From: Osher Doctorow [log in to unmask], Sunday April 28, 2002 10:35AM

In B. N. Kursunuglu et al (Eds.) Quantum Gravity, Generalized Theory of
Gravitation, and Superstring Theory-Based Unification, Kluwer
Academic/Plenum: N.Y. 2000, 89-97, readers will find my paper relating
probability-statistics, fuzzy multivalued logics,
geometry-topology-proximity, and mathematical physics.    It is written in
mostly easy to understand English, and is based on the work of my wife
Marleen and myself since 1980.  I will discuss some of the CID
(Causation-Influence-Dependence) aspects which have become apparent since
that 2000 paper and which are very important for pure and applied statistics
and probability.

It turns out that the universes of mathematics and physics are divided into
3 classes of events/processes/things, which will be designated here for
brevity as Rare Events (RE), Fairly Frequent Events (FFE), and Very Frequent
Events (VFE).   Until 1980, when Marleen and I introduced the
probability-statistics of Rare Events (known as Logic-Based
Probability-Statistics or LBP), nobody had studied Rare Events, despite the
fact that in the mainstream Bayesian analysis of FFE it was widely known
that an anomaly or discontinuity occurs at rare events because of division
by 0 which is prohibited in mathematics.  FFE uses Bayesian Conditional
Probability-Statistics (BCP in my abbreviation), also known as Bayesian or
Conditional Probability-Statics, whose main characteristic is that it
divides probabilities to obtain the *dependence* or *given* nature of two
events, and Markov Chains are examples.  Rare events have probabilities
close to or at zero, and division by zero is not permitted in mathematics,
and in fact BCP *blows up* close to zero.  This was considered to be
satisfactory by Bayesians if it was considered at all.  Marleen and I
replaced division by subtraction, using the same probabilities, and
eliminated the division by zero in that manner.   The third type of events,
VFE, turns out to use Independent Probability-Statistics (IPS), and these
are on the boundary between LBP and BCP.  If the probability of the
*influencing* or *causing* event is taken to to 1 (*certainty,* although
technically certainty is only a subset of these events) in either LBP or
BCP, we get IPS.  Although the *influence* and/or *given/dependence* nature
of IPS is not multiplicative, it gives rise to a multiplicative expression
of probabilities which is related to it. The non-multiplicative expression
is called an identity function, which will also be mentioned again below.

Let me summarize the situation so far.   Rare Events use subtraction of
probabilities, Very Frequent Events use multiplication of probabilities, and
Fairly Frequent Events use division of probabilities, in order to express
influence or dependence of events, although technically for IPS (Independent
Probability-Statistics), dependence goes to what might be called the
opposite extreme of *independence*).    This was the situation in the 1980s,
although neither Bayesians nor Independent (IPS) researchers paid much
attention to Rare Events as has been indicated, and perhaps not more than a
handful of them knew about Rare Events which we had published in
Non-Mainstream journals and various papers presented and discussed with
various researchers by word of mouth.

In the early 1990s, Marleen and I discovered that fuzzy multivalued logics
had the SAME three categories and used the SAME three operations of
subtraction, division, and identity function which is closely related to
multiplication.   In the fuzzy multivalued logical implication x-->y (x
implies y), if x replaces one of the probabilities and y replaces the other
in the corresponding probability-statistics operations, we get the exact
same expressions respectively: y - x +1, y/x, and y or xy depending on
whether identity or multiplication is used.   In fact, the 1 in the first
expression is also found in both probability-statistics and fuzzy
multivalued logics in exactly the same position.   We began to realize that
we had a *tiger by the tail*, and began expanding our presentations and
research and publications.   Pavel Hajek's volume *Metamathematics of Fuzzy
Logics,* Kluwer: Dordrecht 1998 gives the first organized and concise and
thorough exposition of the three types of fuzzy multivalud logics, which are
respectively known as Lukaciewicz/Rational Pavelka, Product/Goguen, and
Godel fuzzy multivalued logics.  Hajek did not work with us and tried to put
probability into his book but knew nothing about Rare Events or LBP, and so
his probabilistic arguments came to naught so to speak.

