Epistemology
Introduction E9F5FC Questions FFFFC0 Software 
Andrius Kulikauskas: Welcome! This is where I record my latest research notes. Consciousness requires the consideration of two different kinds of Bayesian priors  a presumption of knowing and a presumption of not knowing. Consider the varieties of energy: External (global) kinectic and potential. Internal kinectic and potential. Work and free energy. Sixsome? What more? Linear (this is this; that is that) and antilinear (this is that; that is this). Linear and antilinear  Bott periodicity  how the question and answer relate. Does the answer follow the question? Or does the question follow the answer? Video on random walks. Idea: In the more sophisticated setting, can use rules derived from different contexts, thus choose which is the most fruitful context. Thus half of the mind is generating the answer, but half of the mind is choosing the context, as with the division of everything, asking the question. A third mind can oversee this. inference = "way of figuring things out" Structure Active Inference Generative model creates beliefs, the policy creates actions, but our actions are "fated". We are only "free" to choose our beliefs. So our consciousness has to do with the beliefs we take, our will. We don't have to experience our Bayesian inference, we can simply identify with it, take ownership of it.
Karl Friston  your universe is trying to infer you (given that you are trying to infer your universe). Orthogonal polynomials are based on a recurrence relation which is second order thus not a Markov chain, not modeled by Active Inference. List of conceptual structures
Todd Trimble. Buildings and BN pairs. Encyclical letter. Laudato Si. On Care for Our Common Home. John Kellden. Cards for insight. Communication tool for relating concepts. https://ncatlab.org/nlab/show/ambidextrous+adjunction {$\phi(x,t)=\int\frac{d^4k}{(2\pi)^4}[\tilde{\phi}_{(k)}^+a^+(k)e^{ikx}+\tilde{\phi}_{(k)}^a^(k)e^{ikx}]$} In what sense is a particle's location in space and time smeared out in both space and time? Is the {$E_8$} symmetry based on one half of Bott periodicity? Do the Dynkin diagrams of the exceptional Lie groups model counting backwards and forwards inside Bott periodicity? Spinors are how you turn around at the widget of the real orthogonal Dynkin diagrams. Brian C. Hall. An Elementary Introduction to Groups and Representations. {$i$} and {$\bar{i}$} are additive inverses and multiplicative inverses. Orthogonality appears in Bott periodicity. Linear complex structures are orthogonal skewsymmetric matrices (for which {$j^T=j$} so that together {$j^{1}=j=j^T$}. My approach to the wave function is based on orthogonal Sheffer polynomials. What is the relation, if any, between Sheffer polynomials and skewsymmetric matrices? John Baez. Mathematics in the 21stCentury. Slides. Symmetric spaces as the basis for geometry. Leonard Susskind. Dear Qubitzers, GR=QM. Gravity and quantum mechanic. Susskind sphere. Malcolm Lowe. Independent researcher of consciousness and language. Inner monologue https://en.wikipedia.org/wiki/Holographic_principle Relationship between holographic principle, Markov blanket and free energy principle. Chris Fields has a diagram of a quantum reference frame with two chains with arrows in both directions, the measurement chain and the preparation chain, and with arrows passing from locations in the measurement chain to the outcome classifier and then to locations on the preparation chain. Is this related to the fivesome, sixsome or sevensome? The skeleton of an autoencoder. Chris Fields. A reference frame allows for measurement but requires and active participant to do that measurement. And the result is approximate, especially when communicated. By the Spectrum theorem, selfadjoint operators are averages of projections. Thus they can be thought of as manifesting continuity of the Unconscious and physical reality of observables. Similar, neural networks that express weight vectors must be selfadjoint operators. Whereas entropy manifests the discreteness of the Conscious. And with discreteness there is the problem of whether or not there is reversibility, whether information can be forgotten or lost. Noncommuting means incompatible frameworks. So rotations and rotoreflections are incompatible. Unconscious and conscious are incompatible. But there is an isometry between unconscious and conscious. The symmetry of Bott periodicity does not depend on the size of the dimensions but is deeper than whole numbers. If we think in terms of numbers, then it becomes much more complicated, as with the homotopy of spheres. {$\ln 2=1\frac{1}{2}+\frac{1}{3}\frac{1}{4}+\frac{1}{5}\frac{1}{6}\dots$} is an important number in information theory because it allows us to consider and calculate binary choices. In Bott periodicity, generators squaring to +1 give quaternionic structure? How does that equal to the operator +3 for consciousness with linear complex structures? Neural connections, their weights are continuous, analog, not discrete. They are given by the Unconscious, which is not discrete, but continuous. Entropy is discrete, which suggests it is a concept of the conscious, and involves reflection. Whereas in the case of the unconscious, there is no reflection, thus no entropy, thus no symmetry, no possibility of different actions yielding the same state. For the unconscious, we have continuity in evolution, we have rotations. The fact that, for orthogonal matrices, {$O^T=O^{1}$} and thus the inverse can be understood as reversing arrows (similarly for the unitary and compact symplectic cases) is an expression of the symmetry in math. Bott periodicity organizes these symmetries in math itself. From Chris Field. Physics is Information Processing. Lecture 1. Entropy only makes sense as a discrete concept. This reminds me of how entropy is calculated by carving up phase space with a discrete grid, which implies there is an observer imposing a specific coordinate system. Also, the relationship with symmetry, where there are different actions which may yield the same effects.
