Andrius Kulikauskas

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Introduction E9F5FC

Questions FFFFC0


Andrius Kulikauskas: Welcome! This is where I record my latest research notes.

The (nonassociative) octonions are perhaps related to the (associative) split-biquaternions, 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 split-biquaternions. 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 4-dimensional "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.

  • +1 Add one operator. Impose linear complex structure. I. Unconscious.
  • +2 Add two mutually anticommuting operators. Impose quaternionic structure. You. Conscious.
  • +3 Add three mutually anticommuting operators. Impose split-biquaternionic structure. Other. Consciousness.

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. Noise-free 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.

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

  • Quantum mechanical system is completely specified by the wave function, and integrating the square of the wave function yields the probability of being in the delimited state.
  • To every observable in classical mechanics there corresponds a linear Hermitian operator in quantum mechanics
  • The only values ever observed are the eigenvalues of the operator
  • The formula for the average value of an observable
  • The wavefunction or state function of a system evolves in time according to the time-dependent Schroedinger equation
  • The total wavefunction must be antisymmetric with respect to the interchange of all coordinates of one

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 anti-commute 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. 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.

  • It is difficult to write up mathematics when we are solving both forwards and backwards.
  • In analysis, partial, approximate solutions are meaningful when we know what kind of answer we are seeking and we are solving backwards from there.

The Physics of Evolution

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.

Think of matter anti-matter cancellation, creating energy, as a driver for culling? Yielding an imbalance? as anti-matter moves away? by randomness? reply regarding adjunctions

{$\mathbb{R}\oplus\mathbb{R}$} is not a division algebra but is a super division algebra.

Symmetric functions

  • Elementary symmetric functions express fermions (can't repeat indices), homogeneous symmetric functions express boson (can repeat indices). But what happens when we multiply together elementary symmetric functions {$e_i$} and {$e_j$}?
  • SU(1) electromagnetism - twosome - particle-hole C
  • SU(2) weak - threesome - chiral - P
  • SU(3) strong - foursome - time reversal - T
  • Reproduction - dialogue between man and woman, conscious and unconscious - dialogue for raising children


  • What does it mean if operators are anti-commuting?

Understanding, consciousness

  • Penrose on Godel: Our understanding of the rules transcends the following of the rules, as in the case of a rule "this rule is false".
  • Daniel Friedman : What is the relationship between recieved semantics (semiosis), and Consciousness? Would love to explore W Kandinsky, Point and Line to Plane, in terms of Synergetics & "geometry of lumps", to say nothing of "Concerning the Spiritual in Art"
  • Degrees of consciousness relate to attention, as with the representation of the nullsome in terms of directness.


  • Matrix decomposition Compare matrix decompositions with divisions of everything and with the six conceptions of them.
  • Derive the properties of a cube 8, 6, 12 as per Euler's rule from the fractions 24/2, 24/3, 24/4. Compare with chemistry, as with the Lewis diagrams.

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 Zermelo-Frankel axioms of set theory in the language of category theory.




Three Minds

  • {$3\times 8$} theory of {$3$} minds and {$8$} mental states
  • Janet Pauketat
  • Megan M. Callahan, Terre Satterfield, Jiaying Zhao. Into the Animal Mind: Perceptions of Emotive and Cognitive Traits in Animals
  • Kara Weismana, Carol S. Dweck, Ellen M. Markman.Rethinking people’s conceptions of mental life
  • Hideyuki Takahashi, Midori Ban, Minoru Asada. Semantic Differential Scale Method Can Reveal Multi-Dimensional Aspects of Mind Perception.
  • Megan N. Kozak, Abigail A. Marsh, Daniel M. Wegner. What Do I Think You’re Doing? Action Identification and Mind Attribution
  • Bertram F. Malle. How Many Dimensions of Mind Perception Really Are There?
  • Kallie Tzelios, Lisa A. Williams, John Omerod, Eliza Bliss‑Moreau. Evidence of the unidimensional structure of mind perception
  • Consciousness +3 is self-reflection (which switches the direction of the twosome's mental shift). Is it then possible that the fractional charge of quarks indicates the Unconscious (1/3) and the Conscious (2/3) and so we have Consciousness (1) for electrons and protons?

The disembodying mind results from evolutionary pressure to devote more resources to modeling the unknown.

  • A mind that knows answers represents almost 100 billion neurons
  • A mind that does not know but asks questions represents perhaps 100 thousand concepts
  • A third mind that is just 8-fold (thus merely 3-bits) balances the two.

