KQ: However it still leaves some fuzziness in my mind. You admit that a line segment can have a length of sqrt(2).
JG: NO! I claimed that sqrt(2) is an incommensurable magnitude, therefore it can't be measured.
KQ: Yet you say it does not correspond to any number.
KQ: If something has a length of sqrt(2), doesn't that mean that the number sqrt(2) exists?
JG: Nothing has a length of sqrt(2), but the hypotenuse of a right-angled isosceles triangle with unit equal sides has a magnitude of sqrt(2).
KQ: What is a length if not a number?
JG: The length is a number if it is measurable.
KQ: If you mean you can't specfy all the digits beforehand, like you can for any rational (since the digits repeat, even if the repeating group is very long), so what? If you want digit #56908124897987367512 of sqrt(2) it can be produced right at your table if you give the waiter enuf time to run the appropriate program on his computer.