Bayesian inference involves thinking backwards and as such is relevant for the fivesome. Emergence of Random Structure Random structure (such as divisions of everything?) arises at certain thresholds. Bioelectricity differentiates the inside and outside of a bubble and its boundary. It functions like the Unconscious which integrates the whole. Whereas, within this constraint, the semiotic language of molecular change acts like the Conscious and allows for analysis in terms of components. What is Consciousness, which balances the two? Pressure on a boundary can play the role of a holistic mind without centralization. Pressure depends on an orientable surface with inside and outside. It also depends on thermodynamics, thus supposes entropy. Thus this could be half of a mind, and not depend on centralized organization. Compare the Raudys hierarchy of a single layer perceptron with gradations and other hierarchies of conceptual maturation, such as Lawrence Kohlberg's stages of moral development, Piaget's developmental psychology and Habermas's related theories. What does the Yoneda Lemma say about the three minds, considered as What, How, Why, acting on an arrow, Whether there is an arrow? What does this mean if we think of an arrow as a linear complex structure? Is there a category where the arrows are products of linear complex structures and the objects are subgroups of the orthogonal group? Does composition work out? Spheres make prominent the dimensions 0/1, 1/2, 3/4, 7/8 that accord with the normed division algebras. The divisions of everything likewise pair odd and even divisions but also include 5/6. So is there any connection or not? JeanPhilippe Bernardy Shalom Lappin. Unitary Matrices As Compositional Word Embeddings. 2022. Linguistic tool that bridges the goals of transparent compositionality and situational robustness. Energy (where mass is the rest energy of a particle) as a fifth dimension where mass indicates the energy required for the particle to come into being. Thus a particle can be like a sphere where the mass indicates the minimum radius when the particle is manifest in this world. In this way, a particle does not behave as a point but rather as a sphere. The (nonassociative) octonions are perhaps related to the (associative) splitbiquaternions, as the Lie group embedding unfolds, by considering that we don't know whether we Humans are living in the reflected world (based in God's mind) or unreflected world (based in our self), in the chain Human's view of God's view of Human's view... We may not know if we are living in the {$V_+$} or {$V_$} of the splitbiquaternions. This may give rise to an ambiguity, and also, a choice between our God and our self. That choice and that ambiguity may ground nonassociativity of perspectives, distinguishing between stepping out [Child's view of (Father's view of Mother's view)] and stepping in [(Child's view of Father's view) of Mother's view]. When associativity no longer holds, the system collapses, and we have zero objectively, and perhaps the field with one element, subjectively. Todd Trimble: You may know that Wolfgang Pauli was in analysis with a Jungian psychologist, in the early 1930's (with Jung keeping a close watch over the case); he was in some mental distress before the analysis and may have thought he was going insane. It's probably not as well known that the analysis of Pauli's dreams, which Pauli worked very hard on with his analyst, became essentially part II of Jung's Psychology and Alchemy. The course of the analysis more or less concluded with a spontaneous vision (or perhaps it was a very vivid dream) Pauli had, of a kind of 4dimensional "World Clock", which Pauli in his conscious reflections subsequently experienced as a kind of healing synthesis or culmination of the analysis. I imagine that for Pauli, this was connected in some way with quaternions and Pauli matrices and so forth and so on.
The quantum world receives a question and its answer given does not change, thus remeasurement does not yield a new answer. The language is degenerate. Meaning is given to the language by the answer. A basis is thereby chosen.
Linear complex structure {$J_i$} is a perspective. A shift in perspective from {$J_i$} to {$J_j$} is their product {$J_iJ_j$}. Together they define a quaternionic structure. What is the meaning of longer products of perspectives? Basis is a context, supplies a context, provides a context. The number of bases involved is the context. Reflection is not commutative with regard to a linear complex structure. In the orthogonal group, commuting means that we have some eigenvector. Show that geometry is based on eight levels as organized by Bott periodicity. Then show that this means that the possible geometries are very limited as given by the inner products and the classical Lie groups, which participate in Bott periodicity. Slack is modeled by complex conjugacy. {$J_1$} is a linear complex structure which models the unconscious, adding a perspective {$+1$}. {$J_2$} adds an antilinear operator which models reflection and a perspective on a perspective. Together they yield the conscious. {$J_3$} divides the space on which these act into two parallel spaces, thus is a perspective upon a perspective on a perspective. Together they are consciousness. {$J_3J_4$} defines an isometry between the two spaces. Ant semiotics  remove noise like quantum computing. Noisefree environment (isolated subsystem) is the quantum world. Think of divisions of everything as an adjoint functor to the application of a linear complex structure on a symmetric space. In what sense is a linear complex structure the going beyond of oneself? Sean Carroll. Quanta and Fields. https://amzn.to/3R7ahLp Seth Lloyd. Computing universe. Variable based causation (plural) vs. Instance of causation (singular). Assignment is asymmetric (computer science), equality is symmetric (math). Compare this with linear complex structure. Body shape is not determined by the genome but by electromagnetic activity. The genome works locally. Unit quaternions {$SU(2)$} have 2 irreducible representations. 3x3 matrices of real numbers. And 2x2 matrices of complex numbers. The latter are spinors. The even subalgebra of a Clifford algebra {$Cl_{0,n+1}$} is the Clifford algebra {$Cl_{0,n}$}. The unit elements of that Clifford algebra is the spin group {$Sp(n+1)$}. Essentia Foundation Bernardo Kastrup. Accepts submission arguing against metaphysical materialism. Complex numbers metaphysically: "This" (even = 1) vs. "that" (odd = i). Dialogue beween two people. "This" refers to what they have, "that" refers to what the other has. Similarly with "I" and "You". Six postulates of quantum mechanics
fermion with those of another. Electronic spin must be included in this set of coordinates. Freeman J. Dyson. The Threefold Way. Algebraic Structure of Symmetry Groups and Ensembles in Quantum Mechanics. J. Math. Phys. 3, 1199–1215 (1962) Symmetry breaking distinguishes vectors and spinors. We can carve up mental space from either end, in the (+1) and (1) directions, or both ends (+2), arriving at {$Cl_{4,0}\cong Cl_{0,4}$}. How to understand (+3)? Compare Clifford algebras with chain complexes. {$A_1$} is the building block for Lie theory and its Lie algebra serves both {$SU(2)$} and {$SO(3)$}, thus relates vectors and spinors, and seems relevant for Bott periodicity. Could the exceptional Lie groups {$E_k$} be generated by Bott periodicity? A symmetric space has, at each point, a (global) isometry that (locally) inverts each tangent vector. Compare that with symmetry where we reflect across the center of a space. In John Baez's talk on the symmetric space, the functor acts on the category of representations of Clifford algebras, which is the category relevant for Morita equivalence. Doubling, halving, dividing a perspective (a space, a module) into two perspectives, yielding a perspective on a perspective. In a complex Clifford algebra, the coefficient {$i$} commutes with all of the generators {$e_k$} whereas the generators anticommute with each other. If the coefficient {$i$} anticommuted with all of the generators, then this would simply be a real Clifford algebra with one more generator. So this is a very deep fact that distinguishes complex and real Clifford algebras. Think of Clifford algebra generators that square to {$1$} as spinors, and those that square to {$+1$} as vectors. Each generator can be thought of as yielding a turn of {$\pi=180^\circ$}. Does that make sense? Thomas Metzinger book. The Ego Tunnel. https://herbspencer.academia.edu Herb Spencer Collecting meaningful experiences in life Associativity means that there is no need for an external observer to place parentheses. There is internal consistency. There is no need for an external sense of time. Time makes sense internally.
