Journal D+iz+a+n+k+Meister's Journal: You are misunderstanding the denotation for the connotation
If you've never watched Joeseph Campbell interviewed by Bill Moyers, you are missing out. But one of the most universal things he talks about in one of the segments was how people often misplace the denotation for the connotation . Which is quite understandable, given the relatively lazy attitude average joe has for thinking about such things. Even special joe probably never thinks outside of practical experience to see the meaning of practical experience.
But why is this important? Well, one conclusion I draw, not necessarily from Joeseph Campbell, is that if you truly believe that, say, Mary was an actual virgin who actually had a baby who actually was the son of God, as in God actually put his sperm, or whatever a deity of that caliber would put, into Mary, you are missing some seriously important literary points that the Bible is trying to make.
Denotation: Mary is the virgin mother of God
Connotation: Well who cares? I sure don't, but I think you get my point. The point is, it's not important to having faith that Mary actually gave a virgin birth to Jesus in the year 1. There is no reason to expect that archeological digs will support the virgin birth as a "fact," nor is there any purpose to having forensic evidence of that fact. "Be excellent to each other" is as true today as the day it was written.
Now how does this apply to the question: does 0.999~ really equal 1?
First off: read the aforementioned thread. I promise, it's relatively low key. No trolls, all genuine individuals struggling against their ignorance of the concepts of compact sets and a radix.
Maybe it's just sooo obvious to me, since I have spent probably months thinking about that very question, making sure I understood everything about it and never got the answer wrong. But thinking about it like "well, if there's no real number between reals X and Y, then X=Y" shows that you don't understand the disconnection between numbers and decimal representation. Or even the more subtle distinction between numbers and quantities.
Denotation: 0.999~, 1
Connotation: Multiplicative Identity Element
That's all that's really necessary to understand. That's the driving force behind:
X = 0.999~
10X = 9.999~
10X - X = 9X = 9
X = 1
You pick 10X because of how we represent numbers in any radix. Basically, we start with X as the repeating "decimal" of the largest symbol for radix n. Then we multiply X by the radix n and subtract X, which then gives us the largest symbol times X equals the largest symbol. If we were using hexadecimal digits, the same would hold:
X = 0.fff~
10X = f.fff~ (Remember 10 in hex is 16 in decimal)
10X - X = fX = f
X = 1
But that brings me to my bigger question. About society in general. Are we teaching math completely wrong when we ask little children, "If I have 5 apples and I give you 3 apples, how many apples do I have left?" Isn't that the start of the confusion between the denotation and the conotation?
Abstraction is more than a programming visualization tool to me. It is a way of life: keep searching for a bigger abstraction to explain new experiences, yet also cover old experiences. So I have thoughts along the lines of:
Existentialism -> Religion and strictly not the other way around(religion does not make room for existential beliefs), ergo existentialism is a more "correct" theory, or religion could be considered a subset, a "kernel" if you will, of the set of existential ideas. But in general, they are both ideas, so I have to abstract a model for ideas, so really, the relative "correctness" of existentialism vs religion "doesn't matter." Much in the same way as the "distance" between 0.999~ and 1 "doesn't matter," or more appropriately, "doesn't exist."
Isn't tying mathematical knowledge to practical experience, specifically tying numbers to quantities that one can count, preventing others from being able to make their own abstractions? Isn't that why algebra is such a turning point in adolescent education? Isn't that why people have such a difficult time with calculus, and in general, believe calculus/differential equations to be the end-all be-all of "hard" math? Isn't that why dumbasses, like meteorologists say, who can't guarantee a forecast within +-4 degrees F, will laugh at the math major who mistakes 32 for 16 when multiplying 8 by 4? Doesn't this type of "practical" thinking reinforce things that shouldn't necessarily be reinforced, like the importance of the bottom line or how fast your car can go?
Or is this really the best of all possible worlds? And I'm just lucky to see what most people can't see in math.
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You are misunderstanding the denotation for the connotation
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