Theory of Completeness of Brain: A Narrative

Gödel incompleteness theorem gives an insight about the nature of a theory that can well describe the dynamics of neurons, keeping in consideration the complexity and incapacity of reductionism. Though there have remained many controversies about his viewpoint- a tussle between optimists and pessimists but there are few philosophical attributes associated with the theorem that I believe can help us to interpret the nature of theorizing neural dynamics due to its complex nature. David Hilbert recognized that axioms tend to be self consistent if we cannot prove that a statement S and its negation ~S are both true theorems. It can be considered complete if for every statement S we can prove either S or ~S is a true theorem (in terms of language) This statement of Professor Hilbert When taken along with Kurt Gödel we can have an intuition of the theory or a mathematical structure completely defining the neural dynamic. Naturally there is a two different dynamics being active at tubular level of neuron where voltage between membrane define the action and at behavioural and cognitive level the stimuli seems different or I may say almost complementary! We have always sought this diverging formalism as a real impediment but I expect that it may prove a greater significant edge in developing or initiating a definitive mathematical structure for brain dynamics. Let somehow we define a theory T that defines the action potential and the dynamics at the tubular level and then formulate somehow some other theory H that define the behaviour, cognitive and psychological parameters at aggregate level then there needs to be two condition whose completeness can ‘to some extend’ initiate a theory and a mathematical formalism of neural dynamics.  Let G be the group (in group theory formalism) that forms all object of theory T and R be a group that forms all objects of theory H. Then,

·  If the intersection of the two groups is an identity group (for not violating group theory), technically  a null set, T ꓵ H =Ф meaning the two theories or the two groups are adjoint groups. and

·   Despite of the ergodicity in the system, there is manifestation of only one theory (Say H) with some special set (Later may be defined as braiding Category ) (Hilbert view on Completeness)

Then it may be believed that the inaccessible law of reality as posed by the theorem may ‘to some extent’ be solved. Our epistemology shifts from arithmetics (posed by Gödel) towards groups or more specifically category.  Gödel proved mathematics to be inexhaustible in that finite set of axioms cannot encompass the whole mathematical world. I believe that it is due to our inaccessible conduction to know what an ‘intrinsic property’ of matter really is! We mostly understand the responses of matter to different scenario and statistically or based on coherent and hierarchical structure of literature we make a step in explaining it. Maybe it is just another way towards reality! Mathematics may now leap from integration, summation and arithmetics towards structures and diagrams like arrows in category theory.  Now in order that we define what theory can be devised for brain dynamics I feel the two theories that almost satisfies the above posed condition and discussion is Quantum field theory and Category theory. The manifestation tough at philosophical level seems equivalent but these are two adjoint theories satisfying the first posed condition and secondly that at functional level topology gets manifested satisfying the Hilbert view of completeness.

·         The dynamics at the tubular level can be well defined by Quantum field theory (I had discussed it in my previously sent proposal) we use Quantum field mathematics to deal with action, learning and dynamics at the membrane of the cells. The Hebb’s rule can be well defined by Green’s propagator, neural state network to fermionic state, neural state superposition to fermionic braiding, synergetic order parameter to partition function and Spatio-temporal integration of neuronal signals to Feynman Schrodinger equation. It defines theory T in our case.

·         The behaviour of the neural at the nerve conduction level is in its very essence topological in nature. It ‘illuminates’ the specific nerves among the bunch of fibres. To every emotion, understanding, response and cognition there is ‘something’ happening at the tubular level in terms of QFT but ‘whatever happens there’ it shows a ‘special’ topology at conduction level that shall be defined by topological field theory as proposed by MR and EM. It defines theory H in our case.  So whatever happens in fundamental using QFT the theory but at the conduction our analysis regime it is theory H that dominates (Hilbert view of Completeness)

Now to link these two adjoint perspectives, we need to device a comprehensive mathematical structure or at least initiate the mapping the parameters of QFT with that of topology. It may not be possible if we talk in terms of pure arthematics, I feel this mapping and corresponding can be done via Category theory.  As an intuitive feeling, what is persistent homology (PH) in terms of QFT?  In PH we let the structure evolve with time and at the end get structures for analysis, at fundamental level it may be regarded as creation-annihilation operators in specific time span, yielding a structure after some time! (It is a radical statement to be made at this point but it was for feeling the third point)

If this Idea works then we have to prove the two stated theories must be adjoint in nature and complementary to each other and that too one of the theory proving right at some level of analysis. There is need to map the parameters of two theories and yielding a ‘new’ mathematical structure. Though it is not an easy task but may be taken for initiation of understanding brain.