My New Favourite Math Abstraction - Lawvere's Fixed Point Theorem

Table of Contents

When someone’s asked to think about the most famous and revolutionary math ideas in recent centuries, there’s a high probability that they will think of either Cantor’s diagonal argument, proving that some infinities are bigger than others, or Gödel’s incompleteness theorem, which shows that a formal system can contain statements that are not provable within the system, or Russell’s paradox, which suggests that some sets are actually not allowed to exist, or Turing’s halting problem, which in essence proves that not all problems have an algorithmic solution. Indeed, these theorems directly or indirectly lead to the development of entirely new branches of mathematics and questioned our basic understanding of this field. Personally, I was always fascinated that such simple observations can tell us so much about the limitations and perplexities of logic. I was also intrigued that all these theorems are structurally very similar to each other when you think about it - each of them contains a self-reference that eventually leads to an unexpected conclusion. It almost feels like they’re special cases of some deeper truth underpinning the very foundations of mathematics.

A few months ago, during a casual learning session on category theory, I’ve stumbled upon Lawvere’s fixed point theorem. At first, I didn’t really see the connection between this seemingly uninteresing theorem and the aformentioned paradoxes. Later, however, I was directed by this mathoverflow answer to the phenomenal paper by Noson S. Yanofsky - “A Universal Approach to Self-Referential Paradoxes, Incompleteness and Fixed Points”. I highly recommend giving it a read. It’s not an particularly easy read, by certainly digestible after some time. I was stunned by how one theorem can caputrue in such an elegant way the richness of so many aspects of logic and math in general. For me, this is a strong argument in favour of category theory (and other meta-matemathics approaches) being a fundamental unification of not only most of mathematics, but also computer science and possibly physics (more on that subject here).

I will not go into the technical details of this theorem in this brief post, since I don’t feel yet qualified enough to explain it to a general public in a comprehensible way, and also because others already did a phenomenal job in this regard.

Instead, I recommend you to go through these resources: