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The Project Gutenberg eBook of Natural Philosophy, by Wilhelm Ostwald.
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<div>*** START OF THE PROJECT GUTENBERG EBOOK 43791 ***</div>
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<h1>NATURAL PHILOSOPHY</h1>
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BY<br />
<span class = "xl">WILHELM OSTWALD</span>
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<span class = "xs">TRANSLATED</span><br />
<span class = "xs">BY</span><br />
THOMAS SELTZER
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<span class = "s"><i>With the author's special revision for the American edition</i></span>
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NEW YORK<br />
HENRY HOLT AND COMPANY<br />
1910
</p>
<p class = "center p6 spaced">
<span class="smcap s">Copyright</span>, 1910,<br />
<span class="s">BY</span><br />
<span class = "l">HENRY HOLT AND COMPANY</span><br />
</p>
<hr class="short" />
<p class = "center underspaced s"><i>Published November</i>, 1910
</p>
<p class = "p6 center spaced underspaced xs break-after">THE QUINN & BODEN CO. PRESS<br />
RAHWAY N. J.
</p>
<p class = "p6 underspaced">
The original of this book was published
as volume I in Reclam's <span class="smcap">Bücher der
Naturwissenschaft</span>.
</p>
<hr class="chap" />
<p><span class="pagenum"><a name="Page_v" id="Page_v">[v]</a></span></p>
<h2><a name="PREFACE" id="PREFACE">PREFACE</a></h2>
<p>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
<i>natural philosophy</i>. 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.</p>
<p>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<span class="pagenum"><a name="Page_vi" id="Page_vi">[vi]</a></span>
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.</p>
<p>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 <i>is</i> meant to uphold a certain method, that is, the
scientific (or, if you will, the <i>natural</i> 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.</p>
<hr class="chap" />
<p><span class="pagenum"><a name="Page_vii" id="Page_vii">[vii]</a></span></p>
<h2><a name="CONTENTS" id="CONTENTS">CONTENTS</a></h2>
<div class="center">
<table border="0" cellpadding="4" cellspacing="0" summary="Contents">
<tr>
<td align="right"> </td>
<td align="right"> </td>
<td align="right">PAGE</td>
</tr>
<tr>
<td align="left" colspan="2"><span class="smcap">Introduction</span></td>
<td align="right"><a href="#Page_1">1</a></td>
</tr>
<tr>
<td align="center" colspan="3" class = "xl">PART I</td>
</tr>
<tr>
<td align="left" colspan="2"><span class="smcap">General Theory of Knowledge</span></td>
<td align="right"><a href="#Page_11">11</a></td>
</tr>
<tr>
<td align="right">1.</td>
<td align="left">The Formation of Concepts</td>
<td align="right"><a href="#Page_11">11</a></td>
</tr>
<tr>
<td align="right">2.</td>
<td align="left">Science</td>
<td align="right"><a href="#Page_13">13</a></td>
</tr>
<tr>
<td align="right">3.</td>
<td align="left">The Aim of Science</td>
<td align="right"><a href="#Page_13">13</a></td>
</tr>
<tr>
<td align="right">4.</td>
<td align="left">Concrete and Abstract</td>
<td align="right"><a href="#Page_16">16</a></td>
</tr>
<tr>
<td align="right">5.</td>
<td align="left">The Subjective Part</td>
<td align="right"><a href="#Page_17">17</a></td>
</tr>
<tr>
<td align="right">6.</td>
<td align="left">Empirical Concepts</td>
<td align="right"><a href="#Page_18">18</a></td>
</tr>
<tr>
<td align="right">7.</td>
<td align="left">Simple and Complex Concepts</td>
<td align="right"><a href="#Page_19">19</a></td>
</tr>
<tr>
<td align="right">8.</td>
<td align="left">The Conclusion</td>
<td align="right"><a href="#Page_24">24</a></td>
</tr>
<tr>
<td align="right">9.</td>
<td align="left">The Natural Laws</td>
<td align="right"><a href="#Page_28">28</a></td>
</tr>
<tr>
<td align="right">10.</td>
<td align="left">The Law of Causation</td>
<td align="right"><a href="#Page_31">31</a></td>
</tr>
<tr>
<td align="right">11.</td>
<td align="left">The Purification of the Causal Relation</td>
<td align="right"><a href="#Page_34">34</a></td>
</tr>
<tr>
<td align="right">12.</td>
<td align="left">Induction</td>
<td align="right"><a href="#Page_38">38</a></td>
</tr>
<tr>
<td align="right">13.</td>
<td align="left">Deduction</td>
<td align="right"><a href="#Page_40">40</a></td>
</tr>
<tr>
<td align="right">14.</td>
<td align="left">Ideal Cases</td>
<td align="right"><a href="#Page_44">44</a></td>
</tr>
<tr>
<td align="right">15.</td>
<td align="left">The Determinateness of Things</td>
<td align="right"><a href="#Page_47">47</a></td>
</tr>
<tr>
<td align="right">16.</td>
<td align="left">The Freedom of the Will</td>
<td align="right"><a href="#Page_50">50</a></td>
</tr>
<tr>
<td align="right">17.</td>
<td align="left">The Classification of the Sciences</td>
<td align="right"><a href="#Page_53">53</a></td>
</tr>
<tr>
<td align="right">18.</td>
<td align="left">The Applied Sciences</td>
<td align="right"><a href="#Page_57">57</a></td>
</tr>
<tr>
<td align="center" colspan="3" class = "xl">PART II</td>
</tr>
<tr>
<td align="left" colspan="2"><span class="smcap">Logic, the Science of the Manifold, and Mathematics</span></td>
<td align="right"><a href="#Page_61">61</a></td>
</tr>
<tr>
<td align="right">19.</td>
<td align="left">The Most General Concept</td>
<td align="right"><a href="#Page_61">61</a></td>
</tr>
<tr>
<td align="right">20.</td>
<td align="left">Association</td>
<td align="right"><a href="#Page_63">63</a></td>
</tr>
<tr>
<td align="right"><span class="pagenum"><a name="Page_viii" id="Page_viii">[viii]</a></span>21.</td>
<td align="left">The Group</td>
<td align="right"><a href="#Page_65">65</a></td>
</tr>
<tr>
<td align="right">22.</td>
<td align="left">Negation</td>
<td align="right"><a href="#Page_68">68</a></td>
</tr>
<tr>
<td align="right">23.</td>
<td align="left">Artificial and Natural Groups</td>
<td align="right"><a href="#Page_69">69</a></td>
</tr>
<tr>
<td align="right">24.</td>
<td align="left">Arrangement of the Members</td>
<td align="right"><a href="#Page_75">75</a></td>
</tr>
<tr>
<td align="right">25.</td>
<td align="left">Numbers</td>
<td align="right"><a href="#Page_78">78</a></td>
</tr>
<tr>
<td align="right">26.</td>
<td align="left">Arithmetic, Algebra, and the Theory of Numbers</td>
<td align="right"><a href="#Page_79">79</a></td>
</tr>
<tr>
<td align="right">27.</td>
<td align="left">Co-ordination</td>
<td align="right"><a href="#Page_80">80</a></td>
</tr>
<tr>
<td align="right">28.</td>
<td align="left">Comparison</td>
<td align="right"><a href="#Page_82">82</a></td>
</tr>
<tr>
<td align="right">29.</td>
<td align="left">Numbers</td>
<td align="right"><a href="#Page_85">85</a></td>
</tr>
<tr>
<td align="right">30.</td>
<td align="left">Signs and Names</td>
<td align="right"><a href="#Page_86">86</a></td>
</tr>
<tr>
<td align="right">31.</td>
<td align="left">The Written Language</td>
<td align="right"><a href="#Page_89">89</a></td>
</tr>
<tr>
<td align="right">32.</td>
<td align="left">Pasigraphy and Sound Writing</td>
<td align="right"><a href="#Page_92">92</a></td>
</tr>
<tr>
<td align="right">33.</td>
<td align="left">Sound Writing</td>
<td align="right"><a href="#Page_96">96</a></td>
</tr>
<tr>
<td align="right">34.</td>
<td align="left">The Science of Language</td>
<td align="right"><a href="#Page_97">97</a></td>
</tr>
<tr>
<td align="right">35.</td>
<td align="left">Continuity</td>
<td align="right"><a href="#Page_101">101</a></td>
</tr>
<tr>
<td align="right">36.</td>
<td align="left">Measurement</td>
<td align="right"><a href="#Page_107">107</a></td>
</tr>
<tr>
<td align="right">37.</td>
<td align="left">The Function</td>
<td align="right"><a href="#Page_109">109</a></td>
</tr>
<tr>
<td align="right">38.</td>
<td align="left">The Application of the Functional Relation</td>
<td align="right"><a href="#Page_112">112</a></td>
</tr>
<tr>
<td align="right">39.</td>
<td align="left">The Law of Continuity</td>
<td align="right"><a href="#Page_113">113</a></td>
</tr>
<tr>
<td align="right">40.</td>
<td align="left">Time and Space</td>
<td align="right"><a href="#Page_118">118</a></td>
</tr>
<tr>
<td align="right">41.</td>
<td align="left">Recapitulation</td>
<td align="right"><a href="#Page_124">124</a></td>
</tr>
<tr>
<td align="center" colspan="3" class = "xl">PART III</td>
</tr>
<tr>
<td align="left" colspan="2"><span class="smcap">The Physical Sciences</span></td>
<td align="right"><a href="#Page_127">127</a></td>
</tr>
<tr>
<td align="right">42.</td>
<td align="left">General</td>
<td align="right"><a href="#Page_127">127</a></td>
</tr>
<tr>
<td align="right">43.</td>
<td align="left">Mechanics</td>
<td align="right"><a href="#Page_128">128</a></td>
</tr>
<tr>
<td align="right">44.</td>
<td align="left">Kinetic Energy</td>
<td align="right"><a href="#Page_132">132</a></td>
</tr>
<tr>
<td align="right">45.</td>
<td align="left">Mass and Matter</td>
<td align="right"><a href="#Page_136">136</a></td>
</tr>
<tr>
<td align="right">46.</td>
<td align="left">Energetic Mechanics</td>
<td align="right"><a href="#Page_138">138</a></td>
</tr>
<tr>
<td align="right">47.</td>
<td align="left">The Mechanistic Theories</td>
<td align="right"><a href="#Page_140">140</a></td>
</tr>
<tr>
<td align="right">48.</td>
<td align="left">Complementary Branches of Mechanics</td>
<td align="right"><a href="#Page_144">144</a></td>
</tr>
<tr>
<td align="right">49.</td>
<td align="left">The Theory of Heat</td>
<td align="right"><a href="#Page_147">147</a></td>
</tr>
<tr>
<td align="right">50.</td>
<td align="left">The Second Fundamental Principle</td>
<td align="right"><a href="#Page_150">150</a></td>
</tr>
<tr>
<td align="right">51.</td>
<td align="left">Electricity and Magnetism</td>
<td align="right"><a href="#Page_154">154</a></td>
</tr>
<tr>
<td align="right">52.</td>
<td align="left">Light</td>
<td align="right"><a href="#Page_156">156</a></td>
</tr>
<tr>
<td align="right"><span class="pagenum"><a name="Page_ix" id="Page_ix">[ix]</a></span>53.</td>
<td align="left">Chemical Energy</td>
<td align="right"><a href="#Page_159">159</a></td>
</tr>
<tr>
<td align="center" colspan="3" class = "xl">PART IV</td>
</tr>
<tr>
<td align="left" colspan="2"><span class="smcap">The Biologic Sciences</span></td>
<td align="right"><a href="#Page_163">163</a></td>
</tr>
<tr>
<td align="right">54.</td>
<td align="left">Life</td>
<td align="right"><a href="#Page_163">163</a></td>
</tr>
<tr>
<td align="right">55.</td>
<td align="left">The Storehouse of Free Energy</td>
<td align="right"><a href="#Page_168">168</a></td>
</tr>
<tr>
<td align="right">56.</td>
<td align="left">The Soul</td>
<td align="right"><a href="#Page_171">171</a></td>
</tr>
<tr>
<td align="right">57.</td>
<td align="left">Feeling, Thinking, Acting</td>
<td align="right"><a href="#Page_174">174</a></td>
</tr>
<tr>
<td align="right">58.</td>
<td align="left">Society</td>
<td align="right"><a href="#Page_179">179</a></td>
</tr>
<tr>
<td align="right">59.</td>
<td align="left">Language and Intercourse</td>
<td align="right"><a href="#Page_182">182</a></td>
</tr>
<tr>
<td align="right">60.</td>
<td align="left">Civilization</td>
<td align="right"><a href="#Page_184">184</a></td>
</tr>
<tr>
<td align="left" colspan="2"><span class="smcap">Index</span></td>
<td align="right"><a href="#Page_187">187</a></td>
</tr>
</table></div>
<hr class="chap" />
<p><span class="pagenum"><a name="Page_1" id="Page_1">[1]</a></span></p>
<h2><a name="INTRODUCTION" id="INTRODUCTION">INTRODUCTION</a></h2>
<p>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 <i>all</i>
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, <i>natural philosophy is the most general
branch of natural science</i>.</p>
<p><span class="pagenum"><a name="Page_2" id="Page_2">[2]</a></span></p>
<p>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.</p>
<p>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, <i>the lack of complete and precise
boundary lines is a general characteristic of all
natural things</i>, 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.</p>
<p>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<span class="pagenum"><a name="Page_3" id="Page_3">[3]</a></span>
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 <i>better fitted
to its purpose the less frequently</i> 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.</p>
<p>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<span class="pagenum"><a name="Page_4" id="Page_4">[4]</a></span>
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.</p>
<p>Thus we see that these questions of classification
are not questions of the so-called "essence" of the
thing, <i>but pertain merely to purely practical arrangements
for an easier and more successful mastery of
scientific problems</i>. This is an extremely important
point of view, much more far-reaching than is apparent
here at its first application.</p>
<p>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<span class="pagenum"><a name="Page_5" id="Page_5">[5]</a></span>
has been fundamentally modified, signifies that <i>the
quality of incompleteness in human learning is no
hindrance to its efficiency</i>. 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 <i>true elements</i>
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<span class="pagenum"><a name="Page_6" id="Page_6">[6]</a></span>
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.</p>
<p>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<span class="pagenum"><a name="Page_7" id="Page_7">[7]</a></span>
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.</p>
<p>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.</p>
<p>The same is true of an individual. No matter
how limited the circle of his knowledge, <i>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</i>. The man who thus enters the
realm of science acquires advantages which may be
compared to those of a telephone in his residence.<span class="pagenum"><a name="Page_8" id="Page_8">[8]</a></span>
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.</p>
<p>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.
<i>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.</i>
The truths a man has once grasped he need never
learn afresh so far as their <i>actual content</i> is concerned,
though not infrequently—especially in newer
sciences—he may have to see the <i>form</i> 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<span class="pagenum"><a name="Page_9" id="Page_9">[9]</a></span>
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.</p>
<p>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<span class="pagenum"><a name="Page_10" id="Page_10">[10]</a></span>
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.</p>
<p>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.</p>
<hr class="chap" />
<p><span class="pagenum"><a name="Page_11" id="Page_11">[11]</a></span></p>
<h2><a name="PART_I" id="PART_I">PART I</a><br />
GENERAL THEORY OF KNOWLEDGE</h2>
<h3>1. The Formation of Concepts.</h3>
<p class ="p0">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 <i>being familiar</i>. The familiarity
is due to our <i>recalling</i> 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 <i>the oftener any
process has taken place in a living organism the more
easily it is repeated</i>. Here is not yet the place to show
how almost all the characteristic qualities of living<span class="pagenum"><a name="Page_12" id="Page_12">[12]</a></span>
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.</p>
<p>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.</p>
<p><span class="pagenum"><a name="Page_13" id="Page_13">[13]</a></span></p>
<h3>2. Science.</h3>
<p class ="p0">The prophecy of future events based
upon the knowledge of the details of recurring
events is called <i>science</i> 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 <i>past</i>
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 <i>play</i>. 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.</p>
<h3>3. The Aim of Science.</h3>
<p class ="p0">These views are deliberately
opposed to a very widespread idea that science
should be cultivated "for its own sake," and not for<span class="pagenum"><a name="Page_14" id="Page_14">[14]</a></span>
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 <i>play</i> (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.</p>
<p>The element of truth hidden in that misunderstood
phrase is, that in a higher state of culture it is found
better to disregard the <i>immediate</i> 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<span class="pagenum"><a name="Page_15" id="Page_15">[15]</a></span>
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.</p>
<p>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 <i>border line</i> 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 <i>perfect</i> an elaboration
as possible of <i>all</i> the relations conceivable, and
in this practical necessity lies the foundation of the
general or <i>theoretical</i> elaboration of science.</p>
<p><b>The Science of Concepts.</b> 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 <i>logic</i>, which
name, etymologically speaking, hints suspiciously at
the <i>word</i>, and the word, as is known, steps in where
ideas are wanting. Here, however, we have to deal<span class="pagenum"><a name="Page_16" id="Page_16">[16]</a></span>
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 <i>memory</i>, 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.</p>
<h3>4. Concrete and Abstract.</h3>
<p class ="p0">Such coinciding or repeated
parts of similar experiences we call, as already
stated, <i>concepts</i>. 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 <i>all</i> the coinciding
experiences; in other words, all the similar experiences.
