Understanding two of the weirdest theorems in math: Gödel’s incompleteness

November 17, 2023

Gödel's incomplete theorems are famously profound, strange, and interesting pieces of math. But it's hard to understand them, and especially hard to understand why they are true. I've never been quite satisfied with the explanations I've seen for the general public, so I wanted to take a crack at explaining what these theorems say and give a flavor of why they hold. See what you think: Incompleteness Theorem 1 is about whether all mathematical truths can be proven. The first inco...

Are Mathematical Truths Discovered Or Invented?

June 26, 2020

It’s long been debated whether mathematical truths are “discovered” or “invented”. The answer becomes clearer when you dig into what math actually is. In my opinion, math is a language for describing patterns, relationships and rules. It’s just like any language except (1) it’s far more precise than normal language, and (2) unlike human languages, its domain is limited to only describing patterns and relationships and rules. In English, you can talk about birds AND equations related to bird ...

Is Math True?

January 18, 2009

Mathematics is often thought to be universally and unassailably true. Some people even argue that even an omnipotent God couldn’t make math false. But can mathematicians actually prove that math is true? If they can’t, does the fact that math is so useful in solving real-world problems provide evidence of its truth? And, if mathematics is not true, then does that imply that conclusions drawn from it are faulty or suspect? Let’s explore those questions. The first attempt we might take to p...