be said that the whole of pure mathematics has been shown, or is on the
verge of being shown, to form a body of conclusions rigidly deduced from
a few unproved postulates which are of a purely logical character.
Descartes has proved to be right in his view that the exceptional
certainty men have always ascribed to mathematical knowledge is not due
to the supposed restriction of the science to relation of number and
magnitude--there is a good deal of pure mathematics which deals with
neither--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