| Topic: |
Science > Physics |
| User: |
"OsherD" |
| Date: |
14 Jun 2005 02:16:42 AM |
| Object: |
Causation as a Set |
From Osher Doctorow
COPYRIGHT NOTICE
Causation as a Set
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
In Probable Influence (PI), Causation or Influence of set/event A on
set/event B is symbolized by (A-->B) defined as:
1) (A-->B) = (AB')' = (by set theory) A' U B
where A' is the part of the universe outside A (the complement of A)
and U is "and/or" (set union).
If you draw a Venn diagram with A, B as intersecting circles (the
intersection could always be empty or null) inside a rectangle that
we'll call the universe S, then it turns out that (A-->B) is the whole
rectangle (universe) except for AB', which is to say except for A
occurring without B occurring.
Except for time, is this what we mean by causation? Let's see. If A
causes B, then every time that A occurs, B has to occur. So the
intersection AB of A and B has to be in "causation", and it is
(remember, only AB' is excluded). On the other hand, it's possible
that neither A nor B will occur, which means that A'B' occurs (both
"not A" and "not B", or both "outside A" and "outside B" occur), so
this has to be in "causation", and it is since it's not AB' which is
the only excluded part of causation, and it doesn't intersect AB' as a
little thought will show. Now only two parts of the universe haven't
been examined: AB' and BA'. BA' means that B occurs but A doesn't
occur. This is legitimate because even though A causes B, something
else could also cause B. For example, a pneumonia can cause a high
temperature, but so can the flu. So BA' should be part of "causation",
and it is since it doesn't intersect AB' which is the only thing
missing from (A-->B).
Finally, let's examine AB' separately without assuming that it should
be absent from causation. AB' means that A occurs but B doesn't
simultaneously occur. But this can't happen because if A causes B,
then the occurrence of A must be followed by or accompanied by B. So
AB' is not in causation, which is to say it is not in (A-->B). No
other sets are left.
So (A-->B) does signify causation, provided that A does not occur at a
time after the occurrence of B in the expression. This is because
causes don't occur after effects but before or simultaneously with
them.
You will not find any of this explained anywhere other than in my wife
Marleen's and my writings since 1980. Why? Roughly speaking,
algebraists de-emphasized sets when Category theory (which is Saunders
MacLane's algebraic invention) came in a little after the mid-1900s.
Category theory claims to generalize sets to "objects" without paying
attention to the counter-argument that any object can be regarded as a
set. It is something like generalizing physics to include art without
specifying how physics relates to art and without regarding it
necessary to find out how they're related.
It might be thought that variables, random or not, and processes, can't
be expressed as sets. But they can. For example, a continuous random
variable X is completely determined by the sets A_X(x) defined as the
set of elements w of the probabiity space such that X(w) < = x for each
real x, which are the sets in the definition of cumulative distribution
functions (cdfs). For processes, you might want to put a time index or
subscript on a set A at time t, such as A_t or for short At, or you can
use an "index convention" which Marleen and I use such that subscripts
are understood at any time in the appropriate time interval.
But isn't there a difficulty with the "set of sets", which leads to
Bertrand Russell's paradox? Yes, but calling the "set of sets" an
"object composed of sets" can easily be shown to lead to the same
paradox once the relationship between the object and the sets is
specified in a way that enables them to be used. There simply are
certain ideas or expressions that can't be expressed in set theory
mathematics or any form of mathematics, such as division by 0. That
doesn't prevent us from dividing by non-zero numbers. Neither should
the set of sets paradox prevent us from using sets. Incidentally, sets
are also called events and vice versa.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation as a Set |
14 Jun 2005 02:30:24 AM |
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From Osher Doctorow
What does the probability of (A--->B) mean?
It means the same thing that the probability of any event means, taking
into consideration the particular nature of the event in question.
Intuitively, probability is often thought of as the "chance" of
occurrence of an event, on a scale of 0 to 1. For example, if we
consider that an event's chance of occurrence is "half the time", we
write P(A) = 0.5 where A is the event and P(A) is its probabiity.
But haven't we ruled out probability by saying that (A-->B) includes "B
must occur when A occurs"? No. A probability can be assigned to any
set provided that it is done consistently and it is understood that the
occurrence of the event is not necessarily "determined or inevitable".
By assigning a probability P(A-->B) to the set (A-->B) of causation,
we're actually putting a measure on the universe outside AB' . It
makes sense that for different events A, B, the "size" of AB' can vary
in the population of all events, or if we regard A, B as time-indexed
sets, the "size" of AB' can vary in time. If P(A-->B) is large (near 1
but slightly below it for example), this roughly means that our measure
of AB' is small, or roughly speaking that AB' is small. Again roughly
speaking and intuitively, there are very few cases or scenarios where A
occurs and B doesn't occur. Similarly with other sizes for P(A-->B).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation as a Set |
14 Jun 2005 03:06:05 AM |
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From Osher Doctorow
An alert reader will notice that I contradicted myself, since I said
that in (A-->B) we can't have A occurring and B not occurring, and yet
later I said that if P(A-->B) is near 1, then there are very few
scenarios where A occurs and B doesn't occur. Which is it, "none" or
"very few", or what?
