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| ``Re: Equivalence of equations''
by Xtraeme on 2009-11-03 04:11:57 |
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| This is more of an epistemological question IMHO.
For instance, if 1 is removed from the domain how can anyone say 2 = 2 when necessarily 2 * 1 = 2 / 1? The underlying component that makes the statement true doesn't exist and furthermore if the identity is gone no cancellation is possible. Put another way 2 * 3 != 6 because 6 != 6 since 6 / 6 != 1. Meaning 6 is not a component of itself, because 1 is undefined. So unless 1 holds no number holds. This is somewhat different than saying the number 7 is removed from the domain. For instance,
6*5 = 30 =>
30 = 30 =>
30 / 30 = 1
The statement holds no matter how you break it up, even if you evaluate various constituent factors (ie/ 15 * 2, 10 * 3, ...). So even though 7 can be a factor of 30 it isn't a _necessary_ factor for the operation to hold when rebalancing the equation. 1 is a necessary factor for all values > and < 1 to equal themselves. The only number this isn't true of is 0.
Ontologically the question can be rephrased, "is 1, as the multiplicative identity, a special property of multiplication or a special property of the number?"
This argument can also be used for the additive identity which seems to be at the heart of Philidor's question. Though the way Philidor phrased it (x*(1/x)=1) he's effectively making 0 a function of 1 which is why we see the hole at 0 because intuitively 1 is a product of 0 not the other way around. |
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