Does a "good explanation" constitute a proof?

Question

I am learning linear algebra and I have heard from $3$Blue$1$Brown that:

good explanation $\gt$ symbolic proof.

He said it when showing that matrix multiplication is associative.

Is it true in mathematics in general, that a solid explanation of why a statement is true would be a good proof?

Answer 1

This is a subtle question. What constitutes a proof actually depends on the relation between the mathematician writing the proof and the intended audience.

In a research paper, the author must provide an argument that's sufficient to convince a reader who is reasonably knowledgable about the field that a claim is true. That may involve some symbolic reasoning, but it's usually mostly words, using mathematical notation for the objects in question, perhaps some algebraic manipulation. It's rarely anything that might be called a "symbolic proof".

As a teacher, I want my students to provide proofs that convince me that they have convinced themselves of some mathematical truth for good reasons. I don't need them to convince me, since I already know. Again, that's almost always best done with words. For example, I prefer an English sentence with the words "for all ..." to expressions using $\forall$.

I haven't watched the video, so I can't comment on whether that particular explanation/proof is good enough.

Answer 2

A good explanation is not the same as a proof, and which is appropriate depends on what you're writing, and for whom. If you're writing something that intends to educate someone so that they understand an existing area of mathematics conceptually, then a good explanation is far superior to a proof. This is what 3b1b is trying to do, and a fair proportion of mathematics textbooks/lectures/seminars in cases where the proof should be easy if you have a good enough conceptual understanding. However, if you're trying to develop an area of mathematics, writing papers etc, then a proof is absolutely required, and whist a conceptual explanation is useful it is not required. This is because, when presenting new results to your piers, it is important to convince them you are correct, and if they genuinely are your piers then the theorum and proof should be sufficient for them to understand it.