All the Way Down

The problem with visual proofs is that there are perfectly similar looking proofs that are false: https://math.stackexchange.com/questions/12906/the-staircase...

They’re great and cool for things you already know to be true, but they can be tricky.

I animated these and related 'proofs' some time ago

    1/3 = 1/4 + 1/16 + 1/64 + ...

https://evanberkowitz.com/images/2014-03-15-quarters/Triangl...

    1/7 = 1/8 + 1/64 + 1/512 + ...

    1/8 = 1/9 + 1/81 + 1/729 + ...

These are neat! I guess you have to be comfortable with geometric proofs for them to really pop as obvious visual proofs, and certainly Archimedes was. I would have just started summing until it got close to 1/3, which is brutish by comparison to these beauties.

A proof (visual or otherwise) shows "how" some statement is true, as in how it is built by the preceding truths. But I always wanted to know "why" something is true. For example, a biological cell grows and division happens. I could find tons of literature which talks about "how" this happens, but not "why" this happens. What's the motivation or goal? And why that goal is pursued? What is the force behind seeking of that goal?