Cognitive Control in Mathematics

The nature of mathematical theorizing underwent a dramatic transformation in the late 19th and early 20th centuries. Mathematicians are prone to describe this transformation by saying that mathematics became more 'conceptual' and that, consequently, we have come to enjoy more and better 'understanding' in mathematics. The purpose of my dissertation is to introduce a configuration of philosophical notions that allows us to analyze the epistemic significance of these changes. In order to arrive at such a configuration, I conduct a case study in which I compare two approaches to the solvability of polynomial equations by radicals, one characteristic of 19th century mathematics, another characteristic of 20th century mathematics. I use the pre-philosophically visible differences between the two approaches to motivate a new epistemological notion I call cognitive control. To have cognitive control over an epistemic process such as reading or writing a proof is to have epistemic guidance for the process in virtue of having an epistemic scaffolding. To have epistemic guidance at a given juncture in a process is to have a constellation of cognitive resources that allows one to represent the different possible ways of pursuing the process further; to have an epistemic scaffolding for a process is to have a suitably organized representation of the epistemically possible facts in the range of facts one has chosen to examine. I apply the notion of cognitive control to two proofs of the fact that there is no general formula for a solution by radicals for polynomial equations of degree 5, again one characteristic of 19th century mathematics, another characteristic of 20th century mathematics. I argue that we enjoy much better cognitive control over the process of reading the 20th century proof than we do over the process of reading the 19th century proof. This suggests that the epistemic significance of the said changes in the nature of mathematical theorizing consists, at least in part, in the circumstance that the conceptual resources of 20th century mathematics allow us to enjoy more and better cognitive control over the epistemic processes in mathematical research and learning