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- Pythagoras is giving yet another lecture that All is Number... (Here, a number is a rational number p/q)
- All is Number.
- ...while using invalid proof methods yet again.
- Why? Consider any quantity. We know we can represent the whole part as a number trivially. For the fractional component, we can approximate the quantity with arbitrary precision as follows: call the desired quantity x (x 1). For each integer i 0, if x = 2^(-i), set x to x - 2^(-i) and add 1/2^i, a number. We know that a number plus a number is a number so we have only used numbers so far. After a given i, we know x 2^(-i), so any quantity can be approximated to an arbitrary precision. Therefore, it is absurd to think a quantity exists that is not a number. Not only that, but numbers are also intrinsic in nature. Consider musical notes produced from a vibrating string. A string 1/2 as long produces an octave, 2/3 produces a fifth, and 3/4 produces a fourth. Thus, numbers are a fundamental part of nature. If not all is number, what is it...
- Hippasus pursued the truth. He toiled for many days, eventually disproving the statement.
- All is not number! I have discovered a truly marvelous proof of this which this margin is too narrow to contain.
- So I'll use another book I guess...
- Hippasus slowly realized the other implications, including the painful truth that what he believed his whole life was false.
- I have discovered that sqrt(2) is not a number. Call quantities that make use of the terms x^(1/y), where y is a natural number, algebraic numbers. Are those all quantities that exist? What if there exist quantities that transcend even that? Transcendentals?
- More importantly, this means everything I know is false. Has my life until now really been for anything? And the lives of my fellow students as well...
- However, now I can see the ray of sunlight at the end of the cave! We can correct our doctrine. All we have to do is place our absolute trust in proofs!
- Hippasus returned and presented his revelation.
- I will prove sqrt(2) is not a number by contradiction. Let sqrt(2) = p/q, where p and q are coprime. qsqrt(2) = p. 2q^2 = p^2. This implies p is even, so let 2k = p. 2q^2 = (2k)^2. 2q^2 = 4k^2. q^2 = 2k^2. This implies q is even. But since p is also even, p and q share a factor, namely 2, a contradiction. QED.
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