Rulead space generated by three minds, the three possible rules on the eight cycle. Higgs boson flips chirality of fermions, which yields their mass. Just like frequency yields energy. So they are flipping between universes. Neutrinos do not flip in this way. They are stuck in our universe. Active inference explains how people are inclined to confirm their biases. The purpose of consciousness is to force people to confront their biases, to investigate, to take a sober, honest look, which way is it. twosome: linear vs. antilinear. Antilinear swaps i with its conjugate i in going from inside to outside or vice versa. Thus antilinear is "opposites coexist" and models free will. https://www.pmwiki.org/wiki/PmWiki/IncludeOtherPages
Think of matter antimatter cancellation, creating energy, as a driver for culling? Yielding an imbalance? as antimatter moves away? by randomness? https://mathoverflow.net/questions/7155/famousmathematicalquotes https://mathoverflow.net/questions/13832/analogiesbetweenanalogies reply regarding adjunctions {$\mathbb{R}\oplus\mathbb{R}$} is not a division algebra but is a super division algebra. Symmetric functions
Operator
Understanding, consciousness
Divisions
Category theory The notion of a singleton, and more broadly, membership in a set can be reworked in the language of category theory. Similarly, rework all of the ZermeloFrankel axioms of set theory in the language of category theory. Math
Geometry Adjunctions Three Minds
The disembodying mind results from evolutionary pressure to devote more resources to modeling the unknown.
If we focus on user requirements than on neural implementation, then it makes sense to talk about left and right hemispheres as champions of these two mindsets. Orthogonal Sheffer polynomials
Biochemistry
Random matrices
Randomness
Linear regression
Entropy
Entropy = nondeliberateness. How is this related to unconscious (nondeliberate) and conscious (deliberate)?
Maximum entropy distribution = least informative distribution. Consider all permutations of all lengths. (In considering the moments.) This is similar to {$O(\infty)$} or {$U(\infty)$} which considers all rotations. There is a theorem that relates them. And consider how to take the Fourier transform of all of that. And how does that relate to the Fourier transform of finite groups? Knowledge engineering
Peirce's squirrel. What is the definition of "around"? The same definition > different definitions. Contradiction yields distinction. Explanatory reasoning. Foursome (four causes) as four explanations. Does an index (how) rely on abduction? Geometry describes things in the world. In response to Daniel Ari Friedman's and Eirik Søvik's distinction of forward test and reverse test, consider the hierarchy of evidence which I use. For me, the most important is the limits of imagination as given by my introspection  what I see with my own eyes, that is, my mind's eye. The least important  but of ultimate importance when there is no data  is my aesthetic sense  but perhaps even beyond that is my consciousness of my own ignorance. In between, or perhaps even in a different dimension, is empirical, factual data from science, and the theories that grounds, which I would not want to contradict, and also testimonies about life from others and myself. In general, there is negative evidence  what I would not want to contradict  and positive evidence  what I affirm that I observe. We engage our faces. But imagine a dog who engages, smells my butt. The dog could conclude that my gut's ecosystem is conscious. In general, we could consider how a creature such as a computer would analyze our consciousness. Consider the ways that ants figure things out  as the basis for consciousness. Schwitzgebel  consciousness of the United States Daniel Ari Friedman https://cognitivesecurity.us Interpret the combinatorics of Associated Legendre polynomials, and substituting {$\cos\;\theta$} and {$\sin\;\theta$}, the related spherical harmonics. Consider how they integrate with the Laguerre polynomials. Understand how they describe the possible states of the hydrogen atom and the periodic table of elements. Think of a series of low energy pits. Consider quantum tunneling as an outcome of Heisenberg's uncertainty principle, that because of this uncertainty, there is a definite probability of a particle appearing in another pit. Also, think about what Heisenberg's uncertainty principle means for the particle itself, as with regard to itself, it is completely definite. What does that imply? The Kreb cycle functions as a switch, turning forwards or backwards. Does one direction (energy?) represent the unconsious +1, the other direction (growth?) represent the conscious +2, with consciouness functioning as the switch? Can we foster alternation between stepping in and stepping out? Think about characterizing the chemical soup in which interactions take place. There is the solvent (like water) and the solute forming a uniformly distributed solution. Can there be a gradient? Then pH is an important characteristic which can lead to an electric gradient if there is a boundary with a zone with a different charge or different pH. Causality is about the granularity of explanation. Determinism would be total granularity and supposed God as an observer. Reproduction  replication  is a type of flow, recurring activity, a form of persistence in terms of flow rather than material Kevin Mitchell, Henry Potter. Naturalising Agent Causation
How does matter antimatter asymmetry relate to the two branches of {$O(\infty)$} and the weak force? How do the growth structures (causal trees) of orthogonal polynomials relate to group representations (such as pairs of Young tableaux  pairs of causal trees)? How are pairs of permutations (which are pairs of involutions) related to pairs of involutions (as with the Hermite polynomials)? Do the pairs of Hermite polynomials relate to the RobinsonSchensted algorithm? Electron shells are given by twice the square {$2n^2$} which yields {$2,8,18,32,50...$} And the square number is understood as the sum of odd numbers: {$1+3+5+7+...$}. The {$2$} is the spin of the electron which we can think of as two sides of the square (as a sheet of paper). The shells (their odd number portions) are filled in a zigzag pattern. This is based on the radial model of the hydrogen atom.
