Objectivity with Objects. A Modal Explanation of the Infinite

Abstract

The objectivity of mathematics has been questioned for two reasons: Benacerraf’s challenge and the emergence of several limitation theorems in mathematics. This has led to the search for new explanations of mathematical knowledge and alternatives to universist platonism. Mainly, mathematical knowledge has been reduced to logical knowledge. Against this, this dissertation argues that mathematical knowledge cannot be reduced to logical knowledge. The two best proposals defending this reduction, fictionalism and multiversism, run into serious problems. Both will be challenged to explain how mathematical language acquires its content. Then, it will be argued that they do not satisfy this test. Faced with this problem, an alternative explanation of mathematical language and mathematical knowledge will be developed using abstraction principles. As a result, mathematical knowledge will be explained through linguistic competence. Finally, the proposal will be applied to ZF set theory.