Mathematical understanding and cognition

This is the home page for a seminar that was taught by Jeremy Avigad ( at Carnegie Mellon in the fall of 2012. The course numbers were 80-513/813.

The seminar met on Fridays from 10:00 to 12:20 in Baker Hall 150, the Philosophy Department Seminar Room.



The philosophy of mathematics has long been concerned with the nature of mathematical knowledge, and the appropriate justification for knowledge claims. Mathematical logic provides us with a formal account of the latter: we know that a mathematical statement is true when we have a proof from appropriate axioms, such as the axioms of set theory. On this view, mathematical knowledge consists of a body of definitions and theorems, and mathematical reasoning amounts to applying valid rules of inference to obtain new theorems from old.

This simple model has been a great success. It has helped clarify the standards of mathematical proof, and has provided an framework for philosophical analysis. It has also had numerous applications within mathematics, and to computer science.

But when it comes to mathematical knowledge, there is a broader class of normative judgements that are worthy of philosophical attention. Mathematical concepts can be fruitful; theories can be powerful; theorems can provide fresh insights; a new proof can provide a better explanation as to why a mathematical fact is true. The simple model described above has little to say about what underlies such judgements.

To that end, more elaborate models of mathematical activity are appropriate. Think of mathematical knowledge not as a body of definitions and theorems, but, rather, as a body of concepts and methods that guide our thought. Mathematical understanding involves knowing how to proceed in mathematical contexts, being able to represent mathematical concepts in useful ways, being able draw on background knowledge, and being able to reason effectively. On this view, mathematical thought is not just a matter of choosing inferences and applying them, but relying on a broad base of cognitive resources to navigate a wild and unruly search space.

The goal of this seminar is to bring a number of disciplinary perspectives to bear on filling out this view. In particular, we will consider philosophical, cognitive, and logical approaches to obtaining more robust models of mathematical thought.