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

  • Sheffer: Two particles have their own independent clocks but when they interact it synchronizes their clocks. Then when the synchronization collapses, one indicates the forward time and the other the backwards time so they are conjugates.
  • How does the combinatorics of orthogonality manifest itself? Orthogonality converts compartments into causal trees in the case of orthogonal Sheffer polynomials. Can we make combinatorial sense of the impositions of this orthogonality constraint? As with Favard's theorem or Meixner's classification?
  • Consider the generating function for (associated) Legendre polynomials and compare it with the generating function for orthogonal Sheffer polynomials.
  • Consider how forgetful functors take us through the five zones for the moments of orthogonal Sheffer polynomials. We start with ordered set partitions. We forget the order and get set partitions. They have an intrinsic order, starting with the largest (and on the inside starting with the smallest). So you should get a permutations. Every permutation is a pair of involutions. If you forget the one involution, then you should still have another involution. And what would you forget to get the alternating permutations?


  • Consider the octo-cycle form of sulfur (the eight-cycle) as a template for the arisal of life.
  • Do the complements of the divisions of everything express the reverse Kreb cycle?
  • Biochemijos paskaitos
  • A cell has a boundary. If it is a single-celled organism, then its boundary can be thin. If it is multi-cellular, then there is no way to push out the protons, so that has to take place inside, into mitochondria, yielding inner boundaries. And this grounds self-awareness, bringing the outside inside of us.

Random matrices

  • Random matrix ensembles are based on orthogonal matrices (of various kinds) because they are symmetric matrices (in the relevant sense) whose individual entries can be changed arbitrarily without affecting that condition, which they continue to satisfy. That is perhaps the whole point. Orthogonality grounds the symmetry, which respects itself upon multiplication, which can thus support arbitrary changes. Thus this is the condition for compatibility with arbitrary change in a particular entry. How does this relate to divisions of everything?
  • How can random matrices model biological processes such as molecular collisions?
  • Philip Cohen, Fabio Deelan Cunden, Neil O’Connell. Moments of discrete orthogonal polynomial ensembles. Related to random matrix theory.


  • How is randomness related to the Riemann Hypothesis?
  • Forms of matter express geometry as uniformity and give rise to mass behavior even randomness.
  • Entropy - physics is related to symmetry - Shannon entropy is related to information. And how does that relate to randomness?
  • Symmetry breaking - choosing one possibility. From symmetry breaking randomness appears and information is constructed. Deterministic is replaced by irreversibility.
  • Randomness as derived from a wall that allows for independent events, as with the other, or with transcendence.
  • Randomness as lack of knowledge.
  • Note that the weak force is a link between the two universes. It grounds radiation which is a source of randomness and is essential for evolution. We can consider how an adjunctions with a randomness functor could relate the two universes. Randomness is based on ignorance. A mirror universe establishes a basic level of ignorance.
  • Choice in an eight-track mind is modeled by the random matrix ensembles.

Linear regression

  • In multiple regression the constant {$b_0$} acts like free space, the initial compartment. The random variables are like compartments.


  • Entropy can be understood independently of a field or with regard to a field. Without a field, everything being in one place is highly deliberate. But with a gravitational field, it is highly non deliberate, whereas having everything spread out uniformly would be highly deliberate. Where is this discussed? Is this a fundamental ambiguity? What is the significance of this ambiguity? What does Penrose say about this?
  • Why does deletion of information increase entropy as heat? How is this related to the Turing machine rule {$X\rightarrow\epsilon$}?
  • Maximum entropy. Distinguish what is known from what is not known and model the latter using nondeliberateness, thus maximum entropy. What is known is understood as deliberate.
  • Entropy is the expected value of surprise, that is, the expected surprise.

Entropy = nondeliberateness. How is this related to unconscious (nondeliberate) and conscious (deliberate)?

  • Conscious = nondeliberate nondeliberateness. Not nondeliberate.
  • Consciousness = nondeliberate nondeliberate nondeliberateness = deliberate. Not not nondeliberate.
  • Nondeliberate = not not not nondeliberate.

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

  • How to not just make knowledge available but inspire people to run with it.

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

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

  • (1) thermodynamic autonomy - isolated from disruptive environment, (2) persistence - the origin of purpose, (3) endogenous activity, (4) holistic integration, (5) low‐level indeterminacy, (6) multiple realisability, (7) historicity, (8) agent‐level normativity
  • A scientific experiment eliminates noise which simplifies the analysis of causality but is unreal. Noise, as in John's dynamics, is normal.
  • Causation is interpretative - looking for difference makers - neural populations do that. There can be multiple responsible parties but many of them are assumed and unexamined.

