Relativity: The Special and General Theory HTML version

Part I: The Special Theory of Relativity
building of Euclid's geometry, and you remember — perhaps with more respect than love
— the magnificent structure, on the lofty staircase of which you were chased about for
uncounted hours by conscientious teachers. By reason of our past experience, you would
certainly regard everyone with disdain who should pronounce even the most out-of-the-
way proposition of this science to be untrue. But perhaps this feeling of proud certainty
would leave you immediately if some one were to ask you: "What, then, do you mean by
the assertion that these propositions are true?" Let us proceed to give this question a little
consideration.
Geometry sets out form certain conceptions such as "plane," "point," and "straight line,"
with which we are able to associate more or less definite ideas, and from certain simple
propositions (axioms) which, in virtue of these ideas, we are inclined to accept as "true."
Then, on the basis of a logical process, the justification of which we feel ourselves
compelled to admit, all remaining propositions are shown to follow from those axioms,
i.e. they are proven. A proposition is then correct ("true") when it has been derived in the
recognised manner from the axioms. The question of "truth" of the individual geometrical
propositions is thus reduced to one of the "truth" of the axioms. Now it has long been
known that the last question is not only unanswerable by the methods of geometry, but
that it is in itself entirely without meaning. We cannot ask whether it is true that only one
straight line goes through two points. We can only say that Euclidean geometry deals
with things called "straight lines," to each of which is ascribed the property of being
uniquely determined by two points situated on it. The concept "true" does not tally with
the assertions of pure geometry, because by the word "true" we are eventually in the habit
of designating always the correspondence with a "real" object; geometry, however, is not
concerned with the relation of the ideas involved in it to objects of experience, but only
with the logical connection of these ideas among themselves.
It is not difficult to understand why, in spite of this, we feel constrained to call the
propositions of geometry "true." Geometrical ideas correspond to more or less exact
objects in nature, and these last are undoubtedly the exclusive cause of the genesis of
those ideas. Geometry ought to refrain from such a course, in order to give to its structure
the largest possible logical unity. The practice, for example, of seeing in a "distance" two
marked positions on a practically rigid body is something which is lodged deeply in our
habit of thought. We are accustomed further to regard three points as being situated on a
straight line, if their apparent positions can be made to coincide for observation with one
eye, under suitable choice of our place of observation.
If, in pursuance of our habit of thought, we now supplement the propositions of
Euclidean geometry by the single proposition that two points on a practically rigid body
always correspond to the same distance (line-interval), independently of any changes in
position to which we may subject the body, the propositions of Euclidean geometry then
resolve themselves into propositions on the possible relative position of practically rigid
bodies.1) Geometry which has been supplemented in this way is then to be treated as a
branch of physics. We can now legitimately ask as to the "truth" of geometrical
propositions interpreted in this way, since we are justified in asking whether these
propositions are satisfied for those real things we have associated with the geometrical
ideas. In less exact terms we can express this by saying that by the "truth" of a