"We can take an arbitrary fixed point"
"As long as it doesn't move"
"No, as long as it doesn't move when we are making measurements"
"And if any fixed point gives us the same measurements as any other fixed point"
I felt the lemony tug on the others' lines. We were going to catch eels it seemed...
Wait a minute...
"It's not the measurements, it's the comparisons of the measurements, it's an equivalence."
At which point things that weren't eels started to appear aplenty.