If I draw the diagonal of a square, do you think that this line has no distance, or that the distance along the line is not a quantity, or something like that? Incommesurate ratios were studied by Euclid.
Post by twetch#3375
Euclid could show that his GCD algorithm would not always halt for any two lines that he could draw. But we still have a concrete representation of the irrational quantity in the diagonal of a square. https://en.wikipedia.org/wiki/Euclidean_algorithm
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Actually draw a diagonal on a square, and you will see that, in fact, the diagonal is composed of a finite, countable number of points.
Euclidean geometry mistakenly assumes that space is continuous. With discrete space, there is no incommensurability.