophy does not lead away from life, but
aims to form a part of our life, and has a right to.

CONTENTS

PAGE

INTRODUCTION 1

PART I

GENERAL THEORY OF KNOWLEDGE 11

1. The Formation of Concepts 11

2. Science 13

3. The Aim of Science 13

4. Concrete and Abstract 16

5. The Subjective Part 17

6. Empirical Concepts 18

7. Simple and Complex Concepts 19

8. The Conclusion 24

9. The Natural Laws 28

10. The Law of Causation 31

11. The Purification of the Causal Relation 34

12. Induction 38

13. Deduction 40

14. Ideal Cases 44

15. The Determinateness of Things 47

16. The Freedom of the Will 50

17. The Classification of the Sciences 53

18. The Applied Sciences 57

PART II

LOGIC, THE SCIENCE OF THE MANIFOLD, AND MATHEMATICS 61

19. The Most General Concept 61

20. Association 63

21. The Group 65

22. Negation 68

23. Artificial and Natural Groups 69

24. Arrangement of the Members 75

25. Numbers 78

26. Arithmetic, Algebra, and the Theory of Numbers 79

27. Co-ordination 80

28. Comparison 82

29. Numbers 85

30. Signs and Names 86

31. The Written Language 89

32. Pasigraphy and Sound Writing 92

33. Sound Writing 96

34. The Science of Language 97

35. Continuity 101

36. Measurement 107

37. The Function 109

38. The Application of the Functional Relation 112

39. The Law of Continuity 113

40. Time and Space 118

41. Recapitulation 124

PART III

THE PHYSICAL SCIENCES 127

42. General 127

43. Mechanics 128

44. Kinetic Energy 132

45. Mass and Matter 136

46. Energetic Mechanics 138

47. The Mechanistic Theories 140

48. Complementary Branches of Mechanics 144

49. The Theory of Heat 147

50. The Second Fundamental Principle 150

51. Electricity and Magnetism 154

52. Light 156

53. Chemical Energy 159

PART IV

THE BIOLOGIC SCIENCES 163

54. Life 163

55. The Storehouse of Free Energy 168

56. The Soul 171

57. Feeling, Thinking, Acting 174

58. Society 179

59. Language and Intercourse 182

60. Civilization 184

INDEX 187

INTRODUCTION

Natural science and natural philosophy are not two provinces mutually
exclusive of each other. They belong together. They are like two roads
leading to the same goal. This goal is the domination of nature by man,
which the various natural sciences reach by collecting all the
individual actual relations between the natural phenomena, placing them
in juxtaposition, and seeking to discover their interdependence, upon
the basis of which one phenomenon may be foretold from another with more
or less certainty. Natural philosophy accompanies these specialized
labors and generalizations with similar labors and generalizations, only
of a more universal nature. For instance, while the science of
electricity, as a branch of physics, deals with the relation of
electrical phenomena to one another and to phenomena in other branches
of physics, natural philosophy is not only concerned with the question
of the mutual connection of _all_ physical relations, but also endeavors
to include in the sphere of its study chemical, biological,
astronomical, in short, all the known phenomena. In other words,
_natural philosophy is the most general branch of natural science_.

Here two questions are usually asked. First, how can we define the
boundary line between natural philosophy and the special sciences,
since, obviously, sharp lines of demarcation are out of the question?
Secondly, how can we investigate and teach natural philosophy, when it
is impossible for any one person to master all the sciences completely,
and so obtain a bird's-eye view of the general relations between all the
branches of knowledge? To the beginner especially, who must first learn
the various sciences, it seems quite hopeless to devote himself to a
study that presupposes a command of them.

Since a discussion of the two questions will afford an excellent
preliminary survey of the work in hand, it will be well to consider them
in detail. In the first place, _the lack of complete and precise
boundary lines is a general characteristic of all natural things_, and
science is a natural thing. If, for instance, we try to differentiate
sharply between physics and chemistry, we are met with the same
difficulty. So also in biology if we try to settle beyond the shadow of
a doubt the line of separation between the animal and the vegetable
kingdoms.

If, despite this well-known impossibility, we consider the division of
natural things into classes and orders as by no means useless and do not
discard it, but regard it as an important scientific work, this is
practical proof that such classification preserves its essential
usefulness, even if it does not attain ideal definiteness. For, this
imperfection notwithstanding, classification reaches its end, which is
a comprehensive view, and thus a mastery, of the manifoldness of
phenomena. For example, with the overwhelming majority of organic beings
there is no doubt whether they are animals or plants. Similarly, most
phenomena of inorganic nature can readily be designated as physical or
chemical. For all such cases, therefore, the existing classification is
good and useful. The few cases presenting difficulty may very well be
considered by themselves wherever they occur, and we need merely take
cognizance of them here. It follows from this, to be sure, that
classification will be all the _better fitted to its purpose the less
frequently_ such doubtful cases arise, and that we have an interest in
repeatedly testing existing classifications with a view to finding out
if they cannot be supplanted by more suitable ones.

In these matters it is much the same as when we look upon the waves on
the surface of a large body of water. Our first glance tells us that a
number of waves are rolling there; and from a point giving us a
sufficiently wide outlook, we can count them and gauge their width. But
where is the line of division between one wave and the next? We
undoubtedly see one wave following another, yet it is impossible for us
to indicate precisely the end of one and the beginning of the next. Are
we then to deduce that it is superfluous or unfeasible to designate the
waves as different? By no means. On the contrary, in strictly
scientific work we will endeavor to find some suitable definition of the
boundary line between two consecutive waves. It may then be called an
arbitrary line, and in a degree arbitrary it will certainly be. But to
the investigator this does not matter. What concerns him is, if, with
the help of this definition, wave lengths can be unequivocally
determined, and if this is possible, he will use the definition as
suitable to the purposes of science, without dismissing from his mind
the idea that possibly some other definition may provide an even easier
or sharper determination. Such an one he would instantly prefer to the
old one.

Thus we see that these questions of classification are not questions of
the so-called "essence" of the thing, _but pertain merely to purely
practical arrangements for an easier and more successful mastery of
scientific problems_. This is an extremely important point of view, much
more far-reaching than is apparent here at its first application.

As to the second objection, I will admit its validity. But here, too, we
have a phenomenon appearing in all branches and forms of science.
Therefore we must familiarize ourselves with it in advance. Science was
created by man for man's purposes, and, consequently, like all human
achievements, possesses the indestructible quality of imperfection. But
the mere fact that a successful working science exists, with the help of
which human life has been fundamentally modified, signifies that _the
quality of incompleteness in human learning is no hindrance to its
efficiency_. For what science has once worked out always contains a
portion of truth, hence a portion of efficiency. The old corpuscular
theory of light, which now seems so childishly incomplete to us, was
adequate, none the less, for satisfactorily explaining the phenomena of
reflection and refraction, and the finest telescopes have been built
with its help. This is due to the _true elements_ in it, which taught us
correctly to calculate the direction of rays of light in reflection and
refraction. The rest was merely an arbitrary accessory which had to fall
when new, contradictory facts were discovered. These facts could not
have been taken into consideration when the theory was propounded,
because they were not yet known. But when the corpuscular theory of
light was replaced by the theory of waves of an elastic ether, geometric
optics at first remained quite unchanged, because the theory of straight
lines of rays could be deduced from the new views also, though not so
easily and smoothly. And geometric optics was then concerned with
nothing but these straight lines, in no wise with the question of their
propagation. It did not become clear until recently that this conception
of straight lines of rays is incomplete, though, it is true, it made a
first approach toward the presentation of actual phenomena. It fails
when it comes to characterize the behavior of a pencil of rays of large
aperture. The old idea of a straight line of rays was to be replaced by
a more complex concept with more varied characteristics, namely, the
wave-surface. The greater variety of this concept renders possible the
presentation of the greater variety of the optical phenomena just
mentioned. And from it proceed the very considerable advances that have
been made, since the new theory was propounded, in optical instruments,
especially the microscope and the photographic objective, for the
purposes of which pencils of rays of large aperture are required. The
astronomic objective with its small angle of aperture has not undergone
particularly important improvements.

Experience in every province of science is the same as in this. Science
is not like a chain which snaps when only a single link proves to be
weak. It is like a tree, or, better still, like a forest, in which all
sorts of changes or ravages go on without causing the whole to pass out
of existence or cease to be active. The relations between the various
phenomena, once they become known, continue to exist as indestructible
components of all future science. It may come to pass, in fact, does
come to pass very frequently, that the form in which those relations
were first expressed prove to be imperfect, and that the relations
cannot be maintained quite generally. It turns out that they are
subjected to other influences which change them because they had been
unknown, and which could not have been taken into consideration at the
discovery and first formulation of these relations. But no matter what
changes science may undergo, a certain residue of that first knowledge
will remain and never be lost. In this sense, a truth that science has
once gained has life eternal, that is, it will exist as long as human
science exists.

Applying this general notion to our case, we have the following. How far
and how generally at any given time the relations of the various
phenomena are summed up in fixed forms, that is, in natural laws, will
depend upon the stage attained by each of the special sciences. But
since science has been in existence it has yielded a certain number of
such general laws, and these, though they have been filed down a good
deal in form and expression, and have undergone many corrections as to
the limits of their application, nevertheless have preserved their
essence, since they began their existence in the brains of human
investigators. The net of the relations of phenomena grows ever wider
and more diversified, but its chief features persist.

The same is true of an individual. No matter how limited the circle of
his knowledge, _it is a part of the great net, and therefore possesses
the quality by virtue of which the other parts readily join it as soon
as they reach the consciousness and knowledge of the individual_. The
man who thus enters the realm of science acquires advantages which may
be compared to those of a telephone in his residence. If he wishes to,
he may be connected with everybody else, though he will make extremely
limited use of his privilege, since he will try to reach only those with
whom he has personal relations. But once such relations have been
established, the possibility of telephone communication is
simultaneously and automatically established. Similarly, every bit of
knowledge that the individual appropriates will prove to be a regular
part of the central organization, the entire extent of which he can
never cover, though each individual part has been made accessible to
him, provided he wants to take cognizance of it.

The mere beginner in learning, therefore, when receiving the most
elementary instruction in school, or from his parents, or even from his
personal experiences in his surroundings, is grasping one or more
threads of the mighty net, and can grope his way farther along it in
order to draw an increasing area of it into his life and the field of
his activity. _And this net has the valuable, even precious quality of
being the same that joins the greatest and most comprehensive intellects
in mankind to one another._ The truths a man has once grasped he need
never learn afresh so far as their _actual content_ is concerned, though
not infrequently--especially in newer sciences--he may have to see the
_form_ of their presentation and generalization change. For this reason
it is of such especial importance for each individual from the first to
perceive these unalterable facts and realize that they are unalterable
and learn to distinguish them from the alterable forms of their
presentation. It is in this very regard that the incompleteness of human
knowledge is most clearly revealed. Time and again in the history of
science form has been taken for content, and necessary changes of
form--a merely practical question--have been confused with revolutionary
modifications of the content.

Thus, each presentation of a science has its natural philosophic
portion. In text-books, whether elementary or advanced, the chapter on
natural philosophy is found usually at the beginning of the book,
sometimes at the end, in the form of a "general introduction," or
"general summary." In the special works in which the latest advances of
science are made known by the investigators, the natural philosophic
portions are usually to be found in the form of theses, of principles,
which are not discussed, often not even explicitly stated, but upon the
acceptance of which depend all the special conclusions that are drawn,
in the case in hand, from the new facts or thoughts imparted. Whether at
the beginning or at the end of the book, these most general principles
do not quite occupy the place that befits them. If at the introduction
of the text-book, they are practically devoid of content, since the
facts they are meant to summarize are yet to be unfolded in the course
of the presentation. If at the end, they come too late, since they have
already been applied in numerous instances, though without reference to
their general nature. The best method is--and a good teacher always
employs this method, whether in the spoken or the written word--to let
the generalizations come whenever the individual facts imparted require
and justify them.

Thus, all instruction in natural sciences is necessarily interspersed
with natural philosophy, good or bad, according to the clearheadedness
of the teacher. If we wish to obtain a perfect survey of a complex
structure, as, for instance, the confusion of streets in a large city,
we had better not try to know each street, but study a general plan,
from which we learn the comparative situation of the streets. So it is
well for us in studying a special science to look at our general plan,
if for no other reason than to keep from losing our way when it may
chance to lead through a quarter hitherto unknown. This is the purpose
of the present work.

PART I

GENERAL THEORY OF KNOWLEDGE

=1. The Formation of Concepts.= To the human mind, as it slowly awakens
in every child, the world at first seems a chaos consisting of mere
individual experiences. The only connection between them is that they
follow each other consecutively. Of these experiences, all of which at
first are different from one another, certain parts come to be
distinguished by the fact that they are repeated more frequently, and
therefore receive a special character, that of _being familiar_. The
familiarity is due to our _recalling_ a former similar experience; in
other words, to our feeling that there is a relation between the present
experience and certain former experiences. The cause of this phenomenon,
which is at the basis of all mental life, is a quality common to all
living things, and manifesting itself in all their functions, while
appearing but rarely or accidentally in inorganic nature. It is the
quality by virtue of which _the oftener any process has taken place in a
living organism the more easily it is repeated_. Here is not yet the
place to show how almost all the characteristic qualities of living
beings, from the preservation of the species to the highest intellectual
accomplishments, are conditioned by this special peculiarity. Suffice it
to say that because of this quality all those processes which are
repeated frequently in any given living organism, assume spontaneously,
that is, from physiologic reasons, a character distinguishing them
essentially from those which appear only in isolated instances, or
sporadically.

If a living being is equipped with consciousness and thought, like man,
then the conscious recollections of such uniform experiences form the
enduring or permanent part in the sum-total of his experiences. Each
time a complex event, like the change of seasons, for example, which we
know from experience repeats itself--each time a part of such an event
reaches our consciousness, we are prepared also for the other parts that
experience teaches are connected with it. This makes it possible for us
to foresee future events. What significance the foreseeing of future
events has for the preservation and the development of the individual as
well as the species can only be indicated here. To give one instance, it
is our ability to foretell the coming of winter with the impossibility
of obtaining food directly during the winter that causes us to refrain
from at once using up all the food we have and to preserve it for the
day of need. The ability to foretell, therefore, becomes the foundation
of the whole structure of economic life.

=2. Science.= The prophecy of future events based upon the knowledge of
the details of recurring events is called _science_ in its most general
sense. Here, as in most cases in which language became fixed long before
men had a clear knowledge of the things designated, the name of the
thing is easily associated with false ideas arising either from errors
that had been overcome or from other, still more accidental, causes.
Thus, the mere knowledge of _past_ events is also called science without
any thought of its use for prophesying future events. Yet a moment's
reflection teaches that mere knowledge of the past which is not meant
to, or cannot, serve as a basis for shaping the future is utterly
aimless knowledge, and must take its place with other aimless activities
called _play_. There are all sorts of plays requiring great acumen and
patient application, as for example the game of chess; and no one has
the right to prevent any individual from pursuing such games. But the
player for his part must not demand special regard for his activity. By
using his energies for his personal pleasure and not for a social
purpose, that is, for a general human purpose, he loses every claim to
the social encouragement of his activity, and must be content if only
his individual rights are respected; and that, too, only so long as the
social interests do not suffer by it.

=3. The Aim of Science.=These views are deliberately opposed to a very
widespread idea that science should be cultivated "for its own sake,"
and not for the sake of the benefits it actually brings or may be made
to bring. We reply that there is nothing at all which is done merely
"for its own sake." Everything, without exception, is done for human
purposes. These purposes range from momentary personal satisfaction to
the most comprehensive social services involving disregard of one's own
person. But in all our actions we never get beyond the sphere of the
human. If, therefore, the phrase "for its own sake" means anything, it
means that science should be followed for the sake of the immediate
pleasure it affords, that is to say, as _play_ (as we have just
characterized it), and in the "for-its-own-sake" demand there is hidden
a misunderstood idealism, which, on closer inspection, resolves itself
into its very opposite, the degradation of science.

The element of truth hidden in that misunderstood phrase is, that in a
higher state of culture it is found better to disregard the _immediate_
technical application in the pursuit of science, and to aim only for the
greatest possible perfection and depth in the solution of its individual
problems. Whether this is the correct method of procedure and when it is
so, is solely a question of the general state of culture. In the early
stages of human civilization such a demand is utterly meaningless, and
all science is necessarily and naturally confined to immediate life. But
the wider and more complex human relations become, the wider and surer
must the ability to predict future events become. Then it is the
function of prophesying science to have answers ready for questions
which as yet have not become pressing, but which with further
development may sooner or later become so.

In the net-like interlacing of the sciences, that is, of the various
fields of knowledge, described in the introduction, we must always
reckon with the fact that our anticipation of what kind of knowledge we
shall next need must always remain very incomplete. It is possible to
foresee future needs in general outline with more or less certainty, but
it is impossible to be prepared for particular individual cases which
lie on the _border line_ of such anticipation, and which may sometimes
become of the utmost importance and urgency. Therefore it is one of the
most important functions of science to achieve as _perfect_ an
elaboration as possible of _all_ the relations conceivable, and in this
practical necessity lies the foundation of the general or _theoretical_
elaboration of science.

=The Science of Concepts.= Here the question immediately arises: how can
we secure such perfection? The answer to this general preliminary
question of all the sciences belongs to the sphere of the first or the
most general of all the sciences, a knowledge of which is presupposed
for the pursuit of the other sciences. Since its foundation by the Greek
philosopher Aristotle it has borne the name of _logic_, which name,
etymologically speaking, hints suspiciously at the _word_, and the word,
as is known, steps in where ideas are wanting. Here, however, we have to
deal with the very science of ideas, to which language bears the
relation only of a means--and often an inadequate means--to an end. We
have already seen how, through the physiologic fact of _memory_,
experiences are found in our consciousness which are similar, that is,
partially coinciding with one another. These coinciding parts are those
concerning which we can make predictions, for the very reason that they
coincide in every single instance, and they alone, therefore, constitute
that part of our experience which bears results and hence has
significance.

