Philosophy of Mathematics seminar

Carl Posy, Hebrew University

Model theory, intuitionism, and paradox  

Brouwerian intuitionism and contemporary classical model theory are generally, and rightly, thought to be opposed approaches to the foundations of mathematics. At the very least, Brouwer rejected the formalism and inherent infinitary attitude that lie at the heart of model theory.

1. However, I will point to a paradox in intuitionistic mathematics whose solution requires attributing to intuitionism a pattern of thought that is characteristic of model theory.

2. Having made this point, I will turn to the question: what then is the deep difference between these two approaches to the foundations of mathematics? I will show that a network of intertwined ontological, epistemological and semantic grounds emerges from the practice of intuitionistic mathematics, and that each of these grounds differs from its classical counterpart. But, I will argue that, surprisingly, the shared pattern of thought does not reduce, but rather amplifies the philosophical differences between intuitionism and classical model theory.

3. I will indicate at the end how this line of thought extends to other forms of constructivism. If there is time I will show how it provides a unified solution to a series of finite-realm paradoxes and bears on yet broader issues inside and beyond the foundations of mathematics.