• View
• Edit
• History
• Print

Modeling Introspection

---

Andrius Kulikauskas: I am writing a paper about my investigation of introspection. I wish to give a talk at the Models of Consciousness Conference.

Modeling Introspected Contexts With Mutually Anticommuting Linear Complex Structures

Abstract

A linear complex structure Jₖ preserves and thereby distinguishes the unitary group U(n) of matrix actions from the symmetric space O(2n)/U(n) of compatible contexts. Jₖ models an introspected perspective, bridging an unconscious mind M₁ that knows answers personally, and a conscious mind M₂ that asks questions, contextually. Applying such mutually anticommuting structures J₁, J₂, ..., J₈ to O(16r), r∈N, yields a chain of Lie group embeddings, O(16r) ⊃ U(8r) ⊃ Sp(4r) ⊃ Sp(2r)×Sp(2r) ⊃ Sp(2r) ⊃ U(2r) ⊃ O(2r) ⊃ O(r)×O(r) ⊃ O(r), with period 8, manifesting Bott periodicity. J₁, J₂, ..., Jₖ generate a Clifford algebra. They divide the global workspace into k perspectives which together define a mental context. Fundamentally, they structure k=2 existential attitudes (free will, fate), k=3 modes of a learning cycle (take a stand, follow through, reflect), and k=4 levels of knowledge (whether, what, how, why). Unconscious M₁ adds a perspective, Conscious M₂ adds a perspective on a perspective, and Consciousness M₃ adds a perspective upon this perspective on a perspective, all modulo 8, inhabiting an eight-cycle of metaphysical contexts, imposing complex, quaternionic and split-biquaternionic structure, respectively.

Outline

Introduction

• Academic context for colleagues. Modeling our inner realm as an inner realm.
• Historical context for idealists. Abstracting and forgetting.
• Evolutionary context for materialists. Modeling the known and the unknown.

Preliminaries

• Perspective and mental reflection.
• Mathematically modeling reflection.
• Clifford algebras of perspectives.
• Classical Lie groups.

Mathematical Model

• Lie group embeddings.
• Embracing knowledge of the knower.
• Rooting out reflective knowledge leads to failure.

Philosophical Interpretation

• Clifford algebras as divisions of everything
• Three minds in human culture.
• Divisions of everything in human culture.

Discussion

• Implications for theories of consciousness.

Academic context for colleagues. Modeling our inner realm as an inner realm.

Mathematical models of consciousness generally assume that we observe and interact with phenomena situated in a preexisting logic, space and time. Inevitably, such assumptions have us conclude that such a framework is prevalent and inherent in all of our thinking. Alternatively, we may attempt to model how our minds grapple with a mental blank state, how we bring sense to utter abstraction, how we imagine a primordial god, prior to everything, including logic, space, time, thought and mathematics. How can we model contradiction itself?

Bott periodicity is a wonder of mathematics, which describes the way holistic structures fit together, as if manifesting the symmetry inherent in mathematical thinking itself. Bott periodicity, both eightfold real and twofold complex, has found application in condensed matter physics, where it coordinates symmetries which are profound not just physically but metaphysically: whether time is reversible, whether holes are particles, and whether we can distinguish a mirror world from our own. It is worthy of consideration as a model of how we experience our own inner life, our most abstract introspections, our various levels of consciousness. It from how we observe and interact with phenomena situated in a preexisting logic, space and time, to the ways in which we can abstractly carve up a mental blank slate to conjure up our very notions of logic, space and time. Bott periodicity suggests itself naturally as a new subject for a tradition of philosophical mathematicians, including Grassmann, Clifford, Peirce, Whitehead, Spencer-Brown, Varela, Kauffman, Bohm, Hiley and Goertzel, who have sought transcendent meaning in logic and algebra, and more recently, Clifford algebra.

Raoul Bott astounded algebraic topologists in 1957 when he proved that

If nothing else, this paper presents a wonder of mathematics, Bott periodicity, eightfold real and twofold complex, as they manifest in Lie group embeddings, very concretely, through the linear algebra of rotations and reflections. My purpose is to show how this may serve to model mental reflection. I myself interpret this Lie group embedding specifically as an abstract dialectic which insistently rejects reflection in favor of the lack of reflection, rejects belief in favor of evidence, and periodically, ultimately, totally fails!

Main theories are...

• higher-order theories
• global workspace theories
• re-entry and predictive processing theories
• integrated information theory

Model extension rather than intension.

Reflexivity. Louis Kauffman.

Karl Jung and Wolfgang Pauli. World clock.

Wikipedia: Hiley is known for his work with Bohm on implicate orders and for his work on algebraic descriptions of quantum physics in terms of underlying symplectic and orthogonal Clifford algebras.

Wikipedia: Hiley expanded on the notion of a process algebra as proposed by Hermann Grassmann and the ideas of distinction of Louis H. Kauffman

• Wikipedia: In 1984, Hiley and Frescura discussed an algebraic approach to Bohm's notion of implicate and explicit orders: the implicate order is carried by an algebra, the explicate order is contained in the various representations of this algebra, and the geometry of space and time appear at a higher level of abstraction of the algebra.
• As Bohm and his colleagues emphasized, in such an algebraic approach operators and operands are of the same type: "there is no need for the disjoint features of the present mathematical formalism [of quantum theory], namely the operators on the one hand and the state vectors on the other. Rather, one uses only a single type of object, the algebraic element".
• Hiley spoke of his aim of finding "an algebraic description of those aspects of this implicate order where mind and matter have their origins".
• Hiley emphasises that active information "informs" in the sense of a literal meaning of the word: it "induces a change of form from within", and "this active side of the notion of information [...] seems to be relevant both to material processes and to thought". (Compare with the application of a linear complex structure - think of that as distinguishing information, raw experience.)
• Our proposal is that in the brain there is a manifest (or physical) side and a subtle (or mental) side acting at various levels. ... The mental side involves the subtle or virtual activities that can be actualised by active information mediating between the two sides.