Thus, <i>horse</i> 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 <i>concrete</i>
and <i>abstract</i>, and there is an inclination to attribute
"reality" only to the first, while the other, as "mere<span class="pagenum"><a name="Page_17" id="Page_17">[17]</a></span>
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 <i>momentary experience</i>, as opposed
to the totality of the corresponding <i>memories</i>
and <i>expectations</i>. Hence not so much a difference
in <i>reality</i> as in <i>presence</i>. 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 name="FNanchor_A_1" id="FNanchor_A_1"></a><a href="#Footnote_A_1" class="fnanchor">[A]</a></p>
<h3>5. The Subjective Part.</h3>
<p class ="p0">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<span class="pagenum"><a name="Page_18" id="Page_18">[18]</a></span>
individual, or a <i>subjective</i> element. This, however,
does not consist in the <i>addition</i> by the individual
of new parts not found in the experience, but, on the
contrary, in the different <i>choice</i> 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.</p>
<h3>6. Empirical Concepts.</h3>
<p class ="p0">First and unconditionally
those concepts possess reality which always and
without exception are based on <i>experienced</i> 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<span class="pagenum"><a name="Page_19" id="Page_19">[19]</a></span>
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.</p>
<h3>7. Simple and Complex Concepts.</h3>
<p class ="p0">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<span class="pagenum"><a name="Page_20" id="Page_20">[20]</a></span>
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.</p>
<p>The elimination of the non-coinciding parts from
the concept-forming experience is called <i>abstraction</i>.
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.</p>
<p>By looking back over the ground just traversed,
the less abstract ideas may also be regarded as the
<i>more complex</i> 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.</p>
<p>These relations bear a striking resemblance to
the relations known from chemistry to exist between
substances, namely, between elements and<span class="pagenum"><a name="Page_21" id="Page_21">[21]</a></span>
compounds. From the chaos of all objects of experimentation
(chemistry purposely limits itself to
ponderable bodies) the <i>pure</i> substances are sifted
out—an operation corresponding to the formation
of concepts. The pure substances prove to be either
<i>simple</i> or <i>compound</i>, and the compounds are so constituted
that they can each be reduced to a limited
number of simple substances. The simple substances,
or <i>elements</i>, 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.</p>
<p>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.</p>
<p>Nevertheless since then the relation has been completely
reversed. While the study of the chemical<span class="pagenum"><a name="Page_22" id="Page_22">[22]</a></span>
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.</p>
<p>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<span class="pagenum"><a name="Page_23" id="Page_23">[23]</a></span>
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.</p>
<p><i>Complex concepts</i> 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 <i>horse</i>
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 <i>complex
empirical concept</i>.</p>
<p>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 <i>complex arbitrary
concepts</i>.</p>
<p>The task of science can now be even more
sharply defined than before by the fact that it <i>permits
the construction of arbitrary concepts which in
circumstances to be foreseen become empirical concepts</i>.
This is another expression for <i>prediction</i>,<span class="pagenum"><a name="Page_24" id="Page_24">[24]</a></span>
which we recognized as the characteristic of science.
It goes deeper than the previous definition, because
here the means for its realization are given.</p>
<h3>8. The Conclusion.</h3>
<p class ="p0">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 <i>conclusion</i>.
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.</p>
<p>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 <i>two</i> parts, or in which only
two parts are taken into consideration.</p>
<p>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 <i>invariably</i>, so that one part of the concept is<span class="pagenum"><a name="Page_25" id="Page_25">[25]</a></span>
never experienced unless the other part is also experienced,
then there is the <i>greatest</i> 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 <i>without
exception</i> 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 <i>degrees of certainty</i>,
or, rather, <i>probability</i>, 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<span class="pagenum"><a name="Page_26" id="Page_26">[26]</a></span>
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.</p>
<p>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 <i>general</i> concepts, and
therefore the probability of the conclusions described
will be the greater the more <i>general</i> 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 <i>memory</i> 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<span class="pagenum"><a name="Page_27" id="Page_27">[27]</a></span>
of "nonsense" whatsoever. Only it is impossible
to reproduce such combinations from memory.</p>
<p>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 <i>incomplete</i>, 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<span class="pagenum"><a name="Page_28" id="Page_28">[28]</a></span>
foresee such cases also and include them in our anticipations.</p>
<h3>9. The Natural Laws.</h3>
<p class ="p0">The facts just described
have very frequently found expression in the
doctrine of the <i>laws of nature</i>, 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 <i>inform us what has happened and
what is wont to happen</i>. 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.</p>
<p>In this conception the natural laws indicate what<span class="pagenum"><a name="Page_29" id="Page_29">[29]</a></span>
simpler concepts are found in complex concepts.
The complex concept <i>water</i> contains the simpler
ones <i>liquid</i>, a certain <i>density</i>, <i>transparency</i>, <i>colorlessness</i>,<a name="FNanchor_B_2" id="FNanchor_B_2"></a><a href="#Footnote_B_2" class="fnanchor">[B]</a>
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.</p>
<p>Now what predictions do those natural laws enable
us to make?</p>
<p>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.</p>
<p>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<span class="pagenum"><a name="Page_30" id="Page_30">[30]</a></span>
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.</p>
<p>Of course, it must not be forgotten that such conclusions
are always obtained in accordance with the
following formula: <i>things have been so until now,
therefore we expect that they will be so in the future</i>.
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.</p>
<p>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<span class="pagenum"><a name="Page_31" id="Page_31">[31]</a></span>
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 <i>easiest</i> 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 <i>hope</i> 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.</p>
<h3>10. The Law of Causation.</h3>
<p class ="p0">By reason of its frequency
and importance the mental process above described<span class="pagenum"><a name="Page_32" id="Page_32">[32]</a></span>
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, <i>memory in the most
general sense</i>—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 <i>a priori</i> 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<span class="pagenum"><a name="Page_33" id="Page_33">[33]</a></span>
<i>without</i> such an attribute is shown by the whole
world of the <i>inorganic</i>, 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.<a name="FNanchor_C_3" id="FNanchor_C_3"></a><a href="#Footnote_C_3" class="fnanchor">[C]</a></p>
<p>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<span class="pagenum"><a name="Page_34" id="Page_34">[34]</a></span>
various experiences in the shape of natural laws.
Consequently we must recognize that in addition to
the <i>subjective</i> 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 <i>objective</i> 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
<i>made</i> the pebbles equal.</p>
<h3>11. The Purification of the Causal Relation.</h3>
<p class ="p0">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<span class="pagenum"><a name="Page_35" id="Page_35">[35]</a></span>
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.</p>
<p>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 <i>observation</i>, and which we ourselves
have not the power to change (as, for example,
astronomical phenomena), or with things
which are the objects of our <i>experimentation</i>, 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.</p>
<p>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<span class="pagenum"><a name="Page_36" id="Page_36">[36]</a></span>
constant elements the corresponding concept B´.
The improved proposition will then read: if A´ is,
then B´ is also.</p>
<p>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 <i>existing</i>
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.</p>
<p>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<span class="pagenum"><a name="Page_37" id="Page_37">[37]</a></span>
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.</p>
<p>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
<i>exceptions</i> 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´.</p>
<p>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<span class="pagenum"><a name="Page_38" id="Page_38">[38]</a></span>
temporary rule require a limitation, while an unforeseen
freedom requires an extension, of the accepted
concept.</p>
<h3>12. Induction.</h3>
<p class ="p0">The form of conclusion previously
discussed, <i>because it has been so, I expect it will
continue to be so in the future</i>, 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 <i>inference by induction</i>, and the
sciences in which it is preponderatingly applied are
called <i>inductive sciences</i>. 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.</p>
<p>To return to the inductive method—it is to be
noted that <i>Aristotle</i>, who was the first to describe it,<span class="pagenum"><a name="Page_39" id="Page_39">[39]</a></span>
proposed two kinds of induction, the <i>complete</i> and
the <i>incomplete</i>. The first has this form: since <i>all</i>
things of a certain kind are so, each <i>individual thing</i>
is so. While the incomplete induction merely says:
since <i>many</i> things of a certain kind are so, <i>presumably</i>
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 <i>all</i> 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 <i>superfluous</i>, as it merely repeats knowledge
that we have already directly acquired, since
we have tested <i>all</i> the things of the one kind,
hence the special thing to which the predication
refers.</p>
<p>On the other hand, the <i>incomplete</i> induction affirms
something that has not yet been tested, and
therefore involves as a condition an <i>extension</i> 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 <a href="#Page_29">p. 29</a>, the scientific inductive conclusion<span class="pagenum"><a name="Page_40" id="Page_40">[40]</a></span>
assumes the form: because it has <i>once</i> been found to
be so, it will <i>always</i> 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.</p>
<h3>13. Deduction.</h3>
<p class ="p0">In addition to the inductive
method, science has (<a href="#Page_38">p. 38</a>) 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 <i>deductive</i> 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.</p>
<p>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<span class="pagenum"><a name="Page_41" id="Page_41">[41]</a></span>
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.</p>
<p>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 <i>applied to special instances which,
at the proposition of the principle, had not been
taken into consideration</i>, 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 <i>deduction</i> complete until
he had found <i>confirmation</i> in experience.</p>
<p>Deduction, therefore, actually consists in the
searching out of particular instances of a principle<span class="pagenum"><a name="Page_42" id="Page_42">[42]</a></span>
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 <i>not</i> 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.</p>
<p>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<span class="pagenum"><a name="Page_43" id="Page_43">[43]</a></span>
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.</p>
<p>It is immediately evident that this <i>deductive</i> part
is of all the more weight, the more <i>general</i> the concepts<span class="pagenum"><a name="Page_44" id="Page_44">[44]</a></span>
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.
<i>Kant</i> was keenly alive to the peculiarity of such a
view, which had been widely spread pre-eminently
by <i>Euclid's</i> presentation of geometry, and he gave
expression to his opinion of it in the famous question:
<i>How are a priori judgments possible?</i> We
have seen that it is not always a question of
<i>a priori</i> judgments, but also of inductive conclusions
applied and tested according to deductive
methods.</p>
<h3>14. Ideal Cases.</h3>
<p class ="p0">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
(<a href="#Page_30">p. 30</a>). 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<span class="pagenum"><a name="Page_45" id="Page_45">[45]</a></span>
the broadest assertions qualified by numerous "ifs"
and "buts."</p>
<p>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.</p>
<p>The answer is, that we repeatedly so find or can
form our experiences that certain natural relations
<i>preponderatingly</i> determine the experience, while
the other parts that remain undetermined fall into
the background. <i>The prophecy will cover so considerable
a part of the experience that we can forego
previous knowledge of the rest.</i> 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.</p>
<p>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<span class="pagenum"><a name="Page_46" id="Page_46">[46]</a></span>
will take without extraneous influences, the
following general method is applied.</p>
<p>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.</p>
<p>A case in which none of the extraneous elements
of experience operate is called an <i>ideal case</i>, and the
inference from a series of values leading to the
limit-value is an <i>extrapolation</i>. <i>Such extrapolations
to the ideal case</i> 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.</p>
<p>We here confront the fact that many natural
laws, and among them the most important, are expressed
as, and taken to be, conditions <i>which never
occur in reality</i>. 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, <i>that with them the natural laws take on the
simplest forms</i>. This is the result of the fact that
in ideal cases we intentionally and arbitrarily overlook
every complication of the determining factors,<span class="pagenum"><a name="Page_47" id="Page_47">[47]</a></span>
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.</p>
<p>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.</p>
<h3>15. The Determinateness of Things.</h3>
<p class ="p0">A very
widespread view and a very grave one, because of<span class="pagenum"><a name="Page_48" id="Page_48">[48]</a></span>
its erroneous results, is <i>that by the natural laws
things are unequivocally and unalterably determined
down to the very minutest detail</i>. This is called
<i>determinism</i>, and is regarded as an inevitable consequence
of every natural scientific generalization.
But an accurate investigation of actual relations
produces something rather different.</p>
<p>The most general formulation of the natural law:
<i>if A is experienced, then we expect B</i>, necessarily
refers in the first place only to certain <i>parts</i> 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 <i>partial</i>
repetition of any experience, and the common part
is all the smaller a fraction of the entire experience,
the more <i>general</i> 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<span class="pagenum"><a name="Page_49" id="Page_49">[49]</a></span>
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 <i>approximate</i>
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.</p>
<p>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.</p>
<p>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<span class="pagenum"><a name="Page_50" id="Page_50">[50]</a></span>
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.</p>
<p>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 <i>culs-de-sac</i>.</p>
<p><span class="pagenum"><a name="Page_51" id="Page_51">[51]</a></span></p>
<h3>16. The Freedom of the Will.</h3>
<p class ="p0">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 <i>freely</i>, 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 <i>that a part of the
causal chain lies within our consciousness</i>, 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: <i>that
we can and must assume in our practical attitude
to the world that it is only partially determined</i>.</p>
<p>But if two different lines of thought in the whole<span class="pagenum"><a name="Page_52" id="Page_52">[52]</a></span>
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 <i>one</i> view
can be used as a prop for the <i>other</i> 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<span class="pagenum"><a name="Page_53" id="Page_53">[53]</a></span>
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 <i>along with them false elements also</i>. 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.</p>
<p>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.</p>
<h3>17. The Classification of the Sciences.</h3>
<p class ="p0">From the
preceding observations the means may be drawn
for outlining a complete table of the sciences. However,<span class="pagenum"><a name="Page_54" id="Page_54">[54]</a></span>
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.</p>
<p>The basic thought upon which this classification
rests is that of graded abstraction. We have seen
(<a href="#Page_19">p. 19</a>) 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,
<i>Auguste Comte</i>, since whom it has undergone
a few changes.</p>
<p>Below is the table of the sciences, which I shall
then proceed to explain:</p>
<p><span class="pagenum"><a name="Page_55" id="Page_55">[55]</a></span></p>
<p>
<span style="margin-left: 1em;">I. <i>Formal Sciences.</i> Main concept: order</span><br />
<span style="margin-left: 3.5em;">Logic, or the science of the Manifold</span><br />
<span style="margin-left: 3.5em;">Mathematics, or the science of Quantity</span><br />
<span style="margin-left: 3.5em;">Geometry, or the science of Space</span><br />
<span style="margin-left: 3.5em;">Phoronomy, or the science of Motion</span><br />
<br />
<span style="margin-left: 0.5em;">II. <i>Physical Sciences.</i> Main concept: energy</span><br />
<span style="margin-left: 3.5em;">Mechanics</span><br />
<span style="margin-left: 3.5em;">Physics</span><br />
<span style="margin-left: 3.5em;">Chemistry</span><br />
<br />
III. <i>Biological Sciences.</i> Main concept: life<br />
<span style="margin-left: 3.5em;">Physiology</span><br />
<span style="margin-left: 3.5em;">Psychology</span><br />
<span style="margin-left: 3.5em;">Sociology</span><br />
</p>
<p>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<span class="pagenum"><a name="Page_56" id="Page_56">[56]</a></span>
causes them to seem native qualities of the
mind, or <i>a priori</i> 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 <i>conjugacy</i>
or <i>function</i>, the content and significance of which
will become clear later in a more thorough study of
the special sciences.</p>
<p>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.</p>
<p>The most general concept of the physical sciences
is that of <i>energy</i>, 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.</p>
<p>The third group comprehends all the relations of
living beings. Their most general concept, accordingly,
is that of <i>life</i>. By physiology is understood
the entire science dealing with non-psychic life<span class="pagenum"><a name="Page_57" id="Page_57">[57]</a></span>
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.</p>
<h3>18. The Applied Sciences.</h3>
<p class ="p0">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.</p>
<p>The explanation and justification of this is, that
for purposes of systematization we must distinguish
between <i>pure</i> and <i>applied</i> 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<span class="pagenum"><a name="Page_58" id="Page_58">[58]</a></span>
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.</p>
<p>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.</p>
<p>Thus, side by side with the pure sciences are the<span class="pagenum"><a name="Page_59" id="Page_59">[59]</a></span>
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.</p>
<p>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.</p>
<hr class="chap" />
<p><span class="pagenum"><a name="Page_61" id="Page_61">[61]</a></span></p>
<h2><a name="PART_II" id="PART_II">PART II</a><br />
LOGIC, THE SCIENCE OF THE MANIFOLD,
AND MATHEMATICS</h2>
<h3>19. The Most General Concept.</h3>
<p class ="p0">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 <i>most general</i> 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, <i>memory</i>, 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,
<i>no</i> 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 <i>difference</i>, a
difference between memory and the present, or, to
put the same idea still more generally, between two
memories.</p>
<p>Our experiences, therefore, are divided into two<span class="pagenum"><a name="Page_62" id="Page_62">[62]</a></span>
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 <i>name</i>. 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 <i>experience</i> without making a distinction
between a so-called inner or outer experience.</p>
<p>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.</p>
<p>The aggregate of similar experiences, hence of
experiences conceptually generalized, we shall call
<i>things</i>. <i>A thing, therefore, is an experience which<span class="pagenum"><a name="Page_63" id="Page_63">[63]</a></span>
has been repeated</i>, 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 <i>the
concept of thing itself is the most general concept</i>,
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 <i>the same</i>, or <i>identical</i>. Here we
shall leave undecided the question whether this lack
of differentiation occurs because we <i>cannot</i>, or because
we <i>would not</i>, 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 <i>thing</i>.</p>
<h3>20. Association.</h3>
<p class ="p0">The experience of the <i>connection</i>
or <i>relation</i> 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 <i>vice versa</i>. The
cause of this invariably lies in some experiences in<span class="pagenum"><a name="Page_64" id="Page_64">[64]</a></span>
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 <i>complex
concept</i> 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 <i>vice versa</i>.
