Sidelights on Relativity | Page 6

all ages has agitated
inquiring minds. How can it be that mathematics, being after all a
product of human thought which is independent of experience, is so
admirably appropriate to the objects of reality? Is human reason, then,
without experience, merely by taking thought, able to fathom the
properties of real things.
In my opinion the answer to this question is, briefly, this:--As far as the
laws of mathematics refer to reality, they are not certain; and as far as
they are certain, they do not refer to reality. It seems to me that
complete clearness as to this state of things first became common
property through that new departure in mathematics which is known by
the name of mathematical logic or "Axiomatics." The progress
achieved by axiomatics consists in its having neatly separated the
logical-formal from its objective or intuitive content; according to
axiomatics the logical-formal alone forms the subject-matter of
mathematics, which is not concerned with the intuitive or other content
associated with the logical-formal.
Let us for a moment consider from this point of view any axiom of
geometry, for instance, the following:--Through two points in space
there always passes one and only one straight line. How is this axiom to
be interpreted in the older sense and in the more modern sense?
The older interpretation:--Every one knows what a straight line is, and
what a point is. Whether this knowledge springs from an ability of the
human mind or from experience, from some collaboration of the two or
from some other source, is not for the mathematician to decide. He
leaves the question to the philosopher. Being based upon this
knowledge, which precedes all mathematics, the axiom stated above is,
like all other axioms, self-evident, that is, it is the expression of a part
of this a priori knowledge.
The more modern interpretation:--Geometry treats of entities which are
denoted by the words straight line, point, etc. These entities do not take
for granted any knowledge or intuition whatever, but they presuppose
only the validity of the axioms, such as the one stated above, which are
to be taken in a purely formal sense, i.e. as void of all content of

intuition or experience. These axioms are free creations of the human
mind. All other propositions of geometry are logical inferences from
the axioms (which are to be taken in the nominalistic sense only). The
matter of which geometry treats is first defined by the axioms. Schlick
in his book on epistemology has therefore characterised axioms very
aptly as "implicit definitions."
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