Science > Physics > Complex Multiplication Does Not Preserve Causation
| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
17 Dec 2005 12:42:25 AM |
| Object: |
Complex Multiplication Does Not Preserve Causation |
From Osher Doctorow
COPYRIGHT NOTICE
Complex Multiplication Does Not Preserve Causation
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
Probable Causation/Influence (PI) is:
1) P(A-->B)(x, y) = 1 + y - x, y = P(AB), x = P(A)
with a variant:
2) P' (A-->B)(x,y) = 1 + y - x, y = P(B), x = P(A), P(B) < = P(A)
The product of two such causations/influences is, using a, b, c, d for
convenience:
3) P(A-->B)(a, b) P(C-->D)(c, d) = 1 - a + b - c + d + ac + bd - bc -
ad
The product of complex numbers a + ib and c + id is:
4) (a + ib)(c + id) = ac - bd + i(bc + ad)
Notice that the negative sign of the causing/influencing variable or
set or its representative probability a, c corresponds to the real part
of the respective complex numbers a + ib, c + id, namely a, c which are
unsigned (or have a positive sign implicitly) and similarly the
positive sign of the caused/influenced quantity b or d corresponds to
the imaginary parts b, d of a + ib, c + id which are again unsigned or
implicitly positively signed.
In order to preserve probable causation/influence, the product (a +
ib)(c + id) in (4) should have -bc - ad of (3), which have negative
signs, correspond to ac - bd which is the real part of the product in
(4), but in fact bc + id in (4) is the imaginary part. Similar
comments hold for ac + bd of (3) compared to what actually turns out to
be ac - bd of (4) where the latter should really be the imaginary part
of the product, that is to say i(ac - bd) should be the second term on
the right hand side of (4) instead of i(bc + ad).
Osher Doctorow
.
|
|
| User: "OsherD" |
|
| Title: Re: Complex Multiplication Does Not Preserve Causation |
17 Dec 2005 01:06:30 AM |
|
|
From Osher Doctorow
It doesn't do any good for critics to claim that conditional
probability, for example, would do any better than probable
causation/influence, since it is rather easy to prove that except at
P(A) = 0, both P(A-->B) and P(B|A) = P(AB) divided by P(A) (conditional
probability) increase together monotonically although of course not
usually closely.
What, then, do we make of the entire Hilbert Space formalism with the
complex variables?
The Hilbert Space formalism is an accurate description arguably of
"uncaused" quantities more or less, since it doesn't preserve probable
causation/influence under complex multiplication but does give some
useful results. These are roughly but not precisely "statistically
independent" quantities, so in other words we can roughly say that
Hilbert Space quantum mechanics is approximately the study of
statistically independent random variables. There is a mathematical
technicality in that statistical independence is not precisely the same
as "zero probable causation".
Admittedly, there are some special cases when the complex product and
the product of probable causations/influences are relatively close to
each other in magnitude if not in real/complex terms vs respectively
negative/positive sign terms. For example, if b and d are very close
to 0, then (a + ib) times (c + id) is approximately ac for all
quantities a, b, c, d in [0, 1]. Likewise, (1 + b - c)(1 + d - c) has
ac as its only surviving product term in that case, although it also
has single terms 1 - c + b - a + d so that 1 - c - a would survive as
well as ac.
It is also a fact that the product P(A-->B)(a, b) times P(C-->D)(c, d)
has the "correct" signs -c and -a for the causing/influencing variables
or their representations and the "correct" signs +b + d for the
caused/influenced quantities. The sign of +db is what one would expect
from b and d having + signs, but the sign of +ac is a bit surprising
since -c and -a occur with negative signs. Curiously enough, ac is
the only term that has the same sign in (a + ib)(c + id) and in
P(A-->B)(a, b) P(C-->D)(c, d), so whatever the reason for ac, it
doesn't discriminate between the Hilbert Space and the Probable
Influence "formalisms".
Osher Doctorow
.
|
|
|
| User: "OsherD" |
|
| Title: Re: Complex Multiplication Does Not Preserve Causation |
17 Dec 2005 01:26:03 AM |
|
|
From Osher Doctorow
What would a "statistically independent" Universe look like? It would
be rather awful to say the least. Nothing would relate to anything
else. Arguably it would describe "Materialism" at its worst: illogical
power-materialism, sensation-materialism, greed-materialism,
nepotism-friendship-materialism ("friendship above Ethics"). It could
be described as the Universe's analog of the Law of the Jungle.
Admittedly, there are parts of the Universe where the Law of the Jungle
does apply even in inorganic variables. "Random molecular
bombardments" from cosmic rays for example come quite close to the Law
of the Jungle and are very important in brownian motion, although
brownian motion isn't exactly what science is mostly about (it's
roughly random zigzag motion). I suppose that neutrinos passing
through everything could be described as obeying a sort of Law of the
Jungle in that nothing they do directly affects most things and vice
versa, a sort of "ultimate hippies" (although hippies don't like to
describe themselves as hippies).
I should also mention that social and behavioral and biological
scientists largely rely on "random sampling" since their fields are so
easily biased by various experimenter and subject errors both
deliberate and accidental so to speak, so that some degree of Law of
the Jungle goes into some sciences' scientific research.
While we're on this topic, it may not be amiss to mention that
imaginary i in Special Relativity may have been the wrong choice of
designations for the time (or, in reverse symbols, space) variable or
unit time variable. It also raises a question of whether a quantity
like sqrt(1 - v^2/c^2) which becomes imaginary if v > c is really a
fundamental quantity in the Universe. I suspect that it isn't. But
then, nobody thought of using + and - signs to distinguish variables
instead of real versus imaginary. That's OK. It's only a few hundred
years, no? Or is it a few thousand? I keep forgetting, what with
history being so exciting :>)
Osher Doctorow
.
|
|
|
| User: "OsherD" |
|
| Title: Re: Complex Multiplication Does Not Preserve Causation |
17 Dec 2005 02:01:11 AM |
|
|
From Osher Doctorow
Most readers who've looked up my early postings here and/or in
geometry.research and/or sci.stat.math or elsewhere may notice that I
only occasionally had threads involving products like P(A-->B)
P(C--->D).
Let's take a look at such a product again, beginning from scratch.
1) P(A-->B)(a, b) = 1 + b - a, b = P(AB), a = P(A)
2) P(C-->D)(c, d) = 1 + d - c, d = P(CD), c = P(C)
Then for the product we get:
3) P(A-->B)(a,b)P(C-->D)(c,d) = (1+b-a)(1+d-c) = 1 -a + b - c + d - ad
+ ac + bd - bc
Notice that 1 occurs in the product just as it does in each of the
original probable causations/influences of (1) and (2), and that in
fact the single factor terms are:
4) 1 + b - a + d - c
and that since 1 + b - a is a probability from (1), it must be between
0 and 1, and d - c must be negative or 0.
Readers can figure out some other properties of (3) or its left hand
side as homework.
Osher Doctorow
.
|
|
|
|
|
|
|
Related Articles |
|
|
