Science > Physics > Invariance-Intersection Theory "Collapses" Into PI
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08 Sep 2005 10:01:42 PM |
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Invariance-Intersection Theory "Collapses" Into PI |
From Osher Doctorow
COPYRIGHT NOTICE
Invariance-Intersection Theory "Collapses" Into PI
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
A remarkable thing happens when one does heavy downloading of papers
from Front For the Math arXiv and from arXiv (physics). It is
something which programmers for the "archives" should be concerned
with, since it isn't as easy to notice with "light" or intermittent
downloading. It turns out that Invariance theories go off in all
directions so to speak, that mathematicians and physicists go off very
largely in separate directions, that different types of Invariances
proliferate and the whole bunch seem less and less related both
explicitly and implicitly, that intersections tend to be lost sight of,
and that people like Steven Carlip of U.C. Davis and John Baez of U.C.
Irvine/Riverside whose writings reveal a bias toward Loop Quantum
Gravity (LQG) seem to thrive in this "anarchy" especially with their
modular and diffeomorphism invariances.
As with algebra's endless proliferation of "data" without clues (not
least of all in algebraic geometry and algebraic topology), Invariances
have a similar proliferation except where clues are found in more
valuable fields - and Probability-Statistics turns out to again come to
the rescue.
Let's look at:
1) P(A-->B) = k (k > 0 constant)
Since probability is between 0 and 1 inclusive, k is between 0 and 1.
This is an Invariance of a very simple type. Now look at the
conditional probability analog (which even quantum physicists can do
since they mostly have used conditional probability rather than PI by
default):
2) P(B|A) = k (k > 0 constant)
Unrelated, no? No. Let's write out each, using primes for respective
numberings of (1) and (2):
1') 1 + P(AB) - P(A) = k
2') P(AB)/P(A) = k (if P(A) isn't 0)
From (2') we get:
3) P(AB) = kP(A)
and from (1') we get:
4) P(AB) = k + P(A) - 1
Different, no? No. At least, not with a little thinking. Since
P(A-->B) is probable influence/causation, we might as well optimize it
in invariance, by selecting k = 1 which is its maximum, yielding:
5) P(A-->B) = 1 + P(AB) - P(A) = 1
and therefore:
6) P(AB) = P(A)
But for k = 1, we have from (2') and (3)
7) P(AB) = P(A)
They intersect! Probable Influence (PI) and conditional probability
(cp let's call it) have intersected!
Invariance has led to intersection with a little thinking about
optimizing probable causation. Bayesians, the most entrenched
mathematical statistics users of conditional probability, wouldn't have
thought of this before Shakespeare's monkeys, if then. You really need
competing ideas to do the best Creativity and I should mention the best
intersecting.
Ah, you'll say, what about the third analog of fuzzy multivalued
logics, namely Independent Probability-Statistics (IPS for short,
although this abbreviation is almost unknown in the field since it
never considered that it had any competition!)? Does it also intersect
the other two types? No. And here we come to a mystery worthy of
Sherlock Holmes. Generalized Boolean Logic is generated by any two of
the Fuzzy Multivalued Logic (FML) analogs of these 3 probability types
(extended beyond implications), namely Lukaciewicz/Rational Pavelka,
Product/Goguen, and Godel FMLs. So it's enough to have established
that two of the types intersect. It is, however, interesting to
examine what happens to IPS, which could be written:
8) P(B|A) = P(B)
or in terms of PI:
9) P(A-->B) = 1 + P(A)P(B) - P(A)
Readers can examine this as homework. It shouldn't take anywhere
nearly as long as the literature on Invariances, though it might not be
as helpful to publish-or-perish careers :>)
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Invariance-Intersection Theory "Collapses" Into PI |
08 Sep 2005 10:25:29 PM |
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From Osher Doctorow
The "homework" assignment is really interesting!
Believe it or not, the universe is "invariance-split" with regard to
PI, conditional probability, and independent probability-statistics,
and analogously with regard to the 3 types of fuzzy multivalued logics.
The relevant theorems are.
Theorem 1. P(A-->B) = P(B|A) = P(B) = k (constant) where k is in (0,
1) iff:
1) P(B) = [1 - P(A)]/[1 + P(A)]
Theorem 2. Theorem 1 fails for k = 1 (that is to say, for the optimal
condition for P(A-->B)).
Notice that if k = 1, then by Theorem 1, P(B) = k = 1 so equation (1)
would yield P(A) = 0, but if P(A) = 0 then P(B|A) = P(AB)/P(A) is
undefined!
The proof of Theorem 1 is just algebraic after substituting the
definitions.
Notice that the "worst possible scenario" for P(B|A), namely that
P(B|A) = P(B) (equivalent to statistical independence) is not detected
by conditional probability but by P(A-->B) because the latter is
undefined!