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 Robinson-Schensted 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 zig-zag pattern. This is based on the radial model of the hydrogen atom.

  • Relate these numbers to the Laguerre polynomials and express them as causal trees.

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 first-step, external, weighted, meaningful in the broader environment. 2 are later step, dependent on {$n$} or {$n(n-1)$}, 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 one-step causality. Oxygen reacts with two electrons in a kink, two-step 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:

  • {\displaystyle M_{K_{m,n}}(x)=n!L_{n}^(m-n)(x^{2}).\,}{\displaystyle M_{K_{m,n}}(x)=n!L_{n}^(m-n)(x^{2}).\,}

If G is the complete graph Kn, then MG(x) is an Hermite polynomial:

  • {\displaystyle M_{K_{n}}(x)=H_{n}(x),\,}{\displaystyle M_{K_{n}}(x)=H_{n}(x),\,}

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

  • four structural chemical elements, hydrogen (H), carbon (C), Nitrogen (N), and oxygen (O) - note the latter have valences 4, 3, 2.
  • two more functional elements of phosphorus (P), and sulfur (S) - these have valences 3 and 2 respectively

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

Meixner-Pollaczek 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. 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 all-encompassing, 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:

  • {$[x,[y,z]]-[[x,y],z]=[[z,x],y]$}
  • {$a(bc)-(ab)c=(ca)b$}

An ideal of an algebra models self-awareness. Cohl Furey: A black hole likewise models self-awareness. 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.

  • Consciousness as collective sense making. How senses are in unity, harmony, in alignment. Systematic accounting. Regularity, not necessarily predictabiility.

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?

  • How does {$e^x$} for groupoids {$X$} relate to the connection between Lie algebras and Lie groups?

Understand the analytic symmetry in this expression for a weight function: {$\omega(x)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\sum_{n=0}^{\infty}\omega(x)\sum_{k=0}^{\infty}\frac{(-1)^n(2\pi i \xi x)^{n+k}}{n!k!}dx d\xi$}.

  • Note that {$\sum_{k=0}^{n}\frac{1}{k!(n-k)!}=\frac{1}{n!}\sum_{k=0}^n\binom{n}{k}=\frac{2^n}{n!}$} and {$\sum_{k=0}^n(-1)^k\binom{n}{k}=(1-1)^n=0$}.
  • Why doesn't the formula equal {$0$}?
  • What happens if we differentiate the formula with regard to {$x$} or {$\xi$}?

A new essay concerning the origin of ideas by Antonio Rosmini. A new essay concerning the origin of ideas by Antonio Rosmini.

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 three-cycle? 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)

O coordinate systems1 coordinate system2 coordinate systems3 coordinate systems
fixed dimensiondistanceangleoriented area
proof by contradictionmodelimplication - working backwardsvariables

Walks on binary trees: The Bruhat-Tits tree for the 2-adic 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 up-to-date 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 n-Categories

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: {$\alpha=e_1-e_2$} and {$\beta=(e_2-e_1)+(e_2-e_3)$}. 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.

What can we say about the symplectic form {$x_1y_2-x_2y_1$}? 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 {$x_i-x_j$} and {$x_j-x_i$} where {$i<j$}. Consider how this works for each Lie group, unitary, symplectic, odd and even orthogonal groups.


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 six-dimensional 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 four-dimensional external world of space-time? 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 (left-handed or right-handed) 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 three-cycle - 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.

  • Baars. A cognitive theory of consciousness. 1989.

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 p-adic 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 eight-cycle, 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).

Kevin Michell, "Free agents" - how agency evolved from single cell animals

Consider combinatorial interpretations of the Gaussian binomial coefficients

Combinatorial QFT on Graphs

  • Get adjoint to coboundary operator. Combining them gives the graph Laplacian.

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

  • "Consider what effects that might conceivably have practical bearings you conceive the objects of your conception to have. Then, your conception of those effects is the whole of your conception of the object."
  • The Wikipedia article gives seven related versions.
  • Expresses the Yoneda Lemma.
  • Expresses how an algebraic object may be best understood in terms of its representations.
  • Relates to significance, the conception of the nullsome, what is unencompassable. This is the level Why.
  • Note that I flip pragmatism on its head. In my philosophy, it circumscribes Whether, that is, what is to be treated as real.

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.

John Isbell. General Functorial Semantics.

