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Andrius Kulikauskas: This is where I jot down my notes. In general, in this Exposition, any grey areas are for my personal notes. I need to delete, move, organize these notes Such concepts are typically defined in terms of simpler concepts and yet we neglect to define the simplest concepts. These are mathematical structures which I am working to interpret as frameworks for how cognitive perspectives fit together. I have in mind a very particular set of mathematical structures which I believe express cognitive frameworks for wisdom that also serve as keys for making sense of all of abstract math. Can I provide wonderful insights into mathematics and related subjects? If so, then perhaps you will make an effort to learn my language and make it your own. We can work together to find connections between wisdom and mathematics, to draw insights from mathematics, to express wisdom accurately, to overview the whole of mathematics and make sense of it, and ultimately to apply wisdom in all of life, and foster a shared culture of wisdom. This is why I'm creating Math 4 Wisdom. What is a perspective? No perspective (God), perspective (I), perspective upon perspective (You), perspective upon perspective upon perspective (Other).
Thinkinginparallel Thinking in parallel  the three basic ways
Abbreviations
If it helps to be more sympathetic, you can consider this as my own private language, as science fiction... More questions of interest I will keep switching back and forth between mathematics and wisdom so as to maintain interest among those of you who are primarily interested in advanced mathematics and those who may not understand any of it but are very keen to investigate wisdom. Here are some more subjects in math that are relevant for how perspectives can fit together.
Goals that relate math and wisdom
What is wisdom  knowing everything Wisdom  knowing everything  knowing something or anything about everything  what can we say about how everything is organized?
math includes ... By knowing everything, I mean a Godly, perfect, total, complete, absolute knowledge of the vantage point from which everything makes sense, from which we have the big picture. Of course, from the heights of that Olympus my humansized brain and my nearsighted eyes will not penetrate through all the clouds below, will not engage all the details at once. Therefore by knowing everything, I also mean the further ability to descend into the details, to readily find the answer to any particular question we may have, including any mathematical question, and then to climb back up to that perfect vantage point. Certainly, I would want to be able to explain where all of math comes from, and how it unfolds, all of the branches, concepts, structures, theorems, questions, ideas, challenges, opportunities, in all their possible directions and subdirections. And if I did do that, if I laid out all of mathematics before you, so that you could see with childlike fascination, at every step, the options by which it proceeds, up through the most impossibly profound results, all of the mathematics that there could every possibly be, as if you were exploring a miraculous fractal world, zooming through a Mandelbrot set of ideas, then I would feel comfortable, I would feel emboldened to tell you that my findings about the wisdom of life may likewise be valid. The goal of knowing everything, the proper, useful, beautiful application of that knowledge, is for independent thinkers to foster a community, nurture a science, develop a conceptual language, not just for a map of mathematics but for all of human life, especially that wisdom by which we grow ever more mature, here and now. Math for wisdom is a call around the world to those who are working towards this goal from every imagineable world view. If we want to be sure of exactly what we mean, and be confident that we have uncovered a conceptual language of well formed, well connected and well functioning ideas, and hope that others will trouble themselves to understand, then we should express ourselves in terms of mathematics, or rather, in terms of mathematical ideas, those ideas that mathematics researchers know from their work. Math is the gold standard for utility of expression, which true philosophy should meet, and which sincere poetry admiringly concedes. Math for Wisdom is my way of engaging you with the topics in advanced mathematics that are relevant for expressing to others the conceptual language that I seek for my own thinking but ultimately for thinking along with you. In this very first video, I am providing an overview for myself to make clear why and how I personally take up and pull together a variety of mathematical investigations. For my sake and for your sake I believe that I should start with such a straightforward approach. In the future, depending on what you find interesting, I expect to present my investigations in ways that ... The universal conceptual language that I seek is, like math, an alternative to words. We may use a word like "tree" pragmatically, simply because we want to refer to the chair next to the tree, and so we need not agree as to what we actually mean by tree. If we were to take a tree and make it smaller, then at some point we would surely disagree as to whether it is still a tree or has become a bush. We simply don't need to know or care to know exactly what we mean by tree. End of Introduction
ABC's of ThinkinginParallel 3Track Mind for Learning. Derived functors. 4Track Mind for Knowledge. Yoneda Lemma. https://en.m.wikipedia.org/wiki/Grothendieck%27s_relative_point_of_view Global quantum
2Track Mind for Existence Divisions, Conceptions, Circumstances Six representations Grothendieck's six operations The natural bases of the symmetric functions Hopf algebras.
Twelve circumstances One, all, many Twelvefold way Principles of life  Christopher Alexander  Mandelbrot set  Catalan numbers  Dyck words  push down automata. SU(2) normal form The cube 8 divisions, 6 conceptions, 12 circumstances. Divisions  carving mental space
Divisions of everything
Division of everything into zero perspectives
More Sophisticated ThinkinginParallel 5Track Mind for Deciding. Dual causality and QFT.