Duality of burning and assembling in biochemistry. Duality of flow and structure in the 24 ways of figuring things out in neuroscience. Duality in Christopher Alexander's patterns of structure and recurring activity. Orthogonal Sheffer polynomial recurrence relation has 5 inputs. 3 are firststep, external, weighted, meaningful in the broader environment. 2 are later step, dependent on {$n$} or {$n(n1)$}, meaningful internally. The pair of causal trees  the squaring of the wave function  may be expressing sexual combination. Consider the waste products that biological activity removes such as {$CO_2$} from animals and {$O_2$} from plants and how the waste products relate to inputs. Thus how different living beings or different organs are different parts of the cycle. Consider how these pieces may be put together as divisions of everything. How does that relate to language? Carbon reacts with a single electron in a link, a onestep causality. Oxygen reacts with two electrons in a kink, twostep causality. The wave equation  and waves in general  are expressions of analytic symmetry. The second type of matching polynomial has remarkable connections with orthogonal polynomials. For instance, if G = Km,n, the complete bipartite graph, then the second type of matching polynomial is related to the generalized Laguerre polynomial Lnα(x) by the identity:
If G is the complete graph Kn, then MG(x) is an Hermite polynomial:
where Hn(x) is the "probabilist's Hermite polynomial" (1) in the definition of Hermite polynomials. These facts were observed by Godsil (1981).
4+2 elements for organic molecules
Alpha helix  amino acids relate to those which are three steps above in the chain, thus this gives the operation +3. Lectures on Orthogonal Polynomials and Special Functions Ne per brangi knyga. Think of {$\alpha$} and {$\beta$} as the steps in two frames that are centered on two events. If {$\alpha=\beta=0$}, then the two frames coincide and so the kinematics collapses, the edge statistics collapse. Think of life as an attractor. Express attractors in terms of Wondrous Wisdom. The Story Behind "Silent Spring": How Rachel Carson's Countercultural Courage Catalyzed the Environmental Movement https://www.themarginalian.org/2017/01/27/rachelcarsonsilentspringdorothyfreeman/
MeixnerPollaczek polynomials have complex conjugate weights {$\alpha$} and {$\beta=\bar\alpha$} but actually they appear as link weight {$\alpha \bar\alpha$} which is real, and kink weight {$\alpha\bar\alpha$} which is real. Note that the kink is the full portion (the amplitude) of {$\alpha$} whereas the link is twice the real portion. Thus the kink is a measure of entanglement, of the imaginary portion. Oxygen, nitrogen, carbon are like the twosome, threesome, foursome of organic chemistry. It is not the atoms but their valences that matter. Thus they belong not to the world of matter but to the world of lack of matter, thus of flow. Integrating the square of the wave function with a definite integral  summing over the step function of the orthogonality equation for the Meixner polynomials  is simply a way of grouping together combinatorial objects, and if you like, giving a linear weight to them (if you are dealing with nonintegers). But you can group them in other ways as well  so this interpreation is superior to that of the wave function. It lets you think of measuring (grouping) units of information in different ways. The longer you run an experiment (as at CERN), the more causality you introduce (in terms of labels n). So time is given by {$n$}, where {$n$} indicates the maximal power of {$x$}. This means that time is relative to, for example, energy. Relate Grassmannian minors to elementary symmetric functions of eigenvalues Sheffer polynomial coefficients can be expressed in terms of elementary symmetric functions. How does that relate to the particle clocks? And {$\alpha$} and {$\beta$}? And do those relate to the forgotten symmetric functions with the two different kinds of labels? How do Dynkin diagrams relate to organic molecules such as sugars? Open Learning Initiative. Biochemistry Course. Yale. Freshman Organic Chemistry Scientific American. Christof Koch. What is Consciousness?` Todd Trimble recommended Lawvere on algebraic theories and on hyperdoctrines, helpful for understanding variables. Mark Ronan. Lectures on Buildings. Mark Ronan. Symmetry and the Monster. Tet methylcytosine dioxygenase 1 Tet1 gene Charles Chihara. A Structural Account of Mathematics. Truth is that which cannot be hidden, in other words, what is obvious. This is very much like the Greek word Aletheia. https://en.wikipedia.org/wiki/Aletheia This comes up in Wondrous Wisdom as the negation of Whether, that level of knowledge that considers whether a cup is in a cupboard even if nobody sees it. Truth means there is no such Whether. Similarly, negating What means there is no intermediary, our relationship is Direct. Negating How means there is no becoming, there is Constancy. Negating Why means there is no allencompassing, there is Signficance. True, Direct, Constant, Significant are the negations of the levels of the Foursome and they are also the four conceptions of the Nullsome, which is to say, God. God is True, Direct, Constant, Significant. I suppose that means that God is not a subject of knowledge, is not defined by knowledge, is not evoked by knowledge. Imagination acts through external relationships but there is also internal structure, beyond the imagination. Lie algebra is not associative but rather acts like the production for a Turing machine. A consequence of the Jacobi identity and anticommutativity:
An ideal of an algebra models selfawareness. Cohl Furey: A black hole likewise models selfawareness. Stability of ideas means they survive evolution. Octonions, standard model and unification. 2023. David Spivak. Dynamic Interfaces and Arrangements: An algebraic framework for interacting systems.
Sobczyk. Vector Analysis of Spinors revised How do the colorings and derangements in the linearization coefficients of orthogonal Sheffer polynomials relate to John Baez's interest in colorings and derangements?
Understand the analytic symmetry in this expression for a weight function: {$\omega(x)=\int_{\infty}^{\infty}\int_{\infty}^{\infty}\sum_{n=0}^{\infty}\omega(x)\sum_{k=0}^{\infty}\frac{(1)^n(2\pi i \xi x)^{n+k}}{n!k!}dx d\xi$}.
A new essay concerning the origin of ideas by Antonio Rosmini. A new essay concerning the origin of ideas by Antonio Rosmini. https://www.amazon.com/GodsUndertakerHasScienceBuried/dp/0745953719 For Sheffer polynomials as such, in the implicate order, prior to orthogonality, there are no notions of moments or distribution or weight function. Change away from our direction is a rotation and is given by Curl. It is between Grad (change in all directions) and Div (change in our direction). Linguistic drive: The desire by young people to have their own language which is not understood by others. The same for ethnic tribes or social classes.
