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%%% Primary Title: Physical Meaning of Geometrical Propositions
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\begin{document}
\subsection{Physical Meaning of Geometrical Propositions}
From \htmladdnormallink{Relativity: The Special and General Theory}{http://planetphysics.us/encyclopedia/SpecialTheoryOfRelativity.html} by \htmladdnormallink{Albert Einstein}{http://planetphysics.us/encyclopedia/AlbertEinstein.html}
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 \htmladdnormallink{proposition}{http://planetphysics.us/encyclopedia/Predicate.html} 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 \htmladdnormallink{simple propositions}{http://planetphysics.us/encyclopedia/PositiveProposition.html} (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, {\it 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 \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} ``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" \htmladdnormallink{object}{http://planetphysics.us/encyclopedia/TrivialGroupoid.html}; geometry,
however, is not concerned with the \htmladdnormallink{relation}{http://planetphysics.us/encyclopedia/Bijective.html} 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 \htmladdnormallink{positions}{http://planetphysics.us/encyclopedia/Position.html} on a
practically \htmladdnormallink{rigid body}{http://planetphysics.us/encyclopedia/CenterOfGravity.html} 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.\footnotemark\ 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 geometrical
proposition in this sense we understand its validity for a
construction with rule and compasses.
Of course the conviction of the ``truth" of geometrical propositions in
this sense is founded exclusively on rather incomplete experience. For
the present we shall assume the ``truth" of the geometrical
propositions, then at a later stage (in the \htmladdnormallink{general theory}{http://planetphysics.us/encyclopedia/GeneralTheory.html} of
relativity) we shall see that this ``truth" is limited, and we shall
consider the extent of its limitation.
\footnotetext[1]{It follows that a natural object is associated also with a
straight line. Three points A, B and C on a rigid body thus lie in a
straight line when the points A and C being given, B is chosen such
that the sum of the distances AB and BC is as short as possible. This
incomplete suggestion will suffice for the present purpose.}
\subsection{References}
This article is derived from the Einstein Reference Archive (marxists.org) 1999, 2002. \htmladdnormallink{Einstein Reference Archive}{http://www.marxists.org/reference/archive/einstein/index.htm} which is under the FDL copyright.
\end{document}
