Logic, mathematics, and conceptual structuralism
Solomon Feferman · Cambridge University Press eBooks · 2014
Abstract. Conceptual structuralism is a non-realist philosophy of mathematics according to which the objects of mathematical thought are humanly conceived “ideal-world” structures. Basic conceptions of structures, such as those of the natural numbers, the continuum, and sets in the cumulative hierarchy, differ in their degree of clarity. One may speak of what is true in a given conception, but that notion of truth may be partial. Mathematics proceeds from such basic conceptions by reflective expansion and carefully reasoned argument, the last of which is analyzed in logical terms. The main questions for the role of logic here is whether there are principled demarcations on its use. It is claimed that in the case of a completely clear conception, such as that of the natural numbers, the logical notions are just those of first-order classical logic and hence that that is the appropriate vehicle for reasoning. At the other extreme, in the case of set theory, where each set is conceived of as a definite totality but the universe of “all ” sets is an indefinite totality, it is proposed that the appropriate logic is semi-intuitionistic in which classical logic applies only to (set-) bounded formulas. Certain subsystems of classical set theory in which extensive parts of mathematics can be formalized are reducible to