Gigliola Staffilani. The Schrodinger equations as inspiration of beautiful mathematics. Categorifying Schroedinger's equation
{$S_\infty$} = colimit of symmetric groups = permutations with finite support Is there a connection between the Yoneda lemma and the Barratt–Priddy_theorem {$H_{k}(\Sigma _{n})\cong H_{k}({\text{Map}}_{0}(S^{n},S^{n}))$}? Roelof Koekoek. Inversion formulas involving orthogonal polynomials and some of their applications Richard P. Stanley. A Survey of Alternating Permutations Wilf. Generatingfunctionology. Math Stack Exchange. Expected value of falling factorials from axioms of Poisson process. The ordered Bell numbers for {$n=3$} can be organized like the root system {$G_2$} as pictured here. Patrick Njionou Sadjang. Moments of Classical Orthogonal Polynomials. Are the CPT symmetries related by the threecycle? What combinations are possible? Legendre polynomials appear when solving the Schrödinger equation in three dimensions for a central force. Combinatorics of the residue theorem can provide a combinatorial interpretation of the distributions arising from the moments of the orthogonal Sheffer polynomials. Here is an example of how to consider it combinatorially:
Feinsilver. Lie algebras, Representations, and Analytic Semigroups through Dual Vector Fields. Group theory related to orthogonal Sheffer sequences. Consider threesome for participation expressed as incidence structure, hypergraph, block design. Frequency is the internal, qualitative expression of energy and intensity is the external, quantitative expression of energy. Is this potential energy and kinetic energy? Constructive hypothesis {$A\overset{C}{\rightarrow}B$} for communication yields the 3 minds. Choice frameworks are logics (as with proof by contradiction  which is asymmetric)
Walks on binary trees: The BruhatTits tree for the 2adic Lie group {$SL(2,Q_2)$}. See Building. Dave Gray: Listener and storyteller have similar brainwaves. Reading a book, watching a movie we sublimate our thoughts, suppress disbelief. We accept another's stream of consciousness. This allows us to see what we have in common. Samuel interests: Psychedelic rituals are delegitimized in the modern age. Conscious is not large enough to grasp the unconscious. Classifying space for U(n) is the complex Grassmannian. Classifying space for O(n) is the real Grassmannian. Lawvere: Most important topos is the etale topos of a given scheme of a given geometry. In analytic geometry, there is the implicit function theorem and so can talk of homeomorphisms. But this is lacking in algebraic geometry, where the square root is not invertible. But there is an infinitesimal analogue that the derivative at each point, which is a linear transformation, should be invertible. Lawvere 2012 5:50 Traditional philosophical notions for thousands of years could be given a mathematical formulation sufficiently general using the most uptodate technology we have in mathematics, i.e., topos theory, etc., etc., could be given a formulation which almost fully captures the philosophical content and not just some fragment of it so that philosophical calucations could become open to some degree to calculation as Leibniz, for example, dreamed. Todd Trimble. Combinatorics of Polyhedra for nCategories External relationships  linear algebra  representation theory  how a structure acts on an external space Geometries may be modding out by different dimensions... projective by a line, conformal by ? symplectic by ? Polanyi's tacit and explicit. Forgetful functor yields tacit, free construction yields explicit. Leonard Bernstein, Charles Ives, The Unanswered Question  the space for inquiry between question and answer. Forgetful functor yields Relate the long root (2 to 1) in the simple root diagram of {$G_2$} with John Baez's spans of groupoids. Why is color not part of geometry? Could it be for another civilization? Or time? Or temperature? Or calligraphy? Hieroglyphics? Semiotics? {$G_2$} has two simple roots: {$\alpha=e_1e_2$} and {$\beta=(e_2e_1)+(e_2e_3)$}. The combinations are {$\pm\alpha$} and {$\pm(\beta + n \alpha)$} where {$n=0,1,2,3$} and {$\pm(2\beta + 3\alpha)$} Neil Turok Mirror universe, big bang as a mirror, CPT symmetry. https://en.wikipedia.org/wiki/Lincos_language What can we say about the symplectic form {$x_1y_2x_2y_1$}? and how does it relate to symplectic geometry? {$\textrm{Sp}(1,\mathbb{R})=\{A:\mathbb{R}\rightarrow\mathbb{R}  \omega(Ax,Ay)=\omega(x,y) \textrm{ for all } x,y\in\mathbb{R}^2\}$}. Urs Schreiber. Differential cohomology in a cohesive ∞topos Modal Homotopy Type Theory relates Hegel, String Theory, Cohesive Infinity Topos. {$O(\infty)$} has two parts and that may be reflected in the fact that the simple roots can be considered in two groups {$x_ix_j$} and {$x_jx_i$} where {$i<j$}. Consider how this works for each Lie group, unitary, symplectic, odd and even orthogonal groups. 数学笔记 https://en.wikipedia.org/wiki/Alfred_University Stuart Kauffman: You can't reduce biology to physics because biology has functions (a heart is a pump). But the Yoneda Lemma shows that mathematics can model a notion of How and distinguish it from what is not How. Light can go backwards in time. Double "time slit" experiment. How to interpret this? What would it mean if all of time consisted of such time slits? (By analogy with Feynman's explanation of all of space as made of slits, as the basis of quantum field theory.) Yao. The Shape of Inner Space. Action (one system  wave behavior) vs. force (two systems  particle behavior). Evolution: Generating varieties vs. culling, pruning. Neurology: Arousal: Sympathetic (fight or flight) vs. parasympathetic (relaxed). How do the transpose, conjugate transpose, quaternionic transpose simplify Cramer's rule combinatorially? Examine the root systems for the exceptional Lie algebras. In what sense are they expressing the duality in counting forwards and backwards? Grassmannian G(4,2) has five Plucker coordinates (4 choose 2) minus 1. Because it is up to scaling, which gives an extra dimension. Four points on a circle  if two chords cross, they can be untangled in two ways. Cluster algebra and "mutations" by which one product is the sum of two products. Ptolemy theorem. String theory searches for a sixdimensional Calabi manifold. Could that be related to the sixsome? Could a person be a point whose internal structure is the sixsome which functions in a fourdimensional external world of spacetime? Taken together can they describe 6+4=10? External space describes four scopes and internal structure describe six relationships. Could twistors relate the two branches of {$O(\infty)$}? And how could the (lefthanded or righthanded) chirality of spinors relate to those two branches? How can there be joint intentionality for a culture of individual unfolding. Robin Hartshorne. Algebraic Geometry. Idea: The threesome is what links together the two worlds of O(infinty). The threesome equates a shift in one world with a node in another world and vice versa. And this creates a circle  the threecycle  which moves in one direction  distinguishing what is unconscious and what is conscious and defining a hole for Z. Top down reamplification  higher modules (conscious awareness) ask questions to senses (unconscious) in response to stimulus from senses.
