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Page 6
This view of axioms, advocated by modern axiomatics, purges mathematics
of all extraneous elements, and thus dispels the mystic obscurity
which formerly surrounded the principles of mathematics.
But a presentation of its principles thus clarified makes it also
evident that mathematics as such cannot predicate anything about
perceptual objects or real objects. In axiomatic geometry the words
"point," "straight line," etc., stand only for empty conceptual
schemata. That which gives them substance is not relevant to
mathematics.
Yet on the other hand it is certain that mathematics generally,
and particularly geometry, owes its existence to the need which
was felt of learning something about the relations of real things
to one another. The very word geometry, which, of course, means
earth-measuring, proves this. For earth-measuring has to do with
the possibilities of the disposition of certain natural objects
with respect to one another, namely, with parts of the earth,
measuring-lines, measuring-wands, etc. It is clear that the system
of concepts of axiomatic geometry alone cannot make any assertions
as to the relations of real objects of this kind, which we will
call practically-rigid bodies. To be able to make such assertions,
geometry must be stripped of its merely logical-formal character
by the co-ordination of real objects of experience with the empty
conceptual frame-work of axiomatic geometry. To accomplish this,
we need only add the proposition:--Solid bodies are related, with
respect to their possible dispositions, as are bodies in Euclidean
geometry of three dimensions. Then the propositions of Euclid contain
affirmations as to the relations of practically-rigid bodies.
Geometry thus completed is evidently a natural science; we may in
fact regard it as the most ancient branch of physics. Its affirmations
rest essentially on induction from experience, but not on logical
inferences only. We will call this completed geometry "practical
geometry," and shall distinguish it in what follows from "purely
axiomatic geometry." The question whether the practical geometry
of the universe is Euclidean or not has a clear meaning, and its
answer can only be furnished by experience. All linear measurement
in physics is practical geometry in this sense, so too is geodetic
and astronomical linear measurement, if we call to our help the
law of experience that light is propagated in a straight line, and
indeed in a straight line in the sense of practical geometry.
I attach special importance to the view of geometry which I
have just set forth, because without it I should have been unable
to formulate the theory of relativity. Without it the following
reflection would have been impossible:--In a system of reference
rotating relatively to an inert system, the laws of disposition of
rigid bodies do not correspond to the rules of Euclidean geometry
on account of the Lorentz contraction; thus if we admit non-inert
systems we must abandon Euclidean geometry. The decisive step in
the transition to general co-variant equations would certainly not
have been taken if the above interpretation had not served as a
stepping-stone. If we deny the relation between the body of axiomatic
Euclidean geometry and the practically-rigid body of reality,
we readily arrive at the following view, which was entertained by
that acute and profound thinker, H. Poincare:--Euclidean geometry
is distinguished above all other imaginable axiomatic geometries
by its simplicity. Now since axiomatic geometry by itself contains
no assertions as to the reality which can be experienced, but can
do so only in combination with physical laws, it should be possible
and reasonable--whatever may be the nature of reality--to retain
Euclidean geometry. For if contradictions between theory and
experience manifest themselves, we should rather decide to change
physical laws than to change axiomatic Euclidean geometry. If we
deny the relation between the practically-rigid body and geometry,
we shall indeed not easily free ourselves from the convention
that Euclidean geometry is to be retained as the simplest. Why
is the equivalence of the practically-rigid body and the body of
geometry--which suggests itself so readily--denied by Poincare and
other investigators? Simply because under closer inspection the
real solid bodies in nature are not rigid, because their geometrical
behaviour, that is, their possibilities of relative disposition,
depend upon temperature, external forces, etc. Thus the original,
immediate relation between geometry and physical reality appears
destroyed, and we feel impelled toward the following more general
view, which characterizes Poincare's standpoint. Geometry (G)
predicates nothing about the relations of real things, but only
geometry together with the purport (P) of physical laws can do so.
Using symbols, we may say that only the sum of (G) + (P) is subject
to the control of experience. Thus (G) may be chosen arbitrarily,
and also parts of (P); all these laws are conventions. All that
is necessary to avoid contradictions is to choose the remainder of
(P) so that (G) and the whole of (P) are together in accord with
experience. Envisaged in this way, axiomatic geometry and the part
of natural law which has been given a conventional status appear
as epistemologically equivalent.
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