^{n}-1 manners corresponding to subspaces of the sub-space defined by n-dimensional projector if the density matrix is n-dimensional projector (the outcome corresponding to 0-dimensional subspace and is excluded). If the probability for the outcome of state function reduction is same for all values of the dimension 1≤m ≤n, the probability distribution for outcome is given by binomial distribution B(n,p) for p=1/2 (head and tail are equally probable) and given by p(m)= b(n,m)× 2

^{-n}= (n!/m!(n-m)!)×2

^{-n}. This gives for the average dimesion E(m)= n/2 so that the negentropy would increase on the average. The world would become gradually better.

One cannot avoid the idea that these different degrees of negentropic entanglement could actually give a realization of Boolean algebra in terms of conscious experiences.

- Could one speak about a hierarchies of codes of cognition based on the assignment of different degrees of "feeling good" to the Boolean statements? If one assumes that the n:th bit is always 1, all independent statements except one correspond at least two non-vanishing bits and corresponds to negentropic entanglement. Only of statement (only last bit equal to 1) would correspond 1 bit and to state function reduction reducing the entanglement completely (brings in mind the fruit in the tree of Good and Bad Knowlege!).

- A given hierarchy of breakings of super-symplectic symmetry corresponds to a hierarchy of integers n
_{i+1}= ∏_{k≤ i}m_{k}. The codons of the first code would consist of sequences of m_{1}bits. The codons of the second code consists of m_{2}codons of the first code and so on. One would have a hierarchy in which codons of previous level become the letters of the code words at the next level of the hierarchy.

_{M(n)}, M

_{n}= 2

^{n}-1.

- The hierarchy starting from M
_{2}=3 contains the Mersenne primes 3,7,127,2^{127}-1 and Hilbert conjectured that all these integers are primes. These numbers are almost dimensions of Boolean algebras with n=2,3,7,127 bits. The maximal Boolean sub-algebras have m=n-1=1,2,6,126 bits.

- The observation that m=6 gives 64 elements led to the proposal that it corresponds to a Boolean algebraic assignable to genetic code and that the sub-algebra represents maximal number of independent statements defining analogs of axioms. The remaining elements would correspond to negations of these statements. I also proposed that the Boolean algebra with m=126=6× 21 bits (21 pieces consisting of 6 bits) corresponds to what I called memetic code obviously realizable as sequences of 21 DNA codons with stop codons included. Emotions and information are closely related and peptides are regarded as both information molecules and molecules of emotion.

- This hierarchy of codes would have the additional property that the Boolean algebra at n+1:th level can be regarded as the set of statements about statements of the previous level. One would have a hierarchy representing thoughts about thoughts about.... It should be emphasized that there is no need to assume that the Hilbert's conjecture is true.

One can obtain this kind of hierarchies as hierarchies with dimensions m, 2

^{m}, 2^{2m},... that is n(i+1)= 2^{n(i)}. The conditions that n(i) divides n(i+1) is non-trivial only for at the lowest step and implies that m is power of 2 so that the hierarchies starting from m=2^{k}. This is natural since Boolean algebras are involved. If n corresponds to the size scale of CD, it would come as a power of 2.

p-Adic length scale hypothesis has also led to this conjecture. A related conjecture is that the sizes of CDs correspond to secondary p-adic length scales, which indeed come as powers of two by p-adic length scale hypothesis. In case of electron this predicts that the minimal size of CD associated with electron corresponds to time scale T=.1 seconds, the fundamental time scale in living matter (10 Hz is the fundamental bio-rhythm). It seems that the basic hypothesis of TGD inspired partly by the study of elementary particle mass spectrum and basic bio-scales (there are 4 p-adic length scales defined by Gaussian Mersenne primes in the range between cell membrane thickness 10 nm and and size 2.5 μm of cell nucleus!) follow from the proposed connection between emotions and Boolean cognition.

- NMP would be in the role of God. Strong NMP as God would force always the optimal choice maximizing negentropy gain and increasing negentropy resources of the Universe. Weak NMP as God allows free choice so that

entropy gain is not be maximal and sinners populate the world. Why the omnipotent God would allow this? The reason is now obvious. Weak form of NMP makes possible the realization of Boolean algebras in terms of degrees of "feels good"! Without the God allowing the possibility to do sin there would be no emotional intelligence!

