Does the square root of $2$ exist?

Question

I know there are plenty of posts on proving that it exists. I know you can, for e.g. by defining $x$ to be the supremum of all rationals $r$ such that $r^2<2$ and proving that $x^2=2$, or by applying the intermediate value theorem.

Timothy Gowers asks us to imagine that if you did not know any advanced mathematics and were confronted by somebody who denied the existence of the square root of two. What would you say?

Answer 1

Take two squares of area 1. Cut them diagonally and assemble the 4 pieces into a square of area 2.

What is the side length of the square?

Answer 2

I would first use a minor variant of the Archimedian Property, in which you can create at least one number between any two numbers by taking their average to prove the continuity of the real numbers, then I would use a binary search to find an approximation for the square root of two, then I would bring up the fact that these results look like they are converging to a specific number ($1.414...$), and then I would argue that this converges to a specific value.

Answer 3

I would ask him to prove that a wudget isn't purple. You can't prove anything about something that isn't defined. What is the definition of a number? If you don't have a def'n of "number" you can't prove that one of them is not $\sqrt 2,$ or not purple either.

Answer 4

I might similarly ask, "Does 1/3 exist"

The unspoken argument here appears to be that a number that cannot be written in decimal form by a finite number of digits does not exist.