Coincidences
Understand how math naturally expresses the main concepts of my philosophy.
Express my philosophy's concepts in terms of mathematics and thus better define my concepts as well as understand which mathematical concepts are most central.
Challenges
- How can contradiction be modeled?
- Express God's Ten Commandments as four geometries and six transformations between them.
- Express the four geometries in terms of symmetric functions.
- How do the four classical Lie groups/algebras distinguish between no perspective, perspective, perspective on perspective, and perspective on perspective on perspective, and thus define the four geometries?
- Show how the gradation of six methods of proof organizes the prayer "Our Father" and the language of argumentation.
- How are the various kinds of opposite important in driving God's dance?
- What do the various kinds of duality yield, applied to God? For example, a dual of God (the Center) is the Everything (the Totality).
- Representations are criteria, filters. Can open sets (closed sets) be thought of as filters? A topology of filters?
Some practical projects:
- Learn about the combinatorics and geometry of finite fields and interpret {$F_{1^n}$}.
- Understand infinity. Understand how finite fields and the field with one element express infinity. Understand how zero and infinity get differentiated, how their symmetry gets variously broken. Express infinity in terms of geometries.
Concepts to express
- God
- Perspectives
- Divisions of everything into perspectives
- The nullsome
- The onesome, twosome, threesome, foursome, ...
- The eight-cycle of divisions and the three operations.
- Representations
- Topologies
- Three languages: Argumentation, verbalization, narration
- The eightfold way
- Walks on trees
- Relations between perspectives: dualities
- Chains of perspectives
- Relations between chains of perspectives: Ten (6+4) commandments
- Trinities of perspectives: God's trinity and the three-cycle
God
I am ever trying to imagine everything from God's point of view.
God
- God can be understood as contradiction.
- God is the Center of polytopes (such as the Simplex). Everything is the dual of God, the Totality consisting of all of the vertices and all of the simplexes.
- God may be given by trivial tensor: T00. It is a zero-dimensional array, thus a scalar. An array is a perspective, and so it is having no perspective. A scalar is "spirit". Thus it is spirit with no perspective.
God's trinity
- The field with one element has one element which can be understood as 0, ∞ and 1. One is a lens that relates zero and infinity. 0 makes way for ∞ and 1 is their point of balance.
- Try to use universal hyperbolic geometry to model going beyond oneself into oneself (where the self is the circle).
- Relate God's dance to {0, 1, ∞} and the anharmonic group and Mobius transformations. Note that the anharmonic group is based on composition of functions.
Four combinations of God and Everything generate four infinite families of polytopes and associated geometries and metalogics. I think these are the four representations of God (true, direct, constant, significant):
- The simplexes An have a Center and a Totality. They are the basis for affine geometry where paths are preserved.
- The cross-polytopes Cn have a Center but no Totality. They are the basis for projective geometry where lines are preserved.
- The cubes Bn have no Center but have a Totality. They are the basis for conformal geometry where angles are preserved.
- The coordinate systems Dn have no Center and no Totality. They are the basis for symplectic geometry where areas are preserved.
Equation of life
- The family Dn seems to model the equation of eternal life, namely, that God doesn't have to be good, life doesn't have to be fair.
- Spirit and structure are related by duality, the operation +2.
- A set is the essence of the spirit, the free monoid, that it generates.
Ideas
- God goes beyond himself: 3 dimensions -> 2 dimensions -> (flip to dual) 1 dimension -> 0 dimensions (point: good heart).
Perspectives
Perspective
- Perspectives are defined structurally by algebra and dynamically by analysis and they come together in the four geometries.
- Scalars of fields (or division rings) define perspectives, their freedom.
- The complex numbers offer a dual perspective as opposed to the real numbers' single perspective.
- Category theory defines perspectives and their composition.
- Perspectives may be logical quantifiers.
Divisions of everything into perspectives
Divisions of everything
- Divisions of everything into N perspectives are given by finite exact sequences with N nonzero terms.
- String diagrams portray such exact sequences with divisions of the plane by way of objects.
Twosome:
Threesome
- Jacobi identity.
- The three-cycle of the quaternions.
Foursome: Four levels of knowledge (whether, what, how, why).
- Yoneda lemma.