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.</p>
<p>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 <i>one</i> relation <span class="locked">C : D</span> there arise two new
relations not immediately given, namely, <span class="locked">A : D</span> and<span class="pagenum"><a name="Page_65" id="Page_65">[65]</a></span>
<span class="locked">B : D</span>. 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.</p>
<p>By this simplest and most general example we
recognize the type of the deductive process (<a href="#Page_41">p. 41</a>),
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.</p>
<h3>21. The Group.</h3>
<p class ="p0">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.</p>
<p>From the definition of the group concept follows
the so-called classic <i>process of argumentation</i> of the<span class="pagenum"><a name="Page_66" id="Page_66">[66]</a></span>
syllogism. Its form is: <i>Group A is distinguished by
the characteristic of B</i>. <i>The thing C belongs to
group A. Therefore C has the characteristic of B.</i>
The prominent part ascribed by <i>Aristotle</i> and his
successors to this process is based upon the <i>certainty</i>
which its results possess. Nevertheless, it has been
pointed out, especially by <i>Kant</i>, 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.</p>
<p>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.</p>
<p>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<span class="pagenum"><a name="Page_67" id="Page_67">[67]</a></span>
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.</p>
<p>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.</p>
<p>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<span class="pagenum"><a name="Page_68" id="Page_68">[68]</a></span>
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.</p>
<h3>22. Negation.</h3>
<p class ="p0">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.</p>
<p>This is the meaning and the significance of the
linguistic form of <i>negation</i>. It excludes the thing
negated from any group given in a proposition, and
this relegates it to the second or complementary
group.</p>
<p>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 <i>one</i> of
the characteristics a, b, c, d excludes the incorporation
of the thing into the group A, while the mere<span class="pagenum"><a name="Page_69" id="Page_69">[69]</a></span>
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 <i>all</i> 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.</p>
<p>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.</p>
<h3>23. Artificial and Natural Groups.</h3>
<p class ="p0">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 <i>poorer in characteristics</i>, will, in general, be
<i>richer in members</i>, 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.</p>
<p>If we call such groups related as contain common
characteristics, though containing them in different
members and combinations, so that the definition of<span class="pagenum"><a name="Page_70" id="Page_70">[70]</a></span>
the one group can be derived from the other by the
elimination or incorporation of individual characteristics,
then we can postulate the general thesis
<i>that in related groups those must be richer in members
which are poorer in characteristics, and inversely</i>.
This is the precise statement of the
proposition of the less definite thesis stated above.</p>
<p>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.</p>
<p>This case, however, can be referred to the general
one first described, by treating the characteristics
belonging together as being <i>one</i> 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 <i>classification</i> of the corresponding things.</p>
<p>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<span class="pagenum"><a name="Page_71" id="Page_71">[71]</a></span>
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.</p>
<p>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</p>
<p class = "blockquot">
a) each of the eighty elements by itself<br />
b) all substances containing two elements and no more<br />
<span class="pagenum"><a name="Page_72" id="Page_72">[72]</a></span>c) all substances containing three elements<br />
d, e, f, etc.) the substances containing four, five, and six, etc., elements,<br />
</p>
<p>until finally we reach a group (not existing in experience)
embracing substances formed of <i>all</i> 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.</p>
<p>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.</p>
<p>Of course, neither in chemistry nor in physics
does this mean that each <i>new</i> case will fall within
the scheme obtained by the exhaustive combination
of elementary concepts (whether chemical elements
or kinds of energy) <i>known</i> at the time. It is quite
possible that a new thing under investigation contains
a <i>new</i> elementary concept, so that on account<span class="pagenum"><a name="Page_73" id="Page_73">[73]</a></span>
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.</p>
<p>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 <i>experienced</i>. 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.</p>
<p>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 <i>sequence</i> according to which this combination<span class="pagenum"><a name="Page_74" id="Page_74">[74]</a></span>
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.</p>
<p>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 <i>quantitative
relations</i>, 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 <i>isomeric</i>, 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 <i>several</i>
different things instead of one appear in the
same group of the original scheme, the systematic<span class="pagenum"><a name="Page_75" id="Page_75">[75]</a></span>
classification of which necessitates a further
schematization by the use of other characteristics.</p>
<h3>24. Arrangement of the Members.</h3>
<p class ="p0">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 <i>one</i> definite
member is followed by a <i>single</i> 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 <i>linear</i> 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.</p>
<p>This simplest of all possible forms expresses itself
in the fact <i>that the immediately experienced things
of our consciousness are arranged in this way</i>. 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<span class="pagenum"><a name="Page_76" id="Page_76">[76]</a></span>
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.</p>
<p>The biologic peculiarity here mentioned of the
linear juxtaposition of the contents of our consciousness
has led to the concept of <i>time</i>, which has
been appropriately called a <i>form of inner life</i>. 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
<i>one dimension</i>. Since, on the other hand, we have
all learned to feel space as <i>tri-dimensional</i>, 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<span class="pagenum"><a name="Page_77" id="Page_77">[77]</a></span>
of adaptation, and that because the change has been
extremely slight in the course of centuries it produces
the impression of being unalterable.<a name="FNanchor_D_4" id="FNanchor_D_4"></a><a href="#Footnote_D_4" class="fnanchor">[D]</a></p>
<p>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 <i>arbitrary</i>, 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.</p>
<p>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<span class="pagenum"><a name="Page_78" id="Page_78">[78]</a></span>
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.</p>
<h3>25. Numbers.</h3>
<p class ="p0">An especially important group in
the linear order is that of the <i>integral numbers</i>. Its
origin is as follows:</p>
<p>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 <i>unlimited</i> or <i>infinite</i>.</p>
<p>The groups or types thus obtained are called the
<i>integral numbers</i>. 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<span class="pagenum"><a name="Page_79" id="Page_79">[79]</a></span>
begins, this characteristic is present in a peculiar
form, the preceding group being <i>group zero</i>, that is,
a group without content. This number in consequence
reveals certain peculiarities into which we
cannot enter here.</p>
<p>Now, according to a previous observation
(<a href="#Page_64">p. 64</a>), 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.</p>
<h3>26. Arithmetic, Algebra, and the Theory of Numbers.</h3>
<p class ="p0">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 <i>practical</i> reasons the study of them
must be as <i>general</i> as possible. This science is
called <i>arithmetic</i>.</p>
<p><span class="pagenum"><a name="Page_80" id="Page_80">[80]</a></span></p>
<p>Arithmetic undergoes an important generalization
if the individual numbers in a calculation are
disregarded and <i>abstract signs</i> 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, <i>algebra</i>, that is, calculation
with abstract or general quantities, has developed as
an extensive and important field of general mathematics.</p>
<p>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.</p>
<h3>27. Co-ordination.</h3>
<p class ="p0">So far our discussion has
confined itself to the <i>individual</i> groups and to the
properties which each one of them exhibits <i>by itself</i>.
We shall now investigate the relations which exist
<i>between two or more groups</i>, both with regard to
their several members and to their aggregate.</p>
<p>If at first we have two groups the members of
which are all differentiated from one another, then<span class="pagenum"><a name="Page_81" id="Page_81">[81]</a></span>
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 <i>association</i>.</p>
<p>First, the co-ordination is <i>mutual</i>, 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.</p>
<p>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.</p>
<p>Finally, a group can be co-ordinated with itself,<span class="pagenum"><a name="Page_82" id="Page_82">[82]</a></span>
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 <i>double members</i>,
or <i>double points</i>. The limit-case is <i>identity</i>, in
which every member corresponds to <i>itself</i>. 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.</p>
<h3>28. Comparison.</h3>
<p class ="p0">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 <i>all</i> their members. In
the first case A is called, in the broader sense of the
word, <i>smaller</i> than B, in the second B is called
smaller than A, in the third the two groups are said
to be of <i>equal magnitude</i>. The expression, "B is
greater than A," is equivalent to the expression, "A
is smaller than B," and inversely.</p>
<p>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<span class="pagenum"><a name="Page_83" id="Page_83">[83]</a></span>
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.</p>
<p>By carrying out the co-ordination proof is further
supplied of the following propositions:</p>
<div class="center">
<table border="0" cellpadding="4" cellspacing="0" summary="groups">
<tr>
<td align="right"></td>
<td align="center" rowspan="3"><span class="xxl">{</span></td>
<td align="center">greater than</td>
<td align="center" rowspan="3"><span class="xxl">}</span></td>
</tr>
<tr>
<td align="right">If group A is</td>
<td align="center">equal to</td>
<td align="left">group B</td>
</tr>
<tr>
<td align="right"></td>
<td align="center">smaller than</td>
</tr>
<tr>
<td align="right"></td>
<td align="center" rowspan="3"><span class="xxl">{</span></td>
<td align="center">greater than</td>
<td align="center" rowspan="3"><span class="xxl">}</span></td>
</tr>
<tr>
<td align="right">and group B is</td>
<td align="center">equal to</td>
<td align="left">group C</td>
</tr>
<tr>
<td align="right"></td>
<td align="center">smaller than</td>
</tr>
<tr>
<td align="right"></td>
<td align="center" rowspan="3"><span class="xxl">{</span></td>
<td align="center">greater than</td>
<td align="center" rowspan="3"><span class="xxl">}</span></td>
</tr>
<tr>
<td align="right">then group A is</td>
<td align="center">equal to</td>
<td align="left">group C</td>
</tr>
<tr>
<td align="right"></td>
<td align="center">smaller than</td>
</tr>
</table></div>
<p>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. <i>This order would be unequivocal</i>, that is,<span class="pagenum"><a name="Page_84" id="Page_84">[84]</a></span>
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.</p>
<p>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 <i>equal</i> 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.
<i>The definition of equality is therefore not
completely fulfilled</i>, 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.</p>
<p><span class="pagenum"><a name="Page_85" id="Page_85">[85]</a></span></p>
<p>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.
<i>The entire methodology of all the sciences
is based upon the most manifold and many-sided
application of the process of co-ordination</i>,
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.</p>
<h3>29. Counting.</h3>
<p class ="p0">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 <i>counting</i>, and from the premises
made it follows <i>that we can count all things in so
far as we disregard their differences</i>.</p>
<p>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<span class="pagenum"><a name="Page_86" id="Page_86">[86]</a></span>
the group to be counted is exhausted. The last
number required for the co-ordination is called the
<i>sum</i> of the members of the counted group. Since
the number series continues indefinitely, every given
group can be counted.</p>
<p>Numerals have been co-ordinated with <i>names</i> as
well as with <i>signs</i>. 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.</p>
<p>The purpose of counting is extremely manifold.
Its most frequent and most important application
lies in the fact that the amount affords a <i>measure for
the effectiveness or the value</i> 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.</p>
<h3>30. Signs and Names.</h3>
<p class ="p0">The co-ordination of
names and signs with numbers calls for a few general
remarks on co-ordination of this nature.</p>
<p>The possibility of carrying out the formal operations
effected in one of the groups upon the co-ordinated
group itself facilitates to an extraordinary<span class="pagenum"><a name="Page_87" id="Page_87">[87]</a></span>
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.</p>
<p>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<span class="pagenum"><a name="Page_88" id="Page_88">[88]</a></span>
to the point, and that, on the other hand, it should be
easily "understood," that is, <i>comprehended</i> by the
senses and unmistakably <i>differentiated</i> from other
signs co-ordinated with other things.</p>
<p>Thus, we find that the most frequent concepts of
co-ordinated sound signs form the beginnings of
<i>language</i> 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.</p>
<p>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 <i>co-ordination should be unambiguous</i>. 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,<span class="pagenum"><a name="Page_89" id="Page_89">[89]</a></span>
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
<i>fundamental defects</i> of language, since they destroy
the <i>principle of unambiguity</i> 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.</p>
<h3>31. The Written Language.</h3>
<p class ="p0">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.</p>
<p>For this we turn to another sense, the sense of<span class="pagenum"><a name="Page_90" id="Page_90">[90]</a></span>
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 <i>written language</i> in the widest
sense, and here, too, it is a question of co-ordinating
signs and concepts.</p>
<p>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.</p>
<p>As we have already pointed out, such a case is
furnished by the <i>written signs</i> 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<span class="pagenum"><a name="Page_91" id="Page_91">[91]</a></span>
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:</p>
<p>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.</p>
<p>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.</p>
<p><span class="pagenum"><a name="Page_92" id="Page_92">[92]</a></span></p>
<p>The comparison of the spoken and the written
languages offers a very illuminating proof of the
much greater imperfection of the language of
<i>words</i>. 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, <span class="locked">-teen,</span> <span class="locked">-thousand,</span>
<span class="locked">-hundred,</span> and <span class="locked">-ty.</span> 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
<i>place value</i> 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.<a name="FNanchor_E_5" id="FNanchor_E_5"></a><a href="#Footnote_E_5" class="fnanchor">[E]</a></p>
<h3>32. Pasigraphy and Sound Writing.</h3>
<p class ="p0">There are
two possibilities for co-ordination between concepts
and written signs. Either the co-ordination is <i>direct</i>,
so that it is only a matter of providing every
concept with a corresponding sign, or it is indirect,<span class="pagenum"><a name="Page_93" id="Page_93">[93]</a></span>
the signs serving only the purpose of expressing the
<i>language sound</i>. 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 <i>an unambiguous co-ordination between
sound and sign</i>. The Chinese script follows
the direct process, but all the scripts of the European-American
civilized peoples are based on the indirect
process.</p>
<p>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.<a name="FNanchor_F_6" id="FNanchor_F_6"></a><a href="#Footnote_F_6" class="fnanchor">[F]</a></p>
<p>Fundamentally we need not hesitate to recognize
in <i>concept writing</i> or <i>pasigraphy</i> a more complete
solution of the problem of sign arrangement.
