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Page 5
Recapitulating, we may say that according to the general theory of
relativity space is endowed with physical qualities; in this sense,
therefore, there exists an ether. According to the general theory
of relativity space without ether is unthinkable; for in such space
there not only would be no propagation of light, but also no possibility
of existence for standards of space and time (measuring-rods and
clocks), nor therefore any space-time intervals in the physical
sense. But this ether may not be thought of as endowed with the
quality characteristic of ponderable media, as consisting of parts
which may be tracked through time. The idea of motion may not be
applied to it.
GEOMETRY AND EXPERIENCE
An expanded form of an Address to the Prussian Academy of Sciences
in Berlin on January 27th, 1921.
One reason why mathematics enjoys special esteem, above all other
sciences, is that its laws are absolutely certain and indisputable,
while those of all other sciences are to some extent debatable and
in constant danger of being overthrown by newly discovered facts.
In spite of this, the investigator in another department of science
would not need to envy the mathematician if the laws of mathematics
referred to objects of our mere imagination, and not to objects
of reality. For it cannot occasion surprise that different persons
should arrive at the same logical conclusions when they have already
agreed upon the fundamental laws (axioms), as well as the methods
by which other laws are to be deduced therefrom. But there is another
reason for the high repute of mathematics, in that it is mathematics
which affords the exact natural sciences a certain measure of
security, to which without mathematics they could not attain.
At this point an enigma presents itself which in 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."
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