- If we think of a functor F as going from a category C of our mental notions and association between them to a category D of linguistic expressions and continuations between them, then this particular application may also serve as a universally relevant interpretation and general foundation of category theory. It may indeed be meaningful to speak in category theory of a duality between paradigmatic application and universally relevant interpretation.
- Understand the jump hierarchy and the Yates Index theorem (the triple jump).
- Four geometries: affine (no perspective), projective (perspective), conformal (perspective on perspective), symplectic (perspective on perspective on perspective).
- Four classical families of Lie groups/algebras that define the four geometries.
Sixsome
- See: An Introduction to the K-theory of Banach Algebras
The eight-cycle of divisions
Bott periodicity
- The eight divisions of everything, and the three operations +1, +2, +3, which act on them cyclically, should be expressible in terms of Bott periodicity and the clock shift of Clifford algebras.
Representations (Criteria)?
Grothendieck's 4+2 = 6 operations.
- Consider a function from one algebraic variety or scheme to another. Then we can define accordingly four functors from one category of sheaves to another such category. These functors are defined to make sense across a family of bases, that is, across base changes. Upper and lower star functors are like everything and nothing. Upper and lower shriek functors are like "fibers" within everything, thus: anything and something. The fiber may be identified with the category. Tensor and Hom are defined within the category of sheaves (thus within the input and also within the output). Tensor can be thought of as decreasing slack by filling it out. Hom can be thought of as increasing slack by creating multiplicity of functions. The four functors relate Hom and Tensor in the input category and in the output category. The six operations can be thought of as naturally defined within a higher order category of correspondences. The six operations can also be thought of as a generalization which grounds Poincare duality and its generalizations, Serre duality. Note that these two seem relate to the Snake lemma.
Scopes
- Do scopes express regularity of choice?
- Nothing. A point is nothing (as regards choice - its fiber is nothing).
Topologies
Systems of constraints that may be thought of as defining worlds. Topology is the study of topologies.
Languages of argumentation, verbalization and narration
The gradation of six methods of proof should organize the prayer "Our Father" and the language of argumentation.
Eightfold Way
- The Eightfold Way relates a left exact sequence and a right exact sequence
- Homology and cohomology
- The snake lemma resembles the eightfold way. However, the exact sequence that it defines goes 4->5->6->1->2->3, thus counter to the expected order.
- Try to understand Lawyere's eight-fold Hegelian taco.
Walks on trees
- Julia sets
- The tree of choices (the regularity of choice) given by the three operations +1, +2, +3.
- Walks from A to B in category theory are morphisms and they get mapped to the morphisms from A to B. Relate this to walks on trees.
- Parentheses establish a tree structure. What are walks on these trees? How do they relate to associativity and to walks in categories from one object to another?
- Walks On Trees are perhaps important as they combine both unification, as the tree has a root, and completion, as given by the walk. In college, I asked God what kind of mathematics might be relevant to knowing everything, and I understood him to say that walks on trees where the trees are made of the elements of the threesome.
Relations between perspectives
Duality
- The kinds of duality express the ways that two perspectives can be related, thus the operations on perspectives.
- The kinds of duality express the kinds of opposites.
Automata theory defines a hierarchy of equations.
Chains of perspectives
Four geometries
- Affine geometry models no perspective.
- Projective geometry models perspective.
- Conformal geometry models perspective on perspective.
- Symplectic geometry models perspective on perspective on perspective.
Relations between chains of perspectives
Six pairs of four levels.
- six specifications between the four geometries.
- six ways of thinking about variables.
- six ways of thinking about multiplication.
- six visualizations (restructurings in terms of sequences, hierarchies and networks).
- six qualities of signs.
- six set theory axioms.
- six bases of symmetric functions
- six ways of relating two mental sheets, a logic and a metalogic.
The Zermelo Frankel axioms of set theory are structured by 4+6.
Restructuring
- The calculus world is the "exponential" of the discrete world. One of the reasons that Lie groups and Lie algebras are important is because they link together the "calculus world" (Lie groups are "differentiable manifolds") and the "discrete world" (Lie algebras are based on "root systems" that are geometric reflections).
- The structure of a set is reminiscent of a tree in that there can be sets of sets. It's important that there not be cycles, irregularity.