About the mathematics needed in TGD

The following is a comment from FB discussion. Since the answer developed to a summary of the mathematics needed in TGD, I decided to make it blog post.

I think that TGD is a "problem" for anyone in the sense that it is very difficult to get graps of what it really is. The reason is that TGD have been silenced for 4 decades - censorship in archives started around 93 or so and has had fatal consequences. The idea about the hegemony of M-theory estabilished by censorship was terribly silly.

TGD is an outcome of concentrated effort lasted more than 4 decades and involves 24 books. I guess that the minimum time to get some perspective to TGD is one year. For instance, I have worked with twistors for about 10 years and gradually begin to really understand the twistorialization of TGD. I am skeptic about communication of TGD without a long series of lectures and personal face to face discussions. This has not been possible, and now also my age poses strong limitations.

Concerning math, I am not interested in technical things such as producing a new p-adic variant of self-interacting phi⁴ scalar field theory. Usually mathematically oriented people do this kind of things in lack of wider physics perspective. I have a big developing a vision binding physics, consciousness, and biology, and I want to identify and even develop the mathematics needed if needed.

1. The math involves sub-manifold geometry intended to geometrize field theory by replacing field patterns with 4-surfaces using generalization of the induction procedure for metric, spinor connection, and spinors (also twistors). Kaehler geometry is in central role. Induction is something new for physics, but well-known for mathematicians.
   
   
2. The great vision involves geometrization of quantum theory in terms of infinite-D Kaehler geometry for the world of "classical world". Wheeler with his superspace was pioneer in general relativity and loop spaces are predecessors in string theory. Infinite-D symmetries as generalization of superstringy conformal symmetries are in pivotal role: they guarantee the existence of infinite-D Kaehler geomery.
   
   
3. Number theoretic vision is complementary to geometric vision and number theory including extensions of rationals, p-adic number fields and their extensions induced by extensions of rationals and classical number fields. Extending the physics to a description of also cognition is the great vision and brings in p-adic number fields, adeles, and predicts hierarchy of Planck constants characterizing dark matter in TGD sense. Cognitive representations are fundamental notion and provide a unique number theoretical discretization of classical and quantum physics. One powerful implication is discretization of coupling constant evolution forced solely by number theoretical universality.
   
   
4. The generalization of twistors to 8-D context to solve basic problem of standard twistor approach (only massless particles are allowed) is part of the vision. A new element is the replacement of space-time surfaces with their twistor spaces represented as 6-surfaces in 12-D twistor space of imbedding space. One powerful prediction is that M⁴×CP_2 implied by standard model is the only possible choice for the imbedding space besides M⁸ which is equivalent dual choice: the reason is that only the spaces M⁴, E⁴ and CP₂ allow twistor space with Kaehler structure. TGD is unique from its existence.
   
   
5. I have considered also categories occasionally. I tend to see them as tools to organize and excellent for bureaucratic challenges: they do not seem to code for core ideas. As a physicist I see interpretations of quantum theory as as unsuccessful attempts to get rid of the basic problem of the measurement theory: here zero energy ontology (ZEO) provides the great vision and leads to a theory of consciousness and quantum biology.
   

   The great idea is terribly simple: transform initial value problem to boundary value problem by replacing initial values at t= constant hyperplane with boundary values characterizing deterministic time evolution t₁→>t₂ and by replacing quantum states with superpositions of deterministic time evolutions.
   
   

The mathematics of TGD has not been found by going to math library but developing the physical vision and generalizing the existing mathematical ideas appropriately. I trust to physics based arguments much more than mathematical proofs involving all kinds of technical assumptions.

Those interested in material about TGD can find links from Material about TGD.

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

Articles and other material related to TGD.