Special kind of logic related to Aristotle.
Contradiction. Which is to say, truth is inherently unstable and tentative, the relation of a level with a metalevel. ShuHong's equation for the meaning of probability words in context. {$R=M_p + a^+(CC_p)$}. Rewritten as {$a^+=\frac{RM_p}{CC_p}$}. Where {$R$} is the subject's response (their judgement on the meaning), {$M_p$} is the prototypical meaning, {$C_p$} is the prototypical confidence, {$C$} is the contextual confidence. "The prototypical meaning provides a reference point for the range of meaning, and the range of meaning provides a reference point for confidence, which is compared to the contextual confidence." (pg.143) "Every probability word has a prototypical confidence associated with it." (pg.117) Confidence deals with what we don't know, meaning with what we know? Mobius transformations. {$f(z) = \frac{a z + b}{c z + d}$} Dualities Seveneight kinds of duality. Reps of Sn and GLn. SchurWeyl duality. I would like to understand the various kinds of opposites in math and classify them. 24 Keys For Solving Math Problems Collecting and analyzing such examples could be a collaborative effort. Here is a database I made of almost 200 examples of figuring things out in math. I used my philosophical structures to systematize the recurring patterns. This yielded the following diagram. I would like to sharpen the results. The lower half of the diagram grounds the mathematical thinking which is presystemic. The upper half of the diagram grounds that which takes place within a mathematical system. Map of Math: Affine, Projective, Conformal, Symplectic Geometry Map of Areas of Mathematics
Algebra and geometry
One of my goals is to be able to make a map of how mathematical subjects, concepts and objects become relevant. Such a map would systematize existing mathematics, identify overlooked mathematics, and show the directions in which math can evolve in the future. I started by organizing the subjects listed in the Mathematics Subject Classification by trying to show which areas depend on which other areas. Acknowledging my general ignorance, I was able to draw several conclusions. As expected, there do seem to be two major areas, algebra and analysis. The capstone of math seems to be number theory, which makes use of tools from all of math. Lie theory seems especially central as a bridge between algebra and analysis. Surprisingly for me, geometry seems to be a well spring for math. I studied algebraic combinatorics as "the basement of math" from which I thought mathematical objects arose. Geometry thus seemed rather idiosyncratic. But from the map it seems that geometry is a key ingredient in math, in terms of its content, perhaps in the way that logic is, in terms of its form.
Yates Index Set Theorem Arithmetic hierarchy
Yates Index Set Theorem. (1966) Given r.e. sets C and D such that {$C<_T D$} and {$S\in\Sigma_3^C$} there is a recursive function {$g(k)$} such that for all {$k$}
Corollary (Yates). If C is r.e., then {$\{x:W_x\equiv_T C\}$} is {$\Sigma_3^C$}complete. Corollary. The set of indices of Turing complete r.e. sets, Comp = {$x:W_x\equiv_T K$} is {$\Sigma_4$}complete.
Classifying adjunctions and adjoint strings Four classical Lie families  Four levels of geometry and logic In particular, I am interested in understanding, intuitively, the cognitive foundations for the four classical Lie groups/algebras. I have been learning about the classification through the Dynkin diagrams. But that does not explain intuitively the qualitative distinctions. So instead I have been working backwards, from the Cartan diagrams, trying to understand concretely how to imagine the growth of a chain (how it ever adds a dimension via an angle of 120 degrees) and the possible ways that chain might end. Polytopes  Choice frameworks I am encouraged that I myself have made some mathematical discoveries by focusing on these questions. I have thought a lot about the regular polytopes which the Weyl groups are symmetries of. I was able to come up with an interpretation for the 1 simplex and a novel qanalogue of the simplex. I am also seeing how the polytopes can be thought to a rise by a "center" which ever generates vertices (for simplices), pairs of vertices (for crosspolytopes), planes (for hypercubes) and "coordinate systems". Three infinite families of regular polytopes and an additional family of coordinate systems make for four families of choice frameworks.
Classical Lie families Investigation: 4 classical Lie families, 4 geometries, 4 choice frameworks. Discover Cognitive Foundations for the Classical Lie Groups/Algebras. It is surprising that in mathematics there is a small collection of structures which seem most rich in content. This is a point that Urs Schreiber keeps returning to. Thus one task is to make a list of such structures and try to relate them with a map, and indeed, understand how they fit in a map of all math. Four geometries John Baez and others have pointed out that the classical Lie algebras ground different geometries.
