Science > Physics > Multiplication As Summation Versus As Statistical Independence 2: The xy = 1 vs xy = k Paradox
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Science > Physics |
| User: |
"OsherD" |
| Date: |
07 Sep 2005 03:34:36 PM |
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Multiplication As Summation Versus As Statistical Independence 2: The xy = 1 vs xy = k Paradox |
From Osher Doctorow
We typically do not regard the definition of multiplicative inversion
as a "law of physics":
1) x(1/x) = 1
Yet it solves:
2) xy = 1
for y when x, y are real. If we look at the latter equation, which
says y = 1/x (since neither x nor y can be 0), then it has the form:
3) y = 1/x (x not 0)
which is a rectangular hyperbola (a hyperbola whose branches are in the
first and 3rd quadrants in this case, with axis inclined -45 degrees or
90 + 45 = 135 degrees to the horizontal or positive x axis).
Now jump to the Heisenberg Uncertainty Principle (HUP):
4) xy > k
where k is of the order of Planck's constant h (very small) and x is
position, y is momentum. If k1 > k, then:
5) xy = k1 > k
so xy = k1 satisfies (4). We can regard the Weak Heisenberg
Uncertainty Principle (WHUP) as xy = k1 for k1 > k but k1 having the
same order of magnitude as k.
Now comes the paradox. xy = 1 says y = 1/x, and xy = k1 says y = k1/x
for k1 > k and k1 of the same order of magnitude as k. Since xy = k1
is arguably a Law of Nature at least under certain conditions, and xy =
1 is a generalization of a law of mathematics (the inverse
multiplication law for nonzero reals), the paradox is why the orders of
magnitude of k1 and 1 differ so much.
Let's examine in this regard the equation:
6) xy = n (n integer > 1)
If we take x = 1, then (6) says that the "whole" n is the sum of y of
the x parts, where y is n and x is 1. That is to say, n(1) = n. If we
take x << 1, then it takes more y parts to add up to n. If we replace
n in (6) by k and let k << 1, then for the same x = 1 we get xy = y = k
<< 1 which means that it takes tinier and tinier fractional parts k of
x = 1 to get the "whole" of 1. We have lost the "counting" intuition
of "the whole is the sum of its parts" by making k small in xy = k.
As explained in previous threads, although the WHUP is plausible in
certain scenarios, the above type of paradox indicates that it loses
its physical significance as k in xy = k gets closer and closer to 0.
This has a remarkable parallel for rectangular hyperbolas because xy =
k for k > 0 get closer and closer to an L shaped (and therefore
discontinuous at the origin) curve in the first quadrant as k gets
closer and closer to 0 from above zero. This is because the distance
of the vertex from the origin depends on k.
In summary, the WHUP is arguably roughly speaking a physical and
mathematical singularity because of the small magnitude of k in xy = k.
We appear to have a "mini-black-hole" type of scenario.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Multiplication As Summation Versus As Statistical Independence 2: The xy = 1 vs xy = k Paradox |
07 Sep 2005 03:49:24 PM |
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From Osher Doctorow
Let's briefly review what's happening. In fuzzy multivalued logics,
there isn't any implication of form xy = k for constant k. The only
implications are:
1) (x-->y) = 1 + y - x
2) (x-->y) = y/x (x > 0)
3) (x-->y) = y
Yet xy = k is highly causal (it says y increases causes x to decreases
and vice versa) for k > 0. So we need to figure out how a
"nonexistent" FML implication can be highly causal. In the previous
threads, two ways were found: (a) xy = k means that "the whole is the
sum of its parts" if the whole is k (usually k is an integer > 1) and x
is one of the parts and it takes an integer y number of x magnitude
parts to get the whole k magnitude. (b) P(AB) = P(A)P(B) or F(x,y) =
FX(x)FY(y) for X, Y (or A, B) statistically independent, which although
"causal" is in a sense the lowest type of causation. We now have seen
a plausible (partial) argument that when k is very small and positive,
(a) has either the properties of (b) or is singularity-like.
Osher Doctorow
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