Science > Physics > Universe Partitioned into Dualities, Unities, and (Relative) Constancies
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Science > Physics |
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"OsherD" |
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19 Sep 2005 10:45:46 PM |
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Universe Partitioned into Dualities, Unities, and (Relative) Constancies |
From Osher Doctorow
COPYRIGHT NOTICE
Universe Partitioned into Dualities, Unities, and (Relative)
Constancies
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
The physical universe is partitioned into dualities or inverses
(additive and multiplicative), unities or "same directional"
quantities, and (relative) constancies which remain unchanged relative
to each other or to everything else.
This rather unusual conclusion comes from:
1) (AB) U (A'B') U (AB') U (A'B) = universe
2) {w: X(w) < = x, Y(w) < = y} U {w: X(w) > x, Y(w) > y} U {w: X(w) < =
x, Y(w) > y} U {w: X(w) > y, Y(w) < = y} = universe
3) P(X < = x, Y < = y) + P(X > x, Y > y) + P(X < = x, Y > y) + P(X > x,
Y < = y) = 1
where (2) and (3) are expressed for simplicity with regard to
continuous random variables X, Y. Note carefully that the sets in (2)
are mutually exclusive or disjoint and comprehensive (they constitute
the universe all together), so they are indeed a(comprehensive)
partition of the universe at least up to probability zero events.
Unities are characterized by relationships like X = kY (k > 0), X = kY
+ b (k > 0), dualities are characterized by relationships like X = k -
Y or X = k/Y with k > 0, and constancies are characterized by X = k or
else P(Y|X = x) = k or P(X --> Y)(x,y) = k types or even "X, Y are
deterministically 'independent'" types.
Readers will notice that everything here is in terms of sets in general
or sets characterizing random variables, unlike Lawvere's and
Eilenberg's and Saunders MacLane's category theory which claims to
adopt "objects" which go beyond sets. I have never found an "object"
in category theory which couldn't be regarded as a (generalized) set or
element of a set or something of the sort, and the use of the word
"object" to replace "generalized set" in category theory goes against
the generalizing idea in mathematics and science. Sometimes it is
claimed that "set of sets" leads to Russell's paradox, but changing the
first occurrence of "set" to "object" rather than "generalized set" is
quibbling about needles in a haystack so to speak.
The absence of anything remotely resembling chaos, "conjugate"
variables of quantum theory like (supposedly) position and momentum,
and singularities at least arguably from the above equations suggests
that these "irregularities" are at best probability 0
events/processes/things.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Universe Partitioned into Dualities, Unities, and (Relative) Constancies |
19 Sep 2005 11:01:38 PM |
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From Osher Doctorow
The universe of fuzzy multivalued logics (FMLs) (rather than the
"physical universe" as such) is also partitioned into three types of
implications (x-->y) for x, y taking on values in the interval [0, 1]:
1) (x-->y) = 1 + y - x
2) (x-->y) = y/x (x not 0)
3) (x-->y) = y
where (3) (constancy) is Godel FML implication, (2) is Product/Goguen
FML implication, (1) is Lukaciewicz/Rational Pavelka FML implication.
Notice that the respective "operations" of subtraction, division, and
constancy in (1) to (3) are themselves amalgams or unions or
disjunctions of unities, dualities, and constancies, with
multiplication and addition omitted but indirectly involved.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Universe Partitioned into Dualities, Unities, and (Relative) Constancies |
20 Sep 2005 01:07:50 AM |
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From Osher Doctorow
"Nothing is what it seems" almost seems to be true when we take a look
at Newton's:
1) F = Gm1m2/r^2
If you look through the arXiv and Front For the Mathematics arXiv
papers on probability and statistics and mathematical physics, there's
little hesitation of specialists in making r and F and arguably
anything else in (1) statistical or probabilistic, with or without
adding extra "noise" terms to the right hand side of equation (1). So
let's look at (1) from this viewpoint.
Is (1) a case of duality:
2) F = (1/r^2)g(m1, m2)
for some function g of m1, m2, where (1/r^2) is the multiplicative
inverse dual of r^2? Or is (1) a case of unity:
3) F = (m1m2)h(r^2)
where h is a function of r^2 and both m1m2 and F are in the same
increasing positive direction? And isn't G a case of constancy?
So we have unity, duality, and constancy incorporated into F =
Gm1m2/r^2.
But we also arguably have something else. Recall that for
product/Goguen FML, we have (x-->y) = y/x for x not 0, although y and x
are FML-valued propositions or in the probabilistic analog are
probabilities. The intuition in the probabilistic picture, known as
conditional probability, is that y = P(AB) and x = P(A) so that the
more B intersects A, the closer y/x is to 1, and the less B intersects
A, the closer y/x is to 0, provided x keeps away from 0.
