the infinitesimal method men were
occupied in carrying the new tool of analysis into every field where its
use promised advance. The conceptions of the method were uncritical. Its
applications were the center of attention. The next century undertook to
bring order into the concepts, consistency into the doctrine, and rigor
into the reasoning. The dominating trend of this movement was logical
rather than methodological. The development was in the interest of the
foundations of mathematics rather than in the use of mathematics as a
method for solving scientific problems. Of course this has in no way
interfered with the freedom of application of mathematical technique to
the problems of physical science. On the contrary, it was on account of
the richness and variety of the contents which the use of mathematical
methods in the physical sciences imported into the doctrine that this
logical housecleaning became necessary in mathematics. The movement has
been not only logical as distinguished from methodological but logical
as distinguished from metaphysical as well. It has abandoned a Euclidean
space with its axioms as a metaphysical presupposition, and it has
abandoned an Aristotelian subsumptive logic for which definition is a
necessary presupposition. It recognizes that everything cannot be
proved, but it does not undertake to state what the axiomata shall be;
and it also recognizes that not everything can be defined, and does not
undertake to determine what shall be defined implicitly and what
explicitly. Its constants are logical constants, as the proposition, the
class and the relation. With these and their like and with relatively
few primitive ideas, which are represented by symbols, and used
according to certain given postulates, it becomes possible to bring the
whole body of mathematics within a single treatment. The development of
this pure mathematics, which comes to be a logic of the mathematical
sciences, has been made possible by such a generalization of number
theory and theories of the elements of space and time that the rigor of
mathematical reasoning is secured, while the physical scientist is left
the widest freedom in the choice and construction of concepts and
imagery for his hypotheses. The only compulsion is a logical compulsion.
The metaphysical compulsion has disappeared from mathematics and the
sciences whose techniques it provides.

It was just this compulsion which confined ancient science. Euclidi