^{127}-1. In TGD one has hierarchy of dimensions associated with space-time surface coming as 0,1,2,4 plus imbedding space dimension 8. The abstraction hierarchy associated with space-time dimensions would correspond discretization of partonic 2-surfaces as point set, discretization of 3-surfaces as a set of strings connecting partonic 2-surfaces characterized by discrete parameters, discretization of space-time surfaces as a collection of string world sheet with discretized parameters, and maybe - discretization of imbedding space by a collection of space-time surfaces. Discretization means that the parameters in question are algebraic numbers in an extension of rationals associated with p-adic numbers.

In TGD framework it is clear why imbedding space cannot be higher-dimensional and why the hierarchy does not continue. Could there be a deeper connection between these two hierarchies. For instance, could it be that higher dimensional manifolds of dimension 2×n can be represented physically only as unions of say n 2-D partonic 2-surfaces (just like 3×N dimensional space can be represented as configuration space of N point-like particles)? Also infinite primes define a hierarchy of abstractions. Could it be that one has also now similar restriction so that the hierarchy would have only finite number of levels, say four. Note that the notion of n-group and n-algebra involves an analogous abstraction hierarchy.

For details see the article Good and Evil, Life and Death.

For a summary of earlier postings see Links to the latest progress in TGD.

## 4 comments:

Also polygonal numbers (feel good! :)) and have similar structure of statements about statements on the earlier level, e.g. triangular numbers, tetragonal etc. higher dimensional:

0 0 0 0 0 ...

1 1 1 1 1 ...

1 2 3 4 5 ...

1 3 6 10 15 ...

1 4 10 19 31 ...

etc.

The pattern can be extended also to the negative side, and there's amazing finding connecting Euler's finding about relation of "full" set of pentagonal numbers (http://en.wikipedia.org/wiki/Pentagonal_number_theorem) and sigma function(http://en.wikipedia.org/wiki/Divisor_function) together with sum of all non-cloned Egyptian fractions aka harmonic numbers and logarithmic function, resulting in elementary problem that is equivalent with Riemann hypothesis:

http://www.math.lsa.umich.edu/~lagarias/doc/elementaryrh.pdf

Nice introduction to the subject here:

https://www.youtube.com/watch?v=1mSk3J3GlA8

Correction: "triangular, tetragonal..." -> triangular, tetrahedral...

Scalar feel-good factor allows to feel better and better, which is good. :)

The fundamental problem with leveled empathy is that leveled negentropy between e.g. human-form emotional states lets in both feel-good and feel-bad, and raising barriers (e.g. us-against-them) against the feel-bad aspect and thus the whole holografic empathy of multi-observer negentropic field.

Empathy can thus be stated as strong form of emotional holography, and by not actively filtering out the emotions of other tribes, other species, spirits and gods, we can trust that the pain that comes in is nothing compared to the whole of love (=God). While also fully sympathizing with the fear of opening heart more and more fully also to the suffering of others, and non-judgementally allowing that basic fear the space and time and life-experience that it needs.

The Boolean aspect of what is called 'monogamy' of entangled observables seems to be the core mathematical issue relating to empathy, understood as strong holography of emotional negentropic entanglement, and monogamy of entangled observables (cf. representation theory and epistemology) is related to weak form of NMP. However, we do not need to treat strong NMP and weak NMP as either-or question, when we remember that according to Spinoza's definition, Absolute contains all attributes, and that Spinoza's Ethics is happily smiling feel-better philosophy. :)

This in mind, can we see a way to combine and relate strong NMP and weak NMP in terms of both monogamic and polyamoric negentropic entanglements of observables?

To Anonymous:

At least now I am happy with weak NMP. It allows realisation of emotional Boolean intelligence rather than only the usual cold and academic one;-).

An interesting question: can one map this emotionally represented Boolean algebra to fermionic representation of Boolean algebra: m-D subspace to m-fermion state. One should pair many-fermion states representing n-bit Boolean algebra with the subspaces of space defined by projection operator. One should entangle2^n-1 many-fermion states with these 2^n-1 state functions reductions? Sounds crazy!

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