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U.K. PIONEERS IN DISCOVERING THAT ANALYSIS "WINS OVER" GEOMETRY, AND IMPLICATIONS FOR PROBABLE CAUSATION-INFLUENCE (PCI)

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Osher

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Feb 26, 2015, 2:39:48 PM2/26/15
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I have previously given some comments on the authors of the following paper and an earlier version of it or similar to it:

1) "Analysis as a source of geometry: a non-geometric representation of the Dirac equation," Yan-Long Fong and Dmitri Vassiliev, University College, London, arxiv:1401.3160v3 [math.AP] 19 Jan 2015.

The context of this, although it may look like pure mathematics, is actually physics.

It is somewhat difficult to summarize this paper, even now, but basically the authors convert the geometry-appearing four main operations into analysis, with complex variables quite often involved.

Readers should know or be told that 'analysis' is basically the generalization of calculus in graduate universities or graduate university departments and research papers, e.g., in arxiv. Analysis has several branches:

2) Real Analysis (e.g., Lebesgue measure, Lebesgue integration, Radon-Nikodym derivative).

3) Complex Analysis (complex derivatives of various types, integrals, etc.).

4) Functional analysis (Hilbert spaces, Banach spaces which are generalizations of Hilbert spaces to scenarios where inner products are not found, etc.).

5) Probability (analogs of Lebesgue measure in (2) but in [0, 1] generalized into scenarios (e.g., physics) where events or things may or may not occur or exist or have a quantification of the (predicted) relative frequency of their occurrence, etc.

Note carefully that (5) does not itself require focusing on sampling or samples - statistics is the field that focuses on sampling and samples, although probability can study them if they relate to particular interests otherwise in probability. Statisticians, on the other hand, MUST use probability, while probability may or may not use statistics depending on the theoretical or practical interests of the researchers.

Osher Doctorow

Osher

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Feb 26, 2015, 2:45:55 PM2/26/15
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On Thursday, February 26, 2015 at 2:39:48 PM UTC-5, Osher wrote:
> I have previously given some comments on the authors of the following paper and an earlier version of it or similar to it:
>

The "topper" or "punch line" of this thread, at least so far, is that both Quantum Physics pioneers and GR pioneers through even the 1930s were under the impression, shared by Einstein, that statistics is the "legitimate" field if one has to decide between probability and statistics. Einstein arguably came to this conclusion because he viewed statistics as merely looking at samples, which to a physicist seems intuitively harmless in itself. After all, Newton as he explored physics only began with a small set of data from a larger population, which we could call a sample although the data almost entirely gave identical results in a sense. But statistics would not be a field of research if it did not ADD probability to sampling, since most aspects of sampling other than the very most elementary involve probability. Einstein refused to believe that there is such a thing as probability, but rather believed that if he learned enough about the real world, everything would turn out to be deterministically (non-probability-wise) predictable or describable.

Osher Doctorow

Osher

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Feb 26, 2015, 2:57:06 PM2/26/15
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On Thursday, February 26, 2015 at 2:45:55 PM UTC-5, Osher wrote:
> On Thursday, February 26, 2015 at 2:39:48 PM UTC-5, Osher wrote:
> > I have previously given some comments on the authors of the following paper and an earlier version of it or similar to it:
> >
>

Readers and researchers in general should try to see whether they do or do not believe in probability and understand probability by asking themselves whether, if some accident or unexpected good event (windfall, victory, success, etc.) occurs, they believe that it is completely predictable by a "good enough theorist" or 'good enough experimenter or observer'. It seems silly to give a really 'wild' example, but if you fall off a canoe, do you really think that some equation of the Universe would complete predict that falling off the canoe by you at the exact time and place, the prediction occurring ahead of time?

Some scientists like Max Born, who explained the probability of the wave function in Schrodinger's equation, realized that it is legitimate to assign a measure (probability) to things like (possibly different measures for different things):

1) How subjectively certain we are of some event occurring (e.g., in the future).

2) The relative frequency of occurrence of a certain event relative to a larger class of events such as the class of all experiments or of all events of a certain type.

3) The limit of (2) as the number of events approaches a large finite number N.

4) The limit of (2) as the number of events n --> infinity.

5) Certain "a priori" events such as the event that "I exist," to which most people would assign a probability of 1 ("100%"), or the probability of the event "a unicorn exists," to which most educated people at least would assign a 0 ("does not exist").

You might not have ever seen such an enumeration as above, but if you find that adopting it causes you more pain than believing that falling off a canoe is predicted by an equation of the universe in the mentioned circumstances, then you do not believe in probability.

Osher Doctorow

Osher

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Feb 26, 2015, 3:04:07 PM2/26/15
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On Thursday, February 26, 2015 at 2:57:06 PM UTC-5, Osher wrote:
> On Thursday, February 26, 2015 at 2:45:55 PM UTC-5, Osher wrote:
> > On Thursday, February 26, 2015 at 2:39:48 PM UTC-5, Osher wrote:
> > > I have previously given some comments on the authors of the following paper and an earlier version of it or similar to it:
> > >
> >

There is a second lesson to be learned for now from the U.K. paper above, namely that probability, which is real-valued and has a real domain and range, belongs to the same category of research (analysis) that "wins over" geometry, even though much of that research involves complex variables which at least now are not necessarily real-valued. In PCI, I argue that ultimately complex-non-real variables in physics will arguably give way to an underlying real-valued theory, which probability is. David Bohm already showed that the Schrodinger Wave Function can be decomposed into its real and imaginary components, which breaks a non-real complex number into two real numbers, although the "total breaking up" of non-real complex variables would have to explain why complex multiplication differs from real multiplication and its physics relationships. Bohm was not interested in doing that "last step".

