Challenging the p-adic view of cognition

The progress in the mathematical understanding of TGD has repeatedly challenged various working hypotheses. At this time, the progress in the understanding of holography= holomorphy vision challenged the proposed p-adic view of cognition and intention and raised the question whether ordinary p-adics could emerge from what I call functional p-adics as a generalization of p-adics (see this ).

1. In the TGD based vision about p-adic physics as physics of cognition, the discretizations defining the intersection of realities and p-acidicites as real and p-adic space-time surfaces define cognitive representations consisting of points of H= M⁴× CP₂ with preferred H coordinates in an extension of rationals. One can challenge this picture for CP₂ since it is not a linear space. For M⁸ identifiable as octonions with the inner product defined as the real part of the octonionic product, the situation is linear and the objection can be circumvented since there exist highly unique linear coordinates. The need for preferred coordinates raises however as an objection.
2. One of the motivations for the p-adic vision about cognition was the inherent non-determinism of p-adic differential equations. Could p-adic view of cognition and imagination explain the non-determinism of imagination? Could the realization of an intention as action correspond to a quantum transition in which a p-adic counterpart of real space-time surfaces, obeying the same defining equations, is replaced with its real counterparts.

   The basic objection is that conservation laws might not allow this unless the conserved quantities are in an intersection of reals and the extension of p-adic numbers induced by an algebraic extension of rationals. If one accepts zero energy ontology (see this), in which quantum states are zero energy states identified as pairs of states with opposite total quantum numbers (this expresses classical conservation laws), this objection might be circumvented.

Holography = holomorphy vision (see this and this), which in fact forces zero energy ontology, suggests a re-analysis of these proposals.

1. Space-time surfaces for which the Taylor coefficients appearing in function pairs (f₁,f₂) appearing in holomorphy-holography vision are in an extension of rationals and can be regarded as real numbers or to belong corresponding extensions p-adics. This is the case also for cognitive representations as the discretizations of space-time surfaces defined by points of H with coordinates in an extension of rationals. Are the p-adic → real transitions really needed? The question about the meaning of the intention-to action transition however remains. Is the p-adic variant of the space-time surface replaced with the real variant?
2. Is p-adic physics really something fundamental or does it emerge at the level of WCW? Holography= holomorphy vision leads to a generalization of p-adics to functional p-adics. Functional p-adic primes correspond to polynomials with prime degree p and functional p-adic number would be a power series of a polynomial with degree p with coefficients which have degree smaller than p. Functional p-adics would have space-time surfaces as representations.

   The functional p-adic numbers can be mapped by very-many-to-one morphism to p-adics (see this) by assigning to the functional p-adic number defined in terms of the degrees of the polynomials involved. This would provide a long-sought-for explanation for the p-adic length scale hypothesis and give a direct connection with the vision about cognitive hierarchies. It would also explain why small primes 2 and 3 appearing in the p-adic length scale hypothesis are special: for these primes the roots of the polynomial can be solved analytically. What is also nice that functional p-adics do not require any preferred coordinates and do not pose any restrictions on the coefficients of f₁ and f₂ since only the degree of the polynomial matters.

3. In this framework it is not clear whether ordinary p-adics are needed as a fundamental concept: are only the real variants of H and M⁸ needed? Could ordinary p-adics emerge from the functional p-adics represented at WCW level as real space-time surfaces. Could the crucial p-adic non-determinism correspond to the classical non-determinism at the level of real space-time (see this)?
4. But what could the transformation of intention to action mean in this framework? Is it associated with any small state function reduction (SSFR) in the degrees of freedom corresponding to the classical and p-adic non-determinism replacing superposition of real space-time surfaces with a new one?

Intention means an idea about the future, about a goal. Intention means time directedness. Zero energy ontology replaces 3-surfaces with 4-surfaces with analogs of Bohr orbits as slightly non-deterministic space-time surfaces obeying holography. Could this 4-D character of the fundamental geometric objects be enough to explain intentionality.

See the article A more detailed view about the TGD counterpart of Langlands correspondence or the chapter with the same title.

For a summary of earlier postings see Latest progress in TGD.

For the lists of articles (most of them published in journals founded by Huping Hu) and books about TGD see this.