=4. Concrete and Abstract.= Such coinciding or repeated parts of similar
experiences we call, as already stated, _concepts_. But here, too,
attention must immediately be drawn to a linguistic imperfection, which
consists in the fact that in such a group of coinciding experiences we
designate by the same name both the isolated experience or the object of
a special experience and the totality of _all_ the coinciding
experiences; in other words, all the similar experiences. Thus, _horse_
means, on the one hand, quite a definite thing which for the moment
forms an object of our experience, and, on the other, the totality of
all possible similar objects which have been present in our former
experiences, and which we shall meet in our future experiences. It is
true that these two sorts of contents of consciousness bearing the same
name are distinguished also as _concrete_ and _abstract_, and there is
an inclination to attribute "reality" only to the first, while the
other, as "mere entities in thought," are relegated to a lesser degree
of reality. As a matter of fact, the difference, though important, is of
quite another kind. It is the difference between the _momentary
experience_, as opposed to the totality of the corresponding _memories_
and _expectations_. Hence not so much a difference in _reality_ as in
_presence_. However, our observations have already made it apparent that
presence alone never yields knowledge. A necessary part of knowledge is
the memory of former similar experiences. For without such memory and
the corresponding comparison, it is quite impossible for us to get at
those things which agree and which, therefore, may be predicted; and we
should stand before every one of our experiences with the helplessness
of a new-born babe.[A]

[A] Sometimes on suddenly awaking from a profound sleep a person finds
himself for the moment deprived of his personal stock of memories,
unable to recall where and in what circumstances he is. No one who has
experienced such a condition can ever forget the terrifying sense of
helplessness it brings.

=5. The Subjective Part.= We shall therefore have to recognize realities
in abstract ideas in so far as they must rest upon some experiences to
be at all intelligible to us. Since the formation of concepts depends
upon memories, and these may refer, according to the individual, to very
different parts of the same experience of different individuals,
concepts always possess an element dependent upon the individual, or a
_subjective_ element. This, however, does not consist in the _addition_
by the individual of new parts not found in the experience, but, on the
contrary, in the different _choice_ out of what is found in the
experience. If every individual absorbed all parts of the experience,
the individual, or subjective, differences would disappear. And since
scientific experience endeavors to make the absorption of experiences as
complete as possible, it aims nearer and nearer to this ideal by seeking
to equalize the subjective deficiency of the individual memory through
the collocation of as many and as various memories as possible, thus
filling in the subjective gaps in experience as far as possible and
rendering them harmless.

=6. Empirical Concepts.= First and unconditionally those concepts
possess reality which always and without exception are based on
_experienced_ facts. But we can easily make manifold arbitrary
combinations of concepts from different experiences, since our memory
freely places them at our disposal, and from such a combination we can
form a new concept. Of course it is not necessary that our arbitrary
combination should also be found in our past or future experiences. On
the contrary, we may rather expect that there could be many more
arbitrary combinations not to be found in experience than combinations
later "confirmed" by experience. The former are purposeless because
unreal, the latter, on the contrary, are of the utmost consequence
because upon them is based the real aim of knowledge, prediction. The
former are those which have brought the very "reality" of the concepts
into ill repute, while the latter show that the formation and the mutual
reaction of the concepts practically constitute the entire content of
all science. It is of the greatest importance, therefore, to distinguish
between the two kinds of concept combinations, and the study of this
differentiation forms the content of that most general of all the
sciences which we have characterized as logic, or, better, the science
of concepts.

=7. Simple and Complex Concepts.= The formation of concepts consists, as
we have seen, in the selection of those parts of different but similar
experiences which coincide with one another and in the elimination of
those that are different in kind. The results of such a procedure may
vary greatly according to the number and the difference of the
experiences placed in relation with one another. If, for example, we
compare only a few experiences, and if, moreover, these experiences are
very similar to one another, then the resulting concepts will contain
very many parts that agree. But at the same time they will have the
peculiarity of not being applicable to other experiences, since these
are without some of the coinciding parts of that narrower circle. Thus,
for example, the concept which a rustic chained to the soil all his life
has of human work does not apply to the work of the city man. A concept
will embrace a larger number of individual cases in proportion as it
contains fewer different parts. And by systematically following out this
thought we arrive at the conclusion that the concepts that are simple
and have no different parts at all find the widest application or are
the most general.

The elimination of the non-coinciding parts from the concept-forming
experience is called _abstraction_. Obviously abstraction must be
carried the farther the more numerous and the more varied the
experiences from which the concepts are abstracted, and the simplest
concepts are the most abstract.

By looking back over the ground just traversed, the less abstract ideas
may also be regarded as the _more complex_ in contradistinction to the
simpler ones. Only we must guard against the error of literal
interpretation and not suppose that the less simple concepts have really
been compounded of the simpler ones. In point of origin they actually
existed first, since the experience contains the ensemble of all the
parts, those which have been retained as well as those which have been
eliminated. It is only later, by a characteristic mental operation,
after we have analyzed the more complex concept, that is, after we have
disclosed the simpler concepts existing in it, that we can compound it
again; in other words, execute its synthesis.

These relations bear a striking resemblance to the relations known from
chemistry to exist between substances, namely, between elements and
compounds. From the chaos of all objects of experimentation (chemistry
purposely limits itself to ponderable bodies) the _pure_ substances are
sifted out--an operation corresponding to the formation of concepts. The
pure substances prove to be either _simple_ or _compound_, and the
compounds are so constituted that they can each be reduced to a limited
number of simple substances. The simple substances, or _elements_,
retain this quality of simplicity only until they are recalled; that is,
until it has been proved that they, too, can be resolved into still
simpler elements. The same is true of the simple concepts. They can
claim simplicity only until their complex nature is demonstrated.

With all these similarities we must be extremely careful never to forget
the differences existing alongside the agreements. So hereafter we shall
make no further use of the chemical simile. It was brought into
requisition merely in order to acquaint the beginner the more readily
with the entire method of investigation by means of a more familiar
field of thought and study. It is quite certain, however, that side by
side with the given similarities there are also radical differences.
Moreover, the notion of simple and complex concepts or "ideas" had been
elaborated by John Locke long before chemistry reached its present state
of clearness concerning the concept of the elements.

Nevertheless since then the relation has been completely reversed. While
the study of the chemical elements has in the meantime undergone great
development, so that not only have the elements of all the substances
coming under the observation of the chemist been discovered, but,
inversely, many compound substances have been constructed from their
elements, not even an approach to such a development is apparent in the
study of concepts. On the contrary, the whole matter has remained at
about the same point as that to which John Locke had brought it in the
second half of the seventeenth century. This is due above all to the
opinion of the most influential philosophers, that Aristotle's logic, or
science of concepts, is absolutely true as well as exhaustive and
complete, so that, at the utmost, what is left for later generations to
do is only to make a change in the form in which the matter is
presented. It is true that in more recent times the grave error of this
view is beginning to be recognized. We realize that Aristotle's logic
embraces but a very small part of the entire field, though in this part
he displays the greatest genius. But beyond this general recognition no
great step forward has been made. Not even a provisional table of the
elementary concepts has been propounded and applied since Locke.

Hence in the following investigation we shall have to speak of the
elements or the simpler parts of a complex concept only in the sense
that these concept elements are simpler as compared with the complex
concepts, but not in the sense that the simplest or truly elementary
concepts have already been worked out. It must be left to later
investigators to find these, and it may be expected that the reduction
of some concepts until then considered elementary into still simpler
ones will take place chiefly in times of great intellectual progress.

_Complex concepts_ can, in the first place, be formed from experience,
for in an empirical concept we meet with several conceptual component
parts which can be separated from one another by a process of
abstraction, but are always found together in the given experiences. For
example, the concept _horse_ has originated from a very frequent,
similarly repeated experience. On analysis it is found to contain a vast
number of other concepts, such as quadruped, vertebrate animal,
warm-blooded, hairiness, and so on. Horse, then, is obviously a _complex
empirical concept_.

On the other hand, we can combine as many simple concepts as we please,
even if we did not find them combined in experience, for in reality
there is nothing to hinder us from uniting all the concepts provided by
memory into any combinations we please. In this way we obtain _complex
arbitrary concepts_.

The task of science can now be even more sharply defined than before by
the fact that it _permits the construction of arbitrary concepts which
in circumstances to be foreseen become empirical concepts_. This is
another expression for _prediction_, which we recognized as the
characteristic of science. It goes deeper than the previous definition,
because here the means for its realization are given.

=8. The Conclusion.= First let us consider the scientific import of the
complex empirical concepts. It consists in the fact that they accustom
us to the coexistence of the corresponding elements of a concept. So
that when, in a new experience, we meet with some of these elements
together, we immediately suppose that we shall find in the same
experience the other elements also which have not yet been ascertained.
Such a supposition is called a _conclusion_. A conclusion always exceeds
the present experience by predicting an expected experience. Therefore,
the form of a conclusion is the universal form of scientific
predication.

A conclusion must contain at least two concepts, the one which is
experienced, and the one which, on the basis of this experience, is
expected. Every complex empirical concept makes such a conclusion
possible after it has been separated into simpler concepts. And the
simplest case is naturally the one in which there are only _two_ parts,
or in which only two parts are taken into consideration.

To what extent such a conclusion is valid, that is to say, to what
extent the experience produces the anticipated concept, obviously
depends upon the reply to a very definite fundamental question. If in
experience the union of the two parts of the concept occurs
_invariably_, so that one part of the concept is never experienced
unless the other part is also experienced, then there is the _greatest_
probability that the expected experience will also have the same
character, and that the conclusion will prove valid or true. To be sure,
there is no way of making certain that the coincident occurrence of the
two concepts, which experience has shown to be _without exception_
hitherto, will continue to be so also in the future. For our only means
of penetrating into the future consists in applying that conclusion from
previous experiences to future experiences, and it can therefore by no
means claim absolute validity. There are, however, different _degrees of
certainty_, or, rather, _probability_, attaching to such a conclusion.
In experiences that occur but rarely the probability is that so far we
have experienced only certain combinations of simple concepts, while
others, though occurring, have not yet entered within the limited circle
of our experience. In such a case a conclusion of the kind mentioned
above may be right, but there is also some probability of its being
false. On the other hand, in experiences which happen extremely
frequently and in the most diverse circumstances, and in which we always
find the constant and unexceptional combination, the probability is very
strong that we shall find the combination in future experiences also,
and the probability of the conclusion approaches practical certainty. Of
course, we can never quite exclude the possibility that new relations
never as yet experienced might enter, by which the conclusion which
hitherto has always been true would now become false, whether because
the expectation entertained prove invalid in single instances or in all
cases.

It follows from this that in general, our conclusions will have the
greater probability the more generally and the oftener the corresponding
experiences have occurred and are occurring. Such concepts as are found
consistently in many experiences otherwise different are called
_general_ concepts, and therefore the probability of the conclusions
described will be the greater the more _general_ the concepts to which
they refer. This obtains to such a degree that we feel that certain very
general conclusions must be true always and without exception, and it is
"unthinkable" to us that they can ever in any circumstances prove not
valid. Such a statement, however, is never anything else than a hidden
appeal to experience. For the mere putting of the question, whether the
conclusion can also be false, demonstrates that the opposite of what has
proved to be the experience so far can be conceived, and the assertion
of its "unthinkability" only signifies that such an experience cannot be
evoked in the mind by the _memory_ for the very reason that, as has been
premised, there are no such memories because the experiences did not
exist. But since, on the other hand, there is no hindrance to thinking
any combinations of concepts at will, we have not the least difficulty,
as everybody knows, in thinking any sort of "nonsense" whatsoever. Only
it is impossible to reproduce such combinations from memory.

The scientific conclusion, therefore, first takes the form: if A is,
then B is also. Here A and B represent the two simple concepts which are
known from experience to be found together in the more complex concept
C. The word "is" signifies here some empirical reality corresponding to
the concepts. The conclusion may therefore also be expressed, somewhat
more circumstantially and more precisely, in this form: if A is
experienced, the experience of B is also expected. The evoking of this
expectation, which implies its justification, is due to the recollection
of the coincidence of the two concepts in former experiences, and the
probability depends, in the manner described above, upon the number of
valid cases. Here it must be observed that even individual cases in
which our expectations have been deceived do not for the most part lead
us to regard the conclusion as generally untrue, that is, to abandon the
expectation of B from A. For we know that our experience is always
_incomplete_, that in certain circumstances we fail to notice existing
factors, and that, therefore, our failure to find that relation valid
which, on other occasions, has been found to be valid, may be attributed
to subjective causes. In case, however, of the repeated occurrence of
such disappointments, we will look elsewhere for relations between these
and other elements of experience, in order that thereafter we may
foresee such cases also and include them in our anticipations.

=9. The Natural Laws.= The facts just described have very frequently
found expression in the doctrine of the _laws of nature_, in connection
with which we have often, as in the man-made social or political laws,
conceived of a lawmaker, who, for some reasons, or perhaps arbitrarily,
has ordained that things should be as they are and not otherwise. But
the intellectual history of the origin of the laws of nature shows that
here the process is quite a different one. The laws of nature do not
decree what shall happen, but _inform us what has happened and what is
wont to happen_. The knowledge of these laws, therefore, makes it
possible for us, as I have emphasized again and again, to foresee the
future in a certain degree and, in some measure, also to determine it.
We determine the future by constructing those relations in which the
desired results appear. If we cannot do so either because of ignorance
or because of inaccessibility to the required relations, then we have no
prospect of fashioning the future according to our desires. The wider
our knowledge of the natural laws, that is, of the actual behavior of
things, the more likely and more numerous the possibilities for
fashioning the future according to our desires. In this way science can
be conceived of as the study of how to become happy. For he is happy
whose desires are fulfilled.

In this conception the natural laws indicate what simpler concepts are
found in complex concepts. The complex concept _water_ contains the
simpler ones _liquid_, a certain _density_, _transparency_,
_colorlessness_,[B] and many others. The sentences, water is a liquid,
water has a density of one, water is transparent, water is colorless,
or, pale blue, etc., are so many natural laws.

[B] More precisely, a very pale blue.

Now what predictions do those natural laws enable us to make?

They enable us to predict that when we have recognized a given body as
water by virtue of the above properties, we are justified in expecting
to find in the same body all the other known properties of water. And so
far experience has invariably confirmed such expectations.

Furthermore, we may expect that if in a given specimen of water we
discover a relation which up to that time was unknown, we shall find
this relation also in all the other specimens of water even though they
were not tested for that particular relation. It is obvious how
enormously this facilitates the progress of science. For it is only
necessary to determine this new relation in some one case accessible to
the investigator to enable us to predict the same relation in all the
other cases without subjecting them to a new test. As a matter of fact,
this is the general method that science pursues. It is this that makes
it possible for science to make regular and generally valid progress
through the labors of the most various investigators who work
independently of one another, and often know nothing of one another.

Of course, it must not be forgotten that such conclusions are always
obtained in accordance with the following formula: _things have been so
until now, therefore we expect that they will be so in the future_. In
every such case, therefore, there is the possibility of error. Thus far,
whenever an expectation was not realized, it was almost always possible
to find an "explanation" for the error. Either the inclusion of the
special case in the general concept proved to be inadmissible because
some of its other characteristics were absent, or the accepted
characterization of the concept required an improvement (limitation or
extension). In other words, one way or another, there was a discrepancy
between the concept and the experience, and, as a rule, sooner or later
it becomes possible for us to arrive at a better adjustment between
them.

This general truth has often been interpreted to mean that in the end
such an adjustment must of necessity always be possible to reach,
without exception; in other words, that absolutely every part of an
experience can be demonstrated as conditioned by natural law. Evidently
such an assertion far exceeds the demonstrable. And even the usual
conclusion cannot be applied here, that because it has happened so in
the past it will happen so in the future also. For the part of our
experiences that we can grasp by natural laws is infinitesimally small
in comparison with that in which our knowledge still fails us entirely.
I will mention only the uncertainty in predicting the weather for only
one day ahead. Moreover, when we consider that until now only the
_easiest_ problems had been solved, and naturally so, because they were
most accessible to the means at hand, then we can readily see that
experience offers no basis whatever for such a conclusion. We must not
say, therefore, that because we have been able so far to explain all
experiences by natural laws it will be so in the future likewise. For we
are far from being able to explain all experiences. In fact, it is only
a very small part that we have begun to investigate. We are as little
justified in saying that we have explained all the problems of our
experience that have been subjected to scientific investigation. We have
by no means explained all of them. Every science, even mathematics,
teems with unsolved problems. So we must resign ourselves to the present
status of human knowledge and ability, and may at best express the
_hope_ founded upon previous experience, that we shall be able to solve
more and more of the incalculable number of problems of our experience
without indulging in any illusions as to the perfection of this work.

=10. The Law of Causation.= By reason of its frequency and importance
the mental process above described has been subjected to the most
diverse investigations, and that most general form of the scientific
conclusion (which we apply in ordinary life even much more frequently
than in science) has been raised, under the name of the law of
causation, to a principle anteceding all experience and to the very
condition making experience possible. Of this so much is true, that
through the peculiar physiological organization of man, _memory in the
most general sense_--the easier execution of such processes as have
already repeatedly taken place in the organism, as against entirely new
kinds of processes--the formation of concepts (of the recurring parts in
the constantly changing variety of processes), is especially stimulated
and facilitated. By it the recurring parts of experience step into the
foreground, and on account of their paramount practical importance for
the security of life, it may well be said in the sense of the theory of
evolution and adaptation, that the entire structure and mode of life of
the organism, especially of the human organism, nay, perhaps life
itself, is indissolubly bound up with that foresight and, therefore,
with the law of causation also. Of course, there is nothing in the way
of calling such a relation an _a priori_ relation, if it is so desired.
As far as the individual is concerned it no doubt antedates all his
experience, since the entire organization which he inherits from his
parents had already been formed under such an influence. But that there
can be forms or existence _without_ such an attribute is shown by the
whole world of the _inorganic_, in which, as far as our knowledge goes,
there is no evidence of either memory or foresight, but only of an
immediate passive participation in the processes of the world around
them.[C]

[C] It cannot be objected that inorganic nature also is known to be
subject to the law of causation. The causal mode of regarding inorganic
phenomena is a distinctly human one, and nothing justifies the assertion
that the same phenomena cannot be viewed in an entirely different
manner.

