Two paradoxes: $\pi = 2$ and $\sqrt 2 = 2$ [duplicate]

Solution 1:

If the limit of a sequence of curves $\{\mathcal{C}_n\}$ is a curve $\mathcal{C}$, that does not mean that the limit of the lengths of the $\mathcal{C}_n$ will be the length of $\mathcal{C}$. The presentation makes the assumption that it will, but what validates that assumption, other than it feels intuitively to be true? This is a great example of how intuition, while usually helpful, can occasionally be hurtful. This example can serve as a poster child to champion mathematical rigor.

Take the circle example. After $N$ iterations, the extremely bumpy curve that you have is still much longer than the diameter.

And the staircase example. After $N$ iterations, the extremely jagged curve that you have is still much longer than the diagonal.

As a function from curves of finite length to $\mathbb{R}$, $\operatorname{length}(\phantom{x})$ is not continuous. It may feel like it should be, until you think about examples like the kind brought up here.

Solution 2:

First: It is not a paradox: it is just wrong. The reasoning is wrong.

About $\pi = 2$ he says: "Well clearly we are approaching the diameter of the circle". That is a statement that he doesn't prove and which is false.

The same problem arises with the $\sqrt{2} = 2$ when he says: "Well clearly this geometric construction approaches the diagonal of the square". How does he know that?

All that this proves is that we have to be careful when we talk about finding limits from purely looking at pictures.

"Just because the sun sets in the west doesn't mean that it has to rise in the west as well.

Edit: There are plenty of example of proofs that seem right, but turn out to be wrong when we go over them in more detail. Take for example the proof that for complex numbers $$ 1 = \sqrt{1} = \sqrt{(-1)\cdot(-1)} = \sqrt{-1}\sqrt{-1} = i\cdot i = -1$$ Here again, the argument is invalid because the rule $\sqrt{ab} = \sqrt{a}\sqrt{b}$ doesn't hold for complex numbers.