Epistemology
Introduction E9F5FC Questions FFFFC0 Software 
Andrius Kulikauskas: Welcome! This is where I record my latest research notes. {$i$} and {$\bar{i}$} are additive inverses and multiplicative inverses. John Baez. Mathematics in the 21stCentury. Slides. Inner monologue 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? Complex numbers
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. 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. https://www.pmwiki.org/wiki/PmWiki/IncludeOtherPages
Linear regression
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? The wave equation  and waves in general  are expressions of analytic symmetry. 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. Tet methylcytosine dioxygenase 1 Tet1 gene 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. Octonions, standard model and unification. 2023. Understand the analytic symmetry in this expression for a weight function: {$\omega(x)=\int_{\infty}^{\infty}\int_{\infty}^{\infty}\sum_{n=0}^{\infty}\omega(x)\sum_{k=0}^{\infty}\frac{(1)^n(2\pi i \xi x)^{n+k}}{n!k!}dx d\xi$}.
https://www.amazon.com/GodsUndertakerHasScienceBuried/dp/0745953719 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). {$S_\infty$} = colimit of symmetric groups = permutations with finite support Frequency is the internal, qualitative expression of energy and intensity is the external, quantitative expression of energy. Is this potential energy and kinetic energy? Choice frameworks are logics (as with proof by contradiction  which is asymmetric) Samuel interests: Psychedelic rituals are delegitimized in the modern age. Conscious is not large enough to grasp the unconscious. 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. Relate the long root (2 to 1) in the simple root diagram of {$G_2$} with John Baez's spans of groupoids. {$G_2$} 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 {$\pm(2\beta + 3\alpha)$} https://en.wikipedia.org/wiki/Lincos_language Urs Schreiber. Differential cohomology in a cohesive ∞topos Modal Homotopy Type Theory relates Hegel, String Theory, Cohesive Infinity Topos. https://en.wikipedia.org/wiki/Alfred_University ShingTung Yau, Steve Nadis. The Shape of Inner Space: String Theory and the Geometry of the Universe's Hidden Dimensions. 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? 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. 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?
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 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 tmethat converts any minus sign into a plus sign. 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. 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. 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
Counterquestions are a foundation for learnability. Each counterquestion defines a domain of new knowledge where we had no facts that we could rely on. 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
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
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. John Stillwell. Mathematics and its History. 1989. 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 https://www.researchgate.net/publication/ 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. Homology sets up potential equivalences  they may be actual equivalences, which yield identities  or nonequivalences which are generators. 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. 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?
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"
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. https://en.wikipedia.org/wiki/Theory_U 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. 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. In Cartesian categories you can copy and delete information. (John Baez  Rosetta Stone) How does that relate to Turing machine? 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 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. 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 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. 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. One {$\exists x$}, all {$\forall x$}, many {$\neg\exists x \wedge \neg\forall x$}. Masaki Kashiwara, Pierre Schapira. Categories and Sheaves. 2006 Relative invariance  more global than another Prove that the matrix made up of eigenvectors diagonalizes a matrix.