Further, the circumstance that the causal relation is brought about by
the peculiar manner in which we react upon our experiences, has
sometimes been expressed in this way--the relation of cause and effect
does not exist in nature at all, but has been introduced by men. The
element of truth in this is, that a quite differently organized being,
it is to be supposed, would be able to, or would have to, arrange its
experiences according to quite different mutual relations. But since we
have no experience of such a being, we have no possibility of forming a
valid opinion of its behavior. On the other hand, we must recognize that
it is possible, at least formally, to conceive also of kinds of
experiences with no coinciding parts, or a world in which there are no
experiences at all with coinciding parts. In such, therefore, prediction
is impossible. Such a world will not call forth, even in a being endowed
with memory, a conception and generalization of the various experiences
in the shape of natural laws. Consequently we must recognize that in
addition to the _subjective_ factor in the formation of our knowledge of
the world, or that factor which is dependent upon our physico-psychical
structure, there is also the _objective_ character of the world with
which we must decidedly reckon, or that character which is independent
of us; and that in so far the natural laws contain also objective parts.
To represent the relation clearly to our minds by a figure, we may
compare the world to a heap of gravel and man to a pair of sieves, one
coarser than the other. As gravel passes through the double sieve
pebbles of apparently equal size accumulate between the sieves, the
larger ones being excluded by the first sieve and the smaller ones
allowed to pass by the second. It would be an error to assert that all
the gravel consisted of such pebbles of equal size. But it would be
equally false to assert that it was the sieves that _made_ the pebbles
equal.

=11. The Purification of the Causal Relation.= If by experience we have
found a proposition of the content, If A is, then B is also, the two
concepts A and B generally consist of several elements which we will
designate as a, a´, a´´, a´´´, etc., and as b, b´, b´´, b´´´. Now the
question arises, whether or not all these elements are essential for the
relation in question. It is quite possible, in fact, even highly
probable, that at first only a special instance of the existing
phenomena was found, that is, that the concept A, which has been found
to be connected with the concept B, contains other determining parts
which are not at all requisite to the appearance of B.

The general method of convincing oneself of this is by eliminating one
by one the component parts of the concept A, namely, a, a´, a´´, etc.,
and then seeing whether B still appears. It is not always easy to carry
out this process of elimination. Our greater or less ability to conduct
such investigations depends upon whether we deal with things that are
merely the objects of our _observation_, and which we ourselves have not
the power to change (as, for example, astronomical phenomena), or with
things which are the objects of our _experimentation_, and which we can
influence. In the latter case one or another factor is usually found
which can be eliminated without the disappearance of B, and then we must
proceed in such a way as to form a corresponding new concept A´ from the
factors recognized as necessary (which new concept will be more general
than the former A), and to express the given proposition in the improved
form: If A´ is, then B is also.

Quite similar is the case with the other member of this relation. It
often happens that when a, or a´´, a´´´ is found, somewhat different
things appear, which do not fit the concept as first constructed. Then
we must multiply the experiences as much as possible in order to
determine what constant elements are found in the concept B, and to form
from these constant elements the corresponding concept B´. The improved
proposition will then read: if A´ is, then B´ is also.

This entire process may be called the purification of the causal
relation. By this term we express the general fact that in first forming
such a regular connection, the proper concepts are very seldom brought
into relation with one another at once. The cause of it is that at first
we make use of _existing_ concepts which had been formed for quite a
different purpose. It must therefore be regarded as a special piece of
good fortune if these old concepts should at once prove suited to the
new purpose. Furthermore, the existing concepts are as a rule so vaguely
characterized by their names, which we must employ to express the new
relation, that for this reason also it is often necessary to determine
empirically in what way the concept is to be definitely established.

The various sciences are constantly occupied with this work of the
mutual adaptation of the concepts that enter into a causal relation. By
way of example, we may take the "self-understood" proposition which we
use when we call out to a careless child when it sticks its finger into
the flame of a candle, "Fire burns!" We discover that there are
self-luminous bodies which produce no increase of temperature, and
therefore no sensation of pain. We discover that there are processes of
combustion that develop no light, but heat enough to burn one's
fingers. And, finally, the scientific investigation of this proposition
arrives at the general expression that, as a rule, chemical processes
are accompanied by the development of heat, but that, conversely, such
processes may also be accompanied by the absorption of heat. In this way
that casual sentence which we call out to the child develops into the
extensive science of thermo-chemistry when it is subjected to the
continuous purification of the causal relation, which is the general
task of science.

It remains to be added that in this process of adapting concepts it is
necessary also sometimes to follow the opposite course. This is the case
when _exceptions_ are noticed in a relation as expressed for the time
being; when, therefore, the proposition if A is present, then B is
present also, is in a great many instances valid, but occasionally
fails. This is an indication that in the concept A an element is still
lacking. This element, however, is present in the instances that tally,
but absent in the negative cases, and its absence is not noticed because
it is not contained in A. Then it is necessary to seek this part, and
after it has been found, to embody it in the concept A, which thus
passes into the new concept A´.

This case is the obverse of the former one. Here the more suitable
concept proves to be less general than the concept accepted temporarily,
while in the first case the improved concept is more general. Hence we
formulate the rule: exceptions to the temporary rule require a
limitation, while an unforeseen freedom requires an extension, of the
accepted concept.

=12. Induction.= The form of conclusion previously discussed, _because
it has been so, I expect it will continue to be so in the future_, is
the form through which each science has arisen and has won its real
content, that is, its value for the judgment of the future. It is called
_inference by induction_, and the sciences in which it is
preponderatingly applied are called _inductive sciences_. They are also
called experiential or empirical sciences. At the basis of this
nomenclature is the notion that there are other sciences, the deductive
or rational sciences, in which a reverse logical procedure is applied,
whereby from general principles admitted to be valid in advance,
according to an absolutely sure logical process, conclusions of like
absolute validity are drawn. At the present time people are beginning to
recognize the fact that the deductive sciences must give up these claims
one by one, and that they already have given them up to a certain
extent; partly because on closer study they prove to be inductive
sciences, and partly because they must forego the title and rank of a
science altogether. The latter alternative applies especially to those
provinces of knowledge which have not been used in prophesying the
future or cannot be so used.

To return to the inductive method--it is to be noted that _Aristotle_,
who was the first to describe it, proposed two kinds of induction, the
_complete_ and the _incomplete_. The first has this form: since _all_
things of a certain kind are so, each _individual thing_ is so. While
the incomplete induction merely says: since _many_ things of a certain
kind are so, _presumably_ all things of this kind are so. One instantly
perceives that the two conclusions are essentially different. The first
lays claim to afford an absolutely certain result. But it rests upon the
assumption that _all_ the things of the kind in question are known and
have been tested as to their behavior. This hypothesis is generally
impossible of fulfilment, since we can never prove that there are not
more things of the same kind other than those known to us or tested by
us. Moreover, the conclusion is _superfluous_, as it merely repeats
knowledge that we have already directly acquired, since we have tested
_all_ the things of the one kind, hence the special thing to which the
predication refers.

On the other hand, the _incomplete_ induction affirms something that has
not yet been tested, and therefore involves as a condition an
_extension_ of our knowledge, sometimes an extremely important
extension. To be sure, it must give up the claim to unqualified or
absolute validity, but, to compensate, it acquires the irreplaceable
advantage of lending itself to practical application. Indeed, in
accordance with the scientific practice justified by experience,
described on p. 29, the scientific inductive conclusion assumes the
form: because it has _once_ been found to be so, it will _always_ be so.
From this appears the significance of this method for the enlargement of
science, which, without it, would have had to proceed at an incomparably
slower pace.

=13. Deduction.= In addition to the inductive method, science has (p.
38) another method, which, in a sense, should be the reverse of the
inductive and is claimed to provide absolutely correct results. It is
called the _deductive_ method, and it is described as the method that
leads from premises of general validity by means of logical methods of
general validity to results of general validity.

As a matter of fact, there is no science that does or could work in such
a way. In the first place, we ask in vain, how can we arrive at such
general, or absolutely valid, premises, since all knowledge is of
empiric origin and is therefore equipped with the possibility of error
as ineradicable evidence of this origin. In the next place, we cannot
see how from principles at hand conclusions can be drawn the content of
which exceeds that of these principles (and of the other means
employed). In the third place, the absolute correctness of such results
is doubtful from the fact that blunders in the process of reasoning
cannot be excluded even where the premises and methods are absolutely
correct. In practice it has actually come to pass that in the so-called
deductive sciences doubts and contradictions on the part of the various
investigators of the same question are by no means excluded. To wit,
the discussion that has been carried on for centuries, and is not yet
ended, over Euclid's parallel theorem in geometry.

If we ask whether, in the sense of the observations we have just made of
the formation of scientific principles, there is anything at all like
deduction, we can find a procedure which bears a certain resemblance
with that impossible procedure and which, as a matter of fact, is
frequently and to very good purpose applied in science. It consists in
the fact that general principles which have been acquired through the
ordinary incomplete induction are _applied to special instances which,
at the proposition of the principle, had not been taken into
consideration_, and whose connection with the general concept had not
become directly evident. Through such application of general principles
to cases that have not been regarded before, specific natural laws are
obtained which had not been foreseen either, but which, according to the
probability of the thesis and the correctness of the application are
also probably correct. However, the investigator, bearing in mind the
factor of uncertainty in these ratiocinations feels in each such
instance the need for testing the results by experience, and he does not
consider the _deduction_ complete until he had found _confirmation_ in
experience.

Deduction, therefore, actually consists in the searching out of
particular instances of a principle established by induction and in its
confirmation by experience. This conduces to the growth of science, not
in breadth, but in profundity. I again resort to the comparison I have
frequently made of science with a very complex network. At first glance
we cannot obtain a complete picture of all the meshes. So, at the first
proposition of a natural law an immediate survey of the entire range of
the possible experiences to which it may apply is inachievable. It is a
regular, important, and necessary part of all scientific work to learn
the extent of this range and investigate the specific forms which the
law assumes in the remoter instances. Now, if an especially gifted and
far-seeing investigator has succeeded in stating in advance an
especially general formulation of an inductive law, it is everywhere
confirmed in the course of the trial applications, and the impression
easily arises that confirmation is superfluous, since it results simply
in what had already been "deduced." In point of fact, however, the
reverse is not infrequently the case, that the principle is _not_
confirmed, and conditions quite different from those anticipated are
found. Such discoveries, then, as a rule, constitute the starting-point
of important and far-reaching modifications of the original formulation
of the law in question.

As we see, deduction is a necessary complement of, in fact, a part of,
the inductive process. The history of the origin of a natural law is in
general as follows. The investigator notices certain agreements in
individual instances under his observation. He assumes that these
agreements are general, and propounds a temporary natural law
corresponding to them. Then he proceeds by further experimentation to
test the law in order to see whether he can find full confirmation of it
by a number of other instances. If not, he tries other formulations of
the law applicable to the contradictory instances, or exclusive of them,
as not allied. Through such a process of adjustment he finally arrives
at a principle that possesses a certain range of validity. He informs
other scientists of the principle. These in their turn are impelled to
test other instances known to them to which the principle can be
applied. Any doubts or contradictions arising from this again impel the
author of the principle to carry out whatever readjustments may have
become necessary. Upon the scientific imagination of the discoverer
depends the range of instances sufficing for the formulation of the
general inductive principle. It also frequently depends upon conscious
operations of the mind dubbed "scientific instinct." But as soon as the
principle has been propounded, even if only in the consciousness of the
discoverer, the deductive part of the work begins, and the consequent
test of the proposition has the most essential influence on the value of
the result.

It is immediately evident that this _deductive_ part is of all the more
weight, the more _general_ the concepts in question are. If, in
addition, the inductive laws posited soon prove to be of a comparatively
high degree of perfection, we obtain the impression described above,
that an unlimited number of independent results can be deduced from a
premise. _Kant_ was keenly alive to the peculiarity of such a view,
which had been widely spread pre-eminently by _Euclid's_ presentation of
geometry, and he gave expression to his opinion of it in the famous
question: _How are a priori judgments possible?_ We have seen that it is
not always a question of _a priori_ judgments, but also of inductive
conclusions applied and tested according to deductive methods.

=14. Ideal Cases.= Each experience may generally be considered under an
indefinite number of various concepts, all of which may be abstracted
from that experience by corresponding observations. Accordingly an
indefinite number of natural laws would be required for prophesying that
experience in all its parts. Likewise the indefinite number of premises
must be known through the application of which those natural laws
acquire a certain content. Thus it seems as if it were altogether
impossible to apply natural laws for the determination of a single
experience to come, and in a certain sense this is true (p. 30). For
example, when a child is born, we are quite incapable of foretelling the
peculiar events that will occur in its life. Beyond the statement that
it will live a while and then die, we can make only the broadest
assertions qualified by numerous "ifs" and "buts."

If, in spite of this, we arrange a very great part of our life and
activity according to the prophecies we make in regard to numerous
details in life, basing them upon natural laws, the question arises, how
we get over the difficulty, or, rather, the impossibility just referred
to.

The answer is, that we repeatedly so find or can form our experiences
that certain natural relations _preponderatingly_ determine the
experience, while the other parts that remain undetermined fall into the
background. _The prophecy will cover so considerable a part of the
experience that we can forego previous knowledge of the rest._ We can
foretell enough to render a practical construction of life possible, and
increasing experience, whether the personal experience of the individual
or the general experience of science, constantly enlarges this
controllable part of future experiences.

The procedure of science is similar to that of practical life, though
freer. Whenever an investigator seeks to test a natural law of the form:
if A is so, then B is so, he endeavors to choose or formulate the
experiences in such a way that the fewest possible extraneous elements
are present, and that those that are unavoidable should exert the least
possible influence upon the relation in question. He never succeeds
completely. In order, nevertheless, to reach a conclusion as to the form
the relation will take without extraneous influences, the following
general method is applied.

A series of instances are investigated which are so adjusted that the
influence of the extraneous elements grows less and less. Then the
relation investigated approaches a limit which is never quite reached,
but to which it draws nearer and nearer, the less the influence of the
extraneous elements. And the conclusion is drawn that if it were
possible to exclude the extraneous elements entirely, the limit of the
relation would be reached.

A case in which none of the extraneous elements of experience operate is
called an _ideal case_, and the inference from a series of values
leading to the limit-value is an _extrapolation_. _Such extrapolations
to the ideal case_ are a quite natural procedure in science, and a very
large part of natural laws, especially all quantitative laws, that is,
such as express a relation between measurable values, have precise
validity only in ideal cases.

We here confront the fact that many natural laws, and among them the
most important, are expressed as, and taken to be, conditions _which
never occur in reality_. This seemingly absurd procedure is, as a matter
of fact, the best fitted for scientific purposes, since ideal cases are
to be distinguished by this, _that with them the natural laws take on
the simplest forms_. This is the result of the fact that in ideal cases
we intentionally and arbitrarily overlook every complication of the
determining factors, and in describing ideal cases we describe the
simplest conceivable form of the class of experiences in question. The
real cases are then constructed from the ideal cases by representing
them as the sum of all the elements that have an influence on the
experience or the result. Just as we can represent the unlimited
multitude of finite numbers by the figures up to ten, so we can
represent an unlimited quantity of complicated events by a finite number
of natural laws, and so reach a highly serviceable approximation to
reality.

Thus geometry deals with absolutely straight lines, absolutely flat
surfaces, and perfect spheres, though such have never been observed, and
the results of geometry come the closer to truth, the more nearly the
real lines, surfaces, and spheres correspond to the ideal demands.
Similarly, in physics, there are no ideal gases or mirrors, or in
chemistry ideally pure substances, though the expressed simple laws in
these sciences are valid for only such bodies. The non-ideal bodies of
these sciences, which reality presents in various degrees of
approximation, correspond the more closely to these laws, the slighter
the deviation of the real from the ideal. And the same method is applied
in the so-called mental sciences, psychology and sociology, in which the
"normal eye" and a "state with an entirely closed door" are examples of
such idealized limit-concepts.

=15. The Determinateness of Things.= A very widespread view and a very
grave one, because of its erroneous results, is _that by the natural
laws things are unequivocally and unalterably determined down to the
very minutest detail_. This is called _determinism_, and is regarded as
an inevitable consequence of every natural scientific generalization.
But an accurate investigation of actual relations produces something
rather different.

The most general formulation of the natural law: _if A is experienced,
then we expect B_, necessarily refers in the first place only to certain
_parts_ of the thing experienced. For perfect similarity in two
experiences is excluded by the mere fact that we ourselves change
unceasingly and one-sidedly. Consequently, no matter how accurate the
repetition of a former experience may be, our very participation in it,
an element bound to enter, causes it to be different. Therefore we deal
with only a _partial_ repetition of any experience, and the common part
is all the smaller a fraction of the entire experience, the more
_general_ the concept corresponding to this part. But the most general
and most important natural laws apply to such very general ideas, and
accordingly they determine only a small part of the whole result. Other
parts are determined by other laws, but we can never point out an
experience that has been determined completely and unequivocally by
natural laws known to us. For example, we know that when we throw a
stone, it will describe an approximate parabolic curve in falling to the
ground. But if we should attempt to determine its course accurately, we
should have to take into consideration the resistance of the air, the
rotatory motion of the stone upon being thrown, the movement of the
earth, and numerous other circumstances, the exact determination of
which is a matter beyond the power of all sciences. Nothing but an
_approximate_ determination of the stone's course is possible, and every
step forward toward accuracy and absoluteness would require scientific
advances which it would probably take centuries to accomplish.