Functorial Semantics of Algebraic Theories, William Lawvere

Global Workspace Theory

The Math of Consciousness: Q&A with Kobi Kremnitzer

Barratt-Priddy-Quillen 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

Tai-Danae Bradley. The Tensor Product, Demystified

Laws of form

  • Concatenation is "saying". Saying multiple times is the same as saying once. {$a^2=a$}.
  • Cross is negating. It is crossing. Negating twice is identity. {$b^2=1$}.
  • We can have a different kind of negating or a different kind of negated for which negating twice is reversal. {$c^2=-1$}

Matematikos žinojimo rūmų sparnus sieja

  • algebra veda iš centro į sąrašą
  • analizė vedą iš sąrašo (indukcijos) į centrą (limitą)

Apsikeičia - ar tai padalinimų ratas? Kiekviename sparne požiūriai prisideda +0, +1, +2, +3.

Auxiliary loops in space-time are compatible with the rays in space-time, perhaps in this way the mind is compatible with the brain. That might be relevant for the hierarchy of agency.

John Bolender. The Self-Organizing Social Mind.

Bott periodicity


Yoneda Lemma


  • Geometry arises on a vector space when we place a quadratic form on it, {$Q:V\rightarrow K$}. It gives a notion of length of a vector because {$Q(av)=a^2Q(v)$}, and also a notion of distance because there is a symmetric bilinear form {$B(v,w)=\frac{1}{2}(Q(v+w)-Q(v)-Q(w))$}.
  • Are there niine Cayley-Klein geometries?

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

  • forgetful functors
  • fibrations, lenses


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

  • Modeling agency
  • Improving a model of the environment = Bayesian prior.
  • Controller gets new information and also the system changes over time.
  • Don't care about the previous states, just the prior (t) and the current issue (t+1)
  • Kalman filter: when the prior is a Gaussian then the posterior is also a Gaussian
  • So only the means and variances need to be stored
  • Bart Jacobs 2020: A channle based perspective on conjugate priors - this pops out of an adjunction
  • Unifilar generator. A generator is unifilar if it is deterministic given output. They form a separate category.
  • Forgetful functor: Unifilar Generator to Generator. In BorelStoch this has a right adjoint.
  • Usually forgetful functor has a left (free) adjoint. A right adjoint of a forgetful functor is cofree. In this case the forgetful functor is forgetting both a fact about the objects and the morphisms.
  • How does this relate to lenses?
  • Strongly representable Markov categories are cool.
  • Epistemic model and dynamical model. You want your model of the system to be unified.

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

  • Monoidal Categories MonCat. Presheaf F: L->MonCat and Sheafs. Generalizing Stone's Theorem.

  • Lax functor - may be relevant for allowing perspectives to be not associative yet related.
  • Spivak and Kent: Ologs
  • The volume of a unit sphere in n-dimensions goes up for small n, reaches a maximum at n=5, then goes down.

Posina Venkata Rayudu


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

Visual frameworks

  • Visual frameworks connects visualization with semantic contexts such as gravity. Note that gravity is based on a quadratic power law and yields the fivefold conics.

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 anti-particles that appear from the foam.


  • Explain why we get alternating signs for the boundary

https://researchsemin The many-worlds interpretation of quantum mechanics and the Born rule Lev Vaidman (Tel Aviv University) ( view ) Mon May 22, 19:00-20:30 (7 days from now) Abstract: I will argue that the many-worlds interpretation is the best interpretation of quantum mechanics and discuss the status of the probability assignments in this deterministic

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

  • A category with a zero object has semantic symmetry with regard to choice frameworks. A category without a zero object, but with initial objects and terminal objects which differ, such as the categories Set or Cat, have syntactic asymmetry.

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

  • How are perspectives transformed?
  • How are triangles mapped to triangles ?

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


Alex Codes. Symbolic Differentation in Python from Scratch!


  • Abstract algebra. Dummit and Foote.

  • Meditations by Marcus Aurelius
  • Meditations by Renee Descartes
  • On Liberty by John Stuart Mill
  • Octavia Butler - Parable of the Sower, Parable of the Talents
  • Cixin Liu - Trijų kūnų problema (Kitos knygos)
  • Ursula Le Guin - The Left Hand of Darkness, Dispossessed

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 distinguishes everything and slack, what is whole and what is free, holisticity and laxity.

How to fold circles

Theo Buhler. Exact categories


Exact sequence: Grad, Curl, Div

Group theory

Nathan Carter "Visual Group Theory"

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.

Paul Lockhart's Measurement

Quaternions act like a gauge - 3 dimensions are unspecified - but identified with the complex i.