Even the very incomplete Chinese pasigraphy renders<span class="pagenum"><a name="Page_94" id="Page_94">[94]</a></span>
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.</p>
<p>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 <i>elementary</i> 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<span class="pagenum"><a name="Page_95" id="Page_95">[95]</a></span>
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.</p>
<p>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<span class="pagenum"><a name="Page_96" id="Page_96">[96]</a></span>
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.</p>
<h3>33. Sound Writing.</h3>
<p class ="p0">In point of unambiguity of
co-ordination <i>phonetic writing</i> 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,<span class="pagenum"><a name="Page_97" id="Page_97">[97]</a></span>
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. <i>The fact
that orthographic mistakes can be made in the writing
of any language is direct proof of its imperfection</i>,
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.</p>
<h3>34. The Science of Language.</h3>
<p class ="p0">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,<span class="pagenum"><a name="Page_98" id="Page_98">[98]</a></span>
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.</p>
<p>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 <i>science</i> of language make a sharp distinction
between it and the <i>knowledge</i> 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<span class="pagenum"><a name="Page_99" id="Page_99">[99]</a></span>
as inconceivably futile to future science as the
scholasticism of the middle ages seems to us now.</p>
<p>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 <i>preserve</i> as accurately as possible existing
linguistic usage, with its imperfections, amounting
at times to absurdities. Its real object lies rather in
the appropriate <i>development</i> and <i>improvement</i> 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.</p>
<p>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<span class="pagenum"><a name="Page_100" id="Page_100">[100]</a></span>
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."</p>
<p>A twofold advantage will have been attained by
the introduction of a <i>universal auxiliary language</i>
(<a href="#Page_183">183</a>). 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<span class="pagenum"><a name="Page_101" id="Page_101">[101]</a></span>
great. For it may be asserted that
philosophy, the most general of all the sciences, has
hitherto made such extremely limited progress only
<i>because it was compelled to make use of the medium
of general language</i>. 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.</p>
<h3>35. Continuity.</h3>
<p class ="p0">Up to this point our discussions
have been based on the general concept of the
<i>thing</i>, that is, of the individual experience differentiated
from other experiences. Here the fact of
<i>being different</i>, 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 <i>continuity</i>.</p>
<p>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 <i>we have never felt the
increase</i> of darkness from moment to moment, although<span class="pagenum"><a name="Page_102" id="Page_102">[102]</a></span>
theoretically we are absolutely convinced
that this is the correct conception of the process.</p>
<p>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 <i>Herbart</i>, but its significance
was first recognized by <i>Fechner</i>, and has since then
become generally known in physiology and psychology
under the name of <i>threshold</i>. <i>Next to memory
the threshold determines the fundamental lines of
our psychic life.</i></p>
<p>The threshold therefore means that whatever
state we are in <i>a certain finite amount of difference
or change must be stepped over</i> 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.</p>
<p>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<span class="pagenum"><a name="Page_103" id="Page_103">[103]</a></span>
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 <i>see</i> 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.</p>
<p>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 <i>continuously</i>,
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<span class="pagenum"><a name="Page_104" id="Page_104">[104]</a></span>
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 <span class="locked">A = B</span>
and <span class="locked">B = C,</span> <span class="locked">A = C.</span> This principle is valid for <i>counted</i>
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 <i>magnitudes</i> in the narrower sense, we must bear
in mind that it is just as much a case of an <i>extrapolation
to the non-existing ideal instance</i> (<a href="#Page_46">p. 46</a>) 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.</p>
<p>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 <span class="locked">A = B,</span> <span class="locked">B = C,</span> but <span class="locked">A ≠ C.</span> 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
<span class="locked">A ≠ C</span> can be observed, the difference of both values<span class="pagenum"><a name="Page_105" id="Page_105">[105]</a></span>
cannot be greater than at utmost the sum of the two
threshold values.</p>
<p>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 <i>experience should approximate
them infinitely</i>, 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 (<a href="#Page_84">p. 84</a>).</p>
<p>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<span class="pagenum"><a name="Page_106" id="Page_106">[106]</a></span>
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.</p>
<p>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
<i>continuous</i>.</p>
<p>Now if we turn to <i>experience</i>, we find, as the
sum total of our knowledge, that for the sake of
expediency we approach everything with the presumption
that it is <i>continuous</i>. This aggregate experience<span class="pagenum"><a name="Page_107" id="Page_107">[107]</a></span>
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.</p>
<h3>36. Measurement.</h3>
<p class ="p0">Measuring is in a certain way
the opposite of counting. While, in counting, the
things are regarded in advance as <i>individual</i>, and
the group, therefore, is a body compounded of discontinuous
elements, measuring, on the other hand,
consists in <i>co-ordinating numbers with continuous
things</i>, that is, in applying to continuous things a
concept formed upon the hypothesis of discontinuity.</p>
<p>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.</p>
<p>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 <i>between</i> two unit-marks.<span class="pagenum"><a name="Page_108" id="Page_108">[108]</a></span>
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 <i>between n and <span class="locked">n + 1</span> units</i>, and
even if a definite number is given, the scientifically
trained person will supplement this number by the
sign ± <i>f</i>, in which <i>f</i> denotes the possible errors, that
is, the limit within which the given number may be
false.</p>
<p>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.</p>
<p>The significance of measurement therefore lies
in the fact that it applies the operation of counting
with all its advantages (see <a href="#Page_85">p. 85</a>) to <i>continuous</i>
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 <i>measured</i> with a liter this general process
is carried out physically. In all other less direct
methods of measurement the physical process is substituted<span class="pagenum"><a name="Page_109" id="Page_109">[109]</a></span>
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.</p>
<p>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 <i>preponderatingly as
continuous</i>, we at once see what an important step
forward has been made through the invention of
measurement in the intellectual domination of our
experience.</p>
<h3>37. The Function.</h3>
<p class ="p0">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 (<a href="#Page_31">p. 31</a>).
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<span class="pagenum"><a name="Page_110" id="Page_110">[110]</a></span>
that nature, so that <i>every special value or state of B
corresponds to every special value or state of A</i>.</p>
<p>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 <i>continuous</i> things, there is
nothing to hinder its application to discrete things,
and accordingly we distinguish between continuous
and discontinuous functions.</p>
<p>The intellectual progress involved in the conception
of the reciprocal relation of entire <i>series</i> or
groups to one another, as distinguished from the
conception of the relations between <i>individual</i>
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<span class="pagenum"><a name="Page_111" id="Page_111">[111]</a></span>
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.</p>
<p>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 <i>Higher
Analysis</i>, or <i>Infinitesimal Analysis</i>, was the first
result of this radical advance, and the <i>Theory of
Functions</i>, 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
<i>variable</i>, that is, values which may range through
all possible quantities. If we represent the relation
between two things by the formula <span class="locked">B = f(A),</span>
expressed in spoken language by B <i>is a function</i> 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.</p>
<p>Herein lies the essential advantage of the concept
of continuity. It is true that it also introduces
into calculation the above-mentioned contradictions<span class="pagenum"><a name="Page_112" id="Page_112">[112]</a></span>
which crop up in the ever-recurring discussions concerning
the infinitely great and the infinitely small.
The system introduced by Leibnitz of calculating
with <i>differentials</i>, 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.</p>
<h3>38. The Application of the Functional Relation.</h3>
<p class ="p0">I have already shown (<a href="#Page_34">p. 34</a>) 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.</p>
<p>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 <i>method of the continuous functional relation</i><span class="pagenum"><a name="Page_113" id="Page_113">[113]</a></span>
steps in with its infinitely greater effectiveness.
If in most cases we cannot <i>eliminate</i> the factors one
by one, there are very few instances in which it is
not possible to <i>change</i> 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
<i>all such factors are essential the change of
which involves a change of the result</i>.</p>
<p>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.</p>
<p>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
<i>partly</i>, 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.</p>
<h3>39. The Law of Continuity.</h3>
<p class ="p0">From the fact that
natural phenomena in general proceed continuously
we can deduce a number of important and generally<span class="pagenum"><a name="Page_114" id="Page_114">[114]</a></span>
applicable conclusions which are constantly used
for the development of science.</p>
<p>When a relation of two continuously varying
values of the form <span class="locked">A = f(B)</span> 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.</p>
<p>The application of the law of continuity or steadiness,<span class="pagenum"><a name="Page_115" id="Page_115">[115]</a></span>
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 <i>scientific</i>
one.</p>
<p>The value of this instrument is still greater if it
succeeds in expressing the relation <span class="locked">A = f(B)</span> 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.</p>
<p>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<span class="pagenum"><a name="Page_116" id="Page_116">[116]</a></span>
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
<span class="locked">p. v = K,</span> 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 <span class="locked">A = f(B)</span> becomes the definite
<span class="locked">p = K/v.</span> 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.</p>
<p>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 <span class="locked">p = 0,</span> then <span class="locked">v = infinity,</span> both
of which lie beyond the field of possible experience.</p>
<p>Similar considerations obtain in all such mathematically
formulated natural laws, and each time,<span class="pagenum"><a name="Page_117" id="Page_117">[117]</a></span>
therefore, we must ask what the <i>range of validity</i>
of such an expression is, and answer the question
by experiment.</p>
<p>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?</p>
<p>It lies in the fact that simple formulas are discovered
only <i>when the conceptual analysis of the
phenomenon has advanced far enough</i>. 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,<span class="pagenum"><a name="Page_118" id="Page_118">[118]</a></span>
and thus the Copernican theory has come to
be generally recognized and applied.</p>
<p>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.</p>
<h3>40. Time and Space.</h3>
<p class ="p0">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 <i>three</i>fold manifoldness.</p>
<p>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<span class="pagenum"><a name="Page_119" id="Page_119">[119]</a></span>
time the subjective-human element of the natural law
becomes very clear.</p>
<p>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 <i>phoronomy</i>, a
subject which we shall consider presently. In phoronomy,
however, time appears only in its simplest
form as a one-seried continuous manifoldness.</p>
<p>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 <i>geometry</i>. 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.</p>
<p>We are not conscious of the extreme arbitrariness<span class="pagenum"><a name="Page_120" id="Page_120">[120]</a></span>
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.</p>
<p>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<span class="pagenum"><a name="Page_121" id="Page_121">[121]</a></span>
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.</p>
<p>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<span class="pagenum"><a name="Page_122" id="Page_122">[122]</a></span>
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.</p>
<p>The common application of the concepts of space
and time results in the concept of <i>motion</i>, 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 <i>that in uninfluenced motions equal periods
of time must correspond to the equal changes in
space</i>. 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.</p>
<p><i>Analytic geometry</i>, 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,<span class="pagenum"><a name="Page_123" id="Page_123">[123]</a></span>
that is, by the application of the <i>algebraic</i>
material of symbols we can obtain data concerning
unknown <i>spacial</i> relations. An explanation is
necessary of how by a method apparently so extraneous
such results as these can be attained.</p>
<p>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 <a href="#Page_86">p. 86</a> of operations undertaken with one
group and repeated correspondingly in the co-ordinated
group. And it is only the great difference<span class="pagenum"><a name="Page_124" id="Page_124">[124]</a></span>
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.</p>
<h3>41. Recapitulation.</h3>
<p class ="p0">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.</p>
<p>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,<span class="pagenum"><a name="Page_125" id="Page_125">[125]</a></span>
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.</p>
<p>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, <i>thing</i>, <i>group</i>, <i>co-ordination</i>.
Here were found the fundamentals of logic
or the science of concepts. A special process of abstraction
yielded the concept of <i>number</i>, and with
it the corresponding field of <i>mathematics</i>, arithmetic,
algebra, and the theory of numbers.</p>
<p><span class="pagenum"><a name="Page_126" id="Page_126">[126]</a></span></p>
<p>By means of the second fundamental fact of
physiology, the <i>threshold</i>, another elementary fact
was explained, that of <i>continuity</i>. The co-ordination
of individual things under the influence of this
concept was expanded into the <i>co-ordination of continuous
phenomena-series</i>, and yielded the correspondingly
more general concept of the <i>function</i>.
From the application of the number concept to continuous
things, the idea of <i>measurement</i> resulted.
In mathematics the concept of continuity led to
higher <i>analysis</i> and the <i>theory of functions</i>. 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.</p>
<hr class="chap" />
<p><span class="pagenum"><a name="Page_127" id="Page_127">[127]</a></span></p>
<h2><a name="PART_III" id="PART_III">PART III</a><br />
THE PHYSICAL SCIENCES</h2>
<h3>42. General.</h3>
<p class ="p0">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.</p>
<p>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<span class="pagenum"><a name="Page_128" id="Page_128">[128]</a></span>
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.</p>
<p>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.</p>
<h3>43. Mechanics.</h3>
<p class ="p0">Recently many scientists have
taken exception to the traditional division of
mechanics into <i>statics</i>, or the science of equilibrium,
and <i>dynamics</i>, or the science of motion, because it
does not correspond to the essence of the thing,
equilibrium being only the limit-case of motion.<span class="pagenum"><a name="Page_129" id="Page_129">[129]</a></span>
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.</p>
<p>By <i>work</i> 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.</p>
<p>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 <i>weight</i>.
But a more general concept has been formed for that
property in the mechanical sense, called <i>force</i>, of
which weight constitutes but a special instance.
Whenever there is a resistance combined with a
change of place we speak of a force, <i>and the<span class="pagenum"><a name="Page_130" id="Page_130">[130]</a></span>
product of the force and the distance we call
work</i>.</p>
<p>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 <i>the law of the conservation of work</i>
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 <i>quantity</i>.</p>
<p>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.</p>
<p>Here for the first time we meet with a law expressing<span class="pagenum"><a name="Page_131" id="Page_131">[131]</a></span>
the <i>quantitative</i> 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.</p>
<p>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<span class="pagenum"><a name="Page_132" id="Page_132">[132]</a></span>
which persists under changing conditions by the
name of <i>substance, we encounter in work the first
substance</i> 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.</p>
<h3>44. Kinetic Energy.</h3>
<p class ="p0">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 <i>ideal</i> 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<span class="pagenum"><a name="Page_133" id="Page_133">[133]</a></span>
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.</p>
<p>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. <i>Therefore, there is
a reciprocal relation between the work which it loses
and the velocity which it gains</i>, 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 <i>mass</i>, can be
established in such a way that this function, called
the <i>kinetic energy</i> of the body, increases precisely as
much as the amount of work the body has expended,<span class="pagenum"><a name="Page_134" id="Page_134">[134]</a></span>
and <i>vice versa</i>. The sum of the kinetic
energy of the body and of the <i>work</i> is therefore
<i>constant</i>, and the clearest mode of conceiving of
this relation is by assuming <i>that work can be transformed
into kinetic energy and vice versa</i> 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 <i>causal</i> relation, and in such
a sense we are justified <i>in regarding the disappearing
work as the cause of the kinetic energy that
arises</i>, and to designate this relation summarily as a
transformation.</p>
<p>By the inclusion of cases in which work is converted
into kinetic energy the law of the conservation
of work therefore becomes <i>the law of the conservation
of the sum of work and kinetic energy</i>.
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.</p>
<p>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,<span class="pagenum"><a name="Page_135" id="Page_135">[135]</a></span>
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 <i>energy</i>. The law in question, therefore,
is:</p>
<p class = "blockquot"><i>In all processes the sum of the existing energies
remains unchanged.</i></p>
<p>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 <i>work</i> 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,</p>
<p class = "blockquot">Energy that has disappeared = energy that has
arisen,</p>
<p>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 <i>vice versa</i>, the equation during the first
period is:</p>
<p class = "blockquot">Work that has disappeared = kinetic energy that
has arisen,</p>
<p>and during the second period in which the pendulum
rises again,</p>
<p><span class="pagenum"><a name="Page_136" id="Page_136">[136]</a></span></p>
<p class = "blockquot">Kinetic energy that has disappeared = work that
has arisen.</p>
<p>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 <i>that a
quantity of any energy never disappears unless an
equivalent quantity of another energy arises</i>. 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.</p>
<h3>45. Mass and Matter.</h3>
<p class ="p0">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.<span class="pagenum"><a name="Page_137" id="Page_137">[137]</a></span>
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 <i>mass</i>, 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.</p>
<p>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 <i>held</i> 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.</p>
<p><span class="pagenum"><a name="Page_138" id="Page_138">[138]</a></span></p>
<p>The name given to this concept of the combined
presence of mass and weight in space is <i>matter</i>.