In the context of quantum physics, the role of an observer. The idempotents of Clifford algebras are the counterparts to projection operators in quantum mechanics. two consciousness peculiarities that are : (-) It is an entity that has self-awareness and this is to say that it has in its inner the image of itself. In most cases we speak of self-image to represent such peculiar feature. (-) The other marking property is that it has awareness of an external space -time located abstract entity. In brief consciousness is an abstract entity that at the same entity has self-awareness (self image) of itself and self image of the outside Clifford algebra as explaining why an observer is not needed, but rather there is sentience. (Suisse, Cameron)

M Pitkänen: Gradually it became difficult to say where physics ends and consciousness theory begins since consciousness theory could be seen as a generalization of quantum measurement theory by identifying quantum jump as a moment of consciousness and by replacing the observer with the notion of self identified as a system which is conscious as long as it can avoid entanglement with environment.

Heinz von Foerster, second order cybernetics

V. A. Lefebvre.

When we are modeling the reality of consciousness of our lived experience, where do we take that reality to be? Is that reality in the brain? in the physical world? in our relationship with it? Or is it in some inner realm? As a thought experiment, what would it mean for God to experience consciousness?

Can we consider what is left when we are divorced from our senses, and from our conceptual language? What is there to model? What can we learn from introspecting the limits of our imagination?

A wide search... upwards of 29 theories of consciousness ...

The highest symmetry in math and the high symmetry of discrete mental models.

Jere Northrop's Relational Symmetry Paradigm is another theory

Tyler Goldstein's Sentient Singularity Theory is the only theory which I am aware of for which Bott periodicity plays a role, indeed, a central role. Bott periodicity, also referred to as Holonic periodicity, is attributed to a fundamental hierarchy of assembly, whereby starting with the universal singularity of G*d, as information and energy, hadrons are assembled in space, and from them, in time - atoms, then in space - molecules, in time - cells, in space - organisms, in time - families, in space - castes, in time - technology, leading to a technological singularity, identified with the universal singularity, thus realizing all stages of sentience. The eight assemblies are identified with loop spaces {$\Omega^1,\dots,\Omega^8$} of {$O(\infty)$} where {$\Omega_8O(\infty)\cong O(\infty)$}. The singularity is expressed as a state of contradiction (or a precontradiction of unity) {$0=1=\infty$} which inhabits all eight levels as the general linear group {$GL_\infty(\mathbb{R})$} which contains {$O(\infty)$} and for which the homotopy groups coincide {$\pi_i(O(\infty))\cong GL_\infty(\mathbb{R})$}.

Symmetry breaking by application of a linear complex structure can yield a hierarchy of emergent structure.

Clifford algebra as geometric algebra in 3 or 4 dimensions. Visual thinking.

Gestalt psychology.

Subtleties of opposites, including negation. Topic of logic.

Basil Hiley: The hope of finding a better understanding of Nature through a process philosophy is not new. Already Whitehead carries this analysis much further and proposes that reality is essentially an organism in which the whole determines the properties of the parts rather than the parts determining the whole. Less well known is the work of Bohm who carries these ideas much further in general terms, as well as attempting to articulate them in a mathematical form.

Think of Basil Hiley's process algebra as relating {$[P_0P_i]$} the point 0 (the Whole, God) with point i (ith part, ith perspective). Note that {$[P_0P_0]=1$}, {$[P_iP_i]=1$} and {$[P_0P_i]=e_i$}.

I am pushing their mindset further by considering finite divisions of everything, relationships of parts of the whole.

Clifford algebra hierarchy for modeling wave equations.

Hermann Grassman. Process algebra.

G. Spencer-Brown. Laws of Form.

Louis Kauffman. Francisco Varela. Reaching back to Charles Sanders Peirce.

Ben Goetzel: the belief that the Clifford Algebras, and their quaternionic and octonionic subalgebras, are archetypal structures of fundamental importance in many areas, including physics, psychology and systems theory. It is part of an ongoing interdisciplinary investigation involving the author, Onar Aam, Kent Palmer and F. Tony Smith.

Historical context for idealists. Abstracting and forgetting.

Often stated that a new approach is needed to understand consciousness. But perhaps what is needed is a return to a traditional quest to achieve wisdom, understand why, know everything, discover absolute truth, interact with God. in preacademic philosophy or at least...

The results in this paper arose from a personal journey dating back to the 1980s when, in the academic scientific community, the study of consciousness was considered taboo, but also notions of causality, meditation, animal emotions, plant communication, prayer, extraterrestrials, panpsychism. Introspection is widely considered unreliable. The scientific study of God is still a taboo, to this day, and absolute truth all the more so.

In 1982, as a freshman at the University of Chicago, we were encouraged to ask big questions, but discouraged to think they could have answers. All truth was relative, whereas I sought absolute truth. Divisions of everything.

In 1989, as an independent student at Vilnius University in Soviet-occupied Lithuania. My advisor, Rolandas Pavilionis, told me about Algirdas Greimas's semiotic square, Aristotle's logical square, which I interpreted as a division of everything into seven perspectives. Adding an eighth perspective, such as "All are known and all are unknown", would force the system to be empty. Pavilionis also suggested that I read Immanuel Kant's Critique of Pure Reason. My analysis of his Transcendental Deduction brought me to think that there were operations that would 2+3=5.

Kahneman and Tversky, "Thinking Fast and Slow".

Emotion and cognition. Third mind. Abstracting from experience, forgetting conceptual language, considering their relation, how we rebuild our conceptual language.

