The Partial Unification of Domains, Hybrids, and the Growth of Mathematical Knowledge

Abstract

Relations among distinct areas of mathematical activity are most commonly discussed in terms of the reduction of axiomatized theories, where reduction is defined to be the deductive derivation of the axioms of the reduced theory as theorems of the reducing theory. Such derivation requires that the characteristic vocabulary of the reduced theory be redefined in terms of the vocabulary of the reducing theory; these definitions are called bridge laws. In this century, philosophers of mathematics have discussed relations among arithmetic, geometry, predicate logic, and set theory in these terms, and have claimed variously that geometry may be reduced to arithmetic, arithmetic to predicate logic, and arithmetic and geometry to set theory. Their arguments run parallel to those made by philosophers of science, who claim variously that biology may be reduced to chemistry and chemistry to physics, and that within physics, classical mechanics may be reduced to some combination of relativity theory and quantum mechanics.