Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

112 MATHEMATICAL PHILOSOPHY
infinitude of false ones which, though differing among
themselves psychologically in an endless variety of ways,
are yet but one in point of form, absolutely identical in
logical frame or structure. I know of no other equal
revelation of the truly amazing economic power of Logic
in our world. Think of having to live in a world where no
two doctrines, no two theories, could own an identity of
logical constitution. I suspect that in such a world there
could be no logic, no science, no philosophy, no genuine
life of intellect, no civilization.
Sense in which All Doctrines having HAF or HaF' fot
Their Matrix Are Euclidean.-The adjectives, Euclidean
and non-Euclidean, are, as you are aware, customarily
employed to designate certain types of geometry. In
this use each of the adjectives has two different meanings
-one of them very specific and common, the other
generic and less common. In order to avoid confusion
in reading geometric literature it is important to know
what the two meanings are. In its generic and less cbmmon meaning the adjective " Euclidean" is used to
designate the kind of geometry that is, in all important
or essential respects, identical with the kind found in
Euclid's Elements. Having that meaning of Euclidean
in mind, we should say that a given geometry is nonEuclidean if, for example, it is algebraic (or analytic) in
method, for the method of the Elements is that of so-called
pure (non-algebraic) geometry; or if it is a geortetry of
four or more dimensions, for that of Euclid is threedimensional; or if, like projective geometry, for example,
or inversion geometry or the so-called hyperbolic geometry
of Lobachevski or the so-called elliptic geometry of
Riemann, it uses one or more postulates iriconsistenft with
Euclid's postulates; or if, like the endless series of geomne