Academic presentations.

Bott periodicity.

Wombling.

Private language.

Maja Spener defends introspection. Much of the research she discusses is about introspection of experience. But I want to focus more not on what it means to experience but what it means to have a perspective on experience or on other perspectives. She herself distinguishes between introspection as access and as method. She also distinguishes between inner attention, retrospection and inner apprehension. Such distinctions are meaningful because of introspection in that we can all make them inside ourselves and we can all agree on these distinctions to an extent that is meaningful.

Phenomenological approaches to consciousness are rare.

Evolutionary context for materialists. Modeling the known and the unknown.

Another story... Present from a materialist point of view.

Modeling the known and the unknown.

I will make my approach more plausible for materialists by noting how the central nervous system evolves towards abstraction. A single-celled paramecium engages the world directly by way of its receptors for light and chemicals. But a butterfly lives in a world of flowers, that is, a neural representation of the world in terms of sensorial images which it pays attention to. As neuroscientist Michael Graziano has noted, a mouse furthermore has awareness, in that it utilizes a model of attention which it can identify not only with its own attention, and thus be itself aware, but likewise model a cat's attention, and thus be aware of whether or not the cat is attending to the mouse. Thus the mouse lives an abstract world of indexical and causal relationships.

But humans and perhaps the great apes can moreover be conscious, that is, we can choose what we wish to be aware of. Birute Galdikas has noted how orangutang males go off to live alone, as if they were Zen Buddhists, and how they can choose to ignore people or not. We humans can choose to "step in" and immerse ourselves in a subjective experience, or to "step out" and consider objectively what is going on, what others are experiencing. I will describe us as experiencing cognitive frameworks by which we divide up what neuroscientists call our global workspace into various perspectives they may take up for a particular issue, for example, contemplating "free will" and "fate". Indeed, I will describe our conscious life as shifting amongst eight such cognitive frameworks. They substitute our world with a highly constrained abstract model of options within which we adjust parameters that subsequently trigger the workings of our involuntary, unconscious mind.

From the disembodying mind to the global workspace.

Neural network, enmeshed with the world, conveying and summarizing what is, what is known, what functions as answers. Conceptual language, apart from the world, expressing what is not, what is unknown, in the form of slots, variables, words, concepts, which function as questions.

Perspective and Mental Reflection

The difference between "stepping in", direct experience, and "stepping out", reflected experience.

Perspective is their compatibility in a specific instance.

Felt meaning.

Twin opposites vs. marked opposites

Historical context (me stepping in) vs. evolutionary context (me stepping out).

I love you - consciousness holds both together as compatible.

7+3=2 Self-standing opposite - "good without bad". You model that in two ways. As everything (without a perspective) and everything (with a perspective). Combining the two (unreflected and reflected) gives the twosome. Self-standing disarms the self-standing opposite, provides context allowing for contradiction. Twosome: contradiction vs. noncontradiction.

Complex structure models this consciousness of compatibility - it doesn't matter which is the unreflected or the reflected.

Imposing complex structure is quaternionic? Linear (twin opposites) vs. antilinear (unmarked input and marked output because of conjugation). Throwing out the prejudicial convention of marked opposites.

Mathematically Modeling Reflection

Different ways of modeling mental reflection.

Linear complex structure distinguishes the tangent space from the manifold

The manifold describe locations and the tangent space describes orientations.

The manifold can be understood in terms of group actions. The tangent space is understood in terms of the Lie algebra commutator. When the commutator with J is zero, then elements commute with J.

J is both a rotation (location) and a skew-symmetric matrix (orientation). J belongs to the orthogonal group, unitary group and symplectic group.

Involution

Inverse

Transpose

Defining orthogonal, unitary, compact symplectic groups.

Negative

Note that for {$s\in S\cong O(2m)/U(m)$} we have {$s^{-1}=s^T=-s$}

{$MM^T=I$}

Reflection of a sphere

Inversion of a sphere. Antipodal map.

• If a sphere is centered at the origin, then an inversion symmetry maps any point {$\vec{p}$} to {$-\vec{p}$}. This is the antipodal map.

Inversion of a point. Symmetric space.

More generally, an inversion symmetry (a point inversion) about p is defined as x* = 2p − x.

Symmetric space. For each point p of M, there exists an isometry of M fixing p and acting on the tangent space {$T_pM$} as minus the identity. Globally, this is an isometry such that locally, at p, all tangent vectors end up mapped to tangent vectors in the opposite direction. Thus, locally at point p, there is an inversion of its tangent space.

Complex conjugate (and transpose)

{$z=x+yi\Rightarrow \bar{z}=x-yi$}

Quaternionic conjugate (and transpose)

{$q=x_0 + x_1i + x_2j + x_3k \Rightarrow \bar{q}=x_0 - x_1i - x_2j - x_3k$}

Antilinear operator

An antilinear map {$j$} on a vector space {$V$} over {$\mathbb{C}$} is an additive map for which {$j(\lambda v)=\bar{\lambda}j(v)$} for all {$\lambda\in\mathbb{C}$}, {$v\in V$}. Consider, for example, the quaternion {$j$}, for which {$ji=-ij$}, and the complex scalar {$\lambda = a+bi$}.

{$$j(\lambda v)=j(a+bi)v=(aj+bji)v=(aj-bij)v=(a-bi)j(v)=\bar{\lambda}j(v)$$}

In this context, where {$j^2=-1$}, {$j$} is called a quaternionic structure.

