Logicism is the thesis that mathematics is reducible to logic, or so it is often said. Unfortunately, this characterization is misleading. Frege's logicism concerned arithmetic, not mathematics in general; and, as we shall see, reduction, in the sense of a derivation from axioms, is not essential to logicism. Frege's logicism is the thesis that all arithmetic truths are logical truths. Russell held a more encompassing form of logicism according to which all mathematical truths, including those of analysis and geometry, are logical truths. Russell's more encompassing form is based upon his acceptance of the arithmetization of the branches of nonapplied mathematics. Arithmetization reconstructs the branches as theories of order and structure, not theories of magnitude. The theory of order is part of the logic of relations. Relational order is involved in modern dynamics insofar as it employs conceptions of continuity and change over time in its account of motion in space. For this reason, Russell imagined that logicism plays a central role in the solution of Zeno's famous dynamical paradoxes and Kant's antinomies of space.

Logicism arose in connection with the quest for rigor in the deductive methods of mathematics and logic. The quest uncovered a new quantification theory quite distinct from the categorical systems dating back to Aristotle and medieval logicians. Boole, Peirce, and Schröder's methods construed logic as an algebra. Frege started afresh, importing the notion of variables and functions into logic. Both schools founded modern quantification theory.