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2021.09.02 I decided to first present my research program as straightforwardly as I can. I am working on that below. I plan to complete my written presentation and publish an associated video by the end of October. Then I will seek Patreon supporters for my investigations and for my presentations of what I am learning through this website and videos such as a Math 4 Wisdom Show.

In my draft below I am expanding on my 2017 Research Program for a Big Picture of Mathematics.


Math for Wisdom

This is my introductory video to explain what Math for Wisdom means to me.

My name is Andrius Kulikauskas. I am speaking from my home in Lithuania. Please visit my website, Math4Wisdom.com, where you can read the text for this video, observe my research activities, contact me and perhaps collaborate with me or others.

I am learning, studying, exploring and discovering advanced mathematics at a graduate student level which is relevant for my quest since childhood to know everything and apply that knowledge usefully. 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. I need to learn enough category theory to classify adjoint strings and master the Yoneda lemma, enough algebraic topology to intuitively understand Bott periodicity inside out, enough algebraic geometry to be able to explain why there are four Lie families and how they lay the foundations for four geometries - affine, projective, conformal and symplectic

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 human-sized 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.

One - all - many

My approach is to observe the limits of my imagination, to document the cognitive frameworks by which I and others live and think, to

contradiction

proof by contradiction

Threesome: Jacobi identity, Pauli matrices, derived functor, ABC=CAB linear Turing machine and then consider reals -> complexes -> quaternions -> octonions=reals ?


Notes


My goals

Goals for this video

Summary of goals

  • Address the shortage of materials for appreciating, contemplating, exploring advanced mathematical topics and especially, gaining a sense of the big picture.
    • A sort of guide for mountain climbers.
  • Show the usefulness of wisdom as a guide in making sense of mathematics and the relevance of various mathematical topics to expressing wisdom.
    • A bit like an alternative reality for wizards except its real - Jesus, Buddha, Mohammed, Harry Potter.
  • Show the reality of cognitive frameworks.
  • Way to engage those who might work together.
    • Find support and income to get by.

General idea of my philosophy

  • Relate the definite and the indefinite.
  • Math is most definite. So show how the indefinite relates.
  • For example, how axiomatic mathematics is inspired by nonaxiomatic premathematics.

General approach

  • Ways of figuring things out in math.

Goals for math

  • Overview all of math - the areas, concepts, questions, results...
  • Identify and appreciate the key concepts behind mathematics.
  • Understand and apply the nature of mathematical beauty.

An example: The Threesome

How I came upon it at the University of Chicago. What is happiness?

Philosophy

  • Scientific method
  • Sesame street
  • Army action review
  • Roger Penrose

Math

  • Jacobi identity
  • SU(2) - Pauli matrices
  • Long exact sequence

Foursome

Levels of knowledge.

Yates Index Theorem.

Yoneda lemma.

Peirce's

idealism vs. materialism

holistic way of looking - consider math holistically


House of Knowledge

Ways of figuring things out in mathematics

Goals

  • See more conceptual structures.
  • Get overview of all of math.
  • Distinguish natural, preaxiomatic math.

Study the Ways of Figuring Things Out in Mathematics

I am most encouraged by my study in 2011 of the ways of figuring things out in mathematics which I shared in this letter to the Math Future online group. Here is an extended version of the results which I presented in 2016 at the Lithuanian Mathematics Association Conference: Discovery in Mathematics: A System of Deep Structure.

The basic idea came in considering George Polya's "pattern of 2 loci" by which he solves Euclid's first problem of how to construct an equilateral triangle. We solve this problems in our minds by constructing a lattice of conditions. Given two points, the third point that we want to construct must satisfy two conditions, namely, it must be on both of two circles centered on the two other points, thus it must be at their intersection. The solution is clear as soon as our minds apply the relevant structure, namely, the lattice of conditions.

This example suggests a distinction between deep structure - the natural mathematical structures which we use in our mind to solve problems, and surface structure - the contrived math by which we describe problems on paper. I thus surveyed the problem solving patterns taught by Paul Zeitz, George Polya and others in their books.