Eigenvalue of -1

Consider a matrix {$M$} such that {$M^2=+1$}. Then {$M^2-1=0$}, thus {$(M+1)(M-1)=0$} and {$(M+1)(M-1)v=0$} for all {$v\in V$}, where {$M$} acts on {$V$}. This means that its possible eigenvalues are +1 and -1 and if {$M\neq I$} and {$M\neq -I$}, then {$M$} has at least one eigenvalue of either. Consequently, if {$M$} acts on a real vector space {$V$}, then {$V=V_+\oplus V_-$} where {$Mv=v$} for {$v\in V_+$} and {$Mv=-v$} for {$v\in V_-$}. Thus {$M$} is rotation-free. It acts on {$V_+$} as the identity and it acts on {$V_-$} as a reflection. {$V_+$} and {$V_-$} are reducing subspaces, a spectral decomposition.

Isometry

Time reversal

Charge conjugation

Parity

Skew-symmetric matrix {$AA^T=-I$}

Skew-Hermitian matrix {$AA^H=-I$}

Quaternionic skew-Hermitian matrix {$AA^\dagger=-I$}?

Category theory

Linear complex structure as the context for rotation

We can distinguish context (symmetric space) and action (unitary group). Rotoreflection is associated with symmetric space and rotation with action. Reflection takes us from action to context.

Clifford Algebras of Perspectives

I make good use of the paper, lean on it and flesh out parts so it would be more understandable for those like us with a simpler mathematical background, linear algebra.

Linear complex structure {$J_k$} is an orthogonal matrix such that {$J_k^2=-1$}.

It can be diagonalized as {$AJA^{-1}$} where {$A$} is an orthogonal matrix and {$J$} is a block diagonal matrix of the form

{$$J=\textrm{diag}\begin{bmatrix}\begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}, \dots, \begin{pmatrix} 0 & -1 \\ 1 & 0 \end{pmatrix}\end{bmatrix}$$}

Namely:

{$$J=\begin{pmatrix}0 & -1 & & & \\ 1 & 0 & & 0 & \\ & & \ddots & & \\ & 0 & & 0 & -1 \\ & & & 1 & 0 \end{pmatrix}$$}

Given an orthogonal matrix {$O$}, it either commutes {$OJ=JO$} or anticommutes {$OJ=-JO$} with {$J$}. This is to say, {$OJO^{–1}=J$} or {$OJO^{-1}=-J$}. Conjugation of {$J$} yields {$J$} or its conjugate {$-J$}.

There can be two different linear complex structures {$J_1=A_1JA_1^{-1}$} and {$J_2=A_2JA_2^{-1}$}. This means that the matrix {$J$} can be understood in two different contexts, implemented with regard to two different bases, given by {$A_1$} and {$A_2$}. Indeed, we can have {$J_1$} and {$J_2$} be anticommuting, {$J_1J_2=-J_2J_1$}, which emphasizes that {$J_1\neq J_2$}.

We are interested in mutually anticommuting linear complex structures {$J_1,J_2,\dots, J_k$}. They are the generators for a real Clifford algebra {$Cl_{0,k}$} which has {$2^k$} basis elements. In particular:

{$Cl_{0,0}\cong\mathbb{R}$} has one basis element {$1$}.

{$Cl_{0,1}\cong\mathbb{C}$} has two basis elements {$1,J_1$} where we may identify {$J_1\leftrightarrow i$}.

{$Cl_{0,2}\cong\mathbb{H}$} has four basis elements {$1,J_1,J_2,J_1J_2$} where we may identify {$J_1\leftrightarrow i$}, {$J_2\leftrightarrow j$}, {$J_1J_2\leftrightarrow k$}.

{$Cl_{0,3}\cong\mathbb{H}\oplus\mathbb{H}$} has eight basis elements {$1,J_1,J_2,J_3,J_1J_2,J_1J_3,J_2J_3,J_1J_2J_3$} where we may identify {$1\leftrightarrow (1,1)$}, {$J_1\leftrightarrow (i,i)$}, {$J_2\leftrightarrow (j,j)$}, {$J_1J_2\leftrightarrow (k,k)$} and {$J_3\leftrightarrow (k,-k)$}, {$J_1J_3\leftrightarrow (-j,j)$}, {$J_2J_3\leftrightarrow (i,-i)$}, {$J_1J_2J_3\leftrightarrow (-1,1)$}.

Clifford algebras keep doubling, growing ... write them out ... whereas the Lie groups will be shrinking. They manifest Bott periodicity.

In analyzing Clifford algebras, it is important to note the properties of the products of various sizes, notably {$J_a$}, {$J_aJ_b$}, {$J_aJ_bJ_c$}. What are their squares? We calculate them using the basic facts that {$J_x^2=-1$} and {$J_xJ_y=-J_yJ_x$} when {$x\neq y$}. Then {$(J_a)(J_a)=-I$} and {$(J_aJ_b)(J_aJ_b)=-I$} whereas {$(J_aJ_bJ_c)(J_aJ_bJ_c)=+I$}. A consequence of the latter fact is that the matrix {$M=J_aJ_bJ_c$} acts on the vector space {$V$} by breaking it up {$V=V_+\oplus V_-$} where {$Mv=v$} for {$v\in V_+$} and {$Mv=-v$} for {$v\in V_-$}.

Classical Lie Groups

Compare the norms for the real number vector space {$\mathbb{R}^4$}, complex number vector space {$\mathbb{C}^2$} and quaternion vector space {$\mathbb{H}$}. They are all the same as quadruples {$a,b,c,d$}. Simply, there are different multiplicative rules defining them as algebras.

Complex numbers treat the personal and the contextual as equally valid. Real numbers emphasize one and discard the other. Quaternions accept both yet introduce that same prejudice a step deeper, in context.

The purpose of numbers x is to encode weights {$x_{ij}$} implications (i implies j). Proper numbers encode them in two directions: (i implies j) and (not j implies not i). Thus proper numbers are complex numbers. Real numbers are not proper numbers because they express only half of the information. Quaternions are abbreviated in that they express double the information.