Science, therefore, can by no means determine the exact linear course
that the stone will take in its fall. It can merely establish a certain
broader path within which the stone's movement will remain. And the path
is the wider the smaller the progress science has made in the branch in
question. The same conditions prevail in the case of every other
prediction based upon natural laws. Natural laws merely provide a
certain frame within which the thing will remain. But which of the
infinitely numerous possibilities within this frame will become reality
can never be absolutely determined by human powers.

The belief that it is possible has been evoked merely by a far-reaching
method of abstraction on the part of science. By assuming in place of
the stone "a non-extended point of mass" and by disregarding all the
other factors which in some way (whether known or unknown) exercise an
influence on the stone's movement, we can effect an apparently perfect
solution of the problem. But the solution is not valid for real
experience, merely for an ideal case, which bears only a more or less
profound similarity to the real. It is only such an ideal world, that
is, a world arbitrarily removed from its actual complexity, that has the
quality of absolute determinateness which we are wont to ascribe to the
real world.

We might point to the method of abstraction generally adopted in science
and to the extrapolation to ideal cases which has just been explained,
and regard the assertion of the absolute determinateness of events in
the world as a justified extrapolation to the ideal case. In other
words, we might say that we know all the natural laws and how to apply
them perfectly to the individual instances. In controversion of this it
must be said that the ulterior justification of such ideal extrapolation
is not yet feasible. The justification lies in the demonstration that
the real cases approximate the ideal the more closely the more we
actualize our presumptions. But in this case this is not feasible,
since, for the greater part of our experiences, we do not even know the
approximate or ideal natural laws by the help of which we can construct
such ideal cases. For instance, the whole province of organic life is at
present essentially like an unknown land, in which there are only a few
widely separated paths ending in _culs-de-sac_.

=16. The Freedom of the Will.= This relation explains why, on the one
hand, we assume a far-reaching determinateness for many things, that is,
for all those accessible to scientific treatment and regulation, and
why, on the other hand, we have the consciousness of acting _freely_,
that is, of being able to control future events according to the
relations they bear to our wishes. Essentially there is no objection to
be found to a fundamental determinism which explains that this feeling
of freedom is only a different way of saying _that a part of the causal
chain lies within our consciousness_, and that we feel these processes
(in themselves determined) as if we ourselves determined their course.
Nor can we prove this idea to be false, that, since the number of
factors which influence each experience is indefinitely great and their
nature indefinitely complex, each event would appear to be determined in
the eyes of an all-comprehensive intellect. But to our finite minds an
undetermined residue necessarily remains in each experience, and to that
extent the world must always remain in part practically undetermined to
human beings. Thus, both views, that the world is not completely
determined, and that it really is, though we can never recognize that it
is, lead practically to the same result: _that we can and must assume in
our practical attitude to the world that it is only partially
determined_.

But if two different lines of thought in the whole world of experience
everywhere lead to the same result, they cannot be materially, but
merely formally or superficially, different. For those things are alike
which cannot be distinguished. There is no other definition of
alikeness. Thus, if we see that the age-long dispute between these two
views always breaks out afresh without seeming to be able to reach an
end, this is readily understood, from what has been said, since the very
same essential arguments which can be adduced of _one_ view can be used
as a prop for the _other_ view, because in their essential results the
two are the same. I have discussed this matter because it presents a
very telling example of a method to be applied in all the sciences when
dealing with the solution of old and ever recurrent moot questions. Each
time we encounter such problems, we must ask ourselves: what would be
the difference empirically if the one or the other view were correct? In
other words, we first assume the one to be correct, and develop the
consequences accordingly. Then we assume the second to be correct and
develop the consequences accordingly. If in the two cases the
consequences differ in a certain definite point, we at least have the
possibility of ascertaining the false view by investigating in favor of
which case experience decides on this point. However, we may not
conclude that by this the other view has been proved to be entirely
correct. It likewise may be false, only with the peculiar quality that
in the case in question it leads to the correct conclusions. That such
a thing is possible, every one knows who has attentively observed his
own experiences. How often we act correctly in actual practice, though
we have started out on false premises! The explanation of this
possibility resides in the highly composite nature of each experience
and each assumption. It is quite possible--and, in fact, it is the
general rule--that a certain view contains true elements, but _along
with them false elements also_. In applications of the view where the
true elements are the decisive factors, true results are obtained,
despite the errors present. Likewise, false results will be achieved
where the false elements are decisive, despite the true results that can
be had, or have been had, elsewhere, by means of the true elements.
Hence, in case of the "confirmation," we can only conclude that that
portion of the view essential for the instance in question is correct.

One readily perceives that these observations find application in all
provinces of science and life. There are no absolutely correct
assertions, and even the falsest may in some respect be true. There are
only greater and lesser probabilities, and every advance made by the
human intellect tends to increase the degree of probability of
experiential relations, or natural laws.

=17. The Classification of the Sciences.= From the preceding
observations the means may be drawn for outlining a complete table of
the sciences. However, we must not regard it complete in the sense that
it gives every possible ramification and turn of each science, but that
it sets up a frame inside of which at given points each science finds
its place, so that, in the course of progressive enlargement, the frame
need not be exceeded.

The basic thought upon which this classification rests is that of graded
abstraction. We have seen (p. 19) that a concept is all the more
general, that is, is applicable to all the more experiences, the fewer
parts or elementary concepts it contains. So we shall begin the system
of the sciences with the most general concepts, that is, the elementary
concepts (or with what for the time being we shall have to consider
elementary concepts), and, in grading the concept complexes according to
their increasing diversity, set up a corresponding graded series of
sciences. One thing more is to be noted here, that this graded series,
on account of the very large number of new concepts entering, must
produce a correspondingly great number of diverse sciences. For
practical reasons groups of such grades have been combined temporarily.
Thereby a rougher classification, though one easier to obtain a survey
of, has been made. The most suitable and lasting scheme of this sort was
originated by the French philosopher, _Auguste Comte_, since whom it has
undergone a few changes.

Below is the table of the sciences, which I shall then proceed to
explain:

I. _Formal Sciences._ Main concept: order
Logic, or the science of the Manifold
Mathematics, or the science of Quantity
Geometry, or the science of Space
Phoronomy, or the science of Motion

II. _Physical Sciences._ Main concept: energy
Mechanics
Physics
Chemistry

III. _Biological Sciences._ Main concept: life
Physiology
Psychology
Sociology

As is evident, we first have to deal with the three great groups of the
formal, the physical, and the biological sciences. The formal sciences
treat of characteristics belonging to all experiences, characteristics,
consequently, that enter into every known phase of life, and so affect
science in the broadest sense. In order immediately to overcome a
widespread error, I emphasize the fact that these sciences are to be
considered just as experiential or empirical as the sciences of the
other two groups, as to which there is no doubt that they are empirical.
But because the concepts dealt with by the first group are so extremely
wide, and the experiences corresponding to them, therefore, are the most
general of all experiences, we easily forget that we are dealing with
experiences at all; and our very firmly rooted consciousness of the
unqualified similarity of these experiences causes them to seem native
qualities of the mind, or _a priori_ judgments. Nevertheless,
mathematics has been proved to be an empirical science by the fact that
in certain of its branches (the theory of numbers) laws are known which
have been found empirically and the "deductive" proof of which we have
as yet not succeeded in obtaining. The most general concept expressed
and operative in these sciences is the concept of order, of _conjugacy_
or _function_, the content and significance of which will become clear
later in a more thorough study of the special sciences.

In the second group, the physical sciences, the arbitrariness of the
classification becomes very apparent, since these sciences are among the
best known. We are perfectly justified in regarding mechanics as a part
of physics; and in our day physical chemistry, which in the last twenty
years suddenly developed into an extended and important special science,
thrust itself between physics and chemistry.

The most general concept of the physical sciences is that of _energy_,
which does not appear in the formal sciences. To be sure it is not a
fundamental concept. On the contrary, its characteristic is undoubtedly
that of compositeness, or, rather, complexity.

The third group comprehends all the relations of living beings. Their
most general concept, accordingly, is that of _life_. By physiology is
understood the entire science dealing with non-psychic life phenomena.
It therefore embraces what is called, in the present often chance
arrangement of scientific activities, botany, zoology, and physiology of
the plants, animals, and man. Psychology is the science of mental
phenomena. As such, it is not limited to man, even though for many
reasons he claims by far the preponderating part of it for himself.
Sociology is the science which deals with the peculiarities of the human
race. It may therefore be called anthropology, but in a far wider sense
than the word is now applied.

=18. The Applied Sciences.= It will be remarked that the grouping of the
table gives no place at all in its scheme to certain branches of
learning taught in the universities and equally good technical
institutions. We look in vain not only for theology and jurisprudence,
but also for astronomy, medicine, etc.

The explanation and justification of this is, that for purposes of
systematization we must distinguish between _pure_ and _applied_
sciences. By virtue of their strictly conceptual exclusiveness the pure
sciences constitute a regular hierarchy or graded series, so that all
the concepts that have been used and dealt with in the preceding
sciences are repeated in the following sciences, while certain
characteristic new concepts enter in addition. Thus logic, the science
of the manifold, exercises its dominion over all the other sciences,
while the specific concepts of physics and chemistry have nothing to do
with it, though they are of importance to all the biologic sciences.
Through this graded addition of new (naturally empiric) concepts, the
construction of the pure sciences proceeds in strict regularity, and
their problems arise exclusively from the application of new concepts to
all the earlier ones. In other words, their problems do not reach them
accidentally from without, but result from the action and reaction of
their concepts upon one another.

At the same time there are problems that each day sets before us without
regard to system. These come from our endeavor to improve life and avert
evil. In the problems of life we are confronted by the whole variety of
possible concepts, and under the day's immediate compulsion we cannot
wait, if we are sowing crops or helping a sick man, until physiology and
all the other appropriate sciences have solved all the problems of plant
growth and the changes of the human body and human energy. When other
signs fail, we use the position of the stars for finding our way on the
high seas. In this manner we turn the teaching of the stars, or
astronomy, into an applied science, in which at first mechanics alone
seemed to have a part. Later physics took a share in it, then optics
took a particularly prominent share, and in recent times not only did
chemistry find its way into astronomy, but the specifically biologic
concept of evolution was applied in astronomy with success.

Thus, side by side with the pure sciences are the applied, which are to
be distinguished from the pure sciences by the fact that they do not
unfold their problems systematically, but are assigned them by the
external circumstances of man's life. The pure sciences, therefore,
almost always have a larger or smaller share in the tasks of the applied
sciences. For instance, in building a bridge or railroad, physical
problems have to be taken into consideration as well as sociologic
problems (problems of trade), and a good physician should be a
psychologist as well as a chemist.

But since all the individual questions arising in the applied sciences
may be considered essentially as problems of one or other pure science,
they need not be explicitly enumerated along with the pure sciences,
especially since their development is greatly dependent upon temporary
conditions and is therefore incapable of simple systematization.

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.

PART III

THE PHYSICAL SCIENCES

=42. General.= In the formal sciences we began the specialization of the
object from the most general concept of thing conceivable, possessing no
other characteristic attribute than its capability of being
distinguished from other things; and we carried the specialization so
far that we could follow in its movements an object definite as to time
and space. This object, to be sure, was defined only in that it occupied
a definite space, and accordingly had a definite form. As a matter of
fact, the spacial thing of geometry and phoronomy reveals no further
attributes.

It is here that the physical sciences enter into their dominion one
after the other, and fill the empty space of the geometric thing with
definite attributes. These are the secondary qualities of Locke, of
which he assumed that they do not belong so much to the bodies
themselves as that they merely appear to us so on account of the nature
of our human sense organs. Now that our knowledge concerning the nature
of those properties as well as the structure of our sense organs is
much more thorough, we have more definite ideas also of the subjective
part of the corresponding experiences, and in a large measure are able
to separate it from the objective part.

All properties which physical bodies in contradistinction to geometric
bodies possess can be traced back to a fundamental concept, which, in
conjunction with the concepts explained in the former chapter, serves to
characterize and distinguish the physical structure. For example, the
fact that we can distinguish cubes of equal size but of different
material, different temperature, and different luminosity, can be traced
back always and entirely to the different kinds of energy acting in the
geometric space in question. The concept of energy, therefore, plays
approximately the same rôle in the physical sciences as the concept of
thing in the formal sciences, and the essentials of this new field of
science are the comprehensive knowledge and development of this concept.
Because of its great importance it has long been known and applied in
individual forms. But the systematization of the physical sciences
relative to energy is a matter of only recent date.

=43. Mechanics.= Recently many scientists have taken exception to the
traditional division of mechanics into _statics_, or the science of
equilibrium, and _dynamics_, or the science of motion, because it does
not correspond to the essence of the thing, equilibrium being only the
limit-case of motion. However, the classic presentations of this
science are based on that division, so that it must express an essential
difference. This difference we can clearly recognize through the
application of the concept of energy to mechanics. We then learn that
statics is the science of work, or the energy of position, and that
dynamics is the science of living force, or of the energy of motion.

By _work_ in the mechanical sense we mean the expenditure of force
required for the locomotion of physical bodies. While a cube of lead is
geometrically equal to a cube of glass, we experience a great difference
between them when we lift them from the floor to a table. We call the
cube of lead heavier than the glass cube, and we find it requires more
work to raise the former than the latter. For psychologic reasons this
judgment becomes especially clear when the work required to lift the
lead cube marks the limit of our physical capacity.

Work depends not only upon the difference described above, but also upon
the distance through which it is exerted. It increases in proportion as
the distance increases. In mechanics work is proportional both to the
distance and to that peculiar property which in the given example we
call _weight_. But a more general concept has been formed for that
property in the mechanical sense, called _force_, of which weight
constitutes but a special instance. Whenever there is a resistance
combined with a change of place we speak of a force, _and the product
of the force and the distance we call work_.

The cause of this kind of concept formation is the following: There are
a great number of different machines, all of them possessing the
peculiarity that work can be put into them at a definite place and taken
out at another place. Now, centuries of experience have shown that it is
impossible to obtain more work from such mechanical machines than has
been put into them. As a matter of fact, the work obtained is always
less than the work put in, and the two approach equality as the machine
approaches perfection. It is to such ideal machines, therefore, that
_the law of the conservation of work_ applies. This law states that,
though a given quantity of work may be changed in the most manifold ways
as to direction, force, etc., it is impossible to change its _quantity_.

The reason we can judge of this fact with such certainty is because for
many centuries a number of the ablest mechanicians have sought for a
solution of the problem of perpetual motion, that is, for the
construction of a machine from which more work can be gotten than is put
into it. All such attempts have failed. But the positive result secured
from these apparently futile efforts is the law of the conservation of
work. The greatness and importance of this result will become apparent
in the further course of our study.

Here for the first time we meet with a law expressing the
_quantitative_ conservation of a thing, which may none the less undergo
the most varied qualitative changes. With the knowledge of this fact we
involuntarily combine the notion that it is the "same" thing that passes
through all these transformations, and that it only changes its outward
form without being changed in its essence. Such ideas, it is true, are
widespread, but they have a very doubtful side to them, since they
correspond to no distinct concept. If we want to call the quantitative
magnitude of the product of the force and distance the "essence" of
work, and the determination of the force and the distance according to
magnitude and direction, which come under consideration for each special
value, as its "form," then, of course, there is no objection to be made
to mere nomenclature. But we must bear in mind that the difference
obtaining here lies exclusively in the fact that the amount of work
measured quantitatively remains unchanged, while its factors undergo
simultaneous and opposite changes.

This discovery, that there is a magnitude which can be quantitatively
determined, and which, as experience shows, remains unchanged, however
much its factors may change, invariably results not only in a very
simple and clear formulation of the corresponding natural law, but also
corresponds to the general tendency of the human mind to work out
conceptually "the permanent in change." If, in accordance with the
word-sense, we denote everything which persists under changing
conditions by the name of _substance, we encounter in work the first
substance_ of which we have attained knowledge in our scientific
journeys. In the history of the evolution of human thought this
substance has been preceded by others, especially by the weight and mass
of ponderable bodies (which are also subject to a law of conservation),
so that at present we are inclined to connect with the word substance a
tacit secondary sense of ponderability. But this is a remnant of the
still very widely spread mechanistic theory of the universe, which,
though it has almost finished its rôle in physics, will presumably
continue to persist for a long time to come in the popularly scientific
consciousness in accordance with the laws of collective thought.

=44. Kinetic Energy.= The law of the conservation of work is by no means
true of all cases in which work is expended or converted, but, as has
been said, only of _ideal_ machines, that is, of such cases which do not
exist in reality. But while in imperfect machines there is at least an
approximation to this law, there are besides countless normal cases in
which we cannot even speak of an approximation. When, for example, a
stone falls to the ground from a certain height, a certain quantity of
work is expended, which is equal to that by means of which the stone can
be raised again to its original height. This quantity of work apparently
disappears entirely when the stone remains lying on the ground. We
shall discuss this case later. Or the falling of the stone can be so
guided that it can lift itself again. This happens, for instance, when,
by fastening the stone to a thread, it is forced to move in a curved
path, or to perform pendular oscillations. In that case, it is true, the
stone will fall to the lowest point which the thread permits, and so
will there have lost its work without having done any other work in the
meantime. But it has entered a condition by virtue of which it raises
itself again, so that (as before, only in the ideal limit-case) it
reaches its former height, and so has lost no work. For this moment,
too, then, the law of the conservation of work obtains. But in the
meantime new relations have arisen.