Perspectives One place to look for the cognitive foundations of mathematics is to develop models of attention, for example, in terms of category theory. 6 visualizations and 6 paradoxes
10 axioms of set theory The axioms of Zermelo Frankel set theory (except for the Axiom of Infinity) and the Axiom of Choice are all present in the above system and so I would like to work further to clarify their role. Of special interest to me, currently, is to study the four concepts (in orange) that seem to ground logic but also geometry. These methods apply the concepts of truth (argument by contradiction), model (solving an easier version), implication (working backwards) and variable (classifying the problem). 4 plus 6 generators of the Poincare group, the symmetries for quantum field theory Map of Math Another way to build a map is to use the tags from MathOverflow. The idea is to make a list of the, say, X=100 most popular tags, and also to make a list of the most popular pairs of tags, where pairs are created for any two tags that are used for the same post. In the map, for each popular tag, I would show a link to its most popular pair, and also include, say, the most popular 2X links overall. Study the Process of Abstraction. Thus it is important to study the process of abstraction. One approach is to try to describe, in an elegant way, a theory that is practically complete, such as the geometry of triangles in the plane. Norman Wildberger's book and videos are very helpful for this. It may be that a matrix approach might be insightful. Having stated a theory it may be possible to see in what directions it develops further. Another approach is to identify classic theorems in the history of mathematics and consider how abstraction and generalization drove them to arise and develop further. I would like to learn more about the kinds of equivalences in math  I know that Voevodsky, etc. have studied that deeply  and draw on that and perhaps contribute. Bott Periodicity: Unfolding of ThinkinginParallel Adjoint string of length N  Division of everything into N perspectives Exact strings. Understand intuitively
Bott periodicity  Eightcycle of divisions
Divisions of everything: Exact Sequences Divisions of everything: Adjunctions Exact sequences of length n <=> Divisions of everything into N perspectives Divisions of everything: Spin Elementary particles
Divisions of everything: Bott periodicity Bott Periodicity <=> The eightcycle of divisions of everything Operations Norman Anderson's theory and modeling thinking fast and slow.
Visualizations  unconscious and conscious Poincare Group: Organizing Structures and Perspectives Generation of structures Notwishes {$SU(2)$}  Emotional transformations Snake lemma Unclear Narration 3 Languages Map of Deepest Values Antiwishes: Emotional responses: SU(2) Study the Geometry of Moods In my study of emotions and moods, I have successfully linked my philosophical and mathematical research. My model of basic emotions is based on whether our expectations are satisfied. Of special importance is the boundary between self and world. For example, if we discover that we are wrong about the world, or anything peripheral, then we may feel surprised, but if we learn that we are wrong about ourselves, or something deeply important, then we may feel distraught. See my talk: A Research Program for a Taxonomy of Moods. I did a study of some thirty classic Chinese poems from the Tang dynasty to explain the moods they evoked. (In Lithuanian: Nuotaikų aplinkybės: Tang dinastijos poezija ir šiuolaikinė geometrija.) I discovered that the mood depended on how the poem transformed the boundary between self and world. Each of them applied one of six transformations (reflection, shear, rotation, dilation, squeeze, translation) which shifted the geometry from a cognitively simpler one to a cognitively richer one (path geometry  affine, line geometry  projective, angle geometry  conformal, area geometry  symplectic). 20) I would like to better understand these geometries by learning about the math but also by seeing what they should be given the data from intepreting such poems. I made a related post at Math Stack Exchange: Is this set of 6 transformations fundamental to geometry? 21) This emotional theory describes beauty as arising upon the disappearance of one's inner self whereby disgust becomes impossible. It would be meaningful to study what is beautiful in mathematics and why. 4 geometries & 6 transformations between them <=> The Ten Commandments (4 positive and 6 negative) Nonwishes: Eightfold way: Snake lemma The Snake Lemma <=> The eightfold way Nonwishes: Eightfold way: Octonions The octonions <=> The eightfold way. Big Picture: From Indefinite to Unimaginable {$F_1$}  Division of everything into 0 perspectives Spinor  Going beyond oneself into oneself Perturbation  Intervention, intercession, miracle Map of deepest values Persons The application of the pumping lemma (Hopcroft & Ullman, p.57) is expressed as a dialogue with an adversary where the adversary chooses the existential quantifiers and we choose the universal quantifiers. Thus this is an example of a division of perspectives and the use of persons such as You and I. Standard model  gauge theories  Indefinite / Definite / Imaginable / Unimaginable Operations 1, +1, +2, +3 correspond to time (1) and three dimensions of space (+1,+1,+1). Gravity  r2 law  Yoneda lemma  Universality  Global quantum
God's going beyond himself: {$F_1$} The state of contradiction <=> God The Field with One Element <=> God
Contradiction: Godel's Incompleteness Theorem I would like to learn more what logic is all about, in practice. I have taken the mathematical logics course, am familiar with Goedel's theorems and have done graduate study in recursive function theory. God's Dance: SU(2) Goals for Math4Wisdom Goals for Math 4 Wisdom
General idea of my philosophy
General approach
Theory regarding the Facts
Relevance of God  source for mathematical intuition
Not math for its own sake
Summary of wisdom
Goals for wisdom Goals for wisdom
Goals for language
Goals for community
Personal goals
Credits ''He gave it to me  started a fire burning in me  he gave it to me  Nuotraukas draugų, kurie padėjo.