Complex numbers are most plainly written as {$2\times 2$} matrices, in which case we clearly see that there are two imaginary numbers, twins, where the antidiagonal elements can have 1 above -1 or the other way around. The imaginary numbers are then defined by the fact that {$m^T=-m$}. This relates a twin opposite (given by the transpose) with a marked opposite. So it eliminates the distinction of marked and unmarked opposites.

{$iJ$} expresses the imaginary number in two ways, internally to mathematical structure as {$i$} and externally upon mathematical structure as {$J$}. This squares to {$1$}. Compare this to the quaternions, whose generators all square to {$-1$}.

Natural bases are orthonormal, with basis elements orthogonal and of unit length. There are three notions of length: real, complex, quaternionic. In the real case, we have to distinguish between odd and even. Even is natural and applies to all cases. Odd is unnatural, terminates periodicity and I think generates chaos.

The unitary group, based on the complex numbers, is most natural. The reason is how symmetry pairs up orthonormality equations. When basis elements are different, their relation gets written twice, and so we divide that by two. One complex equation yields two real equations, and so we get one real equation per entry. When basis elements are the same, the we get only one real equation, which is real because we are multiplying complex entries with their complex conjugates. Thus we get one real equation per entry. But if we have the orthogonal or symplectic group, then the diagonal elements count differently than the off diagonal elements. This has to do with the usage of "same" and "different" as regards "to" and "from". "From A to B" and "From B to A" count as two different entries, whereas "From A to A" counts as one entry.

Consider how choice frameworks relate to the widgets at the end of Lie algebra Dynkin diagrams, how they relate going forwards and backwards, how that compares to the relationship between identity and pseudoscalar for Clifford algebras.

Orthogonal group elements can be understood as actions but also as inputs and as outputs.

In the orthogonal group, both rotations and reflections make sense. A rotation is a pair of reflections, which implement a change of basis. In the unitary group, only rotations make sense. In the symplectic group, ...

Symplectic form {$J=\begin{pmatrix}0 & -I_n\\ I_n & 0\end{pmatrix}$} manifests the complex number {$i$}, when {$n=1$}, and can also be thought as the diagonal {$2\times 2$} block matrix. All three reflections coincide: {$J^{-1}=-J=J^T$}. For a symplectic matrix {$M$} we have {$M^TJM=J$}, thus {$JMJ^{-1}=(M^T)^{-1}$}.

Wikipedia: O(2n) is the maximal compact subgroup of GL(2n, R), and U(n) is the maximal compact subgroup of both GL(n, C) and Sp(2n). Thus the intersection O(2n) ∩ GL(n, C) or O(2n) ∩ Sp(2n) is the maximal compact subgroup of both of these, so U(n). From this perspective, what is unexpected is the intersection GL(n, C) ∩ Sp(2n) = U(n).

Wikipedia: Decomposing a Hermitian form into its real and imaginary parts: the real part is symmetric (orthogonal), and the imaginary part is skew-symmetric (symplectic)—and these are related by the complex structure (which is the compatibility). On an almost Kähler manifold, one can write this decomposition as h = g + iω, where h is the Hermitian form, g is the Riemannian metric, i is the almost complex structure, and ω is the almost symplectic structure.

For complex matrices there is the decomposition {$M=UT$} into unitary and upper triangular. The upper triangular matrices are homotopy equivalent to the diagonal matrices and to the identity. So the general linear group over the complex numbers is homotopy equivalent to the unitary group.

Rotations and reflections

• A rotation is a pair of reflections.
• A pair of reflections is distinguished by a rotation of one reflection into the other.
• A rotoreflection is a reflection times a rotation. But the rotation is a pair of reflections. So can the rotoreflection's reflection be chosen so as to undo one of the reflections in the rotation, and leave the remaining reflection?
• All of this shows that bases define context, and rotations relate contexts. Are any two contexts, any two bases related by a rotation? I think yes, if we don't care about the orientation of the basis elements, but no, in that we may need to reflect one of them.

Dimensional analysis

• {$O(n)$} has {$\frac{1}{2}n^2-\frac{1}{2}n = \frac{n(n-1)}{2}$} real dimensions. This is {$n^2-(\frac{n(n-1)}{2}+n)$}.
• {$U(n)$} has {$n^2$} real dimensions. This is {$2n^2-(2\frac{n(n-1)}{2}+n)$}.
• {$Sp(n)$} has {$2n^2+n=n(2n+1)$} real dimensions. This is {$4n^2 - (4\frac{n(n-1)}{2}+n)$}.

The dimension is calculated by calculating the number of independent equations, then subtracting from the total number of variables (matrix entries times field dimension), to get the number of free variables.

In each case, the diagonals manifest a different self-relation, which is the reason for the discrepancies, and the wobbling behind the periodicity.

Unitary is the most regular. Note the Dynkin diagrams.

• {$A_n$} as with {$U(n+1)$}: {$3,2,1$}
• {$B_n$} as with {$O(2n+1)$}: {$3,2,1,0,-1,-2,-3$}
• {$C_n$} as with {$Sp(n)$}: {$3,2,1=-1,-2,-3$}
• {$D_n$} as with {$O(2n)$}: {$3,2,1,-1,-2,-3$}

Lie Group Embeddings

{$\begin{matrix} \mathbf{Lie\; group} & \mathbf{real\; dimensions} \\ U(2r) & 4r^2 \\ U(r)\oplus U(r) & 2r^2 \\ U(r) & r^2 \\ \end{matrix}$}

Complex Bott Periodicity

• Compare {$iJ_1$} with {$J_1J_2J_3$}. They both square to {$+1$}.