The answer is that when we put a probability P on a set (A-->B), we
slightly generalize the ordinary notion of causation precisely as in
the second usage above. P(A-->B) could be taken as 1 if we totally
can't have A occurring and B not occurring, but we also can generalize
it to the cases where we usually can't have A occurring and B not
occurring, or where this only happens half of the time, or where it
almost always happens, etc. We distinguish between all of these cases
by the numerical value of P(A-->B) on a scale from 0 to 1. When
P(A-->B) is 1, we never have A occurring and B not occurring (except
for sets of probability 0 - see below). When P(A-->B) gets smaller and
smaller, and eventually heads toward 0, the proportion of cases where A
occurs and B doesn't occur increases. For P(A-->B) = 0, A occurs and B
never occurs, except for a set of probability 0 (don't worry for now
about the meaning of this last restriction about except for a set of
probability 0 - intuitively it's not the most critical thing).
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation as a Set |
14 Jun 2005 02:50:56 AM |
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From Osher Doctorow
I typed a third post over 5 minutes ago and it didn't appear, so I'll
briefly summarize it here.
A slight modification occurs from the "A can't occur without B"
scenario when we put a probability P(A-->B) on (A-->B). We still can
include the "A can't occur without B" case in the interpretation of AB'
not being in (A-->B) by setting P(A-->B) = 1 which makes P(AB' ) = 0
(up to sets of probability 0 - don't worry about the latter now). But
we also generalize to cases where A usually can't occur without B,
where A can only occur without B half of the time, where A almost
always occurs without B, etc., the magnitude of P(A-->B) on a scale of
0 to 1 indicating the proportion of cases.
Osher Doctorow
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| User: "Robert Kolker" |
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| Title: Re: Causation as a Set |
14 Jun 2005 04:49:13 AM |
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OsherD wrote:
From Osher Doctorow
COPYRIGHT NOTICE
Causation as a Set
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
In Probable Influence (PI), Causation or Influence of set/event A on
set/event B is symbolized by (A-->B) defined as:
1) (A-->B) = (AB')' = (by set theory) A' U B
where A' is the part of the universe outside A (the complement of A)
and U is "and/or" (set union).
Material implication does not model causation particularly well because
there is no necessary connextion between the premis and the conclusion,
whereas in causation there is a generative relation between the cause
the the effect. Likening material implication to causation is a fancy
way of assertion the post hoc ergo propter hoc fallacy.
Bob Kolker
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| User: "Jack Martinelli" |
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| Title: Re: Causation as a Set |
14 Jun 2005 02:46:42 PM |
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"Robert Kolker" <nowhere@nowhere.com> wrote in message
news:fOSdnf_kC8SINTPfRVn-2Q@comcast.com...
OsherD wrote:
From Osher Doctorow
COPYRIGHT NOTICE
Causation as a Set
Copyright By Owner Osher Doctorow Ph.D.
First Published 2005
In Probable Influence (PI), Causation or Influence of set/event A on
set/event B is symbolized by (A-->B) defined as:
1) (A-->B) = (AB')' = (by set theory) A' U B
where A' is the part of the universe outside A (the complement of A)
and U is "and/or" (set union).
Material implication does not model causation
Causation is a fantasy. And as such you can model it however you want and
every model will be just as correct as the next.
Material implication is important, however, in discussing anything with
physical significance. You don't need causation, only observables. And it
is the observables that we anchor our labels (abstractions) to.
Regards
Jack Martinelli
http://www.martinelli.org
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| User: "OsherD" |
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| Title: Re: Causation as a Set |
14 Jun 2005 06:13:00 AM |
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From Osher Doctorow
Bob Kolker wrote:
Material implication does not model causation particularly well because
there is no necessary connextion between the premis and the conclusion,
whereas in causation there is a generative relation between the cause
How nice to hear from you, Bob! The fact that you found it necessary
to reply indicates to me that I must be on the right track :>)
However, I knew that already.
You are describing here the Mainstream AI (artificial intelligence),
Bayesian (conditional probability with emphasis on an entire
conjectured distribution of "givens"), and Computer Engineering
argument which has kept research funds and employment in the hands of
the "young lions" since around 1950 with their dislike of truth and
abstractions and the wisdom of the aged and ages and their Ancient
Roman-like admiration for concrete Ingenious Imitation and the
ignorance and babbling of youth.