Experience shows that there is a law of <i>conservation</i>
for these magnitudes also, according to which <i>whatever
changes we may produce in bodies possessing
weight and mass, no change will occur in the sum of
their weight and mass</i>. 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 <i>scientific
materialism</i>, 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.</p>
<h3>46. Energetic Mechanics.</h3>
<p class ="p0">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<span class="pagenum"><a name="Page_139" id="Page_139">[139]</a></span>
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 <i>vice versa</i>. 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 <i>vice versa</i>. 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.</p>
<p>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.</p>
<p>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
<i>mathematical</i> treatment. It is true that the
mathematical treatment of mechanics was possible
only after idealizing assumptions had been made—<span class="pagenum"><a name="Page_140" id="Page_140">[140]</a></span>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.</p>
<h3>47. The Mechanistic Theories.</h3>
<p class ="p0">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.</p>
<p><span class="pagenum"><a name="Page_141" id="Page_141">[141]</a></span></p>
<p>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.</p>
<p>The impulse to this was first given by the extraordinary
successes which mechanics has attained in
the generalization and prediction of the <i>motions of
the heavenly bodies</i>. 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 <i>atoms</i>, supported these tendencies and
invited the attempt to regard the little world of
atoms as subject to the same laws as had been found<span class="pagenum"><a name="Page_142" id="Page_142">[142]</a></span>
to apply so successfully to the great world of the
stars.</p>
<p>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 <i>insoluble</i>,
and constitute an inexhaustible source of differences
of scientific opinion.</p>
<p>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<span class="pagenum"><a name="Page_143" id="Page_143">[143]</a></span>
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.</p>
<p>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 <i>pre-established
harmony</i> proposed by Leibnitz is in the
ascendant in our day, and is now called the theory
of the <i>psycho-physical parallelism</i>. 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.</p>
<p>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 <i>raison d'être</i> than<span class="pagenum"><a name="Page_144" id="Page_144">[144]</a></span>
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.</p>
<h3>48. Complementary Branches of Mechanics.</h3>
<p class ="p0">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.</p>
<p>If by mechanical energies we understand all
energies in which <i>changes of space are connected
with changes of energy</i>, there are as many different
forms as there are spacial concepts that seem applicable.
<i>Form</i>, <i>Volume</i>, and <i>Surface</i> of bodies
in space are especially recognizable as the field of<span class="pagenum"><a name="Page_145" id="Page_145">[145]</a></span>
action for energy, which shows different properties
or manifoldnesses according to each of these relations.</p>
<p>The <i>energy of form</i> 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 <i>elasticity</i>.
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.</p>
<p>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 <i>liquids</i>, which have a definite volume (corresponding
to the definite shape of solids), the changes
of which in <i>every</i> sense, both compression and expansion,
require work. Secondly, the <i>gases</i> with
volume energy in only one sense of the word, in
which only the compression of volume requires<span class="pagenum"><a name="Page_146" id="Page_146">[146]</a></span>
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
<i>pressure</i>.</p>
<p>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.</p>
<p>Since the seat of this kind of energy is in the
surfaces (or superficies), it is called <i>surface-energy</i>.
The phenomena depending upon it manifest themselves
most clearly at the surface boundaries between
<i>liquids</i> and <i>gases</i>. They are called <i>capillary
phenomena</i>. This strange name, derived from the
word <i>capilla</i>, 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 <i>hair</i>,
a considerable rise can be observed. This is the entire
connection between the name and the thing.</p>
<p><span class="pagenum"><a name="Page_147" id="Page_147">[147]</a></span></p>
<p>The mechanics of liquids is called <i>hydromechanics</i>,
that of gases, <i>aeromechanics</i>, 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.</p>
<h3>49. The Theory of Heat.</h3>
<p class ="p0">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<span class="pagenum"><a name="Page_148" id="Page_148">[148]</a></span>
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.</p>
<p>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."</p>
<p>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.</p>
<p>The same property is found in heat energy itself.
In heat energy we measure the quantity of energy
itself and call it the <i>heat quantity</i>, while in some of
the other kinds of energy, only the factors into
which they can be divided are measured, and no<span class="pagenum"><a name="Page_149" id="Page_149">[149]</a></span>
habitual conception of the energy itself is developed.
A heat quantity is likewise fully indicated by its
measure number.</p>
<p>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 <i>Julius
Robert Mayer</i> of Heilbronn, who published his results
in 1842. Mayer not only showed that the
imperfect machines (<a href="#Page_134">p. 134</a>), 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 <i>heat</i>, 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<span class="pagenum"><a name="Page_150" id="Page_150">[150]</a></span>
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.</p>
<p>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 <i>same unit</i>.
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 <i>erg</i>, an abbreviation of energy. The
amount is very small, and for technical reasons 10<sup>10</sup>
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.</p>
<h3>50. The Second Fundamental Principle.</h3>
<p class ="p0">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<span class="pagenum"><a name="Page_151" id="Page_151">[151]</a></span>
a change occurs, this second law asserts the condition
under which such changes arise, and is therefore
called the <i>second fundamental principle</i>.</p>
<p>The discovery of this law antedates <i>Mayer's</i> discovery
of the law of conservation by about twenty
years, and was made by a French military engineer,
<i>Sadi Carnot</i>, who died soon afterward without having
lived to see the recognition his great work obtained.
<i>Carnot</i> 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 <i>ideal heat engine</i> 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 <i>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</i>. This follows
from the following considerations:</p>
<p>In the first place an ideal engine must be <i>reversible</i>,
that is, it must be capable of working both<span class="pagenum"><a name="Page_152" id="Page_152">[152]</a></span>
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 <i>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</i>. 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.</p>
<p>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.</p>
<p>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<span class="pagenum"><a name="Page_153" id="Page_153">[153]</a></span>
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.</p>
<p>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 <i>all</i> 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 <i>intensity</i>
of the energy. In work, for instance, it is <i>force</i>,<span class="pagenum"><a name="Page_154" id="Page_154">[154]</a></span>
in volume-energy it is <i>pressure</i>. If once the intensity
in a body is equal, its energy is at rest, and
it never again moves of its own accord.</p>
<p>Another form in which to present these relations
is to make a distinction between <i>free</i> energy and
energy <i>at rest</i>. 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 <i>capable
of change</i>, or <i>free</i> 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, <i>free energy is the condition of all phenomena</i>.</p>
<h3>51. Electricity and Magnetism.</h3>
<p class ="p0">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.</p>
<p>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<span class="pagenum"><a name="Page_155" id="Page_155">[155]</a></span>
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
<i>polar</i>. 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.<a name="FNanchor_G_7" id="FNanchor_G_7"></a><a href="#Footnote_G_7" class="fnanchor">[G]</a></p>
<p>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<span class="pagenum"><a name="Page_156" id="Page_156">[156]</a></span>
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.</p>
<h3>52. Light.</h3>
<p class ="p0">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<span class="pagenum"><a name="Page_157" id="Page_157">[157]</a></span>
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.</p>
<p>However that may be, light is an energy which,
according to the known laws, travels through space
with tremendous rapidity. We will call it <i>radiant
energy</i>, 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.</p>
<p>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 <i>ether</i>, of which
we know nothing else. This elastic undulatory
theory has been abandoned in our time in favor
of an <i>electromagnetic</i> theory supported by quite
considerable experiential grounds. Whether it<span class="pagenum"><a name="Page_158" id="Page_158">[158]</a></span>
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.</p>
<p>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.</p>
<p>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<span class="pagenum"><a name="Page_159" id="Page_159">[159]</a></span>
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.</p>
<h3>53. Chemical Energy.</h3>
<p class ="p0">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.</p>
<p>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.</p>
<p>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 <i>chemical elements</i>.
The experiential law, that the elements cannot be<span class="pagenum"><a name="Page_160" id="Page_160">[160]</a></span>
changed into one another,<a name="FNanchor_H_8" id="FNanchor_H_8"></a><a href="#Footnote_H_8" class="fnanchor">[H]</a> 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.</p>
<p>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.</p>
<p>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,<span class="pagenum"><a name="Page_161" id="Page_161">[161]</a></span>
it possesses the peculiar property which enables it to
be <i>preserved</i> for a long time without passing into
other forms and being dissipated. Furthermore,
energy in this form permits of the most powerful
<i>concentration</i>. 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.</p>
<p>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 <i>chemical</i>
energy of, say, coal, the coal <i>alone</i> is not sufficient;
<i>another</i> 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<span class="pagenum"><a name="Page_162" id="Page_162">[162]</a></span>
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 <i>diffusion</i>
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.</p>
<hr class="chap" />
<p><span class="pagenum"><a name="Page_163" id="Page_163">[163]</a></span></p>
<h2><a name="PART_IV" id="PART_IV">PART IV</a><br />
THE BIOLOGIC SCIENCES</h2>
<h3>54. Life.</h3>
<p class ="p0">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.</p>
<p>The first peculiarity is this, that living organisms
are not <i>stable</i> but <i>stationary</i> 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<span class="pagenum"><a name="Page_164" id="Page_164">[164]</a></span>
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.</p>
<p>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.</p>
<p>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 <i>being
healed</i> or of <i>regeneration</i>.</p>
<p>For a body to continue permanently in a stationary
condition the material of which it is composed must
be permanently <i>supplied</i>. 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<span class="pagenum"><a name="Page_165" id="Page_165">[165]</a></span>
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 <i>change
of substance</i>, <i>metabolism</i>, appears as a necessary
property of the stationary body. In order, however,
that metabolism should take place we must
have free <i>energy</i>, 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 <i>nutrition</i>,
and thus we recognize in the capacity for <i>self-acquisition
of nutrition</i> another essential property
of organisms.</p>
<p>A third essential property of organisms is the
capacity for <i>reproduction</i>, 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<span class="pagenum"><a name="Page_166" id="Page_166">[166]</a></span>
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.</p>
<p>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<span class="pagenum"><a name="Page_167" id="Page_167">[167]</a></span>
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.</p>
<p>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 <i>same
species</i> continue to exist even after the existence of
the individual has ceased.</p>
<p>These three properties constitute the essential
characteristics of animate things or organisms.</p>
<p>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<span class="pagenum"><a name="Page_168" id="Page_168">[168]</a></span>
bee or the swimming of a dolphin cannot be conceived
of except as brought about through chemical
energy.</p>
<p>That this chemical energy is essentially that of
<i>carbon</i> 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.</p>
<p>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.</p>
<p>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
<i>essential characteristics of organisms</i>. They constitute
the frame within which everything must be
found which we should recognize as living in the
widest sense.</p>
<h3>55. The Storehouse of Free Energy.</h3>
<p class ="p0">If we ask
whence the organisms obtain the free energy which<span class="pagenum"><a name="Page_169" id="Page_169">[169]</a></span>
they require for the maintenance of their stationary
existence, the answer is that <i>solar radiation</i> 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.</p>
<p>It is comprehensible, therefore, that machines
should have evolved in the organism for <i>transforming
the radiant energy of the sun into a permanent
form</i>, 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 <i>photochemical</i> processes, that is, in the transformation
of radiant energy into chemical energy,
the foundation of life on earth.</p>
<p>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<span class="pagenum"><a name="Page_170" id="Page_170">[170]</a></span>
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.</p>
<p>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.</p>
<p>While the free energy is poured out in such a<span class="pagenum"><a name="Page_171" id="Page_171">[171]</a></span>
stream in one direction, the ponderable substances
of which the organisms are made up <i>circulate</i>
through plants and animals and back again. This
is especially true of <i>carbon</i>, 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.</p>
<p>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.</p>
<h3>56. The Soul.</h3>
<p class ="p0">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<span class="pagenum"><a name="Page_172" id="Page_172">[172]</a></span>
the lifeless machines, and which we have already
encountered in the very beginning of our treatise.</p>
<p>It is the property which we there called <i>memory</i>,
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.</p>
<p>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.</p>
<p>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<span class="pagenum"><a name="Page_173" id="Page_173">[173]</a></span>
inorganic possibilities. This is an important point
of view for what follows.</p>
<p>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.</p>
<p>In its most primitive forms the quality of adaptation
gives rise to the <i>phenomena of reaction</i>, or to
<i>reflex</i> 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<span class="pagenum"><a name="Page_174" id="Page_174">[174]</a></span>
of this is the moth, which flies into the light
and is burned.</p>
<p>As the reactions become more fixed they develop
into longer and more complicated series, which then
appear to us as <i>instinctive actions</i>. But here, too, we
find the characteristic unsuitability when unwonted
circumstances arise, even if the teleologic reactions
to stimuli become more manifold.</p>
<p>Finally, there are the <i>conscious acts</i> 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 <i>concepts</i>,
is made possible only through <i>repetition</i>.
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.</p>
<h3>57. Feeling, Thinking, Acting.</h3>
<p class ="p0">For good reasons
it is generally assumed that the organisms have not
always been what they are now, but have "developed"<span class="pagenum"><a name="Page_175" id="Page_175">[175]</a></span>
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 <i>changed</i> conditions
of existence has yielded the most important factor
of change.</p>
<p>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<span class="pagenum"><a name="Page_176" id="Page_176">[176]</a></span>
of the organization (as for example in
parasites) as backward steps.</p>
<p>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 <i>objective</i>
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 <i>human</i> progress, appearing,
as it does, as the general standard of all civilization.</p>
<p>The perfection of the organism shows itself in
relation to the outer world in the development of
the <i>sense organs</i>. 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<span class="pagenum"><a name="Page_177" id="Page_177">[177]</a></span>
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 <i>conducting paths</i>, 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 <i>discharge</i>,
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.</p>
<p>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<span class="pagenum"><a name="Page_178" id="Page_178">[178]</a></span>
of the nerve process which will solve the question.</p>
<p>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.</p>
<p>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 <i>mechanical</i>,
there is no difficulty in conceiving of it as
<i>energetic</i>, especially since we know that mental work
is connected with expenditure of energy and exhaustion<span class="pagenum"><a name="Page_179" id="Page_179">[179]</a></span>
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.</p>
<h3>58. Society.</h3>
<p class ="p0">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.</p>
<p>Since the second case is true of man in a highly
marked degree, his <i>social</i> 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—<span class="pagenum"><a name="Page_180" id="Page_180">[180]</a></span>
the social problems also occupy an ever
larger place in the organization of human life.</p>
<p>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 <i>self-preservation</i>.
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.</p>
<p>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 <i>one-sidedness</i>, and
as soon as a certain measure is slightly overstepped,<span class="pagenum"><a name="Page_181" id="Page_181">[181]</a></span>
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 <i>by himself</i>. 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.</p>
<p>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.</p>
<p>While the evolution of such superior functions
involves a corresponding differentiation, and therefore
a <i>division</i> and <i>separation</i> of the superior functions
within the social structure, the necessity for
<i>communication</i> and for <i>mutual support</i> results in
an <i>approximation</i> of the individuals and the groups.