Ways of figuring things out in other fields:

  • neuroscience, chess, physics, biology, Jesus, Gaon of Vilna, my philosophy.

Mathematical Goal

  • Overview how all of the branches of math unfold, the questions and answers, problems and solutions, challenges and theorems, concepts (structures, objects, relations, constants), ideas.

Philosophical Goal

  • Language of absolutes - Cognitive frameworks
  • Most interesting, most fruitful and most speculative would be for me to look for connections between the philosophical structures I work with and what seem to be related mathematical structures.

Observations

  • ABC's - twosome, threesome, foursome

Foursome

  • Foursome: Yoneda lemma, embedding. Chomsky hierarchy. Universality.
  • Foursome: Yates Index Set Theorem, the jump hierarchy.

The kinds of variables

Analyze How We Use Variables. My understanding is that there are four levels of knowledge (whether, what, how, why) and that in thinking in a mathematical system we establish a level (surface structure) and a metalevel (deep structure). Our use of variables plays off this distinction. I thus made a diagram of the roles that we imagine variables to play.

I need to validate this further. I expected there to be six kinds of variables but instead found evidence for twelve kinds. It seems that, on the one hand, variables are used to solve problems, but on the other hand, variables are used to create problems.

Indeed, this points to a key aspect missing in my research so far. Much of advanced mathematics is about abstraction, the creation of frameworks. But it is clear that this process of abstraction is not arriving at cognitive foundations but rather is growing ever more rich, complex and distant.


Classical Lie families

Investigation: 4 classical Lie families, 4 geometries, 4 choice frameworks.

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.

I then tried to improve my map by adding more detail. I used the graphic editor yEd. This simply yielded a spaghetti diagram. However, I am hopeful that ultimately it should be possible to discover principles for making a meaningful map and collaborating with others to make it a comprehensive resource related to Wikipedia and MathStackExchange/MathOverflow.

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 pre-systemic. The upper half of the diagram grounds that which takes place within a mathematical system.

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.

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. In particular, I was able to come up with an interpretation for the -1 simplex and a novel q-analogue of the simplex.

I am also seeing how the polytopes can be thought to arise by a "center" which ever generates vertices (for simplices), pairs of vertices (for cross-polytopes), planes (for hypercubes) and "coordinate systems" (for demicubes). This type of process is very relevant for my theological ideas, see: God's Question: Is God Necessary? In particular, I think about the "field with one element" as being interpretable as 0, 1 and infinity.

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. In particular, John Baez and others have pointed out that the classical Lie algebras ground different geometries. I would like to learn the basics of affine, projective, conformal and symplectic geometries so that I could understand how they relate to the four classical groups.

The center of a regular polytope <=> God

The totality of a regular polytope <=> Everything


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

Cortical minicolumns


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).


Fivesome

Sheffer polynomials, quantum physics, Feynman diagrams, quantum field theory.


Sevensome

Shu-Hong's equation. Mobius transformations.

I want to be able to describe the cognitive foundations that account for logic.

Seven-eight kinds of duality. Reps of Sn and GLn. Schur-Weyl duality.

I would like to understand the various kinds of opposites in math and classify them.

Understand the Basics of Logic and Truth. I have made some progress in describing such foundations for truth: Truth as the Admission of Self- Contradiction. Which is to say, truth is inherently unstable and tentative, the relation of a level with a metalevel.


Divisions of everything: Adjunctions

Exact sequences of length n <=> Divisions of everything into N perspectives


Divisions of everything: Bott periodicity

Bott Periodicity <=> The eight-cycle of divisions of everything


Operations

Norman Anderson's theory and modeling thinking fast and slow.

Visualizations - unconscious and conscious


Six representations

Grothendieck's six operations, the natural bases of the symmetric functions, Hopf algebras.


Twelve circumstances

One, all, many

SU(2) normal form


Nonwishes: 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.


