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From: OsherD on 6 May 2010 19:05
From Osher Doctorow

Readers who have followed this thread for a long time are familiar
with Fibonacci numbers and their relationships to the Golden Mean or
Golden Ratio and to Probable Causation/Influence (PI). The basic
equation of Fibonacci Numbers is:

1) Fn = Fn-1 + Fn-2

I've recently pointed out in the last few posts here that if we
slightly change this to the PI-based equation:

2) Gn = Gn-1 - Gn-2

then we obtain a quantity G unifying the 4 Fundamental Interactions/
Forces:

3) G = [x +/- sqrt(3)iy]/z, x = Weak Interaction, y = Strong
Interaction, z = either Electromagnetism or Gravitation.

By "converting" G to PI using u/v --> 1 + u - v and u + iv --> 1 + v -
u, G converts to a "Lagrangian" L unifying the 4 Fundamental
Interactions:

4) L = -x + sqrt(3)y - z + 2

What introduces Multivalued Logics and the Multiverse into this is the
fact that the "derivation" of G and L from Gn contains a "missing
link", namely the fact that although the exactly same derivation or
its analog works for Fn, the corresponding method applied to Gn
requires the apparently or at least "overtly" paradoxical conclusion
that real numbers can approach a nonreal complex number as limit,
which is not true in standard mathematical scenarios.

In a Multiverse, however, or in a space of Multivalued Logics (which
can be "deformed" into each other!), more generalized "limits" can
arise. There are two points of view at least:

5) Just as complex numbers, quaternions, octonians, and similar
numbers are not "real" in the physical intuitive sense of being
directly perceived or perceivable in Euclidean or Euclidean-like
spaces by human observers, so a "generalized limit" of complex numbers
from real numbers may not be "real" in a similar sense but may GIVE
REAL RESULTS OR IMPLICATIONS OR RELATIONSHIPS.

6) "Real" or "Observable" or similar quantities or qualities may be
embedded in "continuous" Multiverse spaces where each point or set or
similar corresponds to a different Multiverse, with the example of
Fibonacci numbers Fn being embedded in a Multiverse space where Gn is
a slight "variation" of Fn but also there are many (infinitely many!)
other slight variations.

For example, Look at this sequence {Hn} which yields a result similar
to Gn:

7) Hn = -Hn-1 - Hn-2

Whereas Gn modified Fn by changing Fn = Fn-1 + Fn-2 to Gn = Gn-1 -
Gn-2 (thus changing a minus to a plus), Hn modifies Gn by then
changing Gn-1 to -Gn-1, which changes a plus to a minus! Readers
can confirm that with a similar assumption to Gn, Hn generates the
equation:

8) H = [-1 +/- sqrt(3)i]/2, which only differs from G in changing the
+1 on the left to -1.

Readers can "derive" H from Hn by dividing both sides of (7) by Hn-1
and assuming (incorrectly in the "real" world, but not necessarily in
the Multiverse space) that Hn/Hn-1 has a finite nonzero limit which
turns out to satisfy:

9) Hn/Hn-1 = -Hn-1/Hn-1 - Hn-2/Gn-1

and "taking limits" yields:

10) H = -1 - 1/H, so multiplying both sides by H yields H^2 + H + 1 =
0, with solutions (8).

Multivalued Logics can similarly be regarded, and turn out to be
analogs of PI, Conditional Probability, Independent Probability-
Statistics, etc. (see earlier posts in this thread).

Osher Doctorow
From: OsherD on 6 May 2010 19:13
From Osher Doctorow

Gn-1 in (9) should be Hn-1. Also, in the text below (7), + is
actually changed to - in both cases, not + to - in one and - to + in
the other, but the second involves two changes from the original.

Osher Doctorow
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