In every society, therefore, the centrifugal and the<span class="pagenum"><a name="Page_182" id="Page_182">[182]</a></span>
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.</p>
<h3>59. Language and Intercourse.</h3>
<p class ="p0">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 <i>have
intercourse</i> 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.</p>
<p>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<span class="pagenum"><a name="Page_183" id="Page_183">[183]</a></span>
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.</p>
<p>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 <i>universal auxiliary language</i> (<a href="#Page_100">p. 100</a>)
by means of which intercourse should be made possible
beyond the language boundaries. There have
already been gratifying results.<a name="FNanchor_I_9" id="FNanchor_I_9"></a><a href="#Footnote_I_9" class="fnanchor">[I]</a></p>
<p><span class="pagenum"><a name="Page_184" id="Page_184">[184]</a></span></p>
<h3>60. Civilization.</h3>
<p class ="p0">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.</p>
<p>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<span class="pagenum"><a name="Page_185" id="Page_185">[185]</a></span>
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.</p>
<hr class="chap" />
<div class="footnotes ">
<h2 class = "s">FOOTNOTES:</h2>
<div class="footnote">
<p><a name="Footnote_A_1" id="Footnote_A_1"></a><a href="#FNanchor_A_1"><span class="label">[A]</span></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.</p></div>
<div class="footnote">
<p><a name="Footnote_B_2" id="Footnote_B_2"></a><a href="#FNanchor_B_2"><span class="label">[B]</span></a> More precisely, a very pale blue.</p></div>
<div class="footnote">
<p><a name="Footnote_C_3" id="Footnote_C_3"></a><a href="#FNanchor_C_3"><span class="label">[C]</span></a> 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.</p></div>
<div class="footnote">
<p><a name="Footnote_D_4" id="Footnote_D_4"></a><a href="#FNanchor_D_4"><span class="label">[D]</span></a> 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.</p></div>
<div class="footnote">
<p><a name="Footnote_E_5" id="Footnote_E_5"></a><a href="#FNanchor_E_5"><span class="label">[E]</span></a> 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<sup>2n</sup>, 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.</p></div>
<div class="footnote">
<p><a name="Footnote_F_6" id="Footnote_F_6"></a><a href="#FNanchor_F_6"><span class="label">[F]</span></a> It is not difficult to perfect musical notation with a view
to unambiguity, a thing which would greatly facilitate the
study of music.</p></div>
<div class="footnote">
<p><a name="Footnote_G_7" id="Footnote_G_7"></a><a href="#FNanchor_G_7"><span class="label">[G]</span></a> 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 <i>exclusively positive magnitude</i>, 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 <i>expended</i> in contradistinction
to the energy <i>received</i>. It is therefore nothing
more than the indication of a mathematical operation.</p></div>
<div class="footnote">
<p><a name="Footnote_H_8" id="Footnote_H_8"></a><a href="#FNanchor_H_8"><span class="label">[H]</span></a> 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.</p></div>
<div class="footnote">
<p><a name="Footnote_I_9" id="Footnote_I_9"></a><a href="#FNanchor_I_9"><span class="label">[I]</span></a> 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.</p></div></div>
<hr class="chap" />
<p><span class="pagenum"><a name="Page_187" id="Page_187">[187]</a></span></p>
<h2><a name="INDEX" id="INDEX">INDEX</a></h2>
<ul class="index">
<li class="ifrst">Above and below, distinction between, <a href="#Page_121">121</a></li>
<li class="indx">Abstract, concrete and, <a href="#Page_16">16</a> ff.</li>
<li class="indx">Abstraction, <a href="#Page_20">20</a></li>
<li class="indx">Action, conscious, <a href="#Page_174">174</a>;</li>
<li class="isub1">instinctive, <a href="#Page_174">174</a></li>
<li class="indx">Adaptation, <a href="#Page_172">172</a> ff.</li>
<li class="indx">Aeromechanics, <a href="#Page_147">147</a></li>
<li class="indx">Algebra, <a href="#Page_80">80</a></li>
<li class="indx">Alikeness, definition of, <a href="#Page_51">51</a> ff.</li>
<li class="indx">Allotropic changes, <a href="#Page_161">161</a></li>
<li class="indx">Analysis, infinitesimal, <a href="#Page_111">111</a></li>
<li class="indx">Analytic geometry, <a href="#Page_122">122</a> ff.</li>
<li class="indx">Analytic judgments, <a href="#Page_66">66</a></li>
<li class="indx">Anthropology, <a href="#Page_57">57</a></li>
<li class="indx">Ants, specialization of, <a href="#Page_181">181</a></li>
<li class="indx">Applied sciences, <a href="#Page_57">57</a> ff.</li>
<li class="indx"><i>A priori</i> judgments, <a href="#Page_44">44</a></li>
<li class="indx">Aristotle, <a href="#Page_38">38</a>, <a href="#Page_66">66</a></li>
<li class="indx">Aristotle's logic, <a href="#Page_22">22</a></li>
<li class="indx">Arithmetic, <a href="#Page_79">79</a> ff.</li>
<li class="indx">Assertions, never absolutely correct, <a href="#Page_53">53</a></li>
<li class="indx">Association, <a href="#Page_63">63</a> ff., <a href="#Page_81">81</a></li>
<li class="indx">Astronomic objective, <a href="#Page_6">6</a></li>
<li class="indx">Astronomy as an applied science, <a href="#Page_58">58</a></li>
<li class="indx">Atomic hypothesis in chemistry, <a href="#Page_142">142</a></li>
<li class="indx">Atoms, <a href="#Page_141">141</a></li>
<li class="ifrst">Bees, specialization of, <a href="#Page_181">181</a></li>
<li class="indx">Biological sciences, <a href="#Page_55">55</a>;</li>
<li class="isub1">life most general concept in, <a href="#Page_56">56</a></li>
<li class="indx">Botany, <a href="#Page_56">56</a></li>
<li class="ifrst">Cæsar, Julius, <a href="#Page_76">76</a></li>
<li class="indx">Capillary phenomena, <a href="#Page_146">146</a></li>
<li class="indx">Capillary theory, <a href="#Page_147">147</a></li>
<li class="indx">Carbon, its circulation through plants and animals, <a href="#Page_171">171</a>;</li>
<li class="isub1">life based on the energy of, <a href="#Page_168">168</a></li>
<li class="indx">Carbonic acid, <a href="#Page_171">171</a></li>
<li class="indx">Carnot, Sadi, <a href="#Page_151">151</a></li>
<li class="indx">Causal relation, purification of, <a href="#Page_34">34</a> ff.</li>
<li class="indx">Causation, the law of, <a href="#Page_31">31</a> ff.</li>
<li class="indx">Chemical combinations, <a href="#Page_71">71</a> ff.;</li>
<li class="isub1">quantitative relations in, <a href="#Page_74">74</a></li>
<li class="indx">Chemical energy, <a href="#Page_159">159</a> ff.;</li>
<li class="isub1">capable of powerful concentration, <a href="#Page_161">161</a>;</li>
<li class="isub1">different forms of, <a href="#Page_159">159</a></li>
<li class="indx">Chemical formulas represent concepts not sounds, <a href="#Page_95">95</a></li>
<li class="indx">Chemistry, <a href="#Page_20">20</a>, <a href="#Page_47">47</a>, <a href="#Page_55">55</a>;</li>
<li class="isub1">significance of, <a href="#Page_160">160</a> ff.</li>
<li class="indx">Chinese script based on direct co-ordination, <a href="#Page_93">93</a></li>
<li class="indx">Civilization, <a href="#Page_184">184</a> ff.</li>
<li class="indx">Classification, not definite, <a href="#Page_2">2</a>;</li>
<li class="isub1">purpose of, <a href="#Page_2">2-4</a></li>
<li class="indx">Classification of the sciences, <a href="#Page_53">53</a> ff.</li>
<li class="indx">Collective activity, <a href="#Page_181">181</a></li>
<li class="indx">Combination, sequence in, <a href="#Page_73">73</a> ff.</li>
<li class="indx">Combinations, theory of, <a href="#Page_71">71</a></li>
<li class="indx">Combinatory schematization, <a href="#Page_73">73</a>;</li>
<li class="isub1">in chemistry, <a href="#Page_71">71</a> ff.;</li>
<li class="isub1">in physics, <a href="#Page_72">72</a></li>
<li class="indx">Communication, <a href="#Page_181">181</a></li>
<li class="indx">Community among plants and animals, <a href="#Page_179">179</a></li>
<li class="indx">Comparison, <a href="#Page_82">82</a></li>
<li class="indx">Comte, Auguste, <a href="#Page_54">54</a></li>
<li class="indx"><span class="pagenum"><a name="Page_188" id="Page_188">[188]</a></span>Concept, the most general, <a href="#Page_61">61</a> ff.</li>
<li class="indx">Concepts, arbitrary, <a href="#Page_23">23</a>;</li>
<li class="isub1">complex, <a href="#Page_23">23</a>;</li>
<li class="isub1">complex empirical, <a href="#Page_23">23</a>;</li>
<li class="isub1">definition of, <a href="#Page_16">16</a>;</li>
<li class="isub1">empirical, <a href="#Page_18">18</a>;</li>
<li class="isub1">formation of, <a href="#Page_19">19</a>;</li>
<li class="isub1">general, <a href="#Page_26">26</a>;</li>
<li class="isub1">in ceaseless flux, <a href="#Page_88">88</a>;</li>
<li class="isub1">science of, <a href="#Page_15">15</a> ff., <a href="#Page_122">122</a>;</li>
<li class="isub1">simple, <a href="#Page_20">20</a>;</li>
<li class="isub1">simple and complex, <a href="#Page_19">19</a> ff.</li>
<li class="indx">Conclusion, the, <a href="#Page_24">24</a> ff.;</li>
<li class="isub1">analytic, <a href="#Page_66">66</a>;</li>
<li class="isub1">scientific, <a href="#Page_27">27</a>, <a href="#Page_30">30</a>, <a href="#Page_66">66</a> ff.</li>
<li class="indx">Concrete and abstract, <a href="#Page_16">16</a> ff.</li>
<li class="indx">Conjugacy, most general concept in formal sciences, <a href="#Page_56">56</a></li>
<li class="indx">Conscious action, <a href="#Page_174">174</a></li>
<li class="indx">Conscious thinking, willing, and acting, <a href="#Page_178">178</a></li>
<li class="indx">Conservation of energy, the law of the, <a href="#Page_135">135</a> ff.</li>
<li class="indx">Conservation of matter, <a href="#Page_138">138</a></li>
<li class="indx">Conservation of the sum of work and kinetic energy, the law of the, <a href="#Page_134">134</a></li>
<li class="indx">Conservation of work, the law of the, <a href="#Page_130">130</a></li>
<li class="indx">Conservation, quantitative, <a href="#Page_131">131</a></li>
<li class="indx">Continuity, <a href="#Page_101">101</a> ff.;</li>
<li class="isub1">the law of, <a href="#Page_113">113</a> ff.</li>
<li class="indx">Co-ordinated signs, change in, <a href="#Page_88">88</a> ff.</li>
<li class="indx">Co-ordination, <a href="#Page_80">80</a> ff.;</li>
<li class="isub1">a means of obtaining facts without dealing directly with the corresponding realities, <a href="#Page_87">87</a>;</li>
<li class="isub1">between concept and word not unambiguous, <a href="#Page_89">89</a>;</li>
<li class="isub1">between concept and written sign, direct and indirect, <a href="#Page_92">92</a> ff.;</li>
<li class="isub1">identity the limit case in, <a href="#Page_82">82</a>;</li>
<li class="isub1">integral numbers the best basis of, <a href="#Page_85">85</a>;</li>
<li class="isub1">in use among primitive men and higher animals, <a href="#Page_87">87</a>;</li>
<li class="isub1">its importance, <a href="#Page_85">85</a>;</li>
<li class="isub1">methodology of the sciences based upon, <a href="#Page_85">85</a>;</li>
<li class="isub1">of numbers with signs, <a href="#Page_90">90</a> ff.;</li>
<li class="isub1">possibility of unambiguous, <a href="#Page_88">88</a></li>
<li class="indx">Copernican theory, <a href="#Page_117">117</a> ff.</li>
<li class="indx">Copernicus, <a href="#Page_117">117</a>, <a href="#Page_141">141</a></li>
<li class="indx">Corpuscular theory of light, <a href="#Page_5">5</a>, <a href="#Page_157">157</a></li>
<li class="indx">Counting, <a href="#Page_85">85</a> ff.;</li>
<li class="isub1">defined, <a href="#Page_85">85</a>;</li>
<li class="isub1">purpose of, <a href="#Page_86">86</a></li>
<li class="indx">Culture, see Civilization</li>
<li class="ifrst">Darwin, his fundamental theory, <a href="#Page_173">173</a></li>
<li class="indx">Deduction, <a href="#Page_40">40</a> ff.;</li>
<li class="isub1">the process of, <a href="#Page_41">41</a> ff.</li>
<li class="indx">Deductive sciences, <a href="#Page_38">38</a></li>
<li class="indx">Determinateness, absolute, only in ideal world, <a href="#Page_50">50</a></li>
<li class="indx">Determinateness of things, the, <a href="#Page_47">47</a> ff.</li>
<li class="indx">Determinism, <a href="#Page_48">48</a>, <a href="#Page_51">51</a></li>
<li class="indx">Differential Calculus, see Differentials</li>
<li class="indx">Differentials, <a href="#Page_112">112</a></li>
<li class="indx">Double numbers or double points in a group, <a href="#Page_82">82</a></li>
<li class="indx">Dynamics, <a href="#Page_128">128</a> ff.;</li>
<li class="isub1">definition of, <a href="#Page_139">139</a></li>
<li class="ifrst">Elasticity, <a href="#Page_145">145</a></li>
<li class="indx">Elastic undulatory theory of light, see Wave theory of light</li>
<li class="indx">Electricity, principal source of, <a href="#Page_156">156</a></li>
<li class="indx">Electricity and magnetism, <a href="#Page_154">154</a> ff.</li>
<li class="indx">Electromagnetic theory of light, <a href="#Page_157">157</a> ff.</li>
<li class="indx">Electrotechnics, <a href="#Page_156">156</a></li>
<li class="indx">Empirical sciences, <a href="#Page_38">38</a></li>
<li class="indx">Energetic mechanics, <a href="#Page_138">138</a> ff.</li>
<li class="indx">Energy, a substance, <a href="#Page_136">136</a>;</li>
<li class="isub1">at rest, <a href="#Page_154">154</a>;</li>
<li class="isub1">free, <a href="#Page_154">154</a>;</li>
<li class="isub1">importance of concept of, <a href="#Page_128">128</a>;</li>
<li class="isub1">in nerves, <a href="#Page_177">177</a>;</li>
<li class="isub1">the most general concept in the physical sciences, <a href="#Page_56">56</a>;</li>
<li class="isub1"><span class="pagenum"><a name="Page_189" id="Page_189">[189]</a></span>of form, <a href="#Page_145">145</a>;</li>
<li class="isub1">of volume, <a href="#Page_145">145</a></li>
<li class="indx">Energy intensity, <a href="#Page_153">153</a></li>
<li class="indx">Erg, definition of, <a href="#Page_150">150</a></li>
<li class="indx">Esperanto, 183, <a href="#Footnote_I_9">note</a></li>
<li class="indx">Euclid, <a href="#Page_44">44</a>, <a href="#Page_140">140</a></li>
<li class="indx">European-American scripts based on indirect co-ordination, <a href="#Page_93">93</a></li>
<li class="indx">Experience, incompleteness of, <a href="#Page_27">27</a>;</li>
<li class="isub1">more limited than the conceivable, <a href="#Page_118">118</a></li>
<li class="indx">Experiences, distinguished by <i>being familiar</i>, <a href="#Page_31">31</a>;</li>
<li class="isub1">limited knowledge of, <a href="#Page_31">31</a></li>
<li class="indx">Experiential sciences, see Empirical sciences</li>
<li class="indx">Extrapolation, <a href="#Page_46">46</a>, <a href="#Page_50">50</a>, <a href="#Page_104">104</a></li>
