Mathematics is often seen by outsiders as an austere discipline of pure reason. The goal of this post is to present an interpretation of mathematics as literal magic. More precisely, the practice of mathematics bears striking similarities (vibes) to the magical tradition known as goetia, or sorcery.

Most of modern mathematics, and as a consequence, science at large assume the existence of real numbers, pretty much as a postulate¹. Given how successful both of these have been, it might seem odd to challenge it. However, when one takes a closer look at exactly how real numbers are defined, a couple of philosophical issues arise. In particular the fact that we (mere mortals) can only meaningfully interact with countably many of them, so that almost all real numbers are beyond our grasp. The goal of this post is to articulate this problem in a mostly self-contained and accessible manner, by constructing the real numbers from the ground up, and then discussing some philosophical consequences of the previously stated fact.

The Julia programming language currently holds the state of the art in terms of ODE solvers, and comes with a variety of ways of specifying ODEs. A common use case is to define modular dynamical systems, where the same pieces occur in multiple places with a bit of variation. Examples of this include running multiple copies of the same system with different parameters for each copy, or dynamical systems on networks like reaction-diffusion systems.

As someone who deeply enjoys mathematics, both on an intellectual and esthetic level, I have inevitably been confronted many times with some remark along the lines of

I never used any of the math I learned at school later in life.

I have had multiple encounters now with people interested or working in Machine Learning/AI Alignment, who wish they had more mathematical background. Most of them come from a Computer Science background, which is not surprising. Despite Machine Learning often being done in CS departments, the typical CS undergrad curriculum contains much less math than what one can find in some ML papers. In my experience, CS people struggle the most with probability and statistics, despite those being an integral part of modern ML (GPT-style models and Diffusion models are both inherently probabilistic, for example).