In addition, P(A-->B) has an advantage over P(B|A) in that the former
is defined when P(A) = 0 in general, unlike P(B|A).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Invariance-Intersection Theory "Collapses" Into PI |
09 Sep 2005 12:01:41 AM |
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From Osher Doctorow
A few more clues are needed before most physicists and mathematicians
can take advantage of the new results.
A. "Independence" is not what "everybody" thinks it is, including the
people who "discovered" it.
B. Fixing something or taking something as "constant" or "given" is not
what "everybody" thinks it is, including the people who "discovered"
it.
Clue A summarizes the situation with regard to Independent
Probability-Statistics (IPS) from the last postings, and Clue B
summarizes the situation with regard to conditional probability (cp,
which I used to call Bayesian Probability-Statistics or BPS) from the
last postings.
Now, one often reads or hears physicists claiming that
probability-statistics is relatively useless (usually in branches that
don't use it), but one seldom reads or hears them say the same thing
about "independence" and "givens" or "fixed values" or "constants".
I suppose that I should add a third clue, although readers who've
followed many of my threads will already know:
C. Causation, Influence, and Probable Causation/Influence are anything
but intuitive at least insofar as their quantification is concerned.
So we begin to see why Invariances and Intersections are so poor guides
to discovery without input from Probability-Statistics or some other
very clue-related field. Invariances set things constant, but we've
just seen that this is nothing like what scientists and arguably
mathematicians think it is. When we throw in "independence" and
"causation/influence", without knowing theorems like the ones in this
thread or anything remotely resembling them, we're asking for trouble.
So what can I say to Sir Michael Atiyah, to John Schwarz, to Michael
Green, to the Smolin-Rovelli-Ashetekars, and to the newer young lions
of superstring theory who have built careers on taking algebraic
geometry and algebraic topology to the nth degree so to speak? And to
Sir Roger Penrose who tries to take twistors from matter to geometry
around black holes and Stephen Hawking who thinks that thermodynamics
(which still uses conditional probability) is more or less all there is
to black holes, and Gott and Li and Steinhardt and Turok and Nathan
Seiberg and Edward Witten who round the squares and square the rounds
and so on?
I would tell them to be devastatingly simple like Chaitin and Einstein
whom they claim to revere and Paul Dirac and Erwin Schrodinger and
Prince Louis de Broglie and Ricci and Levi-Civita and Lorentz and
Fitzgerald and Archimedes and Euclid and Pierre de Fermat and Sir Isaac
Newton and Leibniz.
Follow Probability-Statistics and Engineering and the non-fad parts of
Mathematical Physics and Applied Mathematics and Differential Equations
and Functional Analysis and Real and Complex Analysis and Nonsmooth
Analysis and the deeper insights of fractals and chaos and number
theory without obsessing on their computer parts (and combinatorics
too). If you have a choice between algebra and geometry, choose
geometry. If you have a choice between algebra and topology, either
avoid them both or concentrate on the intersection between topology and
geometry instead. Never lose sight of the Experimental Physics and
Elementary Particle Physics and Cosmology in their "raw" and
"uncombined" forms as sources of inspiration and data. Ditto Condensed
matter Physics. A Phase is worth 1000 words (it used to be a picture,
but we're a bit past that mostly). If you must wander into algebra,
follow Nathan Jacobson with the Jacobson Radical and Lie
groups/algebras and be inspired by John von Neumann but don't follow
him blindly (he was wrong like the rest of us) and Rudolf Haag. Choose
Logic, especially Mathematical Logic, but not logicians - many of them
think that they're in algebra (which I've come to believe is a
particularly annoying form of insanity).
The next time you're tempted to publish or perish, think instead about
reinventing physics with the help of Sir Isaac Newton and Faraday and
Maxwell and some of the others that I've mentioned. At the first hint
of an "anomaly" or "paradox", stop to completely clarify its essence
rather than merely exorcise it mathematically. If you can't, keep
trying. Even if you don't publish another word. You'll probably
invent a Physics free of BS and "strangeness" and mysticism and one
whose Nonlocality entirely is composed of many Localities. If you
find yourself wandering into chemistry, beware - chemistry is algebra
with smells (a slight exaggeration, but in any case chemistry is mostly
for empirical results and a grain of salt elsewhere).