<li class="ifrst">Familiarity due to recalling former similar experiences, <a href="#Page_11">11</a></li>
<li class="indx">Fechner, <a href="#Page_102">102</a></li>
<li class="indx">Feeling, thinking, acting, <a href="#Page_174">174</a> ff.</li>
<li class="indx">Force, <a href="#Page_129">129</a> ff., <a href="#Page_153">153</a></li>
<li class="indx">Formal sciences, <a href="#Page_54">54</a>;</li>
<li class="isub1">are empirical sciences, <a href="#Page_55">55</a>;</li>
<li class="isub1">order most general concept in, <a href="#Page_56">56</a></li>
<li class="indx">Foucault's pendulum experiment, <a href="#Page_121">121</a></li>
<li class="indx">Freedom of the will, <a href="#Page_50">50</a> ff.</li>
<li class="indx">Frequency of process facilitates repetition, <a href="#Page_11">11</a> ff.</li>
<li class="indx">Function, <a href="#Page_109">109</a> ff.;</li>
<li class="isub1">continuous and discontinuous, <a href="#Page_110">110</a>;</li>
<li class="isub1">most general concept in formal sciences, <a href="#Page_56">56</a></li>
<li class="indx">Functional relation, the application of the, <a href="#Page_112">112</a> ff.</li>
<li class="indx">Functions, the theory of, <a href="#Page_111">111</a></li>
<li class="indx">Fundamental principle, the second, <a href="#Page_150">150</a> ff.</li>
<li class="ifrst">Gases, <a href="#Page_145">145</a></li>
<li class="indx">Generalization, suitable place for, in text-books, <a href="#Page_9">9</a> ff.</li>
<li class="indx">Geometry, <a href="#Page_47">47</a>, <a href="#Page_54">54</a>, <a href="#Page_119">119</a>, <a href="#Page_127">127</a>;</li>
<li class="isub1">ancient and modern methods of, <a href="#Page_110">110</a> ff.</li>
<li class="indx">Goethe, <a href="#Page_99">99</a></li>
<li class="indx">Good usage in language, <a href="#Page_100">100</a></li>
<li class="indx">Grammatical correctness, importance attached to, <a href="#Page_99">99</a></li>
<li class="indx">Grammatical rules, <a href="#Page_97">97</a></li>
<li class="indx">Gravitation potential, the, <a href="#Page_112">112</a></li>
<li class="indx">Group, the, <a href="#Page_65">65</a> ff.;</li>
<li class="isub1">double members or double points in, <a href="#Page_82">82</a>;</li>
<li class="isub1">linear arrangement of members of, <a href="#Page_75">75</a> ff.</li>
<li class="indx">Groups, artificial and natural, <a href="#Page_69">69</a> ff.;</li>
<li class="isub1">closed, among animals, <a href="#Page_179">179</a>;</li>
<li class="isub1">infinite, equality of, <a href="#Page_84">84</a>;</li>
<li class="isub1">related, <a href="#Page_69">69</a> ff.;</li>
<li class="isub1">unequivocal order of, <a href="#Page_83">83</a></li>
<li class="ifrst">Heat, mechanical equivalent of, <a href="#Page_149">149</a>;</li>
<li class="isub1">theory of, <a href="#Page_147">147</a> ff.</li>
<li class="indx">Heat energy, <a href="#Page_148">148</a> ff.</li>
<li class="indx">Heat engine, <a href="#Page_151">151</a>;</li>
<li class="isub1">ideal, <a href="#Page_151">151</a> ff.</li>
<li class="indx">Heat quantity, <a href="#Page_148">148</a> ff.</li>
<li class="indx">Heliotrope, <a href="#Page_90">90</a></li>
<li class="indx">Herbart, <a href="#Page_102">102</a></li>
<li class="indx">Heredity, <a href="#Page_180">180</a></li>
<li class="indx">Higher analysis, <a href="#Page_111">111</a></li>
<li class="indx">Homonym, <a href="#Page_89">89</a></li>
<li class="indx">Hydromechanics, <a href="#Page_147">147</a></li>
<li class="ifrst">Ideal cases, <a href="#Page_44">44</a> ff.</li>
<li class="indx">Ideal machines, <a href="#Page_132">132</a></li>
<li class="indx">Identity, the limit case in co-ordination, <a href="#Page_82">82</a></li>
<li class="indx">Ido, 183, <a href="#Footnote_I_9">note</a></li>
<li class="indx">Imperfection, indestructible quality of science, <a href="#Page_4">4</a></li>
<li class="indx">Incompleteness, no hindrance to efficiency of science, <a href="#Page_5">5</a></li>
<li class="indx">Indestructibility of matter, see Conservation of matter</li>
<li class="indx">Indo-Arabic notation, <a href="#Page_91">91</a></li>
<li class="indx">Induction, <a href="#Page_38">38</a>;</li>
<li class="isub1">the complete and the incomplete, <a href="#Page_39">39</a></li>
<li class="indx">Inductive sciences, <a href="#Page_38">38</a></li>
<li class="indx"><span class="pagenum"><a name="Page_190" id="Page_190">[190]</a></span>Inference, by induction, <a href="#Page_38">38</a>;</li>
<li class="isub1">from analogy, <a href="#Page_140">140</a></li>
<li class="indx">Infinitesimal analysis, <a href="#Page_111">111</a></li>
<li class="indx">Inorganic world, lack of memory and foresight in, <a href="#Page_33">33</a></li>
<li class="indx">Insoluble problems, <a href="#Page_142">142</a></li>
<li class="indx">Instinctive action, <a href="#Page_174">174</a></li>
<li class="indx">Intercourse, language and, <a href="#Page_182">182</a> ff.</li>
<li class="indx">Isolation among plants and animals, <a href="#Page_179">179</a></li>
<li class="indx">Isomeric, <a href="#Page_74">74</a></li>
<li class="indx">Isomeric changes, <a href="#Page_161">161</a></li>
<li class="ifrst">Judgments, analytic, <a href="#Page_66">66</a></li>
<li class="ifrst">Kant, <a href="#Page_44">44</a>, <a href="#Page_66">66</a>, <a href="#Page_105">105</a></li>
<li class="indx">Kepler, <a href="#Page_141">141</a></li>
<li class="indx">Kinetic energy, <a href="#Page_132">132</a>;</li>
<li class="isub1">and work, their sum constant, <a href="#Page_133">133</a> ff.;</li>
<li class="isub1">transformed into work and <i>vice versa</i>, <a href="#Page_134">134</a></li>
<li class="indx">Knowledge, aim of, <a href="#Page_19">19</a>;</li>
<li class="isub1">individual, compared to telephone, <a href="#Page_7">7</a> ff.;</li>
<li class="isub1">limited, <a href="#Page_31">31</a>;</li>
<li class="isub1">possibility of error in, ineradicable, <a href="#Page_40">40</a>;</li>
<li class="isub1">social character of, <a href="#Page_183">183</a></li>
<li class="ifrst">Language, beginnings of, <a href="#Page_88">88</a>;</li>
<li class="isub1">defective in co-ordination, <a href="#Page_96">96</a>;</li>
<li class="isub1">distinction between science and knowledge of, <a href="#Page_98">98</a>;</li>
<li class="isub1">good usage in, <a href="#Page_100">100</a>;</li>
<li class="isub1">and intercourse, <a href="#Page_182">182</a> ff.;</li>
<li class="isub1">needless inflections in, <a href="#Page_99">99</a> ff.;</li>
<li class="isub1">of words more imperfect than written language, <a href="#Page_92">92</a>;</li>
<li class="isub1">purpose of its cultivation, <a href="#Page_99">99</a>;</li>
<li class="isub1">the science of, <a href="#Page_97">97</a> ff.;</li>
<li class="isub1">unambiguity the ideal of, <a href="#Page_89">89</a>;</li>
<li class="isub1">a universal auxiliary, <a href="#Page_100">100</a>;</li>
<li class="isub1">written, <a href="#Page_89">89</a> ff.</li>
<li class="indx">Leibnitz, <a href="#Page_88">88</a>;</li>
<li class="isub1">his doctrine of pre-established harmony, <a href="#Page_143">143</a>;</li>
<li class="isub1">inventor of differentials, <a href="#Page_112">112</a></li>
<li class="indx">Life, <a href="#Page_163">163</a> ff.;</li>
<li class="isub1">the most general concept in the biological sciences, <a href="#Page_56">56</a></li>
<li class="indx">Light, <a href="#Page_5">5</a>, <a href="#Page_156">156</a> ff.</li>
<li class="indx">Liquids, <a href="#Page_145">145</a></li>
<li class="indx">Locke, John, <a href="#Page_21">21</a> ff., <a href="#Page_88">88</a>;</li>
<li class="isub1">his elaboration of the notion of simple and complex "ideas," <a href="#Page_21">21</a>;</li>
<li class="isub1">his secondary qualities, <a href="#Page_127">127</a></li>
<li class="indx">Logic, <a href="#Page_54">54</a>, <a href="#Page_67">67</a> ff.;</li>
<li class="isub1">content of, <a href="#Page_19">19</a>;</li>
<li class="isub1">definition of, <a href="#Page_15">15</a> ff.</li>
<li class="indx">Luther, <a href="#Page_99">99</a></li>
<li class="ifrst">Magnetism, electricity and, <a href="#Page_154">154</a> ff.</li>
<li class="indx">Man, compared to pair of sieves, <a href="#Page_34">34</a>;</li>
<li class="isub1">his capacity for perfection, <a href="#Page_180">180</a></li>
<li class="indx">Manifold, the science of the, <a href="#Page_54">54</a></li>
<li class="indx">Mass, <a href="#Page_132">132</a> ff., <a href="#Page_136">136</a> ff.;</li>
<li class="isub1">a substance, <a href="#Page_138">138</a></li>
<li class="indx">Mathematical laws, accuracy of, <a href="#Page_105">105</a></li>
<li class="indx">Mathematics, <a href="#Page_54">54</a>;</li>
<li class="isub1">an empirical science, <a href="#Page_55">55</a>;</li>
<li class="isub1">influence on, of concept of continuity, <a href="#Page_111">111</a>;</li>
<li class="isub1">its progress after introduction of Indo-Arabic numerals and algebraic signs, <a href="#Page_101">101</a></li>
<li class="indx">Matter, definition of, <a href="#Page_138">138</a></li>
<li class="indx">Mayer, Julius Robert, <a href="#Page_149">149</a>;</li>
<li class="isub1">his discovery of the law of conservation, <a href="#Page_151">151</a></li>
<li class="indx">Measurement, <a href="#Page_107">107</a></li>
<li class="indx">Mechanical energies, <a href="#Page_144">144</a></li>
<li class="indx">Mechanics, <a href="#Page_55">55</a>, <a href="#Page_128">128</a> ff.;</li>
<li class="isub1">complementary branches of, <a href="#Page_144">144</a> ff.;</li>
<li class="isub1">definition of, <a href="#Page_138">138</a>;</li>
<li class="isub1">early development of, <a href="#Page_139">139</a>;</li>
<li class="isub1">energetic, <a href="#Page_138">138</a> ff.;</li>
<li class="isub1">the first branch of physics treated mathematically, <a href="#Page_139">139</a>;</li>
<li class="isub1">pure or classical, <a href="#Page_144">144</a></li>
<li class="indx">Mechanistic hypothesis, the, as an interpretation of all
<span class="pagenum"><a name="Page_191" id="Page_191">[191]</a></span>natural phenomena, <a href="#Page_142">142</a>;</li>
<li class="isub1">especially injurious in study of mental phenomena, <a href="#Page_142">142</a></li>
<li class="indx">Mechanistic theories, <a href="#Page_140">140</a> ff.</li>
<li class="indx">Mechanistic theory of the universe, <a href="#Page_132">132</a></li>
<li class="indx">Mechanization of astronomy, <a href="#Page_141">141</a></li>
<li class="indx">Memory, <a href="#Page_16">16</a>, <a href="#Page_32">32</a>, <a href="#Page_180">180</a>;</li>
<li class="isub1">definition of, <a href="#Page_172">172</a>;</li>
<li class="isub1">general characteristic of, <a href="#Page_61">61</a>;</li>
<li class="isub1">lack of, in inorganic world, <a href="#Page_53">53</a></li>
<li class="indx">Metabolism, <a href="#Page_165">165</a></li>
<li class="indx">Methodology of the sciences based upon co-ordination, <a href="#Page_85">85</a></li>
<li class="indx">Microscope, <a href="#Page_6">6</a></li>
<li class="indx">Motion, the science of, <a href="#Page_54">54</a>, <a href="#Page_122">122</a>;</li>
<li class="isub1">uninfluenced, <a href="#Page_122">122</a></li>
<li class="indx">Musical notation, <a href="#Page_93">93</a></li>
<li class="ifrst">Names, arbitrariness of, <a href="#Page_62">62</a>;</li>
<li class="isub1">signs and, <a href="#Page_86">86</a> ff.</li>
<li class="indx">Natural laws, <a href="#Page_28">28</a> ff.;</li>
<li class="isub1">definition of, <a href="#Page_28">28</a>;</li>
<li class="isub1">their extent dependent upon stage of knowledge in each science, <a href="#Page_7">7</a>;</li>
<li class="isub1">their usual origin, <a href="#Page_42">42</a> ff.;</li>
<li class="isub1">prediction from, only approximate, <a href="#Page_48">48</a></li>
<li class="indx">Natural philosophy, definition of, <a href="#Page_1">1</a>;</li>
<li class="isub1">importance of, in study of science, <a href="#Page_10">10</a>;</li>
<li class="isub1">place of, in text-books, <a href="#Page_9">9</a> ff.</li>
<li class="indx">Negation, <a href="#Page_68">68</a> ff.</li>
<li class="indx">Nerves, <a href="#Page_177">177</a></li>
<li class="indx">Nervous discharge, <a href="#Page_177">177</a></li>
<li class="indx">Newton, Sir Isaac, <a href="#Page_141">141</a></li>
<li class="indx">Number groups, unlimited, <a href="#Page_78">78</a></li>
<li class="indx">Numbers, <a href="#Page_78">78</a> ff.;</li>
<li class="isub1">theory of, <a href="#Page_80">80</a></li>
<li class="indx">Numerals, co-ordination of, with signs, <a href="#Page_86">86</a></li>
<li class="indx">Numerical names different in different languages, <a href="#Page_86">86</a></li>
<li class="indx">Numerical signs international, <a href="#Page_86">86</a></li>
<li class="indx">Nutrition, <a href="#Page_165">165</a></li>
<li class="ifrst">Objective, astronomic, <a href="#Page_6">6</a>;</li>
<li class="isub1">photographic, <a href="#Page_6">6</a></li>
<li class="indx">Objective character of the world, <a href="#Page_34">34</a></li>
<li class="indx">Optical telegraph, <a href="#Page_90">90</a></li>
<li class="indx">Optics, geometric, <a href="#Page_5">5</a></li>
<li class="indx">Optic signs, <a href="#Page_90">90</a></li>
<li class="indx">Order, most general concept in formal sciences, <a href="#Page_56">56</a></li>
<li class="indx">Organisms, standard for measuring relative perfection of, <a href="#Page_176">176</a>;</li>
<li class="isub1">stationary forms, <a href="#Page_163">163</a></li>
<li class="indx">Orthography, efforts to improve, <a href="#Page_99">99</a>;</li>
<li class="isub1">English, defective in co-ordination, <a href="#Page_96">96</a>;</li>
<li class="isub1">exaggerated importance of correctness in, <a href="#Page_99">99</a>;</li>
<li class="isub1">mistakes in, <a href="#Page_97">97</a>;</li>
<li class="isub1">reform of, <a href="#Page_97">97</a></li>
<li class="ifrst">Parabolic curve, <a href="#Page_48">48</a></li>
<li class="indx">Paradoxes of the infinite, <a href="#Page_84">84</a></li>
<li class="indx">Pasigraphy, <a href="#Page_92">92</a> ff.;</li>
<li class="isub1">Chinese system of, <a href="#Page_94">94</a></li>
<li class="indx">Permanent in change, the, <a href="#Page_131">131</a></li>
<li class="indx">Perpetual motion, <a href="#Page_130">130</a></li>
<li class="indx">Perpetual motion machine, <a href="#Page_153">153</a></li>
<li class="indx">Philology, <a href="#Page_97">97</a> ff.</li>
<li class="indx">Philosophy, limited progress in, <a href="#Page_101">101</a></li>
<li class="indx">Phonetic writing, <a href="#Page_33">33</a> ff.</li>
<li class="indx">Phoronomy, <a href="#Page_54">54</a>, <a href="#Page_119">119</a>, <a href="#Page_122">122</a>, <a href="#Page_127">127</a></li>
<li class="indx">Photochemical processes, foundation of terrestial life, <a href="#Page_169">169</a></li>
<li class="indx">Photographic objective, <a href="#Page_6">6</a></li>
