• Andrius Kulikauskas
  • ms@ms.lt
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Andrius Kulikauskas: I am writing up my research program. I intend to present it with a set of introductory videos. I expect to write up the text for the first video by the end of January, 2022. At that point I will seek support through Patreon.


My writings include several talks where I present links between wisdom and math, such as: A Geometry of Moods: Evoked by Wujue Poems of the Tang Dynasty

Current Focus

In mathematics, I am currently investigating Adjunction. I am mastering the three ways of defining adjunction in category theory, and how they are related. Then I will try to collect and classify adjoint strings and investigate how they may express cognitive frameworks that organize perspectives and contexts. Also related is my study of how the Yoneda Lemma expresses four levels of knowledge: whether, what, how, why. I zoom every other Sunday with Wenbo Gao and the New York Category Theory and Algebra Meet Up. We are currently studying Allen Hatcher's book on algebraic topology. You are welcome to join us!

In physics, I am relating the combinatorics of Feynman diagrams and Hermite polynomials, with the hope to extend quantum field theory further through five natural coordinate systems based on the fivefold classification of Sheffer polynomials. I am interpreting this classification as a cognitive framework for decision making, for relating a subsystem and a system, based on two kinds of causality, backwards from effect to cause - every effect has had its cause, and forwards from cause to effect - not every cause has had its effects. I zoom every other Monday with John Harland.

In my philosophy, I am working on the big picture of everything, the passage from the Indefinite to the Definite to the Imaginable to the Unimaginable. God manifests himself by becoming God for nobody (for God), for somebody (for Me), for anybody (for You), for everybody (for Other). I am also investigating how to nurture myself and others and God to grow ever more mature. My friends and I published the first newsletter, Laiškai šviesuoliams, in Lithuanian, for a culture of enlighteners (independent thinkers).

Latest Research

2022.01.12 Finally, after five years of effort, I have related the Yoneda Lemma to the four levels of knowledge: Whether, What, How, Why. It's actually the Yoneda Embedding which is relevant. A typical morphism goes from How to What. Why is the perspective that keeps How and What separate, and Whether is the perspective that relates them. Thus a mental shift from How to What is followed by a mental shift from Why to Whether. Why and Whether are contextual and given by the sum of relationships. Why is dynamic consciousness of those relationships whereas Whether preserves those relationships as knowledge. The levels of knowledge also express the scope of understanding, which is to say, the scope of separation: Why separates everything, How - anything, What - something, Whether - nothing, for with Whether it is no longer separated but is related. I will be explaining more of this in the introductory video in the series where I will present my research program.

2022.01.05 In studying Bott periodicity, I realized that for my purposes I should focus on the classification of Clifford algebras. I see that it is straightforward to construct the regular representation of a Clifford algebra by considering how the basis elements act on each other by multiplication. For example, I can define matrices for {$1$}, {$e_1$}, {$e_2$}, {$e_1e_2$} by describing what these basis elements yield when I multiply them on the left of {$1$}, {$e_1$}, {$e_2$}, {$e_1e_2$}. I can investigate what happens when I add a new element {$e_3$} for which {$e_3^2=q_3$}. I can study the combinatorics of the resulting regular representations and understand how they express {$\mathbb{C}$}, {$\mathbb{H}$}, {$\mathbb{H}\oplus\mathbb{H}$}, {$M_2(\mathbb{R})$} and other such structures. I can figure out how the regular representations break down into irreducible representations. My efforts will help me better understand the various readings.

I am also trying to master the adjunction of the loopspace functor and the reduced suspension functor, which is relevant for understanding the K-theory of Bott periodicity. Tai-Danae mentions this adjunction in giving examples of adjunctions and describes it in her exposition of the one-line proof that the fundamental group of a circle is {$\mathbb{Z}$}.

2021.12.26 I have been studying some helpful expositions of Bott periodicity, which I believe expresses the eight-cycle of divisions of everything, a circular tour of mental states. I am growing familiar with the concepts in the Wikipedia article. I appreciate Cameron Krulewski's accounts of Bott periodicity and the classification of topological insulators and superconductors. Her latter article includes a proof of the classification of Clifford algebras, which I wish to focus on. There is also a Wikipedia article for that classification. Other expositions of Bott periodicity are by Aygul Galimova, Anthony Bosman, Carlos Salinas, and in the complex case, Allen Hatcher. There is an important proof by Mark Behrens and a list of proofs on MathOverflow. And there is a paper by Jost Eschenburg and Bernhard Hanke that relates Clifford modules to vector bundles over spheres.

2021.11.05 I gave a talk on the 24 ways of figuring things out in biology. You can read a rather decent machine translation from Lithuanian into English: Adam Named the Animals: There Is Love of Life In the Epistemology of Biology. This investigation yields a definition of life as a system that manages its traits for success.

2021.09.02 I am making some progress understanding the adjunction that relates restriction and induction functors on categories of group representations. I want to master this and use it to illustrate the three definitions of adjunction.

2021.08.01 After some perplexity, I now understand what it means to construct the possible Feynman diagrams given a power of {$J$} and a power of {$\lambda$}. I will be able to relate this to Wick contraction, which is also the basis for the combinatorics of the Hermite polynomials. So I am realizing it will be straightforward to think of the other Sheffer polynomials as extending quantum field theory to model what happens kinematically as a particle passes from one interaction to another. Kinematically, I intend to show that the particle can be taken to be in one of five natural coordinate systems: Hermite (empty space), Charlier (moving packet), Laguerre (bound state), Meixner-Pollaczek (scattering), Meixner (creation and annihilation).

2021.06.21 After more than half a year of trying, I am finally making sense of the three definitions of adjunction in terms of homsets, triangle identities, and the universal mapping property. In sharing them with Wenbo Gao, I variously corrected myself, and finally came upon the revelation I've been seeking:

Adjunctions describe how information stays the same though context changes.

