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Why is it better to invent a weird new class of numbers (irrationals) rather than just identify that there is something wrong with how we think about this issue

by abtinf 2y ago

Why is it better to invent a weird new class of numbers (irrationals) rather than just identify that there is something wrong with how we think about this issue? Put another way, why don’t we reject out-of-hand the notion that sqrt(2) cannot be calculated, given that right isosceles triangles do exist in reality and their hypotenuse has a definite length? Put yet another way, why not just say sqrt(2) equals 1.41 (or however much precision you need) + some infinitesimal amount?

neeleshs 2y ago
We probably have not discovered the unified theory of numbers yet, and these are all patches to the current system
function_seven 2y ago
> 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 :)
xigoi 2y ago
How would that help you? Being able to reason about irrational numbers is useful.
tim333 2y ago
>invent a weird new class of numbers (irrationals) No one is really inventing numbers like 2^(0.5). They just are what they are. Calling them irrationals is just naming something that always existed.

by abtinf 2y ago

neeleshs 2y ago

We probably have not discovered the unified theory of numbers yet, and these are all patches to the current system

function_seven 2y ago

> 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 :)

xigoi 2y ago

How would that help you? Being able to reason about irrational numbers is useful.

tim333 2y ago

>invent a weird new class of numbers (irrationals) No one is really inventing numbers like 2^(0.5). They just are what they are. Calling them irrationals is just naming something that always existed.