Our universe (of structure) and the mirror universe (of activity) are chiral opposites and are present together in all of life as with Christopher Alexander's patterns  structure channels activity and activity evokes structure. "The Kingdom of God is within you". Material and spiritual intertwined. Is padic geometry related to fractions and thus the number of bits of information needed to express that? Is simultaneous CPT transformation the same as switching us from rotations to reflected rotations? Are there two parts of {$O(\infty)$}? And is this like the unconscious and the conscious, where the conscious is a reflected version of the unconscious? Since the classical Lie families present the symmetry in counting forwards and backwards, could the exception Lie groups and algebras present the symmetry in counting on an eightcycle, using and not using a certain number of tracks? Could there be four geometries based on the four normed division algebras? Real, complex, quaternion, octonion? and their projective planes? But the symplectic geometry seems based on quaternions. What would octonion geometry mean? Lie groups and algebras capture the symmetries in math by capturing how basic structures can operate in both algebra and analysis simultaneously. Algebra expresses what is, and analysis (as with homology) expresses what is not (as with holes). https://en.wikipedia.org/wiki/Relational_quantum_mechanics Kevin Michell, "Free agents"  how agency evolved from single cell animals Consider combinatorial interpretations of the Gaussian binomial coefficients
Combinatorial QFT on Graphs
The conjugate of a quaternion flips the sign of all three dimensions. Is this a manifestation of a parity transformation? Anticommutativity makes cross terms cancel. Nilpotents make squares cancel. What about commutativity?
Peirce's pragmatic maxim
How does conversation of energy, momentum, angular momentum require and ground continuity? Where exactly is that enforced in quantum mechanics? Study Noether's theorem. Conceptual Mathematics Posina Venkata Rayudu about William Lawvere. https://ncatlab.org/nlab/show/Functorial+Semantics+of+Algebraic+Theories John Isbell. General Functorial Semantics. Functorial Semantics of Algebraic Theories, William Lawvere The Math of Consciousness: Q&A with Kobi Kremnitzer BarrattPriddyQuillen theorem. The group completion of the monoid of finite sets {$Fin^{gp}$} is the stable homotopy group of spheres. Complex numbers are more natural than real numbers or quaternions because complex numbers have simpler nondegenerate quadratic forms: {$Q(u)=u_1^2+u_2^2+\cdots +u_n^2$}. For we can insert a scalar {$i$} and that converts any minus sign into a plus sign.
Baez, Moeller, Trimble. Schur functors and categorified plethysm.
Posina Venkata Rayudu
nLab: Sphere spectrum The sphere spectrum is the suspension spectrum of the point. The homotopy groups of the sphere spectrum are the stable homotopy groups of spheres. The sphere spectrum is the higher version of the ring Z of integers. See also: nLab: Suspension TaiDanae Bradley. The Tensor Product, Demystified Laws of form
Matematikos žinojimo rūmų sparnus sieja
Apsikeičia  ar tai padalinimų ratas? Kiekviename sparne požiūriai prisideda +0, +1, +2, +3. Auxiliary loops in spacetime are compatible with the rays in spacetime, perhaps in this way the mind is compatible with the brain. That might be relevant for the hierarchy of agency. John Bolender. The SelfOrganizing Social Mind. Bott periodicity
Symmetry Yoneda Lemma Geometry
Wave function arises when two systems interact. As given by orthogonal Sheffer polynomials. In Lie group for rotations, SO(3), the bracket of [x,y] gives you z. In choice frameworks, such as the simplex, the center is the basis for geometry and the vertices are the basis for matter. Together can they ground general relativity? I dreamed of the grouping of examples from branches of mathematics by considering whether they involve, for example, aspects of mathematics, logic, semantics, and so on. Minkowski space. Time {$t^2$} has us step out (thus reversing direction), space {$x^2+y^2+z^2$} has us step in. Jane Loevinger's psychometrics = Maslow's hierarchy of needs. E8 is "worry about the needs of another". E9 is "be perfect". Equality is inherently contradictory. John Baez. Getting to the Bottom of Noether's Theorem. Ivan. Was or was not regional politics helpful for potential democratic transition in Russia? Novosibirsk, Tatarstan. Partial knowledge
Consciousness
Counterquestions are a foundation for learnability. Each counterquestion defines a domain of new knowledge where we had no facts that we could rely on. Mind and Life Institute. Varela. Supporting contemplative research.
Nathaniel Virgo
Supermaps  holes  contexts. Pragmatic approach. Context defines meaning. Robert Brandom. Making It Explicit. {$\textrm{Set}^{op}$} atomic boolean algebras. Map back into Set. Map back out PowerSet. Simplexes observe coordinate systems. df/dx = f. Can be expressed through the notion of infinity (Taylor series) {$e^x$}. Or through periodicity (trigonometry, Euler's equation) {$e^{ix}$} Jim in Oneonta. Adapt, improvise, overcome. Sheaf Representation of Monoidal Categories
Eri
Conformal and analytic is the same. Energy can be defined as the "separation constant" in Schroedinger's equation. If we can separate the wave function into a time dependent function and a position dependent function, then we can segregate the two sides of the equation so that one side depends on time, the other side depends on position, and both sides are constant, and that constant is the energy. Peter Scholze  condensed math
Absolutism based on relativism is good. Relativism based on absolutism is bad. Objectivity based on subjectivity is good. Subjectivity based on objectivity is bad. The mind that does not know is based on the mind that knows and not the other way around. This is the rule of consciousness and the basis for morality. There is a gap between the quantum ether (the quantum foam) and the waves that propagate through it. Particles don't exist, particles are the medium. Waves exist in the medium. Folk psychology. Daniel Dennett suggested studying this. Contact him.
