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Introduction E9F5FC
Questions FFFFC0
Software
Andrius Kulikauskas: Welcome! This is where I record my latest research notes.
Flag varieties and their relationship to field with one element - compare with chains of perspectives PPP and ZDZDZ
Jacob Fielder. Ergodic Theory and Entropy.
Feeling one's way through vs. predictive.
Are the following kinds of theorems related to the three minds? and, in particular, the second mind?
Emotional responses and the geometry of moods are related to Mobius transformations and {$SU(2)$} and the weak force. This suggests that the eightfold way is related to {$SU(3)$} and the strong force, and the doubts and counterquestions are related to {$U(1)$} and electromagnetism, and the needs and operating principles are related to contradiction and gravity.
Stefano Franchi. General Homeostasis, Passive Life and the Challenge to Autonomy
Consider how variety in cybernetics relates to entropy and information theory.
Is Active Inference based on the compatibility of different approaches?
Consciousness considers in parallel {$\pm 1$} which squares to {$+1} (unconscious) and {$\pm i$} which squares to {$-1$} (conscious).
Think of Bott periodicity as starting with (3+3)=humanity in one eight-cycle and continuing with +1, +1, +1 given by charge conjugation, and +1, +2, +3 in the other eight-cycle.
Ivan Di Liberti. Givant, Morley, Zilber.
- Givant 1970. There are precisely 4 finitary stable monads.
- The identity.
- The maybe monad.
- {$R[-]$} for {$R$} a division ring (a noncommutative field)
- {$Aff_R[-]$}, the free affine space over a division ring
- Monad encodes a ring, indeed, a field.
- (A field is a ring whose modules are free.)
Alethonaut - sailor of truth
Amanda Gefter. Enaction for QBists. Similar to Active Inference?
Chris Fuchs. Painting a QBist Picture of Reality.
In neural networks, categories emerge. Monosemantics allows information to be independent, thus vastly reduces the complexity, compared with polysemantics.
Urs Schreiber. Differential cohomology in a cohesive ∞-topos Modal Homotopy Type Theory relates Hegel, String Theory, Cohesive Infinity Topos.
How can there be joint intentionality for a culture of individual unfolding.
Anticommutativity makes cross terms cancel. Nilpotents make squares cancel. What about commutativity?
Laws of form
- Concatenation is "saying". Saying multiple times is the same as saying once. {$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$}
John Bolender. The Self-Organizing Social Mind.
Ivan. Was or was not regional politics helpful for potential democratic transition in Russia? Novosibirsk, Tatarstan.
Counterquestions are a foundation for learnability. Each counterquestion defines a domain of new knowledge where we had no facts that we could rely on.
Supermaps - holes - contexts.
Pragmatic approach. Context defines meaning. Robert Brandom. Making It Explicit.
df/dx = f. Can be expressed through the notion of infinity (Taylor series) {$e^x$}. Or through periodicity (trigonometry, Euler's equation) {$e^{ix}$}
- The volume of a unit sphere in n-dimensions goes up for small n, reaches a maximum at n=5, then goes down. https://en.wikipedia.org/wiki/Volume_of_an_n-ball
Conformal and analytic is the same.
Peter Scholze - condensed math
- Explain why we get alternating signs for the boundary
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.
Can large language models work by simply transforming existing input - taken to be grammatical - to preserve grammaticality.
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
- https://freedomhouse.org/report/freedom-world/2023/marking-50-years
- https://freedomhouse.org/countries/freedom-world/scores?sort=asc&order=Total%20Score%20and%20Status
Differentiation
Alex Codes. Symbolic Differentation in Python from Scratch!
https://www.quantamagazine.org/triangulation-conjecture-disproved-20150113/
https://jeremykun.com/2013/04/10/computing-homology/
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.
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.
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.
Ergodic theorem
- https://terrytao.wordpress.com/2008/01/30/254a-lecture-8-the-mean-ergodic-theorem/
- https://en.wikipedia.org/wiki/Ergodic_theory
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.
What Are The Hidden Rules Of The Universe?
http://www.math.lsa.umich.edu/~wfulton/CurveBook.pdf Fulton Curve Book
trivial tangent bundles on spheres?
Study of variables
- Use of variable as an operator (or an action) as in https://en.wikipedia.org/wiki/Operational_calculus but also group multiplication, group action.
Kervaire-Milnor formula
- {$\Theta = \Pi B$} where {$B=a_m2^{2m-2}(2^{2m-1}-1)B_{2m}/4m$}
Generalized Linear Models
Lattice
8 is special because {$\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.
- {$128=dim(\mathbb{O}\otimes\mathbb{O}^2)$}
Shoelace formula for oriented area of a polygon
Dobinski's formula relates Bell numbers and e. {$B_n=\frac{1}{e}\sum_{k=0}^{n=\infty}\frac{k^n}{k!}$}
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.
Discourse
https://en.wikipedia.org/wiki/Overton_window Window of political discourse: Unthinkable - Radical - Acceptable - Sensible - Popular - Policy
Lambda Calculus
Coxeter. Regular polytopes. Includes prehistory. Boole.
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.
Differentiation changes level. {$x^n$} number of levels of volatility, number of derivatives
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.
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
Creating what you can feel certain about. (Continuity.)
- Building up levels of certainty through topological invariants.
Stone's theorem: continuous implies differentiable
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/emily-riehl/ 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/is-composition-of-covering-maps-covering-map
Kojin Karatani. Architecture as Metaphor: Language, Number, Money (1995)
https://ncatlab.org/nlab/show/cellular+approximation+theorem#applications.
Relative invariance - more global than another