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Einstein

Relativity: The Special and General Theory
Albert Einstein: Relativity
Part I: The Special Theory of Relativity
Part I
The Special Theory of Relativity
Physical Meaning of Geometrical Propositions
In your schooldays most of you who read this book made acquaintance with the noble 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
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