{$\begin{matrix} \mathbf{Lie\; group} & \mathbf{real\; dimensions} \\ O(16r) & 128r^2-8r \\ U(8r) & 64r^2 \\ Sp(4r) & 32r^2 + 4r \\ Sp(2r)\times Sp(2r) & 16r^2 + 4r\\ Sp(2r) & 8r^2+2r \\ U(2r) & 4r^2\\ O(2r) & 2r^2-r\\ O(r)\times O(r) & r^2-r\\ O(r) & \frac{1}{2}r^2-\frac{1}{2}r \end{matrix}$}

Off-kilter will show how minds are off-kilter, requiring three minds and eight mental states.

Note that {$O(16r)\nsupseteq U(8r) \nsupseteq O(8r)\times O(8r)$}

real dimension of {$O(8r)\times O(8r)$} is {$64r^2-4r$}

The dimension of a quotient (a symmetric space) has the same quadratic power as the group in the denominator.

Note that with {$J_7$} there must be unitary matrices which do not commute with {$J_7$} because otherwise we would go from {$O(2r)$} to {$U(r)$}.

The choice of {$J_k$} is given by the symmetric space {$G_k/G_{k+1}$} and {$G_k$} is the subgroup of {$G_0\equiv O(16r)$} that commutes with {$J_1,J_2,\dots J_k$}. Similarly, the choice of {$iJ_k$} is given by the symmetric space {$H_k/H_{k+1}$} where {$H_0\equiv U(2r), H_1\equiv U(r)\times U(r), H_2\equiv U(r)$}.

Each symmetric space {$G_i/G_{i+1}$} is naturally embedded (as a totally geodesic submanifold) in {$G_{i-1}/G_i$}. For {$G_i/G_{i+1}$} parametrizes the set of geodesics {$\gamma=J_i\textrm{exp}^{\pi J_i^{-1}J_{i+1}t}=J_i\cos\pi t+J_{i+1}\sin\pi t$} where {$J_i=\gamma (0)$} and {$J_{i+1}=\gamma(1/2)$} and {$-J_i=\gamma (1)$}.

The classification of symmetric spaces is given by these quotients and shows how fundamental, remarkable and special this chain is.

The set of choices for {$J_k$} is parametrized by {$G_k/H_k$} where {$J_k$} commutes with each element of {$H_k$}. But {$J$} itself belongs to {$H_k$}, I think, for all {$k$}, and I imagine {$J_k=DJD^{-1}$} as well. The parametrization is of the contexts {$D$}, the bases, whereby we have {$J_k=DJD^{-1}$}.

But what does this parametrization mean? How does it relate to the elements of {$G_k$} that {$J_k$} anticommutes with?

Rooting Out Reflective Knowledge Leads to Failure

William Kingdon Clifford. "It is wrong always, everywhere, and for anyone to believe anything on insufficient evidence."

The orthogonal group models all experience, what is known. What is true, which goes hand in hand with what is not true, what is false. Negation serves and functions as reflection.

Apply the linear complex structure {$J_1$} to distinguish the known from the unknown, the true from the false.

(What does it mean if, in four dimensions, we have two reflections?) Reorganizing the dimensions to pair eigenvalues and get rotations.

Active inference distinguishes what is not the organism from what is the organism. Thus distinguishes the conscious from the unconscious.

Apply {$J_2$} to distinguish the known known from the known unknown, which is to say, distinguish the knower. This yields an anti-linear operator.

Apply {$J_3$} to distinguish the known known known from the known known unknown. Then the operator {$J_1J_2J_3$}, which squares to {$+1$}, splits the vector space into subspaces with eigenvalues {$+1$} (nonreflective) and {$-1$} (reflective).

{$J_1, J_2, J_3$} function as a three-cycle on {$V_-$}, and in the opposite direction, on {$V_+$}.

Apply {$J_4$} to distinguish the known known known known from the known known known unknown. But at this point we have that {$J_3J_4$} is an isometry (a reflection) between unreflective {$V_+$} and reflective {$V_-$}, thus relating them.

The transformation {$J_3J_4J_5$} splits the space in two subspaces and on one of them {$J_3J_4=J_5$}. But that is true for every {$J_3J_4J_k$}.

This yields {$Sp(2)\subset Sp(2)\times Sp(2)$} which consists of the identifications of the unreflective and the reflective. The points in the diagonal {$Sp(2)$} are these identifications. How do these identifications evolve further? and are they maintained?

Applying {$J_5$}, {$J_6$}, {$J_7$} further splits the space yielding {$W_+$}, {$X_+$}, {$Y_+$} as we go along.

Note that {$J_3J_4J_5$} splits {$V$} into two spaces {$S_+$} where {$J_3J_4J_5v=v$} and {$S_-$} where {$J_3J_4J_5=-v$}. Then on {$S_+$} we have {$J_5=J_4J_3$} and on {$S_-$} we have {$J_5=J_3J_4$}. Thus we get a reversal of the shift between the two shifts of the foursome. But the same is true of any subsequent operator.

We have isometries {$W_-=J_2W_+$}, {$X_-=J_1X_+$}.

Then we have the isometry {$J_7J_8$} such that {$Y_-=J_7J_8Y_+$}.

Consider how adding more linear complex structures takes us down from {$O(2)$} to {$O(1)\times O(1)$} rather than {$U(1)$}. Note that {$SO(2)\cong U(1)$}.

The collapse arises because to proceed further we need to have the two subspaces {$O(n_1)$} and {$O(n_2)$} be the same size. Not just the same size, but they have to be exactly the same, both identical to the diagonal element.

