I see many people claiming you can’t trust a Lean proof because Lean is buggy. I know someone tried to validate a false proof of Collatz by exploiting a bug. I also know that if there is an error in the Lean statements of the theorem or the axioms, the proof will be invalid. Is there any example of a false proof due to a bug or a reason other than the above?

In most subjects---like physics or biology---it is normal to learn things (mostly) chronologically, in the order that they were discovered. Mathematics is a rare exception where it can make sense to learn in (almost) reverse chronological order. This, I claim, is essentially the case with (introductory) linear algebra.

Why? Linear algebra started off as a subject about one thing---solving linear equations---and eventually turned into something very different---fields, vector spaces, and linear transformations---as more and more fundamental discoveries were made, recontextualizing all that came before. So it now makes sense to start with fields and vector spaces (even if they were discovered thousands of years later) and then start to make the way back to some of those more ancient results, since they are still important.

This article is the beginning of a series on linear algebra. In this first one, I aim to answer three questions: 1) What is the history of linear algebra, in broad strokes? 2) What sort of applications does it have? 3) What is a field, exactly?

Read the full post (for free) on Substack: When Do I Have to Use Linear Algebra?

Little off topic for the chalk snobs. A coworker told me about this chalk and said that they were quite good. Wanted to see what the hype was about. Are there any shops/sellers where you can get them for a reasonable price in Europe (more specifically Benelux/adjacent areas of germany/france)? Shipping fees from east asia are incredibly high, to a point where it's simply not worth it anymore for me.