What distinguishes the stone moving like a pendulum from the stone which
simply falls is, that at its lowest point it has not remained lying
still, but possesses a certain velocity. By means of this it lifts
itself again, and after it has reached its former height, it has lost
its velocity. _Therefore, there is a reciprocal relation between the
work which it loses and the velocity which it gains_, and the question
may therefore be put, How can this relation be represented
mathematically? Experience teaches that in every such case a function of
the velocity and of another property of the body, called _mass_, can be
established in such a way that this function, called the _kinetic
energy_ of the body, increases precisely as much as the amount of work
the body has expended, and _vice versa_. The sum of the kinetic energy
of the body and of the _work_ is therefore _constant_, and the clearest
mode of conceiving of this relation is by assuming _that work can be
transformed into kinetic energy and vice versa_ in such a way that given
amounts of the two magnitudes are equal or equivalent to one another.
Naturally, this is only an abbreviated way of expressing the actual
relations, for it might just as well be assumed that the work really
disappears and the kinetic energy really originates anew, and that the
disappearance of the one substance only happens regularly to coincide
with the origin of the other. But it is this regular conjunction of
phenomena that constitutes the sole ground of every _causal_ relation,
and in such a sense we are justified _in regarding the disappearing work
as the cause of the kinetic energy that arises_, and to designate this
relation summarily as a transformation.

By the inclusion of cases in which work is converted into kinetic energy
the law of the conservation of work therefore becomes _the law of the
conservation of the sum of work and kinetic energy_. We are thereby
compelled to extend the concept of substance, which at first contains
only work, to the sum of both magnitudes, and to introduce a new name
for this enlarged concept.

It will soon appear that all cases of imperfect machines, in which work
disappears without giving rise to an equivalent amount of kinetic
energy, can, with a corresponding enlargement of the concept, be
likewise included in the law of conservation. For experience has shown
that in such cases something else arises, heat, light, or electric
force, etc. This generalized concept, which embraces all natural
processes and permits the sum of all corresponding values to be
expressed by a law of conservation, we call _energy_. The law in
question, therefore, is:

_In all processes the sum of the existing energies remains unchanged._

The principle of the conservation of work in perfect machines proves to
be an ideal special instance of this general law. A perfect machine is
one in which work changes into nothing but _work_ of another kind, and
not into a different kind of energy. Then each side of the equation
which expresses the general law of energy, namely,

Energy that has disappeared = energy that has arisen,

contains only the magnitude of the work, and expresses the law of the
conservation of work. If, on the other hand, as in the case of the
pendulum, the work increasingly changes part by part into kinetic
energy, and _vice versa_, the equation during the first period is:

Work that has disappeared = kinetic energy that has arisen,

and during the second period in which the pendulum rises again,

Kinetic energy that has disappeared = work that has arisen.

Thus, while work can be called a substance only in a limited sense,
since its conservation is limited only to perfect machines, we may call
energy a substance unqualifiedly, since in every instance of which we
know the principle has been maintained _that a quantity of any energy
never disappears unless an equivalent quantity of another energy
arises_. Accordingly, this law of the conservation of energy must be
taken as a fundamental law of the physical sciences. But not only do all
the phenomena of physics, including chemistry, occur within the limits
of the law of conservation, but until the contrary is proved the law of
conservation must also be regarded as operative in all the later
sciences, that is, in all the activities of organisms, so that all the
phenomena of life must also take place within the limits of the law of
conservation. This corresponds to the general fact, which I have
emphasized a number of times, that all the laws of a former science find
application in all the following sciences, since the latter can only
contain concepts which by specialization, that is, by the addition of
further characteristics, have sprung from the concepts of the former or
more general sciences.

=45. Mass and Matter.= It has been noted above that kinetic energy
depends upon another magnitude beside velocity. A conception of its
nature can be obtained when we try to put different bodies in motion.
In doing so the muscles of the arm perform certain quantities of work,
and we feel whether the quantities are greater or smaller. In this way
we obtain a clear consciousness of the fact that different bodies
require quite different quantities of work for the same velocity. The
property which comes into play here is called _mass_, and mass is
proportional to the work which the various bodies require to attain the
same velocity. Since the work and the velocity can be measured very
accurately by appropriate means, mass also lends itself to a
correspondingly accurate measurement.

All known ponderable bodies have mass. That means there is a regular
connection between the property which makes a body tend to the earth
with a certain definite force (called weight) and the property by virtue
of which a body assumes certain velocities under the influences of
motive causes. We can readily conceive that it is possible for us to
learn only of such bodies as are heavy, that is, bodies which are _held_
by the earth, since the others, if they exist at all, would naturally
have left the earth long ago. That all these bodies also have mass is to
be explained in a similar way. For a body of mass zero would at each
impulse assume infinitely great velocity, and could therefore never be
the object of our observation. Consequently, by reason of the physical
conditions obtaining on the earth's surface, the bodies known to us must
combine both properties, mass and weight.

The name given to this concept of the combined presence of mass and
weight in space is _matter_. Experience shows that there is a law of
_conservation_ for these magnitudes also, according to which _whatever
changes we may produce in bodies possessing weight and mass, no change
will occur in the sum of their weight and mass_. According to the
nomenclature previously introduced we must therefore call weight and
mass substances, since they remain the same as to quantity, no matter
what changes they may undergo. However, it is usual to apply the name
substance to the concept of matter composed of mass and weight. In fact,
scientists often go so far as to limit the name to this single instance
of the various laws of conservation, and to take substance to mean
exclusively the combination of mass and weight. This is connected with
the conception which we are about to discuss, that all natural phenomena
can ultimately be conceived as the motion of matter. Through the greater
part of the nineteenth century this conception, called _scientific
materialism_, was accepted almost without opposition. At present it is
being more and more recognized that it was only an unproved assumption,
which the development of science daily proves to be more untenable.

=46. Energetic Mechanics.= In the light of our previous observations the
branch of science traditionally known as mechanics appears as the
science of work and of kinetic energy. Furthermore, statics is shown to
be the science of work, while dynamics, besides treating of kinetic
energy in itself, also treats of the phenomena of the change of work
into kinetic energy, and _vice versa_. We shall find the same relation
again later, only in more manifold forms. Every branch of physics proves
to be the science of a special kind of energy, and to the knowledge of
each kind of energy must be added the knowledge of the relations by
which it changes to the other forms of energy and _vice versa_. It is
true that in the traditional division of physics this system has not
been strictly carried out, since an additional and very influential
motive for classification has been the regard paid to the various human
sense organs.

Nevertheless this ground does not lie in the field of physics, but in
that of physiology, and must, therefore, be abandoned in the interest of
strict systematization.

Of the physical sciences mechanics was the first to develop in the
course of historical evolution. A number of factors contributed to this
end--the wide distribution of mechanical phenomena, their significance
to human life, and the comparative simplicity of the principles of
mechanics, which made it possible to discover them at an early date.
Most to be noted is, that of all departments of physics mechanics is the
first which lent itself to comprehensive _mathematical_ treatment. It is
true that the mathematical treatment of mechanics was possible only
after idealizing assumptions had been made--perfect machines and the
like--so that the results of this mathematical treatment not
infrequently had very little to do with reality. The mistake of losing
sight of the physical problem and of making mechanics a chapter of
mathematics has not always been avoided, and it is only in most recent
times that the consciousness has again arisen that the classical
mechanics, in arbitrarily limiting itself to extreme idealized cases,
sometimes runs the risk of losing sight of the aim of science.

=47. The Mechanistic Theories.= Because the evolution of mechanics
antedates that of the other branches of physics, mechanics has largely
served as a model for the formal organization of the other physical
sciences, just as geometry, which has been handed down to us from
antiquity in the very elaborate form of Euclid, has largely been used as
a model for scientific work in general. Such methods of analogy prove to
be extremely useful at first because they serve as a guide to indicate
when and where new sciences, in which all possibilities are open, can be
got hold of. But later on such analogies are apt to be harmful. For each
new science soon requires new methods, by reason of the peculiar
manifoldness which it has to deal with, and the finding and the
introduction of these new methods are easily delayed, and, as a matter
of fact, often have been delayed, because scientists could not free
themselves soon enough from the old analogy.

By its being based upon memory the human mind is so constructed that it
cannot assimilate something entirely new. The new must in some way be
connected with the known in order that it may be organically embodied in
the aggregate of concepts. Therefore, it is the first involuntary
impulse of our mind, in the presence of new experiences or thoughts, to
look about for such points at which a linking of the unknown to the
known seems possible. In the case of mechanics this necessity for
finding connecting links has acted in such a way that the attempt has
been made, and is still being made, to conceive and represent all
physical phenomena as mechanical.

The impulse to this was first given by the extraordinary successes which
mechanics has attained in the generalization and prediction of the
_motions of the heavenly bodies_. The names of Copernicus, Kepler, and
Newton mark the individual steps in the mechanization of astronomy. The
cause of this lies in the fact that the heavenly bodies actually
approximate very closely the ideal of the purely mechanical form with
which classical mechanics deals. These successes encourage the attempt
to apply these mental instruments that were productive of such rich
results to all other natural phenomena. An old theory, according to
which all physical things are composed of the most minute solid
particles of matter called _atoms_, supported these tendencies and
invited the attempt to regard the little world of atoms as subject to
the same laws as had been found to apply so successfully to the great
world of the stars.

Thus we see how this mechanistic hypothesis, the assumption that all
natural phenomena can be reduced to mechanical phenomena, comes as if it
were a self-understood matter, and with its claim to be a profound
interpretation of nature it scarcely permits the question as to its
justification to be raised at all. And the effects here have been the
same as I described above in cases in which inferences from analogy are
accepted too extensively or too credulously. While it is true, no doubt,
that the mechanical hypothesis at first was fruitful of results in
special research, because it facilitated the putting of the
question--for example, we need think only of the atomic hypothesis in
chemistry--later, the efforts to find further hypothetic help for the
inadequacies of the hypothesis that gradually came to light, have not
infrequently led scientific research to pseudo-problems, that is, to
questions which are questions only in hypothesis, but to which no actual
reality can be shown to correspond. Such problems, therefore, are by
their very nature _insoluble_, and constitute an inexhaustible source of
differences of scientific opinion.

The most flagrant of the injurious consequences of the mechanistic
hypothesis appear in the scientific treatment of the mental phenomena.
Ready as scientists were to represent all other life phenomena, such as
digestion, assimilation, and even generation and propagation, as the
consequence of an extremely complicated play of certain atoms, their
courage never went so far as to apply this principle to mental life and
to consider that by mechanics the last word had been said on the
subject.

It is because of this hesitancy to bring mental phenomena under the same
mechanistic principle as all the other phenomena that the philosophical
systems had to search for some other means to connect the mental world
with the mechanical, and the efforts of the philosophers to bring about
this end have been most varied. Of the various doctrines that have come
down to us, that of the _pre-established harmony_ proposed by Leibnitz
is in the ascendant in our day, and is now called the theory of the
_psycho-physical parallelism_. According to this theory it is assumed
that the mental world exists alongside, and quite independent of, the
mechanical, but that the things have been so prearranged that mental
processes take place simultaneously with certain mechanical processes
(according to some, with all mechanical processes) in such a way that,
although the two series do not influence each other in the least, they
always correspond to each other precisely. How such a relation has come
about and how it is maintained remains unsaid, or is left to future
explanation.

We need only think of the content of this hypothesis with an unbiased
mind to lose all relish for it at once. In fact, it has no other _raison
d'être_ than the presumption that the mental and the mechanical world
are opposed to each other. As soon as we abandon the thesis that the
non-mental world is exclusively mechanical, we acquire the possibility
again of finding for the theory of mental phenomena a constant and
regular connection with the theories of all other phenomena, especially
with the phenomena of life. Therefore it will be found most expedient in
every respect, instead of rendering scientific research one-sided and
almost blind to nonconforming facts by preconceived hypotheses, such as
the mechanistic hypothesis, to seek, as hitherto, from step to step, the
new elements of manifoldness which must be taken account of in the
progressive upbuilding of science and to limit ourselves faithfully to
them in the formation of general ideas.

=48. Complementary Branches of Mechanics.= The field of pure or
classical mechanics is limited to the above two kinds of energy, work
and kinetic energy, though these do not exhaust the manifoldness of the
mechanical energies. Accordingly, other branches of mechanics dealing
with the corresponding phenomena are added to the classical mechanics
described above.

If by mechanical energies we understand all energies in which _changes
of space are connected with changes of energy_, there are as many
different forms as there are spacial concepts that seem applicable.
_Form_, _Volume_, and _Surface_ of bodies in space are especially
recognizable as the field of action for energy, which shows different
properties or manifoldnesses according to each of these relations.

The _energy of form_ is manifested in bodies (solid or rigid bodies)
that maintain a definite shape because every change of shape is
connected with work or with the expenditure of some other energy. If the
changes are small, the bodies are of such a nature that they return to
their former condition of their own accord after the force exerted upon
them has ceased to act. This property is called _elasticity_. However,
the theory of elasticity, which has been extensively and rationally
developed, is regarded as belonging rather to mathematical physics in
general than to mechanics in particular. In greater changes of shape the
energy of form, or elastic energy, passes into other forms, and the body
does not return to its former shape after the force has been removed.

Other bodies have no energy of form (or only in an infinitesimally
slight degree), so that they allow of changes of form without the
expenditure of work, but their volume can be changed only by work. These
are divided into two classes. First, the _liquids_, which have a
definite volume (corresponding to the definite shape of solids), the
changes of which in _every_ sense, both compression and expansion,
require work. Secondly, the _gases_ with volume energy in only one sense
of the word, in which only the compression of volume requires work,
while in expansion a certain amount of work is thrown off. Such bodies
can exist only so long as the expenditure of their volume energy by
spontaneous expansion is prevented by the presence of a counter energy,
as, for example, the elasticity of the walls of a vessel. This tendency
is called _pressure_.

Finally, there are energy qualities at the surfaces between various
kinds of bodies which come into play at the change of these surfaces.
They always lie in such a direction that the enlargement of the surfaces
requires work, and hence, by reason of the law of conservation of
energy, cannot proceed by itself. In cases where there has been an
inverse kind of energy present, that is, one which diminishes with
increasing surface, it also has been active as a rule, thus bringing
about the disappearance of the existing boundaries.

Since the seat of this kind of energy is in the surfaces (or
superficies), it is called _surface-energy_. The phenomena depending
upon it manifest themselves most clearly at the surface boundaries
between _liquids_ and _gases_. They are called _capillary phenomena_.
This strange name, derived from the word _capilla_, hair, has its origin
in the fact that because of surface-energy liquids rise in tubes which
they wet, and the narrower the tube the higher they rise. If the lumen
of the tube is as fine as a _hair_, a considerable rise can be observed.
This is the entire connection between the name and the thing.

The mechanics of liquids is called _hydromechanics_, that of gases,
_aeromechanics_, after the most familiar liquid, water, and the most
familiar gas, air. The study of surface-energy under the name of the
capillary theory forms part of theoretical physics. While formerly this
branch, too, was regarded as a working part, or, rather, as a playing
part, of mathematical problems, in more recent times extensive
experimental research has made its entry in this province also, and has
demonstrated the necessity of passing from the former abstractions or
idealizations, which were carried altogether too far, to a better and
profounder regard for the actually existing complexities.

=49. The Theory of Heat.= The various forms of energies the aggregate of
which is comprehended in physics, have very different special
characters. A systematic investigation has not yet been made of the
characters of manifoldness by which, for example, work is distinguished
from heat, electrical energy from kinetic energy, etc., nor of what are
the essential properties peculiar to each individual energy. We feel
certain that differences do exist, for otherwise the energies could not
be distinguished, and we feel certain that these differences are very
important, for doubt seldom arises as to the kind of energy to which a
certain phenomenon is to be assigned. But just as we have no systematic
table of the elementary concepts, so we are still without a systematic
natural history of the forms of energy in which the peculiarities of
every species are characterized, and in which the entire material is so
arranged according to these characteristics that we can take a general
survey of it.

As regards heat energy, its foremost and most striking characteristic is
its physiological effect. In our skin there are organs for the
perception of heat as well as of cold, that is, for temperatures above
and below the temperature of the skin. However, the temperature that
these organs can bear without injury to themselves is of a very small
range, beyond which physical apparatuses of all kinds must be used, such
as "thermometers."

Heat is the simplest kind of energy from the point of view of
manifoldness. Every heat quantity is marked by a temperature, just as a
kinetic energy is marked by velocity. But while a velocity is determined
in space so that velocities of equal magnitude have in addition a
threefold infinite manifoldness in reference to direction, a temperature
is characterized completely and unambiguously by a simple number, the
degree of temperature. Two temperatures of equal degree can in no wise
be distinguished, since temperature possesses no other possible
manifoldness than degree.

The same property is found in heat energy itself. In heat energy we
measure the quantity of energy itself and call it the _heat quantity_,
while in some of the other kinds of energy, only the factors into which
they can be divided are measured, and no habitual conception of the
energy itself is developed. A heat quantity is likewise fully indicated
by its measure number.

That heat is an energy, that is, that it is developed in equal
quantities from other kinds of energy, and can change back again into
them, is a discovery which, despite its fundamental and general
character, was not made before the forties of the nineteenth century. As
often happens in cases of important scientific advances, the same idea
came simultaneously to a number of investigators. The first to grasp and
fully comprehend this idea was _Julius Robert Mayer_ of Heilbronn, who
published his results in 1842. Mayer not only showed that the imperfect
machines (p. 134), which limit the validity of the law of the
conservation of work, owe this peculiarity to the fact that they
transform a part of the work into _heat_, and that when we take account
of this part, the law of conservation holds perfectly good, but he also
calculated, with extraordinary acumen, the mechanical equivalent of heat
from the then existing data of physics. That is to say, he determined
how many units of heat (in the measure then in use) correspond to a unit
of work (in its specific measure) in the change from one to the other,
and back. And this fundamental knowledge of the existence of a
quantitatively unchangeable substance, arising from work, and capable of
being transformed into it, Mayer did not limit in its application merely
to heat. He was the first to construct a table, which he made as
complete as possible, of all the forms of energy then known, and to
assert and prove the possibility of their reciprocal change into each
other.