But then the argument is that there is no distinction between the two spaces, what is known and what is not known. In the symplectic case we also have this issue but it is removed because it is on the level of knowledge. Note also that what was distinctly different and opposite has, through the symmetric space, become encoded in the same way, represented in the same way. So we are shifting through different kinds of opposites.

In the case of {$U(n_1)$} and {$U(n_2)$} and the diagonal we have the case of Jesus (the diagonal) equating God and human.

Consider how after we have laid down eight linear complex structures it is as if we had not laid down any and they no longer have an effect. In what sense is the system "empty"? In what sense is it true that "all are known and all are known" for what is left? In what sense is there a problem with {$G_7/G_8$}?

These symmetric spaces are Grassmannians of various flavors. They are ways of choosing various {$k$}-dimensional subspaces of various {$n$}-dimensional spaces.

Clifford Algebras as Divisions of Everything

Nullsome and onesome

Twosome

Threesome

Foursome

Fivesome

Sixsome

Sevensome

Three Minds in Human Culture

• Logical square: Knowing Mind vs. Not-Knowing Mind
• Three minds express What, How, Why from the idealist point of view. Whether is God. Why functions like a break, allowing How to shift to What upon release, at which point Why becomes Whether.
• System 1 and System 2
• Preconscious processing and Conscious processing
• Right Hemisphere and Left Hemisphere
• Jung. Anima and animus.
• Yin and yang.
• Freud. Id, Superego, Ego.
• Polanyi: Tacit and Explicit
• Peirce: Firstness, Secondness, Thirdness.
• Plato: Subjects, Warriors, Rulers.
• Hegel: Thesis, Antithesis, Synthesis.
• Irrational and Rational.
• Perspective vs. Perspective on Perspective.
• First, Second, Third Person.
• Beautiful, good, true.
• Compare with a transistor.
• Traditional Understandings of Female and Male Preferences. Two stereotypic genders, two roles in dialogue, in traditional understanding, as pillars essential for robust dialogue. Building on those pillars as the basis for gender diversity. Full variety of nonstereotypic genders can be constructed based on their logical relationships.
• Neural network AI. Fast speed: Run-time inference. Dropping a pin ball and it cascades through. Medium speed: Attention/Modulation. Dopamine as an amplifier or diminisher. Sliding window of doubt that sets the neuromodulatory tone (or attitude or context in which the fast speed is received.) Lets you pop out of your local minimum. Slower scale: Structural topological, the actual connectivities, adding or removing connections.

Clifford

• Receiving evidence. “It is wrong always, everywhere, and for anyone to believe anything on insufficient evidence.”
• Forming beliefs. “It is wrong always, everywhere, and for anyone to ignore evidence that is relevant to his beliefs, or to dismiss relevant evidence in a facile way.”
• Governing how we form, hold, revise, relinquish beliefs over time. [If someone violates such a diachronic obligation by “purposely avoiding the reading of books and the company of men who call in question” his presuppositions, Clifford warns, then “the life of that man is one long sin against mankind”]

Divisions of Everything in Human Culture

Gestalt psychology?

Implications for theories of consciousness

Gives a very clear definition of consciousness which allows for a wide variety of implementations. Much to offer to scientists and scholars pursuing distinct research programs.

Biology

• Ant colony. Ways of figuring things out: not centrally organized (divergent), centrally organized (convergent), and relating the two.
• Bioelectricity and proteins and DNA.
• Mitchell - free agent hierarchy

Neurology

• Evoke consciousness wherever there are two hemispheres. Baby, puppy, octopus.

Artificial intelligence

• Raudys hierarchy of statistical classifiers for neural networks

Global workspace

• There exists a map of the mind which neurology could find.
• The map could be perhaps in the basal ganglia.
• The hemispheres exist

Information integration theory

• There is only one way that k perspectives can assemble.
• Bott periodicity runs through all symmetric spaces. So this seems unavoidable, irresistible.
• Random matrix ensembles.
• Seems to encode the symmetry inherent in mathematics itself, the way that notation is interpreted, the ways that matrices can fold up.
• All systems are the same. The complexity is latent and it is just how much is manifest. Teleological.

Active inference

• The gap between the world and the organism can be understood as the gap between the unconscious and the conscious. For the unconscious is enmeshed in the world and the conscious is not. Sensation takes the form of the emotions by which the unconscious communicates to the conscious the insufficiency of the conceptual language. Action takes the form of cognition by which the Conscious imposes the updated conceptual language upon the Unconscious. Consciousness

Panpsychism

• Look for eight-cycles and Bott periodicity as a sign of consciousness.
• Cohl Furey and particle physics.
• Krebs cycle of metabolism in all cells.
• Octosulfur and the origin of life.

Theological

• Genesis. For God, a day of creation is a division of everything.
• Genesis. You will be like God rather than you will be God.
• God and love.
• Jesus.
• Three kinds of knowledge.

Philosophers

• Gives a ground for introspection.

Wokeism

• Universe is woke. Throws out prejudicial conventions.
• But are there ways of putting prejudices in contexts?

Morality

• {$iJ_1$} is relevant for connecting to people and to God.
• But the three-cycle {$J_1J_2J_3$} likewise plays that role, squares to {$+1$}.