It is also interesting how BS in bureaucracies changes its form from
quantitative to verbal and back depending on how dangerous their
Materialist conformist situation is deemed to be. I give equations,
the "opposition" replies verbally. I use words, the "opposition"
demands equations. I use English, the critics use Latin. I use
brevity for simplicity, the critics use brevity for embedding seeds of
doubt. But do expand your claims rather than have them be buried in
others.
I do appreciate your occasional revival of Consciousness, and keep up
the good efforts. Eventually, there may be success, especially if one
marries a psychologist or psychiatrist and goes through enough stress
and enough learning and enough aging like me :>) It also helps to be
an old Russian in a young German-American world.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation as a Set |
14 Jun 2005 07:02:55 AM |
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From Osher Doctorow
Sets are an interesting nexus among many worlds and many thoughts. Not
even Saunders MacLane with his Category theory could exorcise them with
or without Latin. Take a look at Birkhoff and MacLane's A Survey of
Modern Algebra, Macmillan: N.Y. 1953, which I almost grew up with. How
many major algebraic concepts were defined in terms of sets there?
Integral domains? Yes. Fields? Yes. Groups? Yes. Vectors? Yes.
Rings? Yes. Ideals? Yes. Algebraic number fields? Yes. Modules?
Yes. Lattices? Yes. Boolean algebras? Yes.
Measure Theory in Real, Complex, Functional Analysis and Geometry and
Topology including topological space were built on sets as much as
functions, and in fact functions are relations were defined by sets.
Look at a really old book like Hewitt and Stromberg's Real and Abstract
Analysis, Springer-Verlag: N.Y. 1965 for this.
Probability and Statistics have never abandoned sets. Abandonment of
sets was left for rootless people in windy, overcrowded cities trying
to make a name for themselves without having read Shakespeare. The
French and Belgians were especially happy about that, though instead of
wind they had wine and chocolates. And where did chocolates come from?
Networking :>). Look for a network, and the French have a name for
it. Karl Marx, Adolf Hitler, Charles De Gaulle - they were all steeped
in the French "higher culture". Hitler wouldn't even bomb Paris. It
was too "precious" for him. It was so concrete! Like the guillotine,
like the Bastille, like Devil's Island, like Robespierre, like Moulin
Rouge, like Napoleon, like Waterloo, like the Maginot Line, like fur
trading, like teaching Indians to massacre settlers.
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation as a Set |
14 Jun 2005 08:56:46 AM |
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From Osher Doctorow
I don't mean to insult France any more than Charles Dickens did in A
Tale of Two Cities :>) After all, beyond the guillotine there is
always the Grace of God :>)
In fact, French mathematics and physics were doing fine around the time
of Pierre De Fermat in the early to mid 1600s. This is usually a sign
of a sick society (I kid you not - stress or marrying a psychologist or
psychiatrist are the only ways to rise from Ingenious Imitation to
Creative Genius). That is precisely what Gaulle/Galle means (illness
of the gall bladder :>) Leibniz was educated in Paris during Fermat's
time, as did Pascal and Descartes. None of the classical composer
Creative Geniuses lived in France (something about "heads you win,
tails I lose" ). Painters and culinary artists did (if you paint one,
you can paint them all :>). The Revolution For Conformity (the French
Revolution of the late 1700s) was still far away, but maybe this is
where the Wheeler-Feynman absorber theory is proven :>) Napoleon was
only a sparkle in a French personality's eyes, neither yet quite
actualized nor entirely potentiated after the Fall of Rome scenario.
Women were "liberated" not in the ethical sense but in the sense of an
uncorked bottle of wine. The French language had either already been
or was about to be censored with elimination of hard consonants at the
Courts of the Monarchs (the reverse of Hebrew - it figures), which is
how BS began in Western civilization :>) The grass was greener on the
other side of the colonies, probably a lesson shared in common with
Spain, except in North Africa where tachyons began or else somebody ran
off with the fertilizer. The alcoholic content of the average French
person was already comparable to that of the young lions in USA physics
Academia from the 1950s onward. Soon to be were the worshippers of
primitive "innocence", humanity without individuals, and the
Judeo-Christian tradition without Jews. The betrayals of the Maginot
Line and the collaboration of WW2 Vichy France were not yet present,
not to mention Charles De Gaulle's reward to the USA for its
money-donating Marshall Plan after WWII and its liberation (together
with the U.K.) of France from Hitler (heck, who said they wanted to be
liberated?).
What I think did France in was Sir Isaac Newton's contribution to
calculus and physics. France was in a daze ever since, convinced no
doubt that he did it all with mirrors :>)
Osher Doctorow
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| User: "OsherD" |
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| Title: Re: Causation as a Set |
14 Jun 2005 09:05:58 AM |
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From Osher Doctorow
Actually, French sensation-materialism predated the French Revolution.
French Royalty were so focused on marrying British Royalty that a new
meaning was given to the words Royal Throne :>)
Next question.
Osher Doctorow
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