The Mathematical Mind: How Euclidean Geometry Lives on the Riemannian Manifold of Our Brain

The mathematical mind, at least in some of its fundamental abstractions, is a mental representation of the brain’s inherent hardwired mathematics and geometry—and possibly not vice versa.

A toroidal manifold of stable attractor patterns in neural activity space, visualized via dimensionality reduction, with each attractor pattern colored by the decoded position within the 2D environment in the inset. —Image reworked by The Quantastic Journal.

Several of my previous scientific articles dive into the mathematical and geometrical representations of the self and physical reality, as well as the associated cognitive processes. When we write an article, however, using mathematical concepts—such as the Euclidean or Minkowski metrics, Hilbert space, Riemannian manifolds, Lie groups, topological objects etc.—one may often come across a nexus point, where we put the specific question to ourselves, why our brains imply these concepts and abstractions? Or, to put it more precisely:

How much of our mathematical and geometrical mental representation, and thus reasoning of the physical reality, is predetermined by the inherent mathematics and geometry underlying brain function?

—This question is central to “Our Knowledge of the External World, As a Field for…