Re: Cross-category "conversions" of some interest

Re: Cross-category "conversions" of some interest

[Osher Doctorow]

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From: Osher Doctorow osher@ix.netcom.com, Tues. July 11, 2000, 11:00AM

Dear Colleagues:

The "conversions" from x/y to 1 - x + y and from x/y to x + y - xy which I introduced here on July 8, 2000 not only can be used across probability-statistics and (fuzzy) multivalued logical categories and ring theory and number theory categories (Fermat's Last Theorem) as indicated in that paper, but they are especially useful in special relativity and quantum field theory as well as quantum mechanics as I shall show here.  The categories in the latter two fields will be discussed in detail later, but here I would like to indicate what results are obtained.

Applying the conversion from x/y to 1 - x + y to the Heisenberg Uncertainty Principle in the form xy > k where k is a positive constant (x, y uncertainties in position and momentum respectively, for example, where notice x and y are nonnegative in the standard deviation version of uncertainty) yields xy = (x/y)y^^2 (where ^^ is exponentiation) --> (1 - x + y)y^^2.  For 1 - x + y < 0, which says x > 1 + y, the latter expression is negative, so the converted form reads: (1 - x + y)y^^2 = -/1 - x + y/y^^2 > k and therefore /1 - x + y/y^^2 < -k for k positive.  Since y^^2 is always nonnegative, this conditional holds trivially (always) provided that x > 1 + y.   Since x and y could be selected with exchanged physical roles (uncertainties in momentum and position respectively, for example), the condition that x > 1 + y is rather arbitrary and certainly will be fulfilled for one of the two orders in which x and y are defined.

The conclusion which we must come to is that there are two phases: the phase in which the Heisenberg Uncertainty Principle is satisfied (which phase may for example correspond to an interaction between macroscopic observation and microscopic phenomena) and the phase in which the Heisenberg Uncertainty Principle is not satisfied (which phrase may correspond to two macroscopic or two microscopic observation/phenomena pairs or both or some other regime).

A similar but even more explicit results holds when we make the conversion in special relativity in the beta or 1/beta Lorentz contraction factor sqrt(1 - v^^2/c^^2) where sqrt means square root of.  We get 1 - v^^2/c^^2 = 1 - (v/c)^^2 = 1 - y/x for y = v^^2, x = c^^2 and this goes over to 1 - (1 - x + y = x - y.   So beta or 1/beta (depending on what notation one uses) goes over to sqrt(x - y).  However, it has been pointed out in earlier papers (and it will be pointed out again) that the closure law holds in most of the categories to which these conversions x/y --> 1 - x + y for example apply, and let us assume the same thing here (closure under subtraction means that if x and y are in the category or in the set part of the category other than the morphism, then x - y and y - x are also in the set part).   Therefore, sqrt(y - x) is an alternate form of the result for beta or 1/beta, and since sqrt (y - x) is imaginary when y < x and real when y < x, it must be that the real and imaginary scales are themselves arbitrary.  In more common terminology, they merely measure different phases in the sense of liquid-solid-gas etc.

Thus, not only can the speed of light be exceeded (although the object exceeding it enters a different phase), but the Heisenberg Uncertainty Principle can be violated (but the objects violating it enter different phases).    This is not surprising in view of the superluminal group velocity results obtained in 1997 and later experiments by Nimtz in Cologne/Koln and Berkeley and elsewhere (which according to Nimtz are also applicable to ordinary velocities).  It is also not surprising in view of M. Jammer's comprehensive analysis (The Philosophy of Quantum Mechanics, Wiley: New York 1974) of the Heisenberg Uncertainty Principle in which it is found that the principle does not apply to individual measurements of physical objects but to statistical summaries of their uncertainties (Schrodinger in fact proved that on Hilbert Space self-adjoint operators obey this inequality for the product of their uncertainties, but Banach Space is far more general than Hilbert Space and gives enough room by far to define or contain a second phase sets of objects - which may not be operators in the usual sense).

The implications for special relativity and for quantum field theory (which combines quantum mechanics with special relativity) are obviously serious, although not necessarily catastrophic.   It just means that these theories are again approximations to one or two states or category of states of physical objects but not to more general or different categories of states of physical objects.  Since light itself satisfies both categories (sqrt (1 -(c^^2/c^^2)) is 0 which can be regarded as both real and imaginary), there certainly is at least one non-empty element in each category.