Osher Doctorow

noTthaTguY

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Feb 26, 2015, 10:11:06 PM2/26/15
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quaternions are not commutative
;that is what makes them the aboriginal vectors

Osher

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Feb 27, 2015, 9:15:40 AM2/27/15
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On Thursday, February 26, 2015 at 10:11:06 PM UTC-5, noTthaTguY wrote:
> quaternions are not commutative
> ;that is what makes them the aboriginal vector

Good point. At least, it doesn't make them the "new kid on the block," as we say in the USA. The logical conditional also is non-commutative in general.

Osher Doctorow

Osher

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Feb 27, 2015, 9:29:34 AM2/27/15
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On Friday, February 27, 2015 at 9:15:40 AM UTC-5, Osher wrote:
> On Thursday, February 26, 2015 at 10:11:06 PM UTC-5, noTthaTguY wrote:
> > quaternions are not commutative
> > ;that is what makes them the aboriginal vector
>

Readers will find papers on gauge groups, GUTs, etc. typically using symmetry groups and other groups typically derived from geometry, which raises the question of how PCI (Probable Causation-Influence) relates to such groups. Recall that "group" in mathematics is not an arbitrary collection of objects but a collection having an identity, an inverse, associative law, etc., using written multiplicatively like xy (x "times" y), although they can be written additively (x "+" y).

A new perspective can be seen from an example that I have given before, but which I did not develop thoroughly. In one dimension, e.g., the real line or the segment [0, 1] of the real line, we have:

1) P'(A-->B) = 1 + P(B) - P(A) = 1 - [P(A) - P(B)] for P(B) < = P(A).

The bracketed expression, however, is precisely one-dimensional Euclidean distance d(P(A), P(B)) for P(A) > = P(B), which is |P(A) - P(B)|.

From (1), therefore, we can generalize:

2) GEOMETRY + NON-GEOMETRY = UNIVERSE, where PCI is included in non-geometry even though it does have one "inverse geometry" indication, namely "nearness" as an opposite of geometric "farness".

But we know from my previous and concurrent threads here that PCI is the (random) set-event-measure analog of Lukaciewicz infinite multivalued logic, so we can go even further:

3) LOGIC (at least, Lukaciewicz) is a subset or subcategory of Non-Geometry.

Osher Doctorow

Osher

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Feb 27, 2015, 9:38:28 AM2/27/15
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On Friday, February 27, 2015 at 9:29:34 AM UTC-5, Osher wrote:
> On Friday, February 27, 2015 at 9:15:40 AM UTC-5, Osher wrote:
> > On Thursday, February 26, 2015 at 10:11:06 PM UTC-5, noTthaTguY wrote:
> > > quaternions are not commutative
> > > ;that is what makes them the aboriginal vector
> >

We also find an arguably interesting aspect of PCI that contrasts with geometry, namely:

1) P'(A-->B) = 1 + P(B) - P(A) for P(A) > = P(B),
2) P(A-->B) = 1 + P(AB) - P(A),

where A, B are any two (random) sets/events such that A (the Cause) occurs before B (the Effect) or simultaneously with B. The interesting aspect that I referred to is the negative (minus) sign, which operates above in categorizing and distinguishing between two logical or causal classes or even 'dimensions' in a sense, namely Cause and Effect, unlike the role of the negative sign in geometry where it typically distinguishes between two geometric objects like points and is primarily used to measure the "farness" or distance between those objects as a geometric quantity.

In comparison with group theory, therefore, PCI not only replaces multiplication like xy by addition like x + y, but uses "minus" as in x - y not merely to represent the additive inverse but to put Cause and Effect into different categories, somewhat analogous to complex z = x + iy, where x is referred to as the real part of z and y the imaginary part of z (although y is real-valued; of course, i is not).

Osher Doctorow

Osher

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Feb 27, 2015, 9:44:23 AM2/27/15
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On Friday, February 27, 2015 at 9:38:28 AM UTC-5, Osher wrote:
> On Friday, February 27, 2015 at 9:29:34 AM UTC-5, Osher wrote:
> > On Friday, February 27, 2015 at 9:15:40 AM UTC-5, Osher wrote:
> > > On Thursday, February 26, 2015 at 10:11:06 PM UTC-5, noTthaTguY wrote:
> > > > quaternions are not commutative
> > > > ;that is what makes them the aboriginal vector

You can also see how PCI is generated from ordinary (classical) logic via logical ^ (conjunction, "and") corresponding to set/event "intersection" ("and"), logical V (disjunction, and/or) corresponding to set/event "and/or" symbolized U, logical ~ ("not") corresponding to set/event ' (as in A' = "not A" or "the part of the Universe outside A". Since PCI P(A-->B) = P(A' U B), the argument of the probability P, namely A' U B, is precisely the (random) set/event analog of the logical object ~a V b where proposition a corresponds to set/event A, proposition b to (random) set/event B. The expression ~a V b = ~(a ^ ~b), just as A-->B = (AB')' = A' U B.

Osher Doctorow

noTthaTguY

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Feb 27, 2015, 2:04:28 PM2/27/15
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quaternions is vector mechanics
;the speed of light varies with the speed of the observer, but
he or she cannot tell that
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