But how does m1m2 versus r^2 fit into this picture? Well, maybe it's
not a probability but suppose there's some measure or semi-measure or
partial measure m such that m1 = m(A), m2 = m(B), r = m(C) = m(D) for
some sets A, B, C, D which arguably are sets in the physical universe.
Since r is a distance, m might even possibly be Lebesgue measure, in
which case mass m1 = m(A) and distance or length r = m(C) are arguably
dimensionally related through the constant (or perhaps implausibly
slowly varying) G.
Newton's law of gravitation embodies all of the machinery of the
previous posts of this thread and then some! What looks like a
simple equation compared to the Einstein Field Equation(s) turns out to
be quite complicated in that it involves at least a double partition of
the universe!
I should also mention that (x-->y) = 1 + y - x for y < = x and x, y in
[0, 1] is also the same (of course) as 1 - (x - y) which increases as
y gets closer to x subject to the above conditions, which means as y
increases for fixed x for example. But then (x-->y) and y for fixed x
are unities (increase in the same direction), while for fixed y (x-->y)
decreases as x increases so that (x-->y) and x for fixed y are
dualities additively up to a constant (subtractive dualities). And the
constant 1 is constant for everything. Thus, not only Newton's law of
gravitation but Probable Influence and Lukaciewicz/Pavelka FML
implication involve all the machinery of the previous posts of this
thread and then some.
Things may not be quite what they seem!
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Universe Partitioned into Dualities, Unities, and (Relative) Constancies |
20 Sep 2005 01:31:53 AM |
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From Osher Doctorow
If the measure m (for example, Lebesgue measure) is bounded above by a
positive constant, and F is expressed in terms of the measure say as F
= m(E), then really strange things start to happen.
It turns out in that case that if the masses of two objects are
constant (fixed), they can't approach each other closer than a certain
distance or F will go to infinity! And if the distance between two
varying masses is held fixed, the masses can't increase above a certain
magnitude or else F will go to infinity!
The remarkable thing is that it isn't difficult to find scenarios in
which Lebesgue measure for example is bounded above - just limit
yourself to a finite volume of space!
What does this have to do with quantum and relativistic physics? I'm
tempted to say everything. By not outlawing sets and set operations
include union, intersection, complements, we've discovered a whole
unexplored world of relationships in Newton's law of gravitation! It
involves constraints both on very small objects and very large objects,
on very small distances and very large distances! Shades of unified
field theories! And it doesn't have an h in it that quantum people
can use to claim that the macroscopic world is just a limit of the
basically quantum underlying world.
Next question :>)
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Universe Partitioned into Dualities, Unities, and (Relative) Constancies |
20 Sep 2005 01:52:25 AM |
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From Osher Doctorow
I can't close this thread for now without giving a homework assignment
that is open ended in that I don't know the answer either. I call it
the Hitchhiker Problem in honor of Hitchhiker's Guide to the Galaxy.
Suppose that F = Gm1m2/r^2 is really a conditional probability in some
representation, so that F = P(AB)/P(A) and that:
1) Gm1m2 = P(AB)
2) r^2 = P(A)
for some sets A, B that may "change in time" (in usual
probability-statistics, they'd have time subscripts A_t, B_t, but I
adopt a "subscript convention" for simplicity). Find A and B and AB.
Note that A and B are sets in the physical universe.
Although I don't know the answer to this problem, I've noticed a
curious thing about it. It suggests an interpretation of Gm1m2 as
potentially increasing toward r^2 and not exceeding it. Now, if the
speed/velocity of light c is bounded and finite, then arguably nobody
can fault a bounded force of gravitation F (except in black holes and
perhaps inflation etc.). It also suggests an interpretation of r^2 as
potentially decreasing toward Gm1m2 and then stopping, so that it's
bounded from below!
But wait a moment! Isn't there a scenario where mass is bounded above?
Black holes and the Chandrasekhar limit! But distance squared r^2
bounded from below? Heavens to Betsy! Charges of the same type?
Bouncing contracting universes? This is getting interesting. Maybe I
should have a try at solving the homework myself. But I don't want to
spoil everybody else's fun and also stay up all night for 1001 nights.
I think I'll let the younger people try first. Wake me up if somebody
discovers time travel, matter transmission, wormhole travel, a cure for
insanity or cancer or heart disease, or an honest and competent
politician or lawyer or teacher. Come to think of it, include that in
the homework :>)
Osher Doctorow
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