In view of this relation of the quantitative equivalent of the various
forms of energy when transformed into one another, an attempt is being
made at present to measure them all with the _same unit_. That is, some
easily obtained quantity of energy is arbitrarily chosen as a unit and
it is determined that in every other form of energy the unit shall be
equal to the quantity obtained from that unit on its transformation into
the energy in question. For formal reasons the kinetic energy of a mass
of two grams which moves with the velocity of one centimeter in a second
has been chosen as the unit. It is called _erg_, an abbreviation of
energy. The amount is very small, and for technical reasons 10^{10}
times greater unit is used. To raise the temperature of a gram of water
one degree a quantity of energy equal to 41,830,000 ergs is required.

=50. The Second Fundamental Principle.= Another fundamental discovery
has been made in connection with the heat form of energy, which, like
the law of conservation, relates to all forms of energy, but has found
its first and most important application in heat. While the law of
conservation answers the question, how much of the new form of energy is
developed if a given quantity of energy changes, but gives no clue as to
when such a change occurs, this second law asserts the condition under
which such changes arise, and is therefore called the _second
fundamental principle_.

The discovery of this law antedates _Mayer's_ discovery of the law of
conservation by about twenty years, and was made by a French military
engineer, _Sadi Carnot_, who died soon afterward without having lived to
see the recognition his great work obtained. _Carnot_ asked himself the
question, Upon what does the action of the steam engine, which had just
then come into use, depend? This led him first to the more general
question of the action of heat engines in general. He found that no heat
engine could work unless the heat dropped from a higher to a lower
temperature, just as no water wheel can work unless the water flows from
a higher to a lower level, and he determined the conditions which an
_ideal heat engine_ must fulfil, that is, a machine in which the
greatest possible value in work is obtained from heat. However, an ideal
machine of this nature can be constructed in very different ways, and
Carnot's discovery consists in the recognition of the fact _that the
quantity of work obtained from the heat unit does not at all depend upon
the peculiar construction of the ideal machine, but is determined solely
by the temperature between which the heat transition takes place_. This
follows from the following considerations:

In the first place an ideal engine must be _reversible_, that is, it
must be capable of working both ways, changing heat into work and work
back into heat. Now, if we have two ideal engines between the same
temperatures, and if we assume that engine A produces more work from the
same quantity of heat than engine B, then let A move one way and let B
move the other way with the work obtained from A. Since B produces less
work from a given amount of heat, hence more heat from an equal amount
of work, there will in the end be more heat at the higher temperature
than was originally there. But experience teaches _that there is no
means in nature by which heat in the absence of concomitant change could
be caused to rise to a higher temperature_. Therefore an engine so
constructed as to produce this result is impossible, And B cannot be of
such a nature as to produce less work from the same quantity of heat
than A.

The reverse is also impossible. For then we need merely couple the
engines in the reverse way in order to obtain the same effect.
Therefore, since B can do neither less nor more work than A, the two
must do the same amount of work--which was to be proved.

It is obvious that this process of proof is similar to that by which the
law of conservation was established. Because the arbitrary creation of
energy from nothing is impossible there must be definite and immutable
relations of change between the forms of energy. Because energy at rest
does not spontaneously pass into conditions in which it can do work,
the efficiencies of the machines must have definite and unchangeable
values. If, for example, we could cause heat of its own accord to rise
to a higher temperature, we could also construct a perpetual motion
machine which would always yield work at no expense. But this perpetual
motion would not be one that creates work out of nothing, but one that
extracts it from energy at rest. A perpetual motion machine of this
nature, too, is, according to our experience, impossible, and this
impossibility forms the content of the second fundamental principle.

On the face of it this apparently "self-evident" proposition does not
reveal how fruitful of results it is when applied to the discovery of
simple but not obvious relations. It can only be said here that the
deductions from this principle form the chief content of the extensive
science of thermodynamics, which deals with the changes of heat into
other forms of energy. We must only emphasize the fact that the
application of this law, as was already observed in stating it, is not
confined to the changes of heat alone. It is a law rather which finds
application in _all_ the forms of energy. For in every form of energy
there is a property which corresponds to temperature in heat, and upon
the equality or the inequality of which depends whether the energy in
question is at rest or ready for transformations. This property is
called the _intensity_ of the energy. In work, for instance, it is
_force_, in volume-energy it is _pressure_. If once the intensity in a
body is equal, its energy is at rest, and it never again moves of its
own accord.

Another form in which to present these relations is to make a
distinction between _free_ energy and energy _at rest_. If we have a
heat quantity the temperature of which is higher than that of the
surrounding objects, it can be used to do work only until its
temperature has dropped to that of the surrounding objects. Although
energy in abundance is still present, there is no longer any energy
_capable of change_, or _free_ energy. Since differences of temperature,
like other differences of intensity, have a constant tendency to
diminish, the amount of free energy on earth is constantly decreasing,
and yet it is only this free energy that has value. For since all
phenomena depend upon change of energy, and change of energy is possible
only through free energy, _free energy is the condition of all
phenomena_.

=51. Electricity and Magnetism.= While the knowledge of heat energy goes
back to the most ancient periods of civilization, electrical and
magnetic energies are relatively young acquisitions. The highly
developed technical application of both with the rich harvests they have
yielded belongs exclusively to most recent times.

Both these forms of energy, like those discussed above, are connected in
the main with ponderable "matter," but in a much slighter and less
regular measure. While it is not possible as yet to render any given
body free of heat (although lately the absolute zero point has been
considerably approximated), freedom from electrical and magnetic energy
is the normal condition of most bodies. This is connected with the
peculiarity that electrical and magnetic properties are decidedly
bi-symmetrical or _polar_. This property is not found in any other form
of energy, and can serve as the special scientific characteristic of
electricity and magnetism. This peculiarity shows itself in the concepts
of positive and negative magnetism, and positive and negative
electricity, and is due to the fact that two equal opposite quantities
of electricity or magnetism, when added together, do not produce double
their value, but nullify each other.[G]

[G] For the sake of the layman it must be observed that those
"quantities" are not energy magnitudes but factors of the electrical and
magnetic energies. Energy itself in its various forms is an _exclusively
positive magnitude_, and the result of the additions of their various
amounts is always the sum, never the difference, of their numerical
values. By the negative sign is understood the energy _expended_ in
contradistinction to the energy _received_. It is therefore nothing more
than the indication of a mathematical operation.

The fact that electrical and magnetic energies generally exist only in a
transitory state (with the notable exception of the magnetic condition
of the earth) is probably the cause of our not having developed a sense
organ for them, especially since their phenomena as they occur in nature
have only occasionally and in very rare instances (thunderstorms) an
influence upon us. On the other hand, the modern development of
electrotechnics is based upon that property of electrical energy by
virtue of which large quantities of it can be conducted along a thin
wire over great distances without any considerable loss, and at the
point desired can be easily changed into any other forms of energy. But
since the collection and conservation of large quantities of electrical
energy is hardly possible technically, the electrical apparatus must be
so constructed that the quantities each time required should be produced
at the moment they are used. The chief source of electricity is the
chemical energy of coal, which is first transformed into heat, then into
mechanical energy, and finally into electrical energy. This extremely
roundabout process is necessary because a method technically practicable
of transforming the chemical energy of coal directly into electrical
energy has not yet been invented. On the other hand, mechanical energy
can be easily and completely changed into electrical energy. Upon this
is based the exploitation of much "water power," the energy of which
could not be utilized but for the great capacity for change of the
electrical form.

=52. Light.= The case of light in our day seems to be similar to that of
sound, which, although it has its special sense organ in man, is yet no
particular form of energy, but has been found to be a combination of
mechanical energies in an oscillatory or mutually changing state. It
seems highly probable that light, too, is not a special form of energy,
but a peculiar oscillatory combination of electrical and magnetic
energies. It is true that the circle of proof is not yet quite closed,
but the gaps have become so small that the above conclusion may at any
rate be accepted as highly probable.

However that may be, light is an energy which, according to the known
laws, travels through space with tremendous rapidity. We will call it
_radiant energy_, since the part optically visible, to which alone the
name light in its original sense belongs, represents an extremely small
portion of a vast field, the properties of which change quite
continuously from one end to the other.

Radiant energy is characterized as an oscillatory or wave-like process.
So long as this fact was unknown (up to the beginning of the nineteenth
century) it was thought that light consisted of minute spherical
particles, which shot through space in a straight line with the
tremendous velocity mentioned above. Later, in order to "explain" its
wave nature, which in the meantime has come to be recognized, it was
assumed to be due to the elastic vibrations of an all-pervading thing
called _ether_, of which we know nothing else. This elastic undulatory
theory has been abandoned in our time in favor of an _electromagnetic_
theory supported by quite considerable experiential grounds. Whether it
will be spared the fate that has overtaken the older theories (or rather
hypotheses) of light cannot as yet be predicted with any degree of
certainty.

Radiant energy is of very marked importance in human relations. As light
it serves, with the aid of the corresponding receiving organs, the eyes,
as a more manifold means of intercommunication between our bodies and
the outer world than any other form of energy. The energy quantities
penetrating to us from the extreme limits of the world space mark the
outermost limits of which we have knowledge in any way whatsoever, and
finally the energy quantities radiating to us from the sun constitute
the supply of free energy at the expense of which all organic life on
earth is maintained. Even the chemical energy stored up in coal
represents nothing else than accumulations of former sun radiation,
which had been transformed by the plants into the permanent form of
chemical energy.

Very recently other newly discovered forms of radiant energy have been
added to light. They are produced in manifold circumstances, and some
bodies emit them constantly. The scientific elaboration of these
extremely manifold and unusual phenomena has not yet been carried so far
that they can be reduced to a doubt-free system. But so much, it seems,
is already apparent, that they are presumably not purely new forms of
energy, but rather very composite phenomena which may yield one or more
new energies as component parts. But despite the peculiarity of these
new rays, nothing certain has as yet been proved against the law of
conservation itself.

=53. Chemical Energy.= Since chemical energy is only one of several
forms of energy, there seems to be no justification for allotting it to
a special science, since all the other forms of energy must be
incorporated in physics.

But the actual existence of chemistry as a special science which has
already many subdivisions is justified in the first place by the
external fact that in practical life and in industry chemistry occupies
a very wide field comparable, if not superior, to that of the whole of
physics. In the next place, from the psychological point of view, it is
found that the chemist's methods of reasoning and working are so
different from those of the physicist that a division seems to be in
order for that reason also. Finally, there is in the nature of chemical
energy itself an important distinction which marks it off from the other
forms.

While, for example, there is only one form of heat or of kinetic energy,
and in electricity there are only the two forms of polar opposites,
chemistry, even after the greatest theoretical reduction, possesses at
least about eighty forms. That is, it possesses as many forms as there
are _chemical elements_. The experiential law, that the elements cannot
be changed into one another,[H] also limits the corresponding changes
of the chemical energies into one another, and thus characterizes the
independence of these various forms. From this results a
disproportionately greater manifoldness of relations, which find their
expression in the many thousands of the individualized chemical
substances or combinations.

[H] Lately changes of elements into one another have been observed in
individual instances, but in such peculiar circumstances that for the
present we need not consider these discoveries, which have only just
begun.

This great manifoldness and the slight regularity hitherto found in
connection with the properties and reciprocal relations of the numerous
chemical elements renders modern chemistry more a descriptive than a
rational science. It was no more than twenty years ago that an earnest
and successful attempt was begun to apply the stricter methods of
physics to the investigation of chemical phenomena. These labors, so far
as they have gone, have yielded a great many far-reaching and
comprehensive principles.

The significance of chemistry in human life is twofold. In the first
place the energy of the human body, just as that of all other living
organisms, depends chiefly upon the action of chemical energies in the
most manifold forms. Of all the physical sciences, therefore, chemistry
is the most important for biology, particularly for physiology. In the
second place, as I have emphasized a number of times, it possesses the
peculiar property which enables it to be _preserved_ for a long time
without passing into other forms and being dissipated. Furthermore,
energy in this form permits of the most powerful _concentration_. More
of chemical energy can be stored in a given space than of any other form
of energy. Both these properties, then, may be considered as the reason
why organic beings are constituted chiefly by means of chemical energy.
At any rate, it is due to these two peculiarities that chemical energy
serves as the primary source for almost all the energy used in industry.

Further, the manifoldness of chemical energy is the cause of the
peculiar manner in which it is transformed into other forms. In the
other forms of energy the transformation can be effected by the body
itself. Nothing else is required. If a stone is thrown and it hits
against a wall, it loses its kinetic energy, the greater part of which
changes into heat. But in order to liberate the _chemical_ energy of,
say, coal, the coal _alone_ is not sufficient; _another_ chemical
substance is required, the oxygen of the air. The interaction of the two
substances produces a new substance, and it is only during this process
that a corresponding part of the chemical energy is liberated. There are
a few chemical processes also (allotropic and isomeric changes) in which
a single substance without the co-agency of another substance can give
off energy. But the quantity of energy thus obtained is infinitely
small as compared to that liberated by the interaction of two or more
substances. Because of the necessity of two or more substances to
co-operate in giving off chemical energy, the opportunity for the
transformation of chemical energy is less than for the transformation of
the other forms of energy, and this is the main reason why it can be
conserved so long and so easily. All that is necessary is to prevent
contact with another substance. This is a problem, it is true, which
from the point of view of strict theoretical rigor it is almost
impossible to solve. In practice, however, it can be easily solved for
periods of time long enough at least to require special means to enable
us to recognize that it is only a temporary and not a fundamental
solution. Scientifically expressed, the cause of this is that the
_diffusion_ of the various substances in one another can theoretically
never be completely eliminated, while on the other hand the velocity of
the diffusion over distances measured only by decimeters is extremely
low.

PART IV

THE BIOLOGIC SCIENCES

=54. Life.= Among the bodies in our environment that are ponderable and
have mass the animate beings are so strikingly distinguished from the
inanimate that in most cases we have not the slightest doubt whether a
body belongs to the one kind or to the other, even if in some cases we
happen not to be familiar with its peculiar form. In the first place,
therefore, we must answer the question in a general way and tell what
the distinguishing peculiarities are that mark them off one from the
other.

The first peculiarity is this, that living organisms are not _stable_
but _stationary_ forms. This distinction is based upon the fact that a
stable form is at rest or unchangeable in all its parts, while a
stationary body, though it seems unchangeable in its form, internally
undergoes a constant change of its parts. Thus, a brass faucet is a
stable body, since it not only preserves its form and function
permanently, but consists at all times of the same material and shows
the same peculiarities, such as stains, defects in form, etc. It cannot
be said, it is true, that it will remain completely unchanged for all
time. Its metal suffers a gradual chemical and mechanical deterioration.
But this is not essential to the existence of the faucet, since the
deterioration varies greatly with circumstances, and if conditions are
ideal it can be reduced to zero.

On the other hand, the jet of water flowing from the faucet is a
stationary body. In favorable circumstances it can assume a constant
form, so that at a hasty glance it might be taken for a stable glass
rod. On closer examination it will be found that the parts of water of
which it is formed are not the same at any given instant as the instant
before, each part that has flowed away being replaced by another just as
large following it.

From this difference in the nature of the two bodies results a
difference in their behavior. If I make a mark on the faucet with a
file, the mark remains permanent. But even if I sever the entire water
jet with a knife, the cut is healed the next moment, because by reason
of the continuous flow of the water, the severed place is instantly
eliminated from the body. Owing to this nature peculiar to stationary
bodies, they have the capacity of _being healed_ or of _regeneration_.

For a body to continue permanently in a stationary condition the
material of which it is composed must be permanently _supplied_. If we
turn off the faucet, the water jet immediately disappears or "dies."
Evidently, therefore, a stationary body can subsist by its own means
only if it has the property or capacity to provide itself continually
with the necessary material. This material consists in the main of
ponderable or chemical substances of definite physical and chemical
properties, and thus the _change of substance_, _metabolism_, appears as
a necessary property of the stationary body. In order, however, that
metabolism should take place we must have free _energy_, or energy
having the capacity to work, since it is only free energy that can cause
substances to change, just as every phenomenon in the world implies the
equalization of free energy. For a stationary body to exist
independently, therefore, it must have the property of being able
spontaneously to possess itself of the necessary substances and of free
energy. But since, as we have already said, the energy of organisms is
stored up and used in the main in the form of chemical energy, the two
tasks which a stationary body has to perform, that of meeting the need
for substances and for energy, are as a rule externally combined. In
organisms these two necessities combined are called _nutrition_, and
thus we recognize in the capacity for _self-acquisition of nutrition_
another essential property of organisms.

A third essential property of organisms is the capacity for
_reproduction_, for the bringing forth of similar beings. It is never
impossible that the balance between the receipts and expenditures of a
stationary body should, in consequence of some external causes, be
disturbed, even when under normal conditions it possesses the property
of self-nutrition. If the disturbance remains below a certain point,
then, as we have already stated, regeneration sets in. But the
disturbance may rise above that point, in which case the body ceases to
exist, or dies. Then a similar body will not arise unless the manifold
necessities that have led to the origin of the first will combine again
to produce the second. That such a thing is possible, that, in fact, it
often happens, is shown, for example, by the waves of the ocean, which
have a stationary character since, while they are composed of constantly
changing masses of water, their form remains unchanged. The waves are
destroyed in the breakers, but arise again and again through the action
of the wind upon the surface of the water. But the more complex such
bodies are the less easily they are formed, while once they have been
formed and have found the conditions of their existence, their
preservation is much easier.

Beings, therefore, which have the capacity to form similar bodies out of
themselves regularly and at the right time can preserve their species
much more easily than those in which this property is absent. Death has
to a great extent lost its power over beings capable of reproduction. By
way of illustration let us take another stationary thing, a flame. A
flame is not an organism because it is not self-sustaining. Yet it
multiplies itself. And while a single little flame soon dies out, the
sea of flame of a burning forest, which started from a single small
flame, is well-nigh inextinguishable, and it cannot be fought in any
other way than by letting it die its natural death and burn to the end.

