There's also the intuition that the circumference of the circle must be less than the perimeter of the square, so if the perimeter of the polygon isn't decreasing as it gets closer to the circle, it doesn't approximate it better than the square itself.

I.e., the perimeter doesn't approach the circumference in value because it doesn't change.

It's an interesting thing to think through though, and maybe a good point about how arguments can seem intuitive at first but be wrong. On the other hand, I'm not sure that's any more true of visual proofs than other proofs.

I think your attempt to rebuke the proof is flawed too. The problem in your reasoning is mixing up "arbitrarily many" and "infinitely many".

There's no convergence after a finite number of steps. But at infinity, the canonical limit of this construction method is a circle. And because it is a circle, the circumference at infinity "jumps" to 2*pi. This is quite counterintuitive but perfectly legit in mathematical analysis. It's just one of many wacky properties of infinity.

Well, yes, it is false, hence there are problems.

But imagine this was a domain you weren't familiar with, you didn't know that pi != 4, you didn't know that the proof was false going into it. Could you have come up with a list of problems so quickly?

For what it's worth, I'm not sure that problems 1 and 2 are actually genuine problems with the proof. You can approximate the length of a curve with straight lines by making them successively smaller. This is the first version of calculus that students learn. Problem 3 is the crux.

These are obvious problems to someone who has studied enough math/geometry/calculus to know how one form of "adding boxes together gets a curve" and another "adding boxes together does NOT get you a curve".

These are nice! I had never seen the 1/5 spiral before!

> unresolvable why's like this are what understanding is made of

Certainly the deeper into the why chain one can get personally, the greater understanding one has.

This seems the most concise and precise answer to the question of "why". At some stage, somehow, self replicating molecules appeared in the primordial soup*. Everything since then has just been improvements (or maybe complete overhauls) to that process.

*How (or why) this step occurred is another intriguing topic.

> there's no goal

Try talking about biological operations without invoking “function”. Claiming it’s “convenient” to do so doesn’t cut it: convenient for what?

Why do acorns become oak trees? They must be causally ordered toward that end. That’s telos.

Even efficient causality presupposes telos. Why does striking a match against a matchbox consistently produce fire? Because the match has a causal ordering toward that end. Otherwise, you could not explain why fire consistently results as opposed to random things like a flock of seagulls or a BMW 7 Series…or nothing at all.

Telos is not necessarily a matter of some external purpose or Paley-style watchmaker. That’s mechanistic metaphysics appealing to a watchmaker to explain a purpose things would - under that metaphysics - inherently lack. It is a matter of causal order and directedness.

That's a really bizarre and oddly Platonist take on things. Your grandfather was viewing laws of nature as rules imposed onto reality by some outside force, and which therefore need some "mechanism" to be in place to "enforce" them.

But I think a more reasonable understanding of natural laws is that they're our attempt to describe the cause-and-effect patterns observable within reality itself. They're not being enforced, they're simply manifest.

Construing "nothing can exist" as a rule that has to be enforced, and not just the absence of any patterns of causality that would produce something that exists, seems to be an error. It actually seems to be a more sophisticated version of reifying the concept of "nothing" such that "nothing exists" would be interpreted as describing the positive existence of an entity called "nothing" rather than merely describing the absence of any such entities within the context.

This is interesting, it reminds me of the chain of logic from this article:

https://alwaysasking.com/why-does-anything-exist/

> In a reality containing nothing, there are no things as such — at least no material things. But in such a nothing, there is an abstract thing: zero.

> Zero reflects the number of material things to count. But how many abstract things are there to count? There is at least one. The one number that exists to define the number of material things is zero.

> But if we have one number and it is one thing to count, now another number exists: one. We then have zero and one together as the only numbers. But now we have two numbers. Now two exists…

Your grandfather's explanation seems to echo this in terms appropriate for a 10-year-old - there is something inherently unstable about nothingness.

Here's a video lecture of Graham Priest

Graham Priest - "Everything and Nothing" (Robert Curtius Lecture of Excellence)

https://youtu.be/66enDcUQUK0?si=nAZjkauxg75lvuZm

basically: "life, uh, finds a way"

really? when you were 10?

Then, who or what has the goal that is driving the cell division?