In the book on interpretations of quantum mechanics, there is the question of what is real. For example, in electromagnetism, the gauge can be adjusted by adding any gradient. This can change whether the change is transmitted by the speed of light or whether that speed is infinite and it happens instantaneously. But these two scenarios also raise the question of the reality of the "wiggle". What is real is a moral choice. Is the medium real? Or is the wiggling real? The wiggling creates the wave that moves across the medium. The medium is made of particles and antiparticles that appear from the foam.
https://researchsemin The manyworlds interpretation of quantum mechanics and the Born rule Lev Vaidman (Tel Aviv University) ( view ) Mon May 22, 19:0020:30 (7 days from now) Abstract: I will argue that the manyworlds interpretation is the best interpretation of quantum mechanics and discuss the status of the probability assignments in this deterministic theory.ars.org/seminar/AlgebraParticlesFoundations Applying a boundary map twice gives zero. Applying it twice removes two vertices, and this can be done in different orders, yielding different signs, canceling out. For the Snake Lemma, add a zero vector space before the first kernel and add a zero vector space after the last cokernel. Then we have the eightfold way with seven mappings. Can large language models work by simply transforming existing input  taken to be grammatical  to preserve grammaticality. Choice Frameworks
Introduction to Commutative Algebra Atiyah & Macdonald. Rings, ideals, modules, dimensions Looking at an ellipse in various ways yields all of the conic shapes. We can get a breaking at infinity. Thus this is a way to ground infinity. What about looking at a circle? We look through the point of the cone, which is where our eye is. Mobius transformations
David Corfield's video. Colin McLarty: semiotics as the language of biology  logic in a biological key  trying to categorify this? Tristan Needham. Visual Complex Analysis. John Stillwell. Mathematics and its History. 1989. Moebius transformations
Emotional sphere
Freedom House Report
Differentiation Alex Codes. Symbolic Differentation in Python from Scratch! Algebra
A sum of particle clocks is like a prism operator (in the proof for singular homology that homotopic maps induce the same homomorphism for the homology groups) but without the minus signs. Wisdom Wisdom distinguishes everything and slack, what is whole and what is free, holisticity and laxity. https://researchseminars.org/seminar/AlgebraParticlesFoundations https://www.johnmyleswhite.com/notebook/2013/03/22/modesmediansandmeansanunifyingperspective/ GradCurlDiv
Exact sequence: Grad, Curl, Div
Group theory Nathan Carter "Visual Group Theory" https://www.quantamagazine.org/triangulationconjecturedisproved20150113/ https://jeremykun.com/2013/04/10/computinghomology/ Purcell. Electricity and Magnetism
90 degrees + 90 degrees can equal anything. But specifically can go from the diameter of a cube (standing on its vertex) to the vertex and back on the diameter to any point. But the same is true for 120 + 120. Quaternions act like a gauge  3 dimensions are unspecified  but identified with the complex i. https://math.stackexchange.com/questions/711492/provethatthemanifoldsonisconnected The Hilbert space that models the spin state of a system with spin 𝑠 is a 2𝑠+1 dimensional Hilbert space. And spin can be halfintegered. Think of the Hilbert space as everything divided into 2s+1 perspectives. Simplicial sets Greg Friedman. An elementary illustrated introduction to simplicial sets. Amplituhedron Physics 283b: Spacetime and Quantum Mechanics, Total Positivity & Motives Causality
Bell's inequalities https://www.researchgate.net/publication/ Causal Set Theory Universal concepts such as universal confounders the confounder. Ambiguity is described by equations.
(1) formulate a hypothesis, (2) deduce a testable consequence of the hypothesis, (3) perform an experiment and collect evidence, and (4) update your belief in the hypothesis. Modeling experience from old self to new self  Bayesian analysis  prior belief + new evidence = revised belief Jacobi polynomials have a notion of combinatorial space that may be relevant for thermodynamics as it relateswo disjoint sets malping into their union. https://gilkalai.wordpress.com/2008/12/23/sevenproblemsaroundtverbergstheorem/ Pearl seeing vs. Doing Homology sets up potential equivalences  they may be actual equivalences, which yield identities  or nonequivalences which are generators. Rotation accords with orientation (of a simplex) accords with an imaginary number i or j. Orientation is related to permutation as with the linearization for orthogonal polynomials. Causality Judea Pearl: The Fundamental Equation of Counterfactuals {$Y_X(u)=Y_{M_X}(u)$}. Relate to independent trials  throwing away a sheet of paper (a module). Judea Pearl, Dana Mackenzie. The Book of Why: The New Science of Cause and Effect Choice
Stephen Wolfram. Metamathematics Foundations & Physicalization Ravi Vakil: Main theme of mathematics  convert harder problems to linear algebra Ergodic theorem
Chomsky: Successor function derives from the merge function applied unitarily to a single object. Parsing hierarchy Speculation: The difference in the measurements of the Hubble constant may relate to the history of the universe. Early in the universe the heavier particle families (in the parsing hierarchy) may have been predominant. And they may be the source of the megastructures of the universe. Field with one element Thomas noted the symmetry of {$x^0=1$}. Relate this to {$F_1$}, choosing one out of one, or none out of none. Schroedinger's cat
What Are The Hidden Rules Of The Universe? http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf Fulton Curve Book Multiplying by quaternion j reverses angular momentum for electron. Is spin a clock? Like a particle clock?
Course: Nonlinear Dynamics. Geometry of Chaos. Statistics
trivial tangent bundles on spheres? Hamiltonian is the sum of all the projections onto the energy eigenstates with the energies being the weights. Study of variables
KervaireMilnor formula
In the unfolding of math
https://neo4j.com Neo4J graph database management Johan Commelin: "Breaking the onemindbarrier in mathematics using formal verification"
Representation theory Generalized Linear Models Lattice 8 is special because {$\sqrt{8/4}=\sqrt{2}$} is the distance between neighbors but also the interspersed lattice in constructing the E8 lattice. 240 is the kissing number.
Statistics Shoelace formula for oriented area of a polygon Dobinski's formula relates Bell numbers and e. {$B_n=\frac{1}{e}\sum_{k=0}^{n=\infty}\frac{k^n}{k!}$} Quaternionically differentiable is linear. 18.4 penrose. Hyperbolic length is one half of the rapidity it represents. Localization arises from local shielding by local interactions. That is what weakens global interactions which othwrwise exist. Complexity theory https://en.wikipedia.org/wiki/PCP_theorem Every decision problem in the NP complexity class has probabilistically checkable proofs Discourse https://en.wikipedia.org/wiki/Overton_window Window of political discourse: Unthinkable  Radical  Acceptable  Sensible  Popular  Policy Lambda Calculus Study choice, probability, statistics.