Thus, while the fulfilment of the first two conditions, the stationary
change and the self-supply of food, could produce bodies, which would be
able to exist for a longer or shorter period, but which at some time
would have to give way to other bodies of different form and nature, the
capacity for reproduction creates the condition that forms of the _same
species_ continue to exist even after the existence of the individual
has ceased.

These three properties constitute the essential characteristics of
animate things or organisms.

That the organisms are all constructed upon the basis of chemical energy
is a fact of experience which may be understood to imply that the other
forms of energy are not capable of producing the above-mentioned
conditions. This is due to the properties of chemical energy to which I
have already called attention: its great concentration and, at the same
time, its capacity for prolonged preservation. That chemical energy is
the only form of energy suitable to life is obvious from the fact that
in airship navigation, for example, the kinetic energy required for
steering can be supplied only in the form of gasoline or hydrogen, that
is, in the form of chemical energy, because any of the other forms would
be much too heavy. The flight of a bee or the swimming of a dolphin
cannot be conceived of except as brought about through chemical energy.

That this chemical energy is essentially that of _carbon_ has also been
established by experience, although it is not quite universal, for the
sulphur bacteria found their household upon the energy of sulphur. The
cause of the preference of carbon is again to be sought in its special
fitness for the purpose, due, on the one hand, to its wide distribution,
and, on the other hand, to the exceeding manifoldness of its
combinations.

Finally, the construction of the organisms from a peculiar combination
of solid and liquid substances can be proved to be equally due to
technical relations.

These three last-named peculiarities are therefore to be regarded as the
special characteristics of the organisms with which we are acquainted on
the surface of the earth in the conditions there prevailing. We need not
regard them conceptually as unchangeable or irreplaceable. But the first
three characteristics, namely, the stationary nature, self-supply of
nutrition, and reproduction, we may regard as the _essential
characteristics of organisms_. They constitute the frame within which
everything must be found which we should recognize as living in the
widest sense.

=55. The Storehouse of Free Energy.= If we ask whence the organisms
obtain the free energy which they require for the maintenance of their
stationary existence, the answer is that _solar radiation_ alone
furnishes this supply. Without this permanent supply the free energies
upon the earth, so far as our knowledge goes, would long ago have
reached a state of equilibrium, and the earth's bodies would be stable,
that is, dead and not stationary and living.

It is comprehensible, therefore, that machines should have evolved in
the organism for _transforming the radiant energy of the sun into a
permanent form_, and, as we have already learned, chemical energy is
permanent, while radiant energy is an extremely transitory form of
energy, that is, it changes very readily. The very fact that, owing to
the change from day to night, the supply of radiant energy periodically
ceases, makes the storing-up of energy for the night necessary to the
existence of a form dependent upon it. Thus, we recognize in the
_photochemical_ processes, that is, in the transformation of radiant
energy into chemical energy, the foundation of life on earth.

This work is done by the plants, which thus provide a store of free
energy not only for their own needs but also for all the other organisms
which possess themselves directly or indirectly of the plant-chemical
supplies in order to utilize them for their individual purposes. In this
manner nourishment in the widest sense is secured for all organisms,
being based upon the regular supply of free energy derived from the
sun. This also explains the great chemical similarity of all organisms,
which could not subsist if they were not so constructed as to be able to
utilize the chemical energy in the form in which it is provided by the
plants.

Of the great stream of free energy poured out from the sun into cosmic
space the earth receives an extremely small portion (corresponding to
the bit of space it occupies in the heavenly sphere as seen from the
sun), and the plants collect and store up only a very small fraction of
this portion received by the earth. Measurements have shown that in most
favorable circumstances a plant leaf changes only about 1/50 of the
radiant energy it receives into chemical energy. If we consider that
only a small part of the surface of the earth is covered with plants and
that during the winter no energy from the sun is stored up at all, we
perceive what infinite possibilities for development there still are in
arresting and storing up free energy. The part stored up by the plants
flows from these into the countless streams, brooks, and veins of the
other organisms, to end finally as used-up energy, or energy at rest.
This energy is at rest, it is true, only in relation to the earth's
surface. We do not know whether the radiation from the earth, which at
present amounts to about as much as the radiation from the sun to the
earth, is in its turn somewhere utilized.

While the free energy is poured out in such a stream in one direction,
the ponderable substances of which the organisms are made up _circulate_
through plants and animals and back again. This is especially true of
_carbon_, which is freed from its combination with oxygen, that is, from
carbonic acid, by the sun energy transformed in the plants. While carbon
serves to build up the plant body and represents its supply of chemical
energy, the oxygen is returned to the air. These two substances are
again chemically combined in the various organisms and the quantities of
energy which were necessary for their decomposition are again available
for the manifold functions of life. The product of the chemical
combination, carbonic acid, returns to the air and is ready for renewed
decomposition in the plants.

Thus, the entire mechanism of life can be compared to a water-wheel. The
free energy corresponds to the water, which must flow in one direction
through the wheel in order to provide it with the necessary amount of
work. The chemical elements of the organisms correspond to the wheel,
which constantly turns in a circle as it transfers the energy of the
falling water to the individual parts of the machine.

=56. The Soul.= Our observations so far have shown the organisms to be
extremely specialized individual instances of physico-chemical machines.
Now we have to take into consideration a property which seems markedly
to distinguish them from the lifeless machines, and which we have
already encountered in the very beginning of our treatise.

It is the property which we there called _memory_, and which we will
define in a very general way as the quality by virtue of which the
repetition in organisms of a process which has taken place a number of
times is preferred to new processes, because it originates more easily
and proceeds more smoothly. It is readily apparent that by this property
the organisms are enabled to travel on the sea of physical possibilities
as if equipped with a keel, by which the voyage is made stable and the
keeping of the course is assured.

If we ask whether this is exclusively a quality of organisms the
question cannot be answered affirmatively. Inanimate bodies also have
something like the quality of adaptation. An accurate clock attains its
valuable qualities only after it has been going for some time, and the
best violin is "raw" until it has been "broken in." An accumulator must
be "formed" before it can do its normal amount of work. All these
processes are due to the fact that the repetition of the same process
improves the action, that is, it facilitates or increases it.

Adaptation or memory, then, is not limited to organisms. In inanimate
things, however, this property is comparatively rare. Memory, therefore,
is to be regarded as another property of organisms representing an
extreme specialization of the inorganic possibilities. This is an
important point of view for what follows.

In the first place, this property of adaptation facilitates and assures
nourishment. If we take the fundamental idea developed by Darwin, that
that predominates in the world which by virtue of its properties endures
the longest time, then it is evident that a body which teleologically
preserves and elaborates its nourishment will live longer than a similar
body without this property. Moreover, by the general process of
adaptation, these "teleological" properties come to be more greatly
developed and more readily exercised in the body that lives longer, so
that its long life gives it another advantage over its rival. Thus we
can understand how this property of adaptation, which at first is to be
conceived of as a purely physico-chemical quality is found developed in
all organisms.

In its most primitive forms the quality of adaptation gives rise to the
_phenomena of reaction_, or to _reflex_ actions, that is, to a series of
processes in the organism in response to the stimulus of an outward
energy. This response is made in furtherance of the life of the
organism. Reactions that serve a certain end, that is, teleological
reactions, can naturally be developed to such stimuli alone to which the
organism is frequently and regularly subjected. This is why adaptation
to unusual phenomena is generally lacking, and in relation to them the
organisms are often extremely unfit. The typical example of this is the
moth, which flies into the light and is burned.

As the reactions become more fixed they develop into longer and more
complicated series, which then appear to us as _instinctive actions_.
But here, too, we find the characteristic unsuitability when unwonted
circumstances arise, even if the teleologic reactions to stimuli become
more manifold.

Finally, there are the _conscious acts_ which appear to us to be the
highest degree of the series. It is with the teleologic regulation of
these conscious acts, including the very highest activities of mankind,
that this book deals. They are distinguished from instinctive action by
the fact that they no longer proceed in a single and definite series,
but are combined at need in the most manifold ways. But the fundamental
fact, namely, that actions are based upon the repetition of coinciding
experiences, at once appears here also, since the basis of the entire
conscious life of the soul, the formation of _concepts_, is made
possible only through _repetition_. Thus, we are justified in regarding
the various degrees of mental activity from the simplest reflex
manifestation to the highest mental act as a connected series of
increasingly manifold and purposive actions proceeding from the same
physico-chemical and physiological foundation.

=57. Feeling, Thinking, Acting.= For good reasons it is generally
assumed that the organisms have not always been what they are now, but
have "developed" from previous simpler forms. It is undecided whether
originally there were one or several forms from which the present forms
sprang, nor is it known how life first made its appearance on earth. So
long as the various assumptions with regard to this question have not
led to decisive, actually demonstrable differences in the results, a
discussion of it is fruitless, and therefore unscientific. The usual
word evolution is non-purposive in so far as it signifies the appearance
of something already existing. Another conception is better according to
which the influence of _changed_ conditions of existence has yielded the
most important factor of change.

The change that the organisms undergo is always in a definite direction.
More and more complex and manifold forms are evolved, and the evolution
of these forms is characterized by an ever greater specialization of the
functions of life, so that every specially developed organ comes to
perform but one function. It is true that by this means the organism is
better fitted to perform those functions, but at the same time it grows
more susceptible to injury, since its existence depends upon the proper
simultaneous activity of many different organs. Such an evolution,
therefore, can occur only when the general conditions of life have grown
steadier, so that the danger of disturbance becomes less. We are
accustomed to regard changes in this direction as higher developments,
and the progressive simplifications of the organization (as for example
in parasites) as backward steps.

Since our opinion as to what constitutes a higher and a lower organism
is doubtless arbitrary, let us ask whether it is not possible to find an
_objective_ standard by which to measure the relative perfection of the
different organisms. The question must be answered in the affirmative
when we take into consideration the following. Since the quantity of
available free energy upon the earth is limited, the organism which
transforms the energy at its disposal more completely and with the least
loss into the forms of energy necessary for the function of life, must
be regarded as the more perfect organism. In fact, we observe that with
increasing complexity of the organisms there is for the most part also
an increasing improvement in that direction, and we can therefore speak
of some beings as more perfect than others. This view-point is
especially significant in the evaluation of _human_ progress, appearing,
as it does, as the general standard of all civilization.

The perfection of the organism shows itself in relation to the outer
world in the development of the _sense organs_. While a single-celled
animal reacts almost exclusively to chemical, sometimes also to optical,
stimuli, and receives these with the entire surface of its body, special
parts of the body develop more and more toward perfection. These are the
parts that respond with special ease to the appropriate stimuli, that
is, react to them with an increasingly smaller expenditure of energy.
Then the points at which the stimuli are received are separated from
those in which the reaction occurs, and the two are connected by
_conducting paths_, the nerves, in which an energy process takes place.
Our present knowledge of this process still leaves much to be desired.
It is a process which moves with fairly great but by no means
extraordinary rapidity (about ten to thirty meters per second) along the
conducting paths. At the one end of this path it is caused by actions of
various kinds, chiefly that of the specific energy, for which the sense
organ is developed. At the other end it discharges specific effects.
There is no doubt that here we have in each instance a case of energy
transformation connected with a _discharge_, that is, with the action of
other energies which lie at the ends ready for change. Hence there is no
equivalence between the different kinds of energy, the discharging and
the discharged, mostly not even a proportional relation, although both
increase and decrease simultaneously.

What the form of the energy is that is propagated in the nerves is
unknown. It can be either a special form which arises only under the
conditions here present (just as, for example, a galvanic stream
develops only under definite chemical and spacial conditions), or a
special combination of known energies, as in sound and probably in
light. Some day, it is likely, we shall have a more accurate knowledge
of the nerve process which will solve the question.

When such a process is caused by some energy impulse from without, it
may produce various results. In the simplest case it discharges the
corresponding reaction, just as the leaves of the sensitive plant close
when they are touched. Or it may give rise to a series of processes
following one another like the instinctive actions. Or, finally, it may
bring about a series of inner processes which lead to an extreme
differentiation of slight differences of this stimulus and to a
corresponding graded reaction with the anticipation of success. We call
this conscious thinking, willing, and acting.

Through the age-long effect of the blunder committed by Plato in making
a fundamental distinction between mental life and physical life, we
experience the utmost difficulty in habituating ourselves to the thought
of the regular connection between the simplest physiological and the
highest intellectual acts. Moreover, this contrast has been accentuated
by the mechanical hypothesis. If we abandon the mechanical hypothesis
and adhere to the summarization of experience free from all hypotheses,
as represented in the science of energy, this contrast disappears. For
even if we concede the impossibility of conceiving thought as
_mechanical_, there is no difficulty in conceiving of it as _energetic_,
especially since we know that mental work is connected with expenditure
of energy and exhaustion just as physical work is. However, the
elucidation of this subject lies almost entirely in the future since the
idea just developed has but only begun to influence scientific work in
this field. But judging from the results that have already been obtained
we may hope for a speedy development.

=58. Society.= The external circumstance that as an organism multiplies
the new being must come to life in the proximity of the older one, is in
itself cause for the formation of closed groups confined to certain
localities by animal organisms of the same species. But they become
scattered if the advantage of their living together is not such as to
outweigh the disadvantage of having a narrow field of competition for
the means of sustenance. Thus we see different plants and animals
behaving differently in this respect. While some species live in as
great isolation as possible, others form communities, even if there is
no mechanical tie to hold them together by a common integument.

Since the second case is true of man in a highly marked degree, his
_social_ characteristics and needs form a large and important part of
his life. And since, further, the socialization of man makes continuous
headway with increasing civilization--we need but think of the
development of the former little groups and tribes into states and the
present very active internationalization of the most important affairs
of mankind, especially of the sciences--the social problems also
occupy an ever larger place in the organization of human life.

What distinguishes man most essentially from the other animals, even the
most advanced, is his capacity for perfection, which in the lower animal
can be paralleled at best by its capacity for _self-preservation_. While
the organization of the animals within the short period of which we have
any historical knowledge has to all appearances remained essentially
unchanged, the world of mankind has changed in quite a remarkable way.
This change consists in an increasing subjection of the external world
to human purposes, and rests upon the increasing socialization of his
capacities.

Memory and heredity (the latter being but an extension of memory to the
offspring, which is to be conceived of as a part of the older organism)
secures in the first place only the preservation of the stock and the
renewed development of the new individual in the average type. If a
specially favored individual succeeds in accomplishing greater
achievements, he may in favorable circumstances transmit this capacity
for higher attainments to his offspring. But such individuals gain an
advantage in the struggle for existence only if the other sides of their
activity do not suffer curtailment as a result. With the limited amount
of energy at the individual's disposal every extraordinary
accomplishment involves a corresponding _one-sidedness_, and as soon as
a certain measure is slightly overstepped, it will cause a reduction of
the other functions which will render the individual less fit in the
struggle for existence. But this is true only so long as an individual
must live _by himself_. As soon as he forms part of a social
organization which benefits by his particular activity, the organization
compensates for the personal disadvantages by its collective activity,
and a social community not only finds room for such special
developments, but it even encourages and promotes them.

We have already seen that such manifestations occur within the organism
itself. Higher functions, depending upon the higher susceptibility of
the sense organs, can only be attained at the expense of the general
functions by the organ in question. We observe this fact in all socially
organized beings, like bees and ants, which display a high degree of
specialization in the functions of the individual subordinate groups;
the specialization often being carried so far that the individual groups
can no longer subsist by themselves alone. It is only the organization
as a whole that is capable of permanent existence.

While the evolution of such superior functions involves a corresponding
differentiation, and therefore a _division_ and _separation_ of the
superior functions within the social structure, the necessity for
_communication_ and for _mutual support_ results in an _approximation_
of the individuals and the groups. In every society, therefore, the
centrifugal and the centripetal forces work simultaneously in
co-operation and in opposition to one another. While the extreme
specialization on the one hand seems to make for the best individual
functioning, on the other hand it renders the entire collective
structure much more dependent, and therefore much more subject to
injury, as is shown by the example of the queen bee, whose departure
threatens the existence of the entire hive. Thus a medium degree of
differentiation will as a general rule produce the most permanent social
structure.

=59. Language and Intercourse.= The essential value of the social
organization resides in the fact that the work of the individual, in so
far as it is adapted to it, accrues to the benefit of the collective
whole. For this it is absolutely essential that the members of the
collectivity should be able to _have intercourse_ with one another in
order that every part of the general activity may be communicated to the
others. This intercourse is obtained through language in the most
general sense.

We have already learned that the essence of language consists in the
co-ordination of concept to sign. The social application of language
demands that the signs co-ordinated to the concepts in use should be the
same for all the members of the social organization. Only in this way
can the members make themselves mutually understood. But intelligible
means of communication and division of labor impart to the social
knowledge that is set down in writing a kind of independent existence.
Many centuries ago the possibility ceased for one person to store in his
memory the entire stock of human knowledge. Nowadays we have men who are
versed only in single parts of separate sciences, and the aggregate
knowledge appears at first to be a unity existing only in thought. But
because this knowledge is set down in signs which endure far beyond the
life of the individual and at the appropriate moment can unfold its
entire power even after a long period of inactivity, it has acquired an
existence of a social character independent of the individual. For
although it survives the individual, it cannot survive the death of
human society.

As the socialization of all mankind advances to ever greater unities,
the linguistic limitations sprung from former stages of evolution prove
to be a hindrance. The mother tongue, of course, forms the first and
most important entry for the individual to the common store of
knowledge. But in view of the linguistic limitation of which I have just
spoken the efforts in our day are carried on with renewed zeal to create
a _universal auxiliary language_ (p. 100) by means of which intercourse
should be made possible beyond the language boundaries. There have
already been gratifying results.[I]

[I] At the present time "Ido" is the best. It is a highly practicable
artificial language, and its advocates have succeeded in organizing it
to insure its normal development. An older and still rather widespread
form called "Esperanto" has failed to organize itself so as to insure
its development and it must inevitably die out.

