Reflections on the categorical foundations of mathematics
pp. 171-186
in: Giovanni Sommaruga (ed), Foundational theories of classical and constructive mathematics, Berlin, Springer, 2011Abstract
Most practicing mathematicians see no need for the foundations of their subject. But those who wish to place it on a solid ground usually pick set theory, an axiomatic treatment of the membership relation expressed in first order logic. Some of us feel that higher order logic is more appropriate and, since Russell and Whitehead's Principia Mathematica, such a system has been known as type theory (more precisely, classical impredicative type theory with Peano's axioms). Although type theory has been greatly simplified by works of Alonzo Church, Leon Henkin, and others, and despite its naturalness for expressing mathematics, it was unjustly neglected until quite recently.
