• Andrius Kulikauskas
  • m a t h 4 w i s d o m @
  • g m a i l . c o m
  • +370 607 27 665
  • Eičiūnų km, Alytaus raj, Lithuania

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Andrius Kulikauskas


I invite you to join the Math 4 Wisdom Discussion List. Thank you to Kirby Urner for his letters.

2022.11.25 The State of Math 4 Wisdom is my vision for 2023.

I thank you for your 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

I am working on upcoming videos on a variety of subjects in a variety of formats.

In mathematics, I am currently investigating Adjunction. I am collecting and classifying adjoint strings and investigating 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! I am also attending John Baez's seminar on Thursdays until December 1st about his favorite structures in representation theory. They are very relevant for wondrous wisdom.

Wondrous Andrius

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 the fivesome, 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.

As regards peacemaking, an application of wisdom, I am trying to find others who would like to encourage dialogue in good will with supporters of both Russia and Ukraine.

In wondrous wisdom, I am relating God's questions (Whether, What, How, Why God exists?) and human's questions (What is the difference between God and goodness in eternal life, wisdom, good will, God's will?) In particular, I am seeking a relationship between God's and human's frameworks for wisdom. As data, I am analyzing the wisdom in meaningful experiences in my own life. Perhaps you would likewise share, study and model your meaningful experiences?

For personal growth, I am 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 self learners (independent thinkers, enlighteners).

Latest Research

2022.06.27 John Harland has impressed upon me the signficance of Stone's theorem on one-parameter unitary groups. I am trying to relate that to the generating function for Sheffer polynomials. I would like to claim that the only physical solutions to the Schroedinger equation are orthogonal Sheffer polynomials.

2022.05.27 I gave a talk on "Irony in the Memoirs of Those Who Knew the Perpetrators of the Holocaust" at the Lithuanian Cultural Studies Institute's conference on "Aesthetics, Philosophy of Art and Creative Activity in the Times of Social Transformation". Here is the original in Lithuanian and a machine translation into English.

2022.04.15 I responded to Russia's invasion of Ukraine with a talk, "A Peacemaker's Way: Wake Up, Love Your Enemy, Befriend Evil", which I gave in Vilnius at the Lithuanian Cultural Studies Institute's conference on "Civilizational Imagination and Contemporary Shifts in Geopolitics".

2022.03.28 I have been trying to understand how special relativity enters into quantum field theory. Studying with John Harland, I developed my own understanding of special relativity. I focus on the problem, which follows from the Pythagorean theorem, that the speed of light is different for different observers. And I show that the mathematics becomes elegant if we set {$c^2-v^2=1$} rather than {$c=1$}. My approach suggests to me that physics should be based on pairs of frames rather than single frames.

2022.02.22 Here in Lithuania I am deeply concerned by Russian autocrat Vladimir Putin's bullying of Ukraine. I learned a little bit about Ukrainian patriot Mikhail Kravchuk, whose Kravchuk polynomials I have been studying these last few days. In my understanding, his polynomials are a discrete version of the Meixner polynomials and, for my purposes, probably more important. The Kravchuk polynomials are orthogonal with respect to the (discrete) binomial distribution and I have investigated how they might relate the Hermite polynomials which are orthogonal with respect to the (continuous) normal distribution. The crucial difference is that the mean and the variance of the binomial distribution depend on {$N$} and get ever shifted to the right as {$N$} grows. The shift in the mean is given by {$N(\alpha - \beta) = Np$} and the distribution is given by {$w((\alpha-\beta)n)=(\frac{\alpha}{\beta})^n\binom{N}{n}$}, which only makes sense when {$\alpha\neq\beta$}. Should {$\alpha=\beta$}, then the mean does not move and so we take {$N\rightarrow\infty$} which gives the normal distribution. Thus, curiously, the continuous cases (Hermite, Laguerre) are a degenerate versions of the discrete case. I note that in the (quantum?) discrete case we have movement of the global space but in the (classical?) continuous cases the global space is at rest.

2022.02.02 I'm focusing on classifying adjunctions. I would like to better understand how they relate to how, in my philosophy, I define perspectives and especially, divisions of everything into perspectives. I think I'm making good progress, as with the diagram above. I think it is important to consider the two categories an adjunction is relating. What is the relationship between these two categories? The results seem to show the parallels between category theory (limits, colimits), algebra (defining structure maximally, minimally), logic (existential, universal quantifiers) and programming (currying, tensor, hom). Thus this may be a way of thinking about the Curry-Howard-Lambek correspondence. I still don't understand how it relates to my philosophy but perhaps it relates to the sevensome of dualities below.

2022.01.29 In thinking about the combinatorics of the Sheffer polynomials, I realized that each particle can be interpreted as having a clock. The clock relates two coordinate systems by steps {$\alpha$} going from one to the other and steps {$\beta$} going back. Each step in space can be thought as a tick by the clock. The five kinds of Sheffer polynomials reflects the ways that {$\alpha$} and {$\beta$} can degenerate. Physically, we have five zones in a scattering problem.

Distant pastUnentangled{$\alpha,\beta$}Meixner polynomials
Entering interaction potentialWave packet{$\alpha, 0$}Charlier polynomials
Within interaction potentialBound state{$\alpha,\alpha$}Laguerre polynomials
Leaving interaction potentialFree space (perturbed){$0,0$}Hermite polynomials
Distant futureEntangled{$\alpha,\bar{\alpha}$}Meixner-Pollaczek polynomials

Imagine particles as marbles, and imagine their interaction potential as an invisible bowling ball which they bounce off of. In the distant past, the potential is so weak that by Heisenberg's principle it is nonexistent. Likewise in the distant future. At some point the particle enters the potential, is within the potential, and leaves the potential. Currently, quantum field theory models events in terms of the results of measurements as the particles is leaving the potential. This is the most degenerate situation, where the clock has been reset, which I take to mean the wave function has collapsed. The combinatorics is given by the contractions in Wick's theorem, which is the combinatorics of the coeficients of the Hermite polynomials that describe the quantum harmonic oscillator as well as free space. A particle's clock is a combinatorial elaboration of this contraction which evidently describes not the interaction potential but the kinematics, what the particle experiences as it links two vertices of an edge in a Feynman diagram, which is to say, how it relates the two vertices as two coordinate systems. I suppose that in order to make calculations based not on free space but on some other of the five frameworks, one needs to consider the relevant combinatorics of the clocks in calculating how the particle creation and annihilation operators act not on the null state but, for example, on a bound state. I believe that if this can all be described by particles with clocks, then it would automatically generate the Minkowski space of special relativity, and so there might not be a need to postulate a Lagrangian density manifesting the continuum, which seems to me like overkill, unnecessary and thus unphysical.

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.

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 November 25, 2022, at 05:04 PM