Useful mathematical concepts, like the number line, can linger for millennia before they are rigorously defined.
The ancient Greeks wanted to believe that the universe could be described in its entirety using only whole numbers and the ratios between them — fractions, or what we now call rational numbers. But this aspiration was undermined when they considered a square with sides of length 1, only to find that the length of its diagonal couldn’t possibly be written as a fraction.
The first proof of this (there would be several) is commonly attributed to Pythagoras, a 6th-century BCE philosopher, even though none of his writings survive and little is known about him. Nevertheless, “it was the first crisis in what we call the foundations of mathematics,” said John Bell, a professor emeritus at Western University in London, Ontario.