Osher Doctorow
Doctorow Consultants
Culver City, California USA   

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Re: Cross-Category "conversions" of some interest

[Osher Doctorow]

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From: Osher Doctorow osher@ix.netcom.com, Wed. July 19, 2000, 7:04PM


A list member claims in a private communication to me that 3 things are wrong with the x/y to 1 - x + y conversion. His first claim is that "it does not ring true" that I meant to type k1 or k2. This criticism is outside mathematics and has no meaning either there or in science.

His second claim is that xy = (x/y)y^^2 works one way and (x/y)^^2 (y^^3)/x works another way in the conversion and that the conversion of / but not other operations is anyway of questionable validity. Concerning the second part, the claim is meaningless. Since x/y is the main "animal" in BCP and in Goguen/Product logic implication and both have the form x/y, there is nothing to prevent me from comparing x/y with 1 - x + y in order to compare BCP and Goguen/Product logic implications with LBP and Lukaciewicz Logic. Concerning the first part, we can define a conversion C_n which converts xy as a product of (x/y)^^n g(x,y) where g(x,y) is a rational expression in powers of y and/or x no part of which contains (x/y)^^m as a factor for 1 < = m < = n and m, n are integers > = 1. C_n (C subscript n) for n = 1 gives exactly my conversion, and the critic's objection case involves C_2 which conversion is not being considered.

The third argument is that the conversion restricted to x/y for x, y coprime integers is alarming (because) continuity is lost (that is to say, I should mention, that the rationals are not continuous while the reals are) and because it is unclear how to apply the conversion to expressions of the form xy (is xy to be the product of two coprime integers or what?). This is really two objections except for the word "alarming" which is not a word in mathematics or science. The first objection, concerning continuity, implicitly assumes that the conversion is required to be continuous. Although I used the conversion earlier to attempt to convert xy in the Heisenberg Uncertainty Principle (HUP) xy > k, which perhaps should "ideally" involve all real xy, if the critic admits that the conversion works for rational x and y or even integer x and y, then xy restricted to either rational or integer x and y converts to two different inequalities depending on whether 1 - x + y < 0 or 1 - x + y > 0 (respectively x > 1 + y or x < 1 + y), so the converted form of xy does not obey the HUP, and therefore continuity is not even required to show what I had attempted to show about the HUP. The second objection, concerning whether xy is restricted to the product of two coprime integers, is rather redundant considering the above.

Yours truly,

Osher Doctorow 


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Re: Cross-Category "conversions" of some interest

[Osher Doctorow]

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From: Osher Doctorow osher@ix.netcom.com, Friday July 14, 2000, 8:47AM

Dear Colleagues:

Dr. S. J. Vickers has found two typographical errors in my July 12 contribution which might well lead someone to conclude that an erroneous one-sided operation on an inequality had been made.  The statement "....so the converted form reads: (1 - x + y)y^^2 = -/1 - x + y/y^^2 > k," shoud have a second constant k1 or k2 (it is arbitrary which notation is used) replacing k.  I used the same k by a typographical error and also because I had skipped an intermediate step and in my haste used the same constant k from before.  The intermediate step was merely to consider what happens to xy when the conversion of y/x to 1 - x + y is made.   Then xy converts to (1 - x + y)y^^2 = -/1 - x + y/y^^2 in the case when 1 - x + y < 0, and this is obviously nonpositive, so -/1 - x + y/y^^2 < k2 with k2 = 0 for example.  For 1 - x + y > 0, we have xy converting to /1 - x + y/y^^2 which is nonnegative and therefore /1 - x + y/y^^2 > = k2 with k2 = 0 again.  

Vickers'  criticism turned out to be very fortunate, not only for clarifying the typographical error and avoiding the wrong conclusion that I operated on only one side of an inequality when converting xy to the other form, but also in my developing a detailed argument concerning when the conversion from y/x to 1 - x + y becomes an actual function.   This occurs, for example, when y/x is a reduced proper or improper fraction in the sense that numerator and denominator have no common primes, in which case the unique factorization into primes and consideration of the three cases y/x > 1 and y/x < 1 and y/x = 1 leads to the conclusion that the conversion is a function.  Thus, on the reduced rationals, we have a function.  This is not a bad set to work with mathematically, and certainly provides a nontrivial case where the conversion is a function and is accurate.

I hope that S. J. Vickers will continue to contribute to further discussion in this thread because of his important contributions, provided of course that he continues to emphasize the correction of errors and ways of further applying the conversions.  If somebody finds any further errors in my future writings, please give me the benefit of considering the possibility that I made a typographical error.

