If we could really determine the entropy of the world it would represent a true, absolute measure of time. In this way is best seen the utter tautology of a statement that the entropy of the world increases with the time. Time, and the fact that certain changes take place only in a definite sense, are one and the same thing.
VII. THE SOURCES OF THE PRINCIPLE OF ENERGY.
We are now prepared to answer the question, What are the sources of the principle of energy? All knowledge of nature is derived in the last instance from experience. In this sense they are right who look upon the principle of energy as a result of experience.
Experience teaches that the sense-elements αβγδ... into which the world may be decomposed, are subject to change. It tells us further, that certain of these elements are connected with other elements, so that they appear and disappear together; or, that the appearance of the elements of one class is connected with the disappearance of the elements of the other class. We will avoid here the notions of cause and effect because of their obscurity and equivocalness. The result of experience may be expressed as follows: The sensuous elements of the world (αβγδ...) show themselves to be interdependent. This interdependence is best represented by some such conception as is in geometry that of the mutual dependence of the sides and angles of a triangle, only much more varied and complex.
As an example, we may take a mass of gas enclosed in a cylinder and possessed of a definite volume (α), which we change by a pressure (β) on the piston, at the same time feeling the cylinder with our hand and receiving a sensation of heat (γ). Increase of pressure diminishes the volume and increases the sensation of heat.
The various facts of experience are not in all respects alike. Their common sensuous elements are placed in relief by a process of abstraction and thus impressed upon the memory. In this way the expression is obtained of the features of agreement of extensive groups of facts. The simplest sentence which we can utter is, by the very nature of language, an abstraction of this kind. But account must also be taken of the differences of related facts. Facts may be so nearly related as to contain the same kind of a αβγ..., but the relation be such that the αβγ... of the one differ from the αβγ... of the other only by the number of equal parts into which they can be divided. Such being the case, if rules can be given for deducing from one another the numbers which are the measures of these αβγ..., then we possess in such rules the most general expression of a group of facts, as also that expression which corresponds to all its differences. This is the goal of quantitative investigation.
If this goal be reached what we have found is that between the αβγ... of a group of facts, or better, between the numbers which are their measures, a number of equations exists. The simple fact of change brings it about that the number of these equations must be smaller than the number of the αβγ.... If the former be smaller by one than the latter, then one portion of the αβγ... is uniquely determined by the other portion.
The quest of relations of this last kind is the most important function of special experimental research, because we are enabled by it to complete in thought facts that are only partly given. It is self-evident that only experience can ascertain that between the αβγ... relations exist and of what kind they are. Further, only experience can tell that the relations that exist between the αβγ... are such that changes of them can be reversed. If this were not the fact all occasion for the enunciation of the principle of energy, as is easily seen, would be wanting. In experience, therefore, is buried the ultimate well-spring of all knowledge of nature, and consequently, in this sense, also the ultimate source of the principle of energy.
But this does not exclude the fact that the principle of energy has also a logical root, as will now be shown. Let us assume on the basis of experience that one group of sensuous elements αβγ... determines uniquely another group λμν.... Experience further teaches that changes of αβγ... can be reversed. It is then a logical consequence of this observation, that every time that αβγ... assume the same values this is also the case with λμν.... Or, that purely periodical changes of αβγ... can produce no permanent changes of λμν.... If the group λμν... is a mechanical group, then a perpetual motion is excluded.