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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 eightcycle of metaphysical contexts, imposing complex, quaternionic and splitbiquaternionic structure, respectively. Outline Introduction
Preliminaries
Mathematical Model
Philosophical Interpretation
Discussion
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, SpencerBrown, 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...
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
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 selfawareness and this is to say that it has in its inner the image of itself. In most cases we speak of selfimage 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 selfawareness (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. SpencerBrown. 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 Sovietoccupied 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 singlecelled 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 Selfstanding 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. Selfstanding disarms the selfstanding 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 skewsymmetric 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.
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) {$z=x+yi\Rightarrow \bar{z}=xyi$} 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 {$\lambda = a+bi$}. {$$j(\lambda v)=j(a+bi)v=(aj+bji)v=(ajbij)v=(abi)j(v)=\bar{\lambda}j(v)$$} 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 {$M^21=0$}, thus {$(M+1)(M1)=0$} and {$(M+1)(M1)v=0$} 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 rotationfree. 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 Skewsymmetric matrix {$AA^T=I$} SkewHermitian matrix {$AA^H=I$} Quaternionic skewHermitian 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 skewsymmetric (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
Dimensional analysis
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 selfrelation, which is the reason for the discrepancies, and the wobbling behind the periodicity. Unitary is the most regular. Note the Dynkin diagrams.
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
{$\begin{matrix} \mathbf{Lie\; group} & \mathbf{real\; dimensions} \\ O(16r) & 128r^28r \\ U(8r) & 64r^2 \\ Sp(4r) & 32r^2 + 4r \\ Sp(2r)\times Sp(2r) & 16r^2 + 4r\\ Sp(2r) & 8r^2+2r \\ U(2r) & 4r^2\\ O(2r) & 2r^2r\\ O(r)\times O(r) & r^2r\\ O(r) & \frac{1}{2}r^2\frac{1}{2}r \end{matrix}$} Offkilter will show how minds are offkilter, 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 {$64r^24r$} 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 {$G_{i1}/G_i$}. 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 antilinear 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 threecycle 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
Clifford
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
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Global workspace
Information integration theory
Active inference
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Additional Topics Homotopy Theory 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/s1122901902130y Henry D. Potter, Kevin J. Mitchell. Naturalising Agent Causation. Entropy 2022, 24, 472. https://plato.stanford.edu/entries/ethicsbelief/ 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/10.1016/01401750(87)900066 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, 157170. 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, p109126. 18p. Koehler, G. Qconsciousness: Where is the flow? Nonlinear Dyn. Psychol. Life Sci. 2011, 15, 335–357. Khrennikov A (2015) Quantumlike 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. 79110. Andrius Kulikauskas. Time and Space as Representations of DecisionMaking. Presented at "Space and Time: An Interdisciplinary Approach", September 2930, 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), 375444. http://slehar.com/wwwRel/webstuff/bubw3/bubw3.html Steven Lehar. Clifford Algebra: A Visual Introduction. https://slehar.wordpress.com/2014/03/18/cliffordalgebraavisualintroduction/ 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 201516 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/wpcontent/uploads/2015/11/BasilHileyQuantumtheorytheImplicateOrderandConsciousness.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
Tasks Interpretation
Understanding
References
20240705 Modeling Introspection 