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 half-integered. Think of the Hilbert space as everything divided into 2s+1 perspectives.

Simplicial sets

Greg Friedman. An elementary illustrated introduction to simplicial sets.


Physics 283b: Spacetime and Quantum Mechanics, Total Positivity & Motives

Spirtes, Glymour and Scheines. Causation, Prediction and Search. (Adaptive Computation and Machine Learning).


Bell's inequalities

Causal Set Theory

Universal concepts such as universal confounders the confounder.

Ambiguity is described by equations.

  • Atmosphere has mass of 5.15×10^{18} kg
  • Person breathes 10 tons of oxygen = 10,000,000 grams of oxygen in their lifetime
  • 16 grams of oxygen has 6*10^23 atoms of oxygen
  • 1 gram of oxygen has 0,375*10^23 atoms of oxygen

Paul Humphreys. The chances of explanation : causal explanation in the social, medical, and physical sciences.

(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.

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.


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


  • Is Monty Hall problem related to quantum probability?

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

  • Do the probabilities of a superposition evolve as per the phasor?
  • Do they run through all possibilities between wavelengths?
  • And does that mean that the decision of whether Schroedinger's cat is awake or asleep depends on the exact moment that we make the measurement?
  • And what does that say about time? and superpositions?
  • Is this a valid interpretation of superposition?

What Are The Hidden Rules Of The Universe? 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.


  • Foundation of statistics is models. Distinguish the signal and the noise. Ignore the noise.

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

Kervaire-Milnor formula

  • {$\Theta = \Pi B$} where {$B=a_m2^{2m-2}(2^{2m-1}-1)B_{2m}/4m$}

In the unfolding of math

  • consider math as given by generators and relations
  • the relations are equivalence classes Neo4J graph database management

Johan Commelin: "Breaking the one-mind-barrier in mathematics using formal verification"

  • Selcuk Bayin.

Representation theory

Generalized Linear Models


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.

  • {$128=dim(\mathbb{O}\otimes\mathbb{O}^2)$}


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 Every decision problem in the NP complexity class has probabilistically checkable proofs

Discourse Window of political discourse: Unthinkable - Radical - Acceptable - Sensible - Popular - Policy

Lambda Calculus

Study choice, probability, statistics.

  • Two reflections give you a rotation. So is a reflection the square root of a rotation? And does that relate to spinors?


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?

Grover's algorithm

Roe Goodman. Alice through Looking Glass after Looking Glass: The Mathematics of Mirrors and Kaleidoscopes

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

  • Choosing one out of one: Driving on a winding road, each turn is a choice of one out of one. Whereas a fork is a choice of one out of two, a usual intersection is a choice of one out of three and so on.

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

  • nLab: State
  • Classical bit: line segment [0,1]
  • Qubit: shaped like an American football

{$\begin{pmatrix} a & b+ic \\ b-ic & d \end{pmatrix}$}

Think of probabilities {$a, 1-a$} 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 {$a-a^2$} from 0 to 1 around the a-axis.

It is the MacMahon Master Theorem that unifies the angular momentum properties of composite systems in the binary build-up of such systems from more elementary constituents.

Summer of Math Expostion 2022 playlist

Learning classifier system

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")


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

  • Kapitalas 700,000 EUR, pelnas 43,000 EUR, paskolinta 4,600,000 EUR.

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.


Creating what you can feel certain about. (Continuity.)

  • Building up levels of certainty through topological invariants.

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

  • Having terminal objects
  • Having a product for any pair of objects
  • Having an equalizer for any pair of parallel arrows

These are the building blocks for limits 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.

Mariana M. Odashima, Beatriz G. Prado, E. Vernek. Pedagogical introduction to equilibrium Green's functions: condensed matter examples with numerical implementations.


Source of contradiction

  • We are finite, our system is finite, but the Spirit is infinite dimensional {$\mu$}-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) semi-join lattice semilattice


Think of -1-cell as the center (of all things), the spirit. And think of 0-cell not simply as a point but as a 0-dimensional 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.

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 particle-clocks 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 three-cycle (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 = Compression-Decompression as understanding.

Philosophy of computation

Life in life

Information 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.

Tai-Danae Bradley: Information is on the Boundary

  • Shannon entropy: the amount of surprise

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 {$i\frac{\partial}{\partial y}$} (half-link)

  • Santykis su Dievu yra atgarsis, kaip kad dalelytė turi santykį su savo lauku.
  • Fizikos dėsnių raida yra pavyzdys Dievo įsakymo patobulinimo.

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