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Bott periodicity, Bott periodicity combinatorics

Poster size: 841 mm x 1189 mm

Abstract: Modeling Introspection with Clifford Algebras: 3 Levels of Awareness and 8 Mental Contexts

Kahneman and Tversky distinguished System 1 (our intuitive, prejudicial, unconsciously answering mind) and System 2 (our rational, conceptual, consciously questioning mind). We introspect a System 3 (our cognizant, deliberate, willfully investigating mind) which matches up System 1 (what we know) and System 2 (what we don't know) and then chooses which system to apply.

We model these three minds as three matrix operations (transpose, conjugate and conjugate transpose) acting on the Hamiltonian H of a noninteracting free fermion, a solipsistic Self. H is characterized by the quantum symmetries (time reversal, charge conjugation, parity) that it satisfies, and these symmetries may be analyzed as mappings (reversion, conjugation, involution) fixing a Clifford algebra.

We interpret the Clifford algebra Cl(k) as modeling k perspectives which the Chevalley action pairs up into shifts in perspectives with an additional metaperspective when k is odd. Cl(k) manifest eightfold real Bott periodicity.

These eight structures (k=0,1,2,3,4,5,6,7) define the dynamics of assembling perspectives, the ways of dividing everything into k perspectives, the mental contexts for the most abstract issues: God, order, existence, participation, knowledge, decision-making, morality, logic.

Existence, as a context, supposes two perspectives: the possiblity that opposites coexist (as with free will) and the actuality where they don't (as with fate). Participation supposes a learning cycle of three perspectives: we take a stand, follow through, and reflect, as with the scientific method. Knowledge supposes four levels: How leads to What, and then Why leads to Whether.

Modeling Introspection with Clifford Algebras

3 Levels of Awareness and 8 Mental Contexts

Summary

Three columns: Metaphysics, Insights from Bridge, Mathematics

Metaphysics

Three levels of awareness: Three minds

Introspective method

Letting go of our prejudices (what we know) and preconceptions (what we don't know).

Studying the limits of imagination.

We define complex things in terms of simpler things. But how can we define the simplest things? They must define themselves in terms of themselves, in terms of their internal relationships.

Eight mental contexts: Eight divisions of everything

Bridge

Mathematics

Relating consistency and inconsistency

Distilling pairs of consistent and inconsistent statements.

Algebraic expression in terms of signed sequences

{$$\begin{matrix} & & & & & & & & & 1 & & & & & & & & \\ & & & & & & & & 1 & & -1 & & & & & & & \\ & & & & & & & 1 & & -2 & & -1 & & & & & & \\ & & & & & & 1 & & -3 & & -3 & & 1 & & & & & \\ & & & & & 1 & & -4 & & -6 & & 4 & & 1 & & & & \\ & & & & 1 & & -5 & & -10 & & 10 & & 5 & & -1 & & & \\ & & & 1 & & -6 & & -15 & & 20 & & 15 & & -6 & & -1 & & \\ & & 1 & & -7 & & -21 & & 35 & & 35 & & -21 & & -7 & & 1 & \\ & 1 & & -8 & & -28 & & 56 & & 70 & & -56 & & -28 & & 8 & & 1 \\ \end{matrix}$$}

Periphery

My Lifelong Investigation

Metaphysics

• 1971 dedicated myself before God to know everything and apply that knowledge usefully
• 1982 divisions of everything
• 1989 eight-cycle of divisions, three operations
• 2012 three minds in terms of Thinking, Fast and Slow
• 2024 Theory translator

Mathematics

• 1993 Ph.D. thesis in algebraic combinatorics: Symmetric Functions of the Eigenvalues of a Matrix.
• 2016 returned to math, mapping it out, focusing on Lie theory, noting Bott periodicity - the hunch
• Stone, then Moore, then Hamiltonians
• 2025 divisions of everything in terms of quantum symmetries
• 2026 spin representations