=60. Civilization.= Everything which serves the social progress of
mankind is appropriately called civilization or culture, and the
objective characteristic of progress consists in improved methods for
seizing and utilizing the raw energies of nature for human purposes.
Thus it was a cultural act when a primitive man discovered that he could
extend the radius of his muscle energy by taking a pole in his hand, and
it was another cultural act when a primitive man discovered that by
throwing a stone he could send his muscle energy a distance of many
meters to the desired point. The effect of the knife, the spear, the
arrow, and of all the other primitive implements can be called in each
case a purposive transformation of energy. And at the other end of the
scale of civilization the most abstract scientific discovery, by reason
of its generalization and simplification, signifies a corresponding
economy of energy for all the coming generations that may have anything
to do with the matter. Thus, in fact, the concept of progress as here
defined embraces the entire sweep of human endeavor for perfection, or
the entire field of culture, and at the same time it shows the great
scientific value of the concept of energy.

If we consider further that, according to the second fundamental
principle, the free energy accessible to us can only decrease, but not
increase, while the number of men whose existence depends directly on
the consumption of a due amount of free energy is constantly on the
increase, then we at once see the objective necessity of the development
of civilization in that sense. His foresight puts man in a position to
act culturally. But if we examine our present social order from this
point of view, we realize with horror how barbarous it still is. Not
only do murder and war destroy cultural values without substituting
others in their place, not only do the countless conflicts which take
place between the different nations and political organizations act
anticulturally, but so do also the conflicts between the various social
classes of one nation, for they destroy quantities of free energy which
are thus withdrawn from the total of real cultural values. At present
mankind is in a state of development in which progress depends much less
upon the leadership of a few distinguished individuals than upon the
collective labor of all workers. Proof of this is that it is coming to
be more and more the fact that the great scientific discoveries are made
simultaneously by a number of independent investigators--an indication
that society creates in several places the individual conditions
requisite for such discoveries. Thus we are living at a time when men
are gradually approximating one another very closely in their natures,
and when the social organization therefore demands and strives for as
thorough an equalization as possible in the conditions of existence of
all men.

INDEX

Above and below, distinction between, 121

Abstract, concrete and, 16 ff.

Abstraction, 20

Action, conscious, 174;
instinctive, 174

Adaptation, 172 ff.

Aeromechanics, 147

Algebra, 80

Alikeness, definition of, 51 ff.

Allotropic changes, 161

Analysis, infinitesimal, 111

Analytic geometry, 122 ff.

Analytic judgments, 66

Anthropology, 57

Ants, specialization of, 181

Applied sciences, 57 ff.

_A priori_ judgments, 44

Aristotle, 38, 66

Aristotle's logic, 22

Arithmetic, 79 ff.

Assertions, never absolutely correct, 53

Association, 63 ff., 81

Astronomic objective, 6

Astronomy as an applied science, 58

Atomic hypothesis in chemistry, 142

Atoms, 141

Bees, specialization of, 181

Biological sciences, 55;
life most general concept in, 56

Botany, 56

Cæsar, Julius, 76

Capillary phenomena, 146

Capillary theory, 147

Carbon, its circulation through plants and animals, 171;
life based on the energy of, 168

Carbonic acid, 171

Carnot, Sadi, 151

Causal relation, purification of, 34 ff.

Causation, the law of, 31 ff.

Chemical combinations, 71 ff.;
quantitative relations in, 74

Chemical energy, 159 ff.;
capable of powerful concentration, 161;
different forms of, 159

Chemical formulas represent concepts not sounds, 95

Chemistry, 20, 47, 55;
significance of, 160 ff.

Chinese script based on direct co-ordination, 93

Civilization, 184 ff.

Classification, not definite, 2;
purpose of, 2-4

Classification of the sciences, 53 ff.

Collective activity, 181

Combination, sequence in, 73 ff.

Combinations, theory of, 71

Combinatory schematization, 73;
in chemistry, 71 ff.;
in physics, 72

Communication, 181

Community among plants and animals, 179

Comparison, 82

Comte, Auguste, 54

Concept, the most general, 61 ff.

Concepts, arbitrary, 23;
complex, 23;
complex empirical, 23;
definition of, 16;
empirical, 18;
formation of, 19;
general, 26;
in ceaseless flux, 88;
science of, 15 ff., 122;
simple, 20;
simple and complex, 19 ff.

Conclusion, the, 24 ff.;
analytic, 66;
scientific, 27, 30, 66 ff.

Concrete and abstract, 16 ff.

Conjugacy, most general concept in formal sciences, 56

Conscious action, 174

Conscious thinking, willing, and acting, 178

Conservation of energy, the law of the, 135 ff.

Conservation of matter, 138

Conservation of the sum of work and kinetic energy, the law of the, 134

Conservation of work, the law of the, 130

Conservation, quantitative, 131

Continuity, 101 ff.;
the law of, 113 ff.

Co-ordinated signs, change in, 88 ff.

Co-ordination, 80 ff.;
a means of obtaining facts without dealing directly with the
corresponding realities, 87;
between concept and word not unambiguous, 89;
between concept and written sign, direct and indirect, 92 ff.;
identity the limit case in, 82;
integral numbers the best basis of, 85;
in use among primitive men and higher animals, 87;
its importance, 85;
methodology of the sciences based upon, 85;
of numbers with signs, 90 ff.;
possibility of unambiguous, 88

Copernican theory, 117 ff.

Copernicus, 117, 141

Corpuscular theory of light, 5, 157

Counting, 85 ff.;
defined, 85;
purpose of, 86

Culture, see Civilization

Darwin, his fundamental theory, 173

Deduction, 40 ff.;
the process of, 41 ff.

Deductive sciences, 38

Determinateness, absolute, only in ideal world, 50

Determinateness of things, the, 47 ff.

Determinism, 48, 51

Differential Calculus, see Differentials

Differentials, 112

Double numbers or double points in a group, 82

Dynamics, 128 ff.;
definition of, 139

Elasticity, 145

Elastic undulatory theory of light, see Wave theory of light

Electricity, principal source of, 156

Electricity and magnetism, 154 ff.

Electromagnetic theory of light, 157 ff.

Electrotechnics, 156

Empirical sciences, 38

Energetic mechanics, 138 ff.

Energy, a substance, 136;
at rest, 154;
free, 154;
importance of concept of, 128;
in nerves, 177;
the most general concept in the physical sciences, 56;
of form, 145;
of volume, 145

Energy intensity, 153

Erg, definition of, 150

Esperanto, 183, note

Euclid, 44, 140

European-American scripts based on indirect co-ordination, 93

Experience, incompleteness of, 27;
more limited than the conceivable, 118

Experiences, distinguished by _being familiar_, 31;
limited knowledge of, 31

Experiential sciences, see Empirical sciences

Extrapolation, 46, 50, 104

Familiarity due to recalling former similar experiences, 11

Fechner, 102

Feeling, thinking, acting, 174 ff.

Force, 129 ff., 153

Formal sciences, 54;
are empirical sciences, 55;
order most general concept in, 56

Foucault's pendulum experiment, 121

Freedom of the will, 50 ff.

Frequency of process facilitates repetition, 11 ff.

Function, 109 ff.;
continuous and discontinuous, 110;
most general concept in formal sciences, 56

Functional relation, the application of the, 112 ff.

Functions, the theory of, 111

Fundamental principle, the second, 150 ff.

Gases, 145

Generalization, suitable place for, in text-books, 9 ff.

Geometry, 47, 54, 119, 127;
ancient and modern methods of, 110 ff.

Goethe, 99

Good usage in language, 100

Grammatical correctness, importance attached to, 99

Grammatical rules, 97

Gravitation potential, the, 112

Group, the, 65 ff.;
double members or double points in, 82;
linear arrangement of members of, 75 ff.

Groups, artificial and natural, 69 ff.;
closed, among animals, 179;
infinite, equality of, 84;
related, 69 ff.;
unequivocal order of, 83

Heat, mechanical equivalent of, 149;
theory of, 147 ff.

Heat energy, 148 ff.

Heat engine, 151;
ideal, 151 ff.

Heat quantity, 148 ff.

Heliotrope, 90

Herbart, 102

Heredity, 180

Higher analysis, 111

Homonym, 89

Hydromechanics, 147

Ideal cases, 44 ff.

Ideal machines, 132

Identity, the limit case in co-ordination, 82

Ido, 183, note

Imperfection, indestructible quality of science, 4

Incompleteness, no hindrance to efficiency of science, 5

Indestructibility of matter, see Conservation of matter

Indo-Arabic notation, 91

Induction, 38;
the complete and the incomplete, 39

Inductive sciences, 38

Inference, by induction, 38;
from analogy, 140

Infinitesimal analysis, 111

Inorganic world, lack of memory and foresight in, 33

Insoluble problems, 142

Instinctive action, 174

Intercourse, language and, 182 ff.

Isolation among plants and animals, 179

Isomeric, 74

Isomeric changes, 161

Judgments, analytic, 66

Kant, 44, 66, 105

Kepler, 141

Kinetic energy, 132;
and work, their sum constant, 133 ff.;
transformed into work and _vice versa_, 134

Knowledge, aim of, 19;
individual, compared to telephone, 7 ff.;
limited, 31;
possibility of error in, ineradicable, 40;
social character of, 183

Language, beginnings of, 88;
defective in co-ordination, 96;
distinction between science and knowledge of, 98;
good usage in, 100;
and intercourse, 182 ff.;
needless inflections in, 99 ff.;
of words more imperfect than written language, 92;
purpose of its cultivation, 99;
the science of, 97 ff.;
unambiguity the ideal of, 89;
a universal auxiliary, 100;
written, 89 ff.

Leibnitz, 88;
his doctrine of pre-established harmony, 143;
inventor of differentials, 112

Life, 163 ff.;
the most general concept in the biological sciences, 56

Light, 5, 156 ff.

Liquids, 145

Locke, John, 21 ff., 88;
his elaboration of the notion of simple and complex "ideas," 21;
his secondary qualities, 127

Logic, 54, 67 ff.;
content of, 19;
definition of, 15 ff.

Luther, 99

Magnetism, electricity and, 154 ff.

Man, compared to pair of sieves, 34;
his capacity for perfection, 180

Manifold, the science of the, 54

Mass, 132 ff., 136 ff.;
a substance, 138

Mathematical laws, accuracy of, 105

Mathematics, 54;
an empirical science, 55;
influence on, of concept of continuity, 111;
its progress after introduction of Indo-Arabic numerals and algebraic
signs, 101

Matter, definition of, 138

Mayer, Julius Robert, 149;
his discovery of the law of conservation, 151

Measurement, 107

Mechanical energies, 144

Mechanics, 55, 128 ff.;
complementary branches of, 144 ff.;
definition of, 138;
early development of, 139;
energetic, 138 ff.;
the first branch of physics treated mathematically, 139;
pure or classical, 144

Mechanistic hypothesis, the, as an interpretation of all
natural phenomena, 142;
especially injurious in study of mental phenomena, 142

Mechanistic theories, 140 ff.

Mechanistic theory of the universe, 132

Mechanization of astronomy, 141

Memory, 16, 32, 180;
definition of, 172;
general characteristic of, 61;
lack of, in inorganic world, 53

Metabolism, 165

Methodology of the sciences based upon co-ordination, 85

Microscope, 6

Motion, the science of, 54, 122;
uninfluenced, 122

Musical notation, 93

Names, arbitrariness of, 62;
signs and, 86 ff.

Natural laws, 28 ff.;
definition of, 28;
their extent dependent upon stage of knowledge in each science, 7;
their usual origin, 42 ff.;
prediction from, only approximate, 48

Natural philosophy, definition of, 1;
importance of, in study of science, 10;
place of, in text-books, 9 ff.

Negation, 68 ff.

Nerves, 177

Nervous discharge, 177

Newton, Sir Isaac, 141

Number groups, unlimited, 78

Numbers, 78 ff.;
theory of, 80

Numerals, co-ordination of, with signs, 86

Numerical names different in different languages, 86

Numerical signs international, 86

Nutrition, 165

Objective, astronomic, 6;
photographic, 6

Objective character of the world, 34

Optical telegraph, 90

Optics, geometric, 5

Optic signs, 90

Order, most general concept in formal sciences, 56

Organisms, standard for measuring relative perfection of, 176;
stationary forms, 163

Orthography, efforts to improve, 99;
English, defective in co-ordination, 96;
exaggerated importance of correctness in, 99;
mistakes in, 97;
reform of, 97

Parabolic curve, 48

Paradoxes of the infinite, 84

Pasigraphy, 92 ff.;
Chinese system of, 94

Permanent in change, the, 131

Perpetual motion, 130

Perpetual motion machine, 153

Philology, 97 ff.

Philosophy, limited progress in, 101

Phonetic writing, 33 ff.

Phoronomy, 54, 119, 122, 127

Photochemical processes, foundation of terrestial life, 169

Photographic objective, 6

Physical sciences, 55

Physics, 47, 55;
each branch of, treats of a special kind of energy, 139
the science of the different kinds of energy, 72;

Physiology, 55 ff.

Plato, his distinction between mental and physical life, 178

Polarity of electricity and magnetism, 155

Political organizations, conflicts between, 185

Prediction, 12

Pre-established harmony, 143

Pressure, 146, 154

Progress, depends on collective labor, 185;
economy of energy, 184;
evaluation of, 176

Pseudo-problems in science, 142

Psychology, 47, 55 ff.

Psycho-physical parallelism, 143

Ptolemy's system, 117

Pure science, 57

Quantity, the science of, see Mathematics, 54

Radiant energy, 157;
its importance to man, 158

Rational sciences, see Deductive sciences

Rays, straight lines of, 5

Reaction, teleological, 173

Reality, 16 ff.

Reflection, 5

Reflex action, 173

Refraction, 5

Repetition, basis of conscious life, 174

Reproduction, 165 ff.

Roman notation, 91

Science, aim of, 13 ff.;
comparison of, to a network, 42;
comparison of, to a tree or forest, 6;
definition of, 13;
eternal truth of, 6 ff.;
"for its own sake," 13 ff.;
the facts of, unalterable, 8 ff.;
the function of, 23, 37;
importance of theoretical, 15;
its procedure, 45;
the study of happiness, 28

Sciences, the table of the, 54 ff.

Scientific discoveries, independent simultaneous, 185

Scientific instinct, 43

Scientific materialism, 138

Scientific written language based on direct co-ordination, 93

Self-preservation, 180

Sense organs, 176 ff.

Shakespeare, 99

Signs and names, 86 ff.

Social characteristics, importance of, 179 ff.

Social classes, conflicts between, 185

Socialization of human capacities, 180

Social order still barbarous, 185

Social organization, 180;
how best obtained, 182;
its tendency to equalize conditions, 185;
secures permanence among specialized individuals, 181

Social problems, 179 ff.

Society, 179 ff.;
centrifugal and centripetal forces in, 181 ff.;
division of functions in, 181

Sociology, 47, 55, 57

Solar radiation, 169

Soul, the, 171 ff.

Sound signs, advantage and disadvantage of, 89 ff.

Sound writing, 33 ff., 92 ff.

Space, four-dimensional, 77, note;
homogeneity of, in
horizontal direction, 121;
the science of, 54;
symmetrical and tri-dimensional, 118;
time and, 118 ff.;
tri-dimensional, 76

Specialization, one-sidedness of, 180 ff.

Spelling reform, 97

Stable forms, 163

Statics, 128 ff.;
definition of, 138 ff.

Stationary bodies, capable of regeneration, 164

Stationary forms, 163

Substance, 132

Surface-energy, 146

Syllogism, the, classic method of argumentation, 65 ff.

Synonym, 89

Table of the sciences, 54 ff.

Telegraph, optical, 90

"Teleological" properties of organisms, 173

Teleological reaction, 173

Telescope, 5

Temperature, 148

Theoretical science, importance of, 15

Theory of functions, 111

Theory of numbers, 80

Thermo-chemistry, 37

Thermo-dynamics, 153

Thing, definition of, 62 ff.

Thought conceived of as energetic, 178

Threshold, 102

Time, a form of inner life, 76;
measurement of, 122;
one-seried, or one-dimensional, 118;
and space, 118 ff.

Unambiguity, in language, 89;
of co-ordination of numbers to signs, 90

Universal auxiliary language, 100, 183

Velocity, 133

Volume energy, 145

War, 185

Wave surface, 6

Wave theory of light, 5, 157

Weight, 132, 137 ff.;
a substance, 138

Work, mechanical, 129;
product of the force and the distance, 130;
a substance in a limited sense, 136

Written language, 89 ff.

Written signs, 90

Zoology, 56

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Transcriber's Notes:

Bold text is denoted by =equal signs=. The caret ^ indicates that the following character or [
{expression} is superscripted.

Mid-sentence capital letters are used by the Author to indicate the
beginning of a quote or question, which terminates at the end of the
sentence.

Typographical errors corrected:

p. 100: approprate changed to appropriate (... to a more appropriate
evaluation ...).

p. 108: meassure changed to measure (By the application of the unit
measure ...).

p. 184: correspondng changed to corresponding (... signifies a
corresponding economy ...).

p. 191: A single period deleted from index.

P. 188, 189: limit-case changed to limit case (2 occurrences), to
mirror text (3 occurrences).

Alphabetical sequencing adjusted in index:

P. 189: Two 'Energy' entries moved after Energetic mechanics.

P. 191: Photographic objective moved below Photochemical processes.

P. 191: Physics: The order of the sub-entries swapped.

P. 192: Pure science moved down four places to end of "P" entries.

P. 193: Two 'Teleological' entries moved after Telegraph, optical.

*** End of this LibraryBlog Digital Book "Natural Philosophy" ***

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