Heh. I did one of those things again. I drank too much coffee and came up with a way to prove the two lower branches of calculus (differential and integral) using algebra and, conversely, disprove algebra using the two lower branches of calculus. Also, that all of Xeno’s paradoxes and the whole Euclidean vs. non-Euclidean geometry thing comes down to whether you hold arithmetic or algebra to be absolute (Euclidean geometry depends on algebra being absolute in the face of failing arithmetic and non-Euclidean geometry depends on trigonometry being absolute in the face of failing algebra.)

All this really comes down to our innate inability to conceive of and deal with numbers that aren’t numbers at all. Thinking physically, I’ve concluded that the laws of quantum mechanics apply not only to particle physics but also any form of mathematics. Like that the uncertainty principle applies to numbers as well. I’m trying to think of an example I can use here. A simple one is that the same number can have different values in different circumstances:

8 ÷ 9 − 1 ÷ 9 = 7 ÷ 9 and 1 − 1 ÷ 9 = 8 ÷ 9. But what’s 8 ÷ 9 + 1 ÷ 9? 9 ÷ 9 or 1. But is 9 ÷ 9 really equal to 1? Algebra says yes, because x ÷ x = 1. But what does basic arithmetic say? Is 8 ÷ 9 − 7 ÷ 9 really equal to 7 ÷ 9 − 6 ÷ 9? 8 ÷ 9 can be written with equal precision as 0.̄8 and 7 ÷ 9 can be written as 0.̄7. The difference between these two numbers is 1 ÷ 9 or 0.̄1. That one works. But what is 8 ÷ 9 + 1 ÷ 9? Is is really 1? 8 ÷ 9 can be written as 0.̄8 and 1 ÷ 9 can be written as 0.̄1. The sum of these two repeating decimals isn’t 1 but 0.&772;9. There is an infinitely small difference between 1 and 0.̄9. That can be written as 0.̄01. If two lines are such that one has a slope of 1 and the other a slope of 0.̄9, are they parallel?

So the magic value of 9 ÷ 9 is really *two* values: 1 (x ÷ x) and 0.̄9 (i.e. 0.̄8 + 0.̄1).

I showed this one (and the ones about the Xeno series) to the head of the math department at my high school and he got very angry and his face turned red and he yelled at me that this wasn’t real math but wordplay. I showed this to a college professor and he laughed and told me that modern particle physics depends on numbers having more than one value at any given time like √2*i*, which has four values – √2 can be positive or negative and *i* has both a real dimension (√1 = 1) and an imaginary dimension (√-1). I liked that professor. He was cool.

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