[an email received from Larry Pensinger on 5th June 2003 after I brought to his attention A.S. Karpenko's work relating prime numbers and m-valued Lukasiewicz logics]
 

Dear Matthew:

Thank you very much for finding this and drawing it to my attention. I will discover what I can about A. S. Karpenko on the net and get a copy of the article ASAP. The timing of your message is very interesting as just two days ago I was communicating with a friend, Tony Blake, about this very subject -- and I would say that we and Karpenko are right on the same wave length. Let me quote from my recent letter to Tony:

"Zwick has an article about Godel and quantum theory on his website which I have yet to read. One thing I can state with some confidence is that Godel numbers (which designate logical propositions, i.e., molecules of meaning, semantic memes) will be a central feature of a rigorous account of the decomposition of the m-logically-valued reference space, which in its cosmological incarnation Barbour calls Platonia. Godel numbers are based on primes and the relationship of these numbers to Musculpt is what Derek focuses upon in the last scene of MOON with the issue of "Number the numbering!". . . This is all involved with the Riemann Hypothesis about distribution of prime numbers. The m-logically-valued reference space is n-dimensional Hilbert space under m-valued logics. It applies to the brain-bodymind and it applies to cosmology as Platonia. A Musculpt-type fusion of Pert's receptor-ligand perspective with that emerging from superconductant neuronal and perineural DNA is not fully doable so long as the quantum-gravity issues remain unresolved. These cannot be resolved without treating Hilbert space under m-valued logics with Godel numbering of "molecules of meaning". AND, so long as this reference space (Platonia) and its decomposition and recompostion is not rigorously described, cosmologically and neuroscience-wise, the collective projection of unconscious contents relative to this space into military applications will continue to dominate human affairs."

Karpenko's "rooted trees" is an apt phrase for how the logical value stack on a given point or set of points on the most dense sheet (mth sheet) of the MVRS (m-logically-valued reference space) involutes or decomposes across the universal covering surface (set of multi-sheeted Riemann surfaces) of the MVRS. Karpenko's statement that "A combination of different logical definitions of prime number leads to the construction of algorithm for creation of classes of prime numbers" is to my way of thinking spot on! Remember how in "Some Thoughts From the Pensingers", posted on your website, we state that there are classes of non-self-identical Godel numbers (which correspond to logical propositions in orders of logical-value 3. . .m, all of which violate the law of non-contradiction)? Such non-self-identical numbers are numbered {numbered primes}. Godel numbers are numbered primes; numbered Godel numbers correspond to "different logical definitions of prime numbers" or in other words to numbering Godel numbers under 3. . .m orders of logical-value. Such numbering of numbered primes will generate classes of hyperprimes (more accurately termed) which correspond to propositions in orders of logical-value greater than 1T2 (i.e., 1T2. . .uTm). These classes of primes and hyperprimes, I would bet, array on logical lattices (probably what Karpenko is calling "tableaux of numbers") mapped on sheets and surfaces something like this (it will be interesting to see how Karpenko has done it): Each sheet of each Riemann surface of the universal covering surface will array in the following series: 1T2, 1T3, 1T4. . .1Tm. These are what Derek in MOON calls the decomposed single-valued sheets (scale levels of a ponderable spacetime continuum). Each Riemann surface of the universal covering surface will array in the following series: 1,2T2, 1,2,3T3, 1,2,3,4T4. . .uTm. Cross-sheet logical lattices will involve hyperprime factorials (factorials in hypernumbers beyond Hamilton's quaternions). Just as was said in "Some Thoughts of the Pensingers", the Riemann distribution of primes into a band will have corresponding bands on each of these sheets and surfaces, Riemann's band itself corresponding to the primes arrayed on 1T2.

My argument to Tony was that Candace Pert's (neurochemist who isolated the opiate receptor) receptors map on "holes", these "holes" being physical region singularities in the quantum field of the organism (plotted in m-logically-valued n-dimensional Hilbert space, i.e., the MVRS or Barbour's Platonia configuration space). All such "holes" will correspond to prime or hyperprime numbers. Logical lattices between such arrayed primes and hyperprimes not only constitute propositions in m-logically-valued neural code but correspond to wave-functions governing classes of functional integration (systems, organs, cell groups, et cetera) and the associated functionally-specific prerequisite stereochemical geometrical transformations. Cosmologically, these logical lattices correspond to Regge lattices of spacetime curvature configurations (solutions to Einstein's field equations). As string theory went to superstring theory went to p-branes (membranes of superstrings), GUTS almost entered the realms of the MVRS ("almost" because the set of all p-branes is equivalent to 1,2T2 in the above scheme of things -- given that m-valued logics were not considered in theoretical genesis of p-branes because quantum mechanics was interpreted in relation to probability amplitudes, not m-logical-values).

I will let you know if I have any further thoughts on this. And thank you again. I most certainly must find out more about Karpenko's work.

Until soon, Larry