Osher Doctorow    

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Re: Cross-category "conversions" of some interest

[S.J.Vickers]

Applying the conversion from x/y to 1 - x + y to the Heisenberg Uncertainty
Principle in the form xy > k where k is a positive constant (x, y
uncertainties in position and momentum respectively, for example, where
notice x and y are nonnegative in the standard deviation version of
uncertainty) yields xy = (x/y)y^^2 (where ^^ is exponentiation) --> (1 - x +
y)y^^2.  For 1 - x + y < 0, which says x > 1 + y, the latter expression is
negative, so the converted form reads: (1 - x + y)y^^2 = -/1 - x + y/y^^2 >
k and therefore /1 - x + y/y^^2 < -k for k positive.  Since y^^2 is always
nonnegative, this conditional holds trivially (always) provided that x > 1 +
y.   Since x and y could be selected with exchanged physical roles
(uncertainties in momentum and position respectively, for example), the
condition that x > 1 + y is rather arbitrary and certainly will be fulfilled
for one of the two orders in which x and y are defined. 
:

Osher Doctorow
Doctorow Consultants
Culver City, California USA   

Even more strikingly, consider the identity 0/1 = 0 of classical
mathematics. Applying the conversion we find 0/1 --> 1 - 0 + 1 = 2 and
deduce 2 = 0, thus giving an unbelievably simple explanation of the Pauli
exclusion principle for fermions.
 
Steve Vickers.

Cross-category "conversions" of some interest

[Osher Doctorow]

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From: Osher Doctorow, Ph.D. osher@ix.netcom.com, Sat. July 7, 2000, 9:46AM

Dear Colleagues:

I have been studying the "conversion" from x/y to 1 - x + y and the conversion from x/y to x + y - xy, where x and y are elements of any category (although in practice I have been restricting attention to probability-statistics (x = Pr(A), y = Pr(AB) where AB is the intersection of sets/events A and B) and (fuzzy) multivalued logics such as Goguen/Product (G) and Lukaciewicz (L) and Godel (Go)  (in the case of G, x/y is the non-trivial implication x --> y and in the case of L, 1 - x + y is the non-trivial implication x --> y) and the Jacobson radical in ring theory (which is based on the circle composition product x*y = x + y - xy for x and y elements of the ring) and Fermat's Last Theorem in number theory (x and y are integers and a super-super short proof seems to depend on generalizing x*y to n dimensions and using the conjugate which will be described below).  The conjugate ^ of 1 - x + y is (1 - x + y)^ = 1 + x - y, and the conjugate of x + y - xy is x + y + xy.    An n dimensional generalization would involve x^^n + y^^n  where ^^ is exponentiation (unrelated to ^) since the product of 1 - x + y and its conjugate can be shown to involve x^^2 + y^^2 - x^^2 y^^2 = x^^2 * y^^2 and so on.  The expression "conversion" is used above instead of function because y/x --> 1 - x + y is not a function but a conversion of the division operation (for x non-zero) to subtraction and the addition of 1.     

The probability-statistics reader may recognize x/y and (Bayesian) conditional probability (BCP for short) written Pr(B/A) for Pr(A) non-zero, and it turns out that 1 - x + y for probability-statistics is Pr(A-->B) = Pr(A' U B) = Pr(A' ) + Pr(B) - Pr(A' B) = Pr(A' ) + Pr(AB) = 1 - Pr(A) + P(AB), which latter expression is maximized for very rare events and lower dimensional events when probability distributions are continuous on a volume of space containing A and B (Pr(A) = 0 for those cases) and also when A is a subset of B.  Pr(A-->B) is abbreviated LBP for logic-based probability, which I have been developing since 1980.   BCP applies to frequent/common and independent-like or low influence events (including Markov processes which have many similarities to independent events although they are "slightly" dependent) and LBP applies to rare events or rare-like events (Pr(A) less than epsilon for epsilon small positive) and to n-k dimensional subset events of n-dimensional Euclidean space for k = 0 to n and to highly dependent or highly one or two-way influencing events. 

As for Goguen/Product logic (G) and Lukaciewicz logic, their union or "join" G U L, similarly to the union or join of either of them with Godel logic G, equals BL2, the basic (fuzzy) multi-valued logic which generalizes Boolean logic with the deduction Theorem and the plausible axiom p V ~p (p or ~p) for each proposition p.   Thus, G and L constitute all of the universe of logic in this sense, and they partition it into disjoint "roughly equal" parts.

Osher Doctorow
Doctorow Consultants
Culver City, California USA    

  

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