Next question :>)
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Invariance-Intersection Theory "Collapses" Into PI |
09 Sep 2005 12:46:16 AM |
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From Osher Doctorow
Once upon a time, long, long ago, my wife and I discovered PI (Probable
Influence/Causation Theory), which we originally called "the
probability that A causes B". That was 1980. My wife was then and is
now a psychologist. The Bayesians and conditional
probability-statistics people who with the Independent
Probability-Statistics people dominated the University of California
campuses wouldn't touch it with a 10-foot pole. Neither would the
psychologists. So we found refuge among philosophers, who were used to
accepting refugees from the cruel, cruel world outside (where most of
them started, in fact). We presented our work at meetings of the Far
West Philosophy of Education Society (FWPES), which were peer-reviewed
and then published in their proceedings and/or those of the Philosophy
of Education Society. Various branches of this organization had their
fallings in and fallings out in what was an amazing politicization of
philosophy, but I suppose Socrates teaches us that philosophy is as
deadly and dangerous to the power-hungry as anything else. A few years
ago, I tried to find my FWPES contributions on the internet (they
weren't originally on the internet), and there's something like a blank
wall where it says if I recall: Far West Philosophy of Education
Society (FWPES): located at _____. The Philosophy of Education
Society records themselves were expurgated, or else the newer version
of the organization is not even in the hands of the same people as the
older one. Since I retained copies of almost all our publications, I
know that it was real, and so does my wife Marleen.
If Nature gives you a lemon, be content, since you can use it to make
lemonade. I used it to think about and try to reconstruct physics and
mathematics and apply it to other fields. I became a Nonmaterialist
who values Knowledge, Ethics, and arguably Spirit above Power, Greed,
Nepotism and even Friendship, Sensation. I learned parts that I had
skipped, relearned parts that I hadn't understood entirely.
I heard on today's car radio that in Denmark or Sweden, it's been found
that women who have more stress have less breast cancer, but that
stress causes heart disease and other ailments. Still, I recommend
stress over cancer if you really have to choose. It's very stressful
to stop and understand anomalies and paradoxes and unwind them so to
speak. But the apes should really have done that when their ancestors
left the trees. Apes and monkeys are in too much of a hurry to get
somewhere else. The answer isn't to stop and smell the flowers (you'd
eventually look for a new smell!) but rather to fight the stress
head-on by facing it. The answer isn't to "keep on trucking" but to
"keep on learning." It isn't "publish-or-perish" but "learn or
perish". There's a big difference. Like the Danish or Swedish women
who have less breast cancer with more stress. My guess is that,
assuming that the radio was correct about their news story (it was a
network news program, which doesn't tell us much except that it wasn't
a comedy show), those women arguably spent less time partying and more
time thinking about problem-solving. Which reminds me that John von
Neumann spent too much time partying. As for Richard Feynman who like
John also got fatal cancer, I've always suspected that CalTech isn't
good for growing things, though I haven't been able to prove it. Has
anybody here seen the smog in Pasadena? The "little old ladies from
Pasadena" that they joke about probably didn't go out of their houses
very much, a wise choice in Smogville. I wonder why Edward Witten
stays there after exiting Princeton? Maybe he doesn't leave an
air-conditioned room? Or he stays in Malibu and contacts CalTech by
mobile phone? (Watch out for the ocean pollution, Witten. Wait a
moment - wet, Witten? Eureka! I'm onto something!)
Next question :>)
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Invariance-Intersection Theory "Collapses" Into PI |
09 Sep 2005 01:17:09 AM |
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From Osher Doctorow
Actually, I think I've figured it out. Witten really does his research
at Stanford, which has cool, clean air. CalTech is a cover. Well,
maybe.
Now let's get back to work.
Take a look at:
1) dy/dx = f' ' (x) = lim [f(x + h) - f(x)]/h (lim as h --> 0)
Now take a look at:
2) P(A-->B) = 1 + P(AB) - P(A)
According to Garrett Birkhoff of Harvard, the grand old mathematician
of Harvard who is no longer with us, equations involving things like
dy/dx or partial derivatives, called of course differential equations,
embody causation. I call that Birkhoff Causation. Most people don't
call it anything. Typically Birkhoff was thinking x = t (time), but
heck - space can substitute in a pinch.
Now set A or A(x) or A(x, y) = {w: X(w) < = x}, B or B(y) or B(x,y) =
{w: Y(w) < = y}, in which case we get:
3) P(X-->Y)(x,y) = 1 + P(X < = x, Y < = y) - P(X < = x)
= 1 + F(x, y) - FX(x)
where F(x, y) is the joint cumulative distribution function (cdf) of
random variables X and Y (taken continuous here for simplicity) and
FX(x) is the (marginal) cdf of X.
So now compare (1) and (3). Well, there's something vaguely similar
about them, except for things like h. But heck, what's a little
difference among friends?
Now compare (1) and (2) again. Remember, (1) says P(A-->B) = 1 + P(AB)
- P(A).
