I often ponder whether mathematics is lying around waiting to be discovered or is non-existent until we invent it. One of the most recent posts at Math Rising led me to a similar question concerning the brain. Has the physical structure of the brain led us to create certain mathematical structures or did a fundamental sort of mathematics govern the formation of the brain?
Math Rising blogger Joselle DiNunzio Kehoe combs through mathematical papers as well as popular media and blogs like n-Category Cafe looking for mathematical connections between different disciplines including art, physics, and biology. To find these connections requires one to focus on structure, so category theory is recurring theme. One of her latest posts focuses on how category theory might describe the means by which we mentally process and sort information. In particular, she discusses the 2003 article by Ronald Brown and Timothy Porter Category Theory and Higher Dimensional Algebra: potential descriptive tools in neuroscience The two authors, both mathematicians, propose colimits as a way of describing processes of processes. The paper also invites discussion with neuroscientists concerning how to use mathematics to connect the activities of a single neuron to formation of a “concept” or “emotion”. One intuitive advantage to category theory is that regardless of the choices made along the way, the end product is preserved.
The Universal Property of a Co-limit shown by a diagram — Colimits can be thought of as a way to amalgamate many pieces of information into one.
Ms. Kehoe, who teaches Mathematics at the University of Texas at Dallas also writes for Plus Magazine and Scientific American about mathematics and cognition.
The idea that category theory is not “abstract nonsense” (as it is so fondly referred to by many a mathematician) was also discussed in Science News back in May by science writer Julie Rehmeyer. In particular, she mentioned David Spivak’s recent book “Category Theory for Scientists” which is available for free on arxiv. This book is aimed at a broad scientific audience.
Lastly, the Foundational Questions Institute recently featured the work of John Baez and the quest for the categorification of quantum mechanics.
Another development (related to some of the things you mentioned) is Homotopy Type Theory, abbreviated HoTT. A bunch of people at the IAS produced a book on it (available free on the web on for money from Lulu) and there is an upcoming meeting in Barcelona that I hastily signed up for. NOT because I have always wanted to go to Barcelona anyway.