By 1999, I had delivered a paper at the Quantum Gravity Conference of the
Global Foundation in Fort Lauderdale, Florida (not related to the
Massachussetts Global Foundation which collects funds for political
actions), which consists of a fair number of Nobel Laureates and other
similar level researchers.  The paper was more technical than the resulting
publication in Kursunuglu et al (Editors) 2000, which I *weeded out* of much
of the mathematics to make it more comprehensible across disciplines.

Around that time, I was heavily involved in discussing the theorems on
internet physics and mathematics and other science/engineering forums (I
currently belong to roughly 350 of them) and subsequently presented papers
at various conferences and had a few more publications.

The last stage, which reaches to the present, has been my realization that
geometry-topology has a corresponding division into 3 almost exactly
analogous branches, provided that instead of studying distances or distance
functions (metrics) between things, we study proximity functions.  Distance
is intuitively *farness*.  Proximity is intuitively *nearness*.  It turns
out that if proximity is correctly formulated (not just by dividing 1 by the
distance for example), we get the expressions 1 + y - x, y/x, and y or xy
(or zy with the slight modification of x) in the proximity analogs
respectively to the probability-statistics and fuzzy multivalued logic
analogs.

For the last few months, I have been contributing rather heavily to
http://www.superstringtheory.com/forum under Forums and its subforums
String-M-Theory-Duality, Supersymmetry, Philosophy of Physics, Dimensions,
Time and Causation, etc., and remarkably the triple theory which I have
discussed above makes important clarifications and hopefully contributions
to String/Superstring/Brane (a generalization of strings)/Membrane theory.
I originally used the abbreviation M Theory to mean Memory Theory as an
abbreviation for Marleen and my complete theory, but string and brane
theorists use M Theory in a totally different way to refer to a theory that
unifies 5 types of string theories (although it is largely unexplored), so
the reader needs to keep this in mind.

Since I am running out of space, I will simply conclude by explaining why I
cross-post to mailing lists and discussion groups and forums rather
frequently, a practice which British mailing lists often discourage.   I
explain this not for personal reasons but, as I will conclude my paragraph,
for somewhat more general reasons.  The rate of acceptance and posting time
to internet groups is roughly 1000 times or more greater than the
corresponding rate for either formal paper or even electronic journal
publishing IN MY CASE.   Part of the difficulty is my own background - I
have a Ph.D. and two M.A. Degrees, and have taught mathematics/statistics in
California universities/colleges since 1982 more or less, but these are
California State Universities and Community Colleges, not the more
prestigious Research Universities like UCLA, Berkeley, Stanford, CalTech.
But beyond this is the tendency of Mainstream Disciplines to dislike
Non-Mainstream or extremely new ideas unless they originate from senior
faculty INSIDE the discipline in question.  This may seem to contradict peer
review, but in fact my description of the reactions of
probability-statistics theorists (Bayesians especially) and mathematical
fuzzy multivalued logic theorists to both new and interdisciplinary ideas
has been exactly reflected in most peer-reviewed journals responses.
Extremely new or original ideas are roughly speaking rejected in my opinion
in the peer review process UNLESS they are known by the peer reviewer(s) to
have already arisen by somebody prominent in their own field or unless the
reviewer guesses the identity of the author of the paper (this is not as
difficult as it seems given bibliographies at the ends of papers - it is
rare not to cite oneself for example).   Thus, I come to my more general
reason for explaining this.  In my opinion, every quantitiative and even
qualitative discipline needs to emphasize causation, (probable) influence,
(statistical) dependence, tolerance for new ideas regardless of where they
come from, interdisciplinary research (the more fields it crosses, the
better), Creative Genius rather than Ingenious Imitation.   By Creative
Genius I mean ideas that are many steps ahead of any previous idea in the
field or subfield, not just one step ahead.   Peer-reviewed journals are
full of papers which are one step ahead of the previous person imitated,
often with a claim of elaborate theoretical novelty which turns out to be
more of a sleight of hand or cleverly contrived trick than a deep
fundamental change.  Ingenious Imitation is quite ingenious in giving the
illusion of novelty, and this is partly why causation, influence, and
dependence of events/processes/things give a deeper insight into what is
imitation and what is creative.

Osher Doctorow