For example, a light switch expresses one bit of information, which might also express whether or not to start a nuclear war.

This idea distills my observations that adjunctions separate out context, and thus strings of adjoint functors may express what I mean by divisions of everything, or perhaps, I am now thinking, alternating chains of definite (human) and indefinite (divine) perspectives. So now I am going through various examples of adjunctions to make the theory all concrete for me. I will try to classify adjoint strings.

The revelation also aligns with my contemplation of the undefinite and the definite. I think the definite arises along with knowledge and thus information and context. This brings me back to my attempts to understand the Yoneda Lemma as manifesting four levels of knowledge: whether, what, how and why.

I am also wondering how adjunctions relate to self-adjoint operators, which are central to quantum physics. I will talk today with my friend John Harland about that.

For earlier news see also my diary and the recently changed pages in my Exposition and Research sections.

Let's work together!

Truthful Wisdom

{$M\Rightarrow W$}, "M 4 W" and Math 4 Wisdom stand for Mathematics for Wisdom by Andrius Kulikauskas, ms@ms.lt, aka Living by Truth.

{$M\Rightarrow W$} inspires our imagination by studying its very limits, what we can imagine or not. Let's embrace the cognitive frameworks we can't escape and leverage them for a conceptual language of wisdom that we can express, validate and apply with the help of mathematics and all manner of sciences, creative arts and moral disciplines.

Cranky Disclaimer

Crank alert

You may be at risk of dismissing these pages as an online art show, science fiction, mental poetry, psychological symptoms, products of numerology, metaphysical truths, existential crises, adventures in mysticism, primordial forms... and so on and so on and so on. Just so you know!

Keys to Math and Other Subjects

Coming sooner or later...

Profoundly Speculative Investigations

See all investigations.

Why study math? Here are mathematical concepts that I believe are relevant for wisdom.

Understandingfor modeling
The ways of figuring things out in mathematicsHouse of Knowledge for Mathematics
Four classical Lie familiesFour levels of geometry and logic
PolytopesChoice frameworks
{$SU(2)$}Emotional transformations
Yoneda lemmaDivision of everything into four levels of knowledge (Whether, What, How, Why)
Yates Index theoremDivision of everything into four...
The five conicsDivision of everything into five perspectives for decision making (thus space and time)
Fivefold classification of Sheffer polynomials of A-type zeroDivision of everything into five...
Adjoint string of length NDivision of everything into N perspectives
Bott periodicityEight-cycle of divisions
Grothendieck's six operationsSix representations of divisions
PerturbationIntervention, intercession, miracle
SpinorGoing beyond oneself into oneself
{$F_1$}Division of everything into 0 perspectives
The state of contradictionGod

Wonderfully Curious Phenomena

See all phenomena.

24 ways of figuring things out.

8 states of mind.

6 transformations of mood.

Frequently Questioned Answers

In 1971, as a six-year old child, I went to God (?) with my quest to know everything (?) and apply that knowledge usefully (?)

In 2021, I am about ready to share wisdom (?) But who would (?) how could (?) why should anybody make an effort to comprehend it (?) take it to heart (?) try it out (?) and work together (?)

Math is a standard for truth. Math is verified again and ever again. It is a universal language, accessible to millions who make the effort, regardless of their personal creed. Enthusiasts, practitioners and specialists appreciate profound results and helpful expositions. They will invest themselves to understand, if they have good reason.

I think it best to demonstrate wisdom through math:

  • to apply wisdom by overviewing all of math, showing how it unfolds, with all of its branches, concepts, questions and results.
  • to express in mathematical form the cognitive frameworks that capture wisdom.
  • to develop insightful expositions of the relevant mathematics.
  • to present my work-in-progress as I pursue my ideas in life and math.

Currently, for a year or two, I have the means to spend all of my time on my philosophy and related mathematics. But then I will need to find a way to make a living. My plan is to create this website as a work-in-progress where I present the math that I have learned, am learning or will be learning which is relevant for sharing wisdom. I think that I will find supporters, including through Patreon, who would fund me to learn and present more math because they love advanced mathematics, want new ways of understanding it, wish to see it reach more people, and are excited how it might serve to express answers to life's great questions. I can also create learning materials, crowd source projects at Kickstarter, and work as a coach for self learners. As ever, my creative work is in the Public Domain, for all to use as they think best.

Ultimately, I hope that wisdom grows and we embrace wisdom, the Holy Spirit, in a language, a science, a culture of self learners. This website may reach you in a way that academic peer reviewed articles never would. I may write such articles, but I fear they won't be read, understood, embraced and lived. I can write here to you in a way that I would write to myself.

It seems natural to illustrate my ideas with visual thinking, combining text, images and diagrams, as I learned from John Rogers and from Robert Horn's books. Moreover, art is fun, art is social, art holds us together, art changes the rules, so that you might give me a chance. I want to speak from all of me, my conscious and unconscious, to all of you, your conscious and unconscious. I want you to feel that I am a person, real and true, who stands alone, yet aims to commune with you.

Wisdom is why I write, math is how I write. What I write are the creative arts that speak to your attention. Community with you is whether I succeed in writing to you that Spirit which reaches you, and whether from you it reaches me, so that it dances amongst us, as we step-in and step-out, as inspired.

I am working on a video where I introduce myself along with Math 4 Wisdom.

An Introduction to Everything

Coming soon...



{$M\Rightarrow W$}, Math 4 Wisdom and Math for Wisdom stand for Mathematics for Wisdom by Andrius Kulikauskas.

Keisti - Įkelti - Istorija - Spausdinti - Naujausi keitimai -
Šis puslapis paskutinį kartą keistas January 18, 2022, at 06:59 PM