either--but to the simplicity of its undefined notions and the high
plausibility of its unproved postulates. Bit by bit the bad logic has
been purged out of the Calculus and the Theory of Functions and these
branches of study have been made into patterns of accurate reasoning on
exactly stated premisses. It has appeared in the process that the
alleged contradictions in mathematics upon which the followers of Kant
and Hegel laid stress do not really exist at all, and only seemed to
exist because mathematicians in the past expressed their meaning so
awkwardly. Further, it has been established that the most fundamental
idea of all in mathematics is not that of number or magnitude but that
of _order_ in a series and that the whole doctrine of series is only a
branch of the logic of Relations. From the logical doctrine of serial
order we seem to be able to deduce the whole arithmetic of integers, and
from this it is easy to deduce further the arithmetic of fractions and
the arithmetic or algebra of the 'real' and 'complex' numbers. As the
logical principles of serial order enable us to deal with infinite as
well as with finite series, it further follows that the Calculus and the
Theory of Functions can now be built up without a single contradiction
or breach of logic. The puzzles about the infinitely great and
infinitely small, which used to throw a cloud of mystery over the
'higher' branches of Mathematics, have been finally dissipated by the
discovery that the 'infinite' is readily definable in purely ordinal
terms and that the 'infinitesimal' does not really enter into the
misnamed 'Infinitesimal Calculus' at all. Arithmetic and the theory of
serial order have been shown to be the sufficient basis of the whole
science which, as Plato long ago remarked, is 'very inappropriately
called geometry'. A resume of the work which has been thus done may be
found in the stately volumes of the _Principia Mathematica_ of Whitehead
and Russell, or--to a large extent--in the _Formulario Matematico_ of
Professor Peano. Of other works dealing with the subject, the finest
from the strictly philosophical point of view is probably that of
Professor G. Frege on _The Fundamental Laws of Arithmetic_. The general
result of the whole development is that we are now at last definitely
freed from the haunting fear that there is some hidden contradiction in
the principles of the exact sciences which would vitiate all our
knowledge of universal truths.