Map of Deepest Values


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)


Materialist point of view

Abstraction may relate to the disembodying mind. Lakoff, Nunez and others have collected much evidence to show the importance of "the embodied mind". However, this same evidence can be used to think about a "disembodying mind". Evolutionary processes are favoring central nervous systems which have been developing to live in increasingly abstract worlds: first icons (sensory images), then indices (models of attention, as noted by Graziano) but ultimately symbols (which function by dividing up the global workspace).


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.


Goals

Goals for Math 4 Wisdom

  • Express my concepts in terms of math.
  • Show that my concepts are relevant and fruitful in math.
  • Present a language for
  • Foster a community.

Goals for wisdom

  • Main ways the indefinite is important.
  • Solve technical issues.
  • Relate different houses of knowledge.

Goals for language

  • Create a language for working together as a scientific community.
  • I look to math as a standard of truth for such expression, for such a language.

Goals for community

  • Support inclusive harmony of self-realization of all people - "meaningful inclusion".
  • Shared question, "How to love each other more"
  • Sharing life experiences, deepest values, investigatory questions, etc.
  • My personal goal is to express myself fully as I wish for others as well.
  • So I am expressing myself through arts and with examples from all of life.

Elementary examples

  • Amounts and units. Combine like units, list differing units. Type theory.
  • Comparing the price of a sandwich {$\frac{4}{3}\frac{2}{3}=\frac{8}{9}$} as illustrating that algebra is thinking step-by-step.
  • Trigonometry: Every rectangle is half a triangle.
  • Extending the domain.

Overview and Compare Similar Efforts

Most working mathematicians have not spent a few hundred hours in their life searching for a key by which to understand all of mathematics. Indeed, it has been said that Henri Poincare (1854-1912) was the last person to excel at all of the mathematics of his time.

1) One step is to overview the history of mathematics to glean insights from the research interests of the most profound mathematicans, such as Euclid, Descartes, Leibnitz, Pascal, Hilbert, von Neumann and more recently, Weyl, Atiyah, Conway, Grothendieck, Langlands, Lurie to imagine their perspectives on the big picture.

When I was a graduate student (1986-1993), an interest in the big picture was quite taboo, and moreover, quite impractical, given that the way to learn math was to take classes, read textbooks, do exercises, and read journal articles. However, since then, much has changed which has made it possible to learn advanced mathematics much more personally, intuitively, selectively and comprehensively. I can read a vast mathematical encyclopedia (Wikipedia), watch video lectures (You Tube) by expert thinkers on the most advanced subjects, and ask questions and get answers at Math Stack Exchange or Math Overflow.

Of special importance are math bloggers who are sharing their personal intuitions regarding math. It is most strange that intuition is acknowledged as the key to learning and furthering math, and yet articulating, documenting and studying that intuition is considered out of bounds, as can be seen from the little space devoted to it in any article or textbook. The reason, I suppose, is that we would have to reveal our general ignorance. It is particularly refreshing and encouraging to read blog posts by John Baez, Urs Schreiber, Terrence Tao, Qiaucho Yuan and others who do seem to grapple with the big picture.

2) A further step is to note the areas and structures which such bold thinkers believe to be fruitful. Succinctly, as I learn from thinkers such as Olivia Caramello, Urls Schreiber, John Baez, Roger Penrose, Lou Kauffman, Vladimir Voevodsky, Saunders MacLane, William Lawvere, John Isbell, Harvey Friedmann, Joseph Goguen, Robert May, Kirby Urner, Maria Droujkova it seems that they focus on particular areas, such as category theory, topoi, algebraic geometry, homology, homotopy, string theory, network theory but also that they are intrigued by particular structures which seem exceptionally rich, such as the octonions...

Of course, I do not intend to master these subjects in the usual way. Instead, I hope to be clever enough to find a new way of looking at math which shares and yields mathematical intuition much more readily.

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Šis puslapis paskutinį kartą keistas September 11, 2021, at 06:58 PM