An explainer on norms, metrics, and inner products, and their relationships to each other.

The title is a reference to The Unreasonable Effectiveness of Mathematics in the Natural Sciences. I had a similar question about the number 2 which repeatedly shows up in engineering and science, specifically in the form of the 2-norm of a vector, and seems surprisingly effective at doing what it’s supposed to do.

One of my motivations for starting a blog was Eugenia Cheng’s book The Joy of Abstraction. It’s a surprisingly accessible, gentle introduction to category theory, a topic that is usually only taught to graduate students in math. In this post, I will introduce the main ideas in category theory and show that it offers an elegant way of thinking about mathematics.

I suggested that there is no objective notion of logical truth, that whether a statement is ’true’ can depend on the system of truth that one is operating in. Here we will develop that argument further using the concept of axiomatic systems.

This post discusses the question of whether spoken and written languages like English could be ’logical’ by design. The first part does not require a mathematical background, whereas the second touches on the concepts of axioms, theorems and proofs.