Let's suppose that we're teaching a class here (not far wrong, except
that this may usually be a classroom of one, although it's one of my
best classes :>)
What's right about this picture? Or more precisely, what's similar
about (1), (2), and (3)?
OK, there's a difference involved. Actually, the difference is
fundamental. In fact, if you look very closely at (1), you may be
reminded that now that q-arithmetic is with us (look under that or
similar keywords on the internet), the limit part of (1) may not be
that much more important than the difference ratio. Or in geometric
language, the secant line/segment may not be that far behind the
tangent line.
Oh, oh, a student raises his/her hand! "But there's no ratio in (2) or
(3)!" No, there isn't. Well, not exactly. Well, maybe there is.
These are probabilities, remember. And the Frequency Theorists of
probability-statistics think that everything is ratios or limits of
ratios in probability-statistics. And with our Grand Secant
perspective, we can approximate, can't we? (I wonder, could one form a
new Religion using the Grand Secant persepctive :>)
So we come to the topper of this post:
4) P(A-->B) = (approximately) 1 + N(AB)/h - N(A)/h
where 1 could be written h/h, but let's not get carried away. What's
the h? Well, I'm trying to approximate P(AB) and P(A) by relative
frequencies, so I suppose that I should write N(AB)/n and N(A)/n (or
N(A)/n2) where N(AB) is some estimate of the number of events in some
finite representation of AB and n is some total population, etc. So h
= n. But h --> 0 isn't n --> infinity. Well, nobody said it would be
easy. Anyway, don't worry about the limit since we're way ahead of
where we were a little while ago.
You can see now that the 1 in (4) doesn't change the nice fact that
Birkhoff Causation from differential equations has a remarkable
resemblance to Probable Influence/Causation (PI). Anyway, you can
write 1 as h/h.
Could you do this with conditional probability P(B|A)? No. It's
P(AB)/P(A) for P(A) not 0. It's like trying to put water into a Klein
bottle in our spacetime. If I were one of the judges in Hitchhiker's
Guide to the Galaxy, I would outlaw P(B|A) entirely after this
demonstration. Don't nobody move :>)
Osher DOctorow
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| User: "OsherD" |
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| Title: Re: Invariance-Intersection Theory "Collapses" Into PI |
09 Sep 2005 01:49:09 AM |
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From Osher Doctorow
Notice my typo in signing my name: DOctorow. Admittedly it's late
here, but not that late. However, there is something interesting about
the name Doctorow - it has multiple meanings or sub-meanings. DO, TO,
OCT, ROW, OR, DOC. Curiously enough, my father believed somewhat in
the Kaballah, like Madonna. I say "somewhat" because his talkativeness
was even less than Paul Dirac's, which made even farmers look
vociferous. He wasn't obsessed with the Kaballah, since it just was
handed down from generation to generation (I think). The ones who're
obsessed with it, I think, belonged to another group.
The name Osher is even more Kabbalah-ish. I never could figure out
exactly what it's Kabbalah interpretation is, but it literally means
"fortunate" which to me means "lucky". From the Ancient
Aramaic/Hebrew. However, I've been thinking about the Biblical phrase
over the years: "Adon Haolam, Asher Malach," "Lord of the World, who
maketh Kings," and I think that Asher may mean "King-maker" in calm
Kaballah (as opposed to vociferous Kaballah). My father actually
studied under Bialik (Haim Nachman Bialik, if I recall, the greatest
storyteller of his time and place. You might find him in the internet
somewhere.), who sang "Adon Haolam." Of course, one who makes Kings
can also unmake them, I think. They really don't tell us the rules
nowadays the way they did in the old days :>)
I actually threw this in indirectly partly to explain why I discussed
the Far West Philosophy of Education Society last time. When we
presented our paper at their meeting at U.C. Berkeley (I think in
1981), an expert in probability-statistics who was closer to the
Bayesian camp (hey, everybody has to get promoted!) suggested that I
try to relate the set theory equations to random variable equations.
In fact, he introduced our talk. Well, Bayesians will be Bayesians.
Anyway, little did we suspect at the time that we would be hobnobbing
with Garrett Birkhoff at least conceptually. Birkhoff's father was an
open and terribly prejudiced Nazi, but Garrett went in an opposite
direction to his father. Sort of like Governor Schwartzenegger who
actually believes in government not dominating people. He also doesn't
believe in raising professors' and teachers' salaries when they are (a)
among the most highly paid professions, and (b) California is barely
recovering from near-bankruptcy under the giveaway Governorships of
Gray Davis and the Brown family and Alan Cranston's Senatorship (the
darling of the Academic Lobby, though it works both ways). And under
Enron and similar scandals which our Underdog Democrats were supposed
to protect us against but instead seemed to mysteriously attract :>)
Next question :>)
Osher Doctorow
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