<li class="indx">Physical sciences, <a href="#Page_55">55</a></li>
<li class="indx">Physics, <a href="#Page_47">47</a>, <a href="#Page_55">55</a>;</li>
<li class="isub1">each branch of, <span class="pagenum"><a name="Page_192" id="Page_192">[192]</a></span>treats of a special kind of energy, <a href="#Page_139">139</a></li>
<li class="isub1">the science of the different kinds of energy, <a href="#Page_72">72</a>;</li>
<li class="indx">Physiology, <a href="#Page_55">55</a> ff.</li>
<li class="indx">Plato, his distinction between mental and physical life, <a href="#Page_178">178</a></li>
<li class="indx">Polarity of electricity and magnetism, <a href="#Page_155">155</a></li>
<li class="indx">Political organizations, conflicts between, <a href="#Page_185">185</a></li>
<li class="indx">Prediction, <a href="#Page_12">12</a></li>
<li class="indx">Pre-established harmony, <a href="#Page_143">143</a></li>
<li class="indx">Pressure, <a href="#Page_146">146</a>, <a href="#Page_154">154</a></li>
<li class="indx">Progress, depends on collective labor, <a href="#Page_185">185</a>;</li>
<li class="isub1">economy of energy, <a href="#Page_184">184</a>;</li>
<li class="isub1">evaluation of, <a href="#Page_176">176</a></li>
<li class="indx">Pseudo-problems in science, <a href="#Page_142">142</a></li>
<li class="indx">Psychology, <a href="#Page_47">47</a>, <a href="#Page_55">55</a> ff.</li>
<li class="indx">Psycho-physical parallelism, <a href="#Page_143">143</a></li>
<li class="indx">Ptolemy's system, <a href="#Page_117">117</a></li>
<li class="indx">Pure science, <a href="#Page_57">57</a></li>
<li class="ifrst">Quantity, the science of, see Mathematics, <a href="#Page_54">54</a></li>
<li class="ifrst">Radiant energy, <a href="#Page_157">157</a>;</li>
<li class="isub1">its importance to man, <a href="#Page_158">158</a></li>
<li class="indx">Rational sciences, see Deductive sciences</li>
<li class="indx">Rays, straight lines of, <a href="#Page_5">5</a></li>
<li class="indx">Reaction, teleological, <a href="#Page_173">173</a></li>
<li class="indx">Reality, <a href="#Page_16">16</a> ff.</li>
<li class="indx">Reflection, <a href="#Page_5">5</a></li>
<li class="indx">Reflex action, <a href="#Page_173">173</a></li>
<li class="indx">Refraction, <a href="#Page_5">5</a></li>
<li class="indx">Repetition, basis of conscious life, <a href="#Page_174">174</a></li>
<li class="indx">Reproduction, <a href="#Page_165">165</a> ff.</li>
<li class="indx">Roman notation, <a href="#Page_91">91</a></li>
<li class="ifrst">Science, aim of, <a href="#Page_13">13</a> ff.;</li>
<li class="isub1">comparison of, to a network, <a href="#Page_42">42</a>;</li>
<li class="isub1">comparison of, to a tree or forest, <a href="#Page_6">6</a>;</li>
<li class="isub1">definition of, <a href="#Page_13">13</a>;</li>
<li class="isub1">eternal truth of, <a href="#Page_6">6</a> ff.;</li>
<li class="isub1">"for its own sake," <a href="#Page_13">13</a> ff.;</li>
<li class="isub1">the facts of, unalterable, <a href="#Page_8">8</a> ff.;</li>
<li class="isub1">the function of, <a href="#Page_23">23</a>, <a href="#Page_37">37</a>;</li>
<li class="isub1">importance of theoretical, <a href="#Page_15">15</a>;</li>
<li class="isub1">its procedure, <a href="#Page_45">45</a>;</li>
<li class="isub1">the study of happiness, <a href="#Page_28">28</a></li>
<li class="indx">Sciences, the table of the, <a href="#Page_54">54</a> ff.</li>
<li class="indx">Scientific discoveries, independent simultaneous, <a href="#Page_185">185</a></li>
<li class="indx">Scientific instinct, <a href="#Page_43">43</a></li>
<li class="indx">Scientific materialism, <a href="#Page_138">138</a></li>
<li class="indx">Scientific written language based on direct co-ordination, <a href="#Page_93">93</a></li>
<li class="indx">Self-preservation, <a href="#Page_180">180</a></li>
<li class="indx">Sense organs, <a href="#Page_176">176</a> ff.</li>
<li class="indx">Shakespeare, <a href="#Page_99">99</a></li>
<li class="indx">Signs and names, <a href="#Page_86">86</a> ff.</li>
<li class="indx">Social characteristics, importance of, <a href="#Page_179">179</a> ff.</li>
<li class="indx">Social classes, conflicts between, <a href="#Page_185">185</a></li>
<li class="indx">Socialization of human capacities, <a href="#Page_180">180</a></li>
<li class="indx">Social order still barbarous, <a href="#Page_185">185</a></li>
<li class="indx">Social organization, <a href="#Page_180">180</a>;</li>
<li class="isub1">how best obtained, <a href="#Page_182">182</a>;</li>
<li class="isub1">its tendency to equalize conditions, <a href="#Page_185">185</a>;</li>
<li class="isub1">secures permanence among specialized individuals, <a href="#Page_181">181</a></li>
<li class="indx">Social problems, <a href="#Page_179">179</a> ff.</li>
<li class="indx">Society, <a href="#Page_179">179</a> ff.;</li>
<li class="isub1">centrifugal and centripetal forces in, <a href="#Page_181">181</a> ff.;</li>
<li class="isub1">division of functions in, <a href="#Page_181">181</a></li>
<li class="indx">Sociology, <a href="#Page_47">47</a>, <a href="#Page_55">55</a>, <a href="#Page_57">57</a></li>
<li class="indx">Solar radiation, <a href="#Page_169">169</a></li>
<li class="indx">Soul, the, <a href="#Page_171">171</a> ff.</li>
<li class="indx">Sound signs, advantage and disadvantage of, <a href="#Page_89">89</a> ff.</li>
<li class="indx">Sound writing, <a href="#Page_33">33</a> ff., <a href="#Page_92">92</a> ff.</li>
<li class="indx">Space, four-dimensional, 77, <a href="#Footnote_D_4">note</a>;</li>
<li class="isub1">homogeneity of, in <span class="pagenum"><a name="Page_193" id="Page_193">[193]</a></span>horizontal direction, <a href="#Page_121">121</a>;</li>
<li class="isub1">the science of, <a href="#Page_54">54</a>;</li>
<li class="isub1">symmetrical and tri-dimensional, <a href="#Page_118">118</a>;</li>
<li class="isub1">time and, <a href="#Page_118">118</a> ff.;</li>
<li class="isub1">tri-dimensional, <a href="#Page_76">76</a></li>
<li class="indx">Specialization, one-sidedness of, <a href="#Page_180">180</a> ff.</li>
<li class="indx">Spelling reform, <a href="#Page_97">97</a></li>
<li class="indx">Stable forms, <a href="#Page_163">163</a></li>
<li class="indx">Statics, <a href="#Page_128">128</a> ff.;</li>
<li class="isub1">definition of, <a href="#Page_138">138</a> ff.</li>
<li class="indx">Stationary bodies, capable of regeneration, <a href="#Page_164">164</a></li>
<li class="indx">Stationary forms, <a href="#Page_163">163</a></li>
<li class="indx">Substance, <a href="#Page_132">132</a></li>
<li class="indx">Surface-energy, <a href="#Page_146">146</a></li>
<li class="indx">Syllogism, the, classic method of argumentation, <a href="#Page_65">65</a> ff.</li>
<li class="indx">Synonym, <a href="#Page_89">89</a></li>
<li class="ifrst">Table of the sciences, <a href="#Page_54">54</a> ff.</li>
<li class="indx">Telegraph, optical, <a href="#Page_90">90</a></li>
<li class="indx">"Teleological" properties of organisms, <a href="#Page_173">173</a></li>
<li class="indx">Teleological reaction, <a href="#Page_173">173</a></li>
<li class="indx">Telescope, <a href="#Page_5">5</a></li>
<li class="indx">Temperature, <a href="#Page_148">148</a></li>
<li class="indx">Theoretical science, importance of, <a href="#Page_15">15</a></li>
<li class="indx">Theory of functions, <a href="#Page_111">111</a></li>
<li class="indx">Theory of numbers, <a href="#Page_80">80</a></li>
<li class="indx">Thermo-chemistry, <a href="#Page_37">37</a></li>
<li class="indx">Thermo-dynamics, <a href="#Page_153">153</a></li>
<li class="indx">Thing, definition of, <a href="#Page_62">62</a> ff.</li>
<li class="indx">Thought conceived of as energetic, <a href="#Page_178">178</a></li>
<li class="indx">Threshold, <a href="#Page_102">102</a></li>
<li class="indx">Time, a form of inner life, <a href="#Page_76">76</a>;</li>
<li class="isub1">measurement of, <a href="#Page_122">122</a>;</li>
<li class="isub1">one-seried, or one-dimensional, <a href="#Page_118">118</a>;</li>
<li class="isub1">and space, <a href="#Page_118">118</a> ff.</li>
<li class="ifrst">Unambiguity, in language, <a href="#Page_89">89</a>;</li>
<li class="isub1">of co-ordination of numbers to signs, <a href="#Page_90">90</a></li>
<li class="indx">Universal auxiliary language, <a href="#Page_100">100</a>, <a href="#Page_183">183</a></li>
<li class="ifrst">Velocity, <a href="#Page_133">133</a></li>
<li class="indx">Volume energy, <a href="#Page_145">145</a></li>
<li class="ifrst">War, <a href="#Page_185">185</a></li>
<li class="indx">Wave surface, <a href="#Page_6">6</a></li>
<li class="indx">Wave theory of light, <a href="#Page_5">5</a>, <a href="#Page_157">157</a></li>
<li class="indx">Weight, <a href="#Page_132">132</a>, <a href="#Page_137">137</a> ff.;</li>
<li class="isub1">a substance, <a href="#Page_138">138</a></li>
<li class="indx">Work, mechanical, <a href="#Page_129">129</a>;</li>
<li class="isub1">product of the force and the distance, <a href="#Page_130">130</a>;</li>
<li class="isub1">a substance in a limited sense, <a href="#Page_136">136</a></li>
<li class="indx">Written language, <a href="#Page_89">89</a> ff.</li>
<li class="indx">Written signs, <a href="#Page_90">90</a></li>
<li class="ifrst">Zoology, <a href="#Page_56">56</a></li>
</ul>
<hr class = "full" />
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<p class = " p0 right">
By
<span class="smcap">Charles H. Sternberg</span>,
</p>
<p class = "p0">With introduction by Prof. H. F. Osborn, 48 Illustrations,
$1.60 net.</p>
<p>The most interesting autobiography of the oldest and best
known explorer in this field.</p>
<p class = "s">"One of the most interesting books to be found anywhere."—<i>William
Allen White.</i></p>
<p class = "l"><b>THE FRESH WATER AQUARIUM
AND ITS INHABITANTS</b></p>
<p class = "p0 right">
By <span class="smcap">Otto Eggeling</span> and
<span class="smcap">Frederick Ehrenberg</span>.
</p>
<p>A Guide for the Amateur Aquarist. With 100 illustrations,
large 12mo, $2.00 net.</p>
<p class = "s">"The best guide to the aquarium."—<i>The Independent.</i></p>
<hr class = "full" />
<p class = "xll center wide">
HENRY HOLT AND COMPANY</p>
<p class = "center break-after">PUBLISHERS NEW YORK
</p>
<hr class = "full" />
<p class = "xll center">THE THEORY OF THE THEATRE</p>
<p class = "p0 center"><b>And Other Principles of Dramatic Criticism</b></p>
<p class = "center">By <span class="smcap">Clayton Hamilton</span>. Author of "Materials and Methods
of Fiction." $1.50 net; by mail, $1.60.</p>
<p class = "center">CONTENTS:</p>
<p class = "s"><span class="smcap">The Theory of the Theatre.</span>—What is a Play?—The Psychology
of Theatre Audiences.—The Actor and the Dramatist.—Stage Conventions
in Modern Times.—Economy of Attention in Theatrical Performances.—Emphasis
in the Drama.—The Four Leading Types of
Drama: Tragedy and Melodrama; Comedy and Farce.—The Modern
Social Drama.</p>
<p class = "s"><span class="smcap">Other Principles of Dramatic Criticism.</span>—The Public and the
Dramatist.—Dramatic Art and the Theatre Business.—The Happy Endings
in the Theatre.—The Boundaries of Approbation.—Imitation and
Suggestion in the Drama.—Holding the Mirror up to Nature.—Blank
Verse on the Contemporary Stage.—Dramatic Literature and Theatric
Journalism.—The Intention of Performance.—The Quality of New
Endeavor.—The Effect of Plays upon the Public.—Pleasant and Unpleasant
Plays.—Themes in the Theatre.—The Function of Imagination.</p>
<p class = "xll center">DRAMATISTS OF TO-DAY</p>
<p class = "center"><span class="smcap">Rostand</span>, <span class="smcap">Hauptmann</span>, <span class="smcap">Sudermann</span>,
<span class="smcap">Pinero</span>, <span class="smcap">Shaw</span>, <span class="smcap">Phillips</span>, <span class="smcap">Maeterlinck</span></p>
<p class = "center">By <span class="smcap">Prof. Edward Everett Hale, Jr.</span>, of Union College. With
gilt top, $1.50 net. (By mail, $1.60.)</p>
<p class = "s">An informal discussion of their principal plays and of the performances
of some of them. The volume opens with a paper "On Standards
of Criticism," and concludes with "Our Idea of Tragedy," and
an appendix of all the plays of each author, with dates of their first
performance or publication.</p>
<p class = "s"><i>New York Evening Post:</i> "It is not often nowadays that a theatrical
book can be met with so free from gush and mere eulogy, or so
weighted by common sense ... an excellent chronological appendix
and full index ... uncommonly useful for reference."</p>
<p class = "s"><i>Dial:</i> "Noteworthy example of literary criticism in one of the
most interesting of literary fields.... Well worth reading a second
time."</p>
<p class = "xll center">THE GERMAN DRAMA OF THE
NINETEENTH CENTURY</p>
<p class = "center">By <span class="smcap">Georg Witkowski</span>. Translated by <span class="smcap">Prof. L. E. Horning</span>.
12mo. $1.00.</p>
<p class = "s">Kleist, Grillparzer, Hebbel, Ludwig, Wildenbruch, Sudermann, Hauptmann,
and minor dramatists receive attention.</p>
<p class = "s"><i>New York Times Review:</i> "The translation of this brief, clear, and
logical account was an extremely happy idea. Nothing at the same time
so comprehensive and terse has appeared on the subject, and it is a
subject of increasing interest to the English-speaking public."</p>
<hr class = "full" />
<p class = "xll center wide">
HENRY HOLT AND COMPANY</p>
<p class = "center break-after">PUBLISHERS NEW YORK
</p>
<hr class = "full" />
<hr class = "full" />
<div class = "transnote">
<p>Transcriber's notes</p>
<p>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.</p>
<p>Typographical errors corrected:</p>
<ul class = "tn"><li>p. 100: approprate changed to appropriate (... to a more appropriate evaluation ...).</li>
<li>p. 108: meassure changed to measure (By the application of the unit measure ...).</li>
<li>p. 184: correspondng changed to corresponding (... signifies a corresponding economy ...).</li>
<li>p. 191: A single period deleted from index.</li>
<li>p. 188, 189: limit-case changed to limit case (2 occurrences), to mirror text (3 occurrences).</li>
</ul>
<p>Alphabetical sequencing adjusted in index:</p>
<ul class = "tn"><li>p. 189: Two 'Energy' entries moved after Energetic mechanics</li>
<li>p. 191: Photographic objective moved below Photochemical processes.</li>
<li>p. 191: Physics: The order of the sub-entries swapped.</li>
<li>p. 192: Pure science moved down four places to end of "P" entries.</li>
<li>p. 193: Two 'Teleological' entries moved after Telegraph, optical</li>
</ul>
<p>Index references to footnotes link to the footnote, not the original page. The return link, however, goes to the original page.</p>
</div>
<div>*** END OF THE PROJECT GUTENBERG EBOOK 43791 ***</div>
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