Additional Topics

Homotopy Theory

CPT Symmetry

References

Sattin, D.; Magnani, F.G.; Bartesaghi, L.; Caputo, M.; Fittipaldo, A.V.; Cacciatore, M.; Picozzi, M.; Leonardi, M. Theoretical Models of Consciousness: A Scoping Review. Brain Sci. 2021, 11, 535. https://doi.org/10.3390/brainsci11050535

M Stone, CK Chiu, A Roy. Symmetries, dimensions and topological insulators: the mechanism behind the face of the Bott clock. Journal of Physics A: Mathematical and Theoretical, 2010. iopscience.iop.org https://arxiv.org/abs/1005.3213

Friedman, D.A., Søvik, E. The ant colony as a test for scientific theories of consciousness. Synthese 198, 1457–1480 (2021). https://doi.org/10.1007/s11229-019-02130-y

Henry D. Potter, Kevin J. Mitchell. Naturalising Agent Causation. Entropy 2022, 24, 472.

https://plato.stanford.edu/entries/ethics-belief/ Andrew Chignell. The Ethics of Belief. Stanford Encyclopedia of Philosophy.

https://people.brandeis.edu/~teuber/Clifford_ethics.pdf William K. Clifford. The Ethics of Belief. Contemporary Review, 1877.

Kauffman, L. H. (1987). "The fundamental structures of human reflexion": Comment. Journal of Social & Biological Structures, 10(2), 189–192. https://doi.org/10.1016/0140-1750(87)90006-6

V. A. Lefebvre. Algebra of Conscience. 2nd ed. Springer. 2001.

Elio Conte. A Clifford algebraic analysis gives mathematical explanation of quantization of quantum theory and delineates a model of quantum reality in which information, primitive cognition entities and a principle of existence are intrinsically represented ab initio*. World Journal of Neuroscience, 2013, 3, 157-170. https://www.scirp.org/pdf/WJNS_2013071910551255.pdf

Elio Conte, Ferda Kaleagasioglu, Rich Norman. Algebraic Quantum Theory of Consciousness. Aracne editrice. 2018. https://www.researchgate.net/publication/328879758_ALGEBRAIC_QUANTUM_THEORY_OF_CONSCIOUSNESS

Elio Conte. On the Logical Origins of Quantum Mechanics Demonstrated By Using Clifford Algebra: A Proof that Quantum Interference Arises in a Clifford Algebraic Formulation of Quantum Mechanics. Electronic Journal of Theoretical Physics. May 2011, Vol. 8 Issue 25, p109-126. 18p.

Koehler, G. Q-consciousness: Where is the flow? Nonlinear Dyn. Psychol. Life Sci. 2011, 15, 335–357.

Khrennikov A (2015) Quantum-like modeling of cognition. Frontiers of Physics. 3:77. doi: 10.3389/fphy.2015.00077

Louis H. Kauffman. The Mathematics of Charles Sanders Peirce. Cybernetics & Human Knowing, Vol.8, no.1–2, 2001, pp. 79-110.

Andrius Kulikauskas. Time and Space as Representations of Decision-Making. Presented at "Space and Time: An Interdisciplinary Approach", September 29-30, 2017 in Vilnius, Lithuania https://www.math4wisdom.com/wiki/Research/20170929TimeSpaceDecisionMaking

Tyler Goldstein. Sentient Singularity Theory. https://www.sentientsingularity.com

Michaele Suisse, Peter Cameron. Aims and Intention from Mindful Mathematics: The Encompassing Physicality of Geometric Clifford Algebra. 2017.

Lehar S. (2003b) "Gestalt Isomorphism and the Primacy of the Subjective Conscious Experience: A Gestalt Bubble Model". The Behavioral and Brain Sciences 26(4), 375-444. http://slehar.com/wwwRel/webstuff/bubw3/bubw3.html

Steven Lehar. Clifford Algebra: A Visual Introduction. https://slehar.wordpress.com/2014/03/18/clifford-algebra-a-visual-introduction/

Steven Lehar. The Boundaries of Human Knowledge: A Phenomenological Epistemology or Waking Up in a Strange Place. http://slehar.com/wwwRel/webstuff/book2/Boundaries.pdf

https://www.youtube.com/channel/UCj1M0P2I8zLLKlVUHb8xf2w/videos

Ben Goertzel. On the Algebraic Structure of Consciousness. Dynamical Psychology. 1996. https://www.goertzel.org/dynapsyc/1996/consalg.html

Ben Goertzel. Ons: An Algebraic Foundation for Being and Time, Explaining the Emergence of Clifford Algebra Structure. December 1997. (Rough Draft, Not for Distribution) https://www.goertzel.org/papers/OnsAlgebra.html

Ben Goertzel, Onar Aam, Tony Smith, Kent Palmer. Ons Algebra: The Emergence of Quaternionic, Octonionic and Clifford Algebra Structure From Laws of Multiboundary Form (Rough Draft, not for distribution) https://www.goertzel.org/papers/Multi.html

Ben Goertzel. Ons: a theory of truly elementary particles, explaining the emergence of structure from void in physics and psychology (Rough Draft, for comments only) November 1996.

Frank Dodd (Tony) Smith, Jr. - 2015 LHC 2015-16 and E8 Physics. https://vixra.org/abs/1508.0157

Basil Hiley. Quantum theory, the Implicate Order and Consciousness. Interview with Richard Bright (Editor: Interalia Magazine). Published in With Consciousness in Mind (Part 3) – November 2015. https://www.interaliamag.org/wp-content/uploads/2015/11/Basil-Hiley-Quantum-theory-the-Implicate-Order-and-Consciousness.pdf

B. J. Hiley. Process, Distinction, Groupoids and Clifford Algebras: an Alternative View of the Quantum Formalism. https://arxiv.org/pdf/1211.2107

Matti Pitkänen. Mathematical Aspects of Consciousness Theory

• An interesting special aspect of 8-dimensional Clifford algebra with Minkowski signature isthat it allows an octonionic representation of gamma matrices obtained as tensor products ofunit matrix 1 and 7-D gamma matrices γk and Pauli sigma matrices by replacing 1 and γk by octonions. This inspires the idea that it might be possible to end up with quantum TGD from purely number theoretical arguments. One can start from a local octonionic Clifford algebra in M8.