Functions Richard Southwell describes how mathematical functions can be visualized by: (1) elements and arrows (2) Wiring diagrams (3) fibres (4) bouquets (5) graphs (6) ontology logs (7) categories Dirac's plate trick Plate trick Summer of Math Exposition 2022 Results Coxeter. Regular polytopes. Includes prehistory. Boole. Coxeter diagram {$D_n$} symmetry group of demicube: every other vertex of a hypercube. Is that related to a coordinate space? Combinatorially, can we flip the vectors of the demicube to get a coordinate system? Cube reflections given by vectors u, v, w from the center of the cube to the center of a face, the center of an edge, and the center of another edge. And the angles between the vectors are pi/2, pi/3 and pi/4. And the two edge midpoints are separated by pi/3 so rotating through six such edges gets you back. And that is the chain for the Dynkin diagram. Conjugation is an example of reflection. Finite field with one element
https://en.wikipedia.org/wiki/Theory_U Locality is the whole achievement of the continuum. Local means low overhead and the actual global time frame is even lower overhead. Locality arises with orthogonality, assumes measurement, observers, space time wrapper. Differentiation changes level. {$x^n$} number of levels of volatility, number of derivatives Spaces of states
{$\begin{pmatrix} a & b+ic \\ bic & d \end{pmatrix}$} Think of probabilities {$a, 1a$} and mediator {$b \pm ic$}. We have {$a^2+b^2+c^2\leq a$} and {$a^2+b^2+c^2 = a$} for pure states. Rotate {$aa^2$} from 0 to 1 around the aaxis. It is the MacMahon Master Theorem that unifies the angular momentum properties of composite systems in the binary buildup of such systems from more elementary constituents. Summer of Math Expostion 2022 playlist Semilocally path connected avoids Zeno's paradox. Universal covering as naming schemes. Robert Gilmore. Group Theory. XIV. Group Theory and Special Functions. Relates Lie groups and orthogonal polynomials. Local  reversible, global (default) not reversible ("Not every cause has had its effects") Modeling How are games in game theory (with incomplete information, partial information) characterized by probability distributions. Antonio Jesus
In Cartesian categories you can copy and delete information. (John Baez  Rosetta Stone) How does that relate to Turing machine? Alytaus kredito unija
Bose statistics  can't assign labels. Fermi statistics  can assign labels to particles. Information capacity is zero if probability is the same for all cases but also if one case is given 100%. Information transmission requires asymmetry. Otherwise you cannot define choice. Probability
Creating what you can feel certain about. (Continuity.)
A few of the best math explainers from this summer Stone's theorem: continuous implies differentiable "belt trick", aka the "Dirac scissors" or "Balinese candle dance When two events happen (the measurement of spins) there is a frame where one happens before the other. So if they are causally connected (as with spin measurements) there needs to be a distinguished frame. But that could be the frame in which they were initially entangled. So entanglement posits the existence of such a distinguished frame. Path integrals depend on the number of points in space, or the number of interactions. But my approach suggests that this number is actually given by the degree of x in the relevant polynomial. Quantum computing
Amelia: [The axiom of function extensionality is] inconsistent with many axioms of a more "computational" nature. For example, "formal Church's thesis" says that for any function N→N, there is a "program" (we call it a realizer) that realizes it. You can kinda see what goes wrong: this would be able to tell e.g. "λ x → x" and "λ x → x + 0" apart. You could imagine an assignment of realizers that sidesteps this, though, so to see that it's actually inconsistent takes slightly more work. Have all finite limits is equivalent to
These are the building blocks for limits
https://ww3.math.ucla.edu/dls/emilyriehl/ video about contractibility An isomorphism is a special morphism but truly it is a pair of morphisms that are inverses to each other. There may be many such pairs relating two objects but in each pair the inverses are unique with respect to each other. So it is similar to complex conjugation. https://math.stackexchange.com/questions/989083/iscompositionofcoveringmapscoveringmap Įvairūs
Source of contradiction
https://en.wikipedia.org/wiki/%CE%9C_operator {$\mu$}operator A qubit specifies the relation between affirmation and negation of probabilities. In matrix form, it provides a complex number which is the coefficient that gets multiplied to the negation (in calculating the new affirmation) and whose conjugate gets multipled to the affirmation (in calculating the new negation). In classical bits, this coefficient is simply zero. Bekaert, Boulanger. The unitary representations of the Poincare group in any spacetime dimension Kojin Karatani, Sabu Kohso  Architecture as Metaphor_ Language, Number, Money (1995) semijoin lattice semilattice Simplex Think of 1cell as the center (of all things), the spirit. And think of 0cell not simply as a point but as a 0dimensional open arc (the point shell) with regard to that center (the spirit). The point shells are glued onto the spirit, and similarly, open arcs are glued onto point shells, and so on, inductively. https://ncatlab.org/nlab/show/cellular+approximation+theorem#applications https://en.wikipedia.org/wiki/Koopman%E2%80%93von_Neumann_classical_mechanics https://www.amazon.com/QuantumChallengeFoundationsMechanicsAstronomy/dp/076372470X In what sense are Feynman diagrams relativistic given that they have directions for time and for space? Instead of thinking of speed of light, think of a clock that doesn't tick, so that t=0 always. And this is the case for the quantum harmonic osciallator and for the particleclocks with no steps. One {$\exists x$}, all {$\forall x$}, many {$\neg\exists x \wedge \neg\forall x$}. Masaki Kashiwara, Pierre Schapira. Categories and Sheaves. 2006 Bohm Pilot Wave, Thomas Spencer Relative invariance  more global than another Relate the threecycle (taking a stand, following through, reflecting) to Kan extension as defined by filtering objects and morphisms, acting on them, and then taking the colimit or limit on that filter. Alan Turing, Cybernetics and the Secrets of Life Gregory Chaitin = Shannon + Turing = CompressionDecompression as understanding. Life in lifeInformation is what you learn. What you learn grows at the boundary, has the shape of the boundary. A shape can be thought of as being created by integrating over these boundaries as they increase. TaiDanae Bradley: Information is on the Boundary
Prove that the matrix made up of eigenvectors diagonalizes a matrix. Potential energy source {$\frac{y}{2}$} (adding free cell) is in balance with kinetic energy source {$i\frac{\partial}{\partial y}$} (halflink)
