> some infinitesimal amount? What does that even mean? If we're rejecting the notion of an irrational, then the statement, "some infinitesimal amount" might as

> some infinitesimal amount? What does that even mean? If we're rejecting the notion of an irrational, then the statement, "some infinitesimal amount" might as well be "some gorkly boggleboop". Sure, we can approximate to whatever precision is required for building a wall or calculating an orbit, but math itself would be hobbled by trying to make discoveries with the handicap of only allowing rationals. > given that right isosceles triangles do exist in reality and their hypotenuse has a definite length? I might be agreeing with you in a sideways manner, but right isosceles triangles don't exist in reality. Nor do any of the simple shapes like squares and circles. We have physical things that approximate those ideal shapes, but even the most precise triangle will not have a perfect right or 45 deg angle. Nor will the real-world hypotenuse be precisely sqrt(2). These physical items are made of a countable amount of molecules each of which is in some quantized state. Hell, the length of each side of the most perfect triangle we can make will be in constant flux. So for practical everyday purposes, sure. We can't work directly with irrationals, and there's no need to. But for making new discoveries in math, we must work out how to deal with "weird new" classes of numbers, like 0, or the negatives, or the complex, etc. Each one of those classes of numbers has survived because it has proven useful. If you can identify the "something wrong with how we think about this issue", you would probably win a big old prize for that :)