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Research program, Research program introduction

[Change of clothes - and change of curtains to white - and open them to see flowers.]

One, all, many are a conception of a deeper framework, the learning cycle of taking a stand, following through and reflecting. In an upcoming episode I will relate this three-cycle to the Zig Zag Lemma in homological algebra. I think of this three-cycle as the threesome, a division of everything into three perspectives. Similarly, of fundamental importance are the twosome and the foursome. The twosome is the framework for existence, where in one perspective opposites coexist, as with free will, and in the other perspective all is the same, as with fate. I will explore how the twosome is modeled by the tensor-hom adjunction but many other structures as well. The foursome is the framework for the four levels of knowledge - whether, what, how, why - and I will show how the Yoneda Lemma serves as a knowledge switch in expressing that, and likewise, I imagine, the Univalence Axiom. I will keep returning to the twosome, threesome and foursome because they are the fundamental building blocks of Math 4 Wisdom.

And yet we will get familiar with all eight divisions of everything.

Mu: I have decided: Sing the Awesome song Now!

Mu: Nullsome, onesome, twosome, threesome, foursome, fivesome, sixsome, Sevensome! Nullsome, onesome, twosome, threesome, foursome, fivesome! ... is awesome!

Announcer: Fivesome for deciding. Every effect has had its cause but not every cause has had its effects so the critical point is decisive.

Mu: That's awesome!

The fivesome has two conceptions, in terms of time and in terms of space, and I have been exploring how it yields physics. Scattering problems distinguish five zones - before the interaction, entering it, within it, leaving it, and after the interaction. This fivefold distinction manifests as a fivefold classification of the orthogonal Sheffer polynomials. I have been investigating how the combinatorics of these polynomials expands upon Wick's theorem to enrich the edges of Feynman diagrams kinematically with particle clocks in five different ways.

The sixsome for morality distinguishes relative learning and absolute learning. It seems to appear in the Snake lemma. The sevensome defines a self standing logical system which balances what is known with what is not known. It consists of four corners and three sides of the logical square. In mathematics, it organizes seven kinds of duality.

If we added an eighth perspective, then we would have a system where all is known and all is unknown, and so the system would be empty, with no perspective. Thus the eightsome collapses into the nullsome.

Each division of everything describes our state of mind. There are three operations that shift our mind from one state to another. We can step-in with a perspective, as when we retrieve an answer, what we know, from our unconscious, and we can step-out with a perspective on a perspective, as when formulate a question, what we don't know, with our conscious. We can control ourselves, whether we step-in or step-out, with consciousness, by taking up a perspective on a perspective on a perspective. Thus consciousness is an operation which adds three perspectives to our state of mind. Here is an example: the twosome plus three perspectives is the fivesome, which means, that consciousness of the twofold framework for existence is modeled by the fivefold framework for decision-making, which we conceive in time or space. I think that is the wondrous wisdom in Immanuel Kant's transcendental deduction. There should be 24 such equations and we'll be seeking out in math and in life.

The crucial point about the three operations +1, +2 and +3 is that they describe our experience of cognitive frameworks as taking place within an eight-cycle of frameworks of perspectives. This is why I am very interested to intuit the Bott periodicity of period eight, which furthermore is a key to the homotopy groups of spheres, which describe how spheres of various dimensions can wrap around each other, and quite possibly could model the algebra of perspectives which I seek. Bott periodicities of period eight and of period two together establish the tenfold way for classifying topological insulators, which is very philosophical because of its relation to charge, parity and time reversal symmetry. Becoming, as a concept, distinguishes forward and backward, thus negates time-reversal symmetry. Being, as a concept, distinguishes the presence and the absence of a particle, and thus negates charge conjugation. I am trying to learn and understand Bott periodicity very concretely in terms of representations of Clifford algebras.

The divisions of everything are fundamental but elusive because we don't conceive them directly but only through conceptions, which in the past I have called representations. Thus when we conceive four levels of knowledge we must either conceive them from the observer's point of view as four questions - why? how? what? whether? - from the observed's point of view as four answers - why! how! what! why! The foursome, fivesome, sixsome, sevensome each have two such conceptions in terms of increasing slack and decreasing slack, as with questions and answers. Whereas the nullsome, onesome, twosome, threesome each have four conceptions, which I think of as four scopes: everything, anything, something, nothing. The learning three-cycle consists of taking a stand, following through and reflecting but we must conceive that in one of four ways, depending on our vantage point, as necessary-actual-possible, object-process-subject, one-all-many, or being-doing-thinking.

My hunch is that the system of six conceptions is, mathematically, the yoga of Grothendieck's six functor formalism. Therefore I am learning about adjunctions and derived functors. Adjunctions describe how the same information appears in different contexts. They capture various mathematical intuitions as to what is trivial, and so I am trying to classify them and understand what they say about perspectives. Another hunch is that the six conceptions are made concrete, combinatorially, by the Hopf algebras and the six natural bases of the symmetric functions.

Holistically, we conceive of the entirety of a division of everything by way of the six conceptions, thus by a *perspective* upon taking a perspective. We can also imagine an individual perspective, as taking a perspective upon a *perspective*, by way of twelve mental circumstances, namely necessary-actual-possible, object-process-subject, one-all-many, being-doing-thinking. Which is to say, locally a perspective is understood as taking part in a conception of the three-cycle for learning. The twelve circumstances are the vocabulary of the imagination. In mathematics, they bring to mind Gian-Carlo Rota's twelvefold way in combinatorics, which, however, seems rather degenerate. Immanuel Kant conceived of twelve categories but I think instead that the twelve circumstances are not the things we imagine but rather the backdrops, the contexts by which we imagine. Kant thought they derived from the logical form X implies Y, but I argue they arise as mind games which are triggered by the four representations of the nullsome - true, direct, constant, significant - which themselves are negations of the four levels of knowledge - what is true or obvious cannot be hidden and thus negates whether; what is direct is not represented and thus negates what; what is constant does not change and thus negates how; what is significant cannot be encompassed and thus negates why. You get a sense of how the the system is its own metasystem in that they are all built from the same building blocks.

Mu: One, all, many arise from the search for constancy. Listen to this!

Rap: I'm searching for constancy, I'm focused on this fantasy, I'm trying to arrange to go without change, that'll be the sign that you truly are divine, I'd really like to meet ya, Princess Incognita, I'm searching for constancy, darling would you dance with me, slowly, if you please, we could go into a freeze... thinking through our chances in three lovely circumstances, I find you in the one, the all, the many.

Mu: What's with all the static?

Yes, so far it is mostly static. The eight divisions, the six representations and the twelve circumstances are all static structures, so on top of that we do need dynamic languages. My working hypothesis since 1987 is that we experience the dynamics of life as three conceptual languages. Argumentation is how things come to matter and takes us from twelve circumstances to six conceptions. Verbalization is how things acquire meaning and takes us from eight divisions to twelve circumstances. Narration is how things happen and takes us from the six conceptions to the eight divisions. I have worked out a theory of narration but am still working on verbalization and argumentation.

This entire system is very compact, and feeds upon itself, and the 4x2=8 divisions, 2x3=6 conceptions, 3x4=12 twelve circumstances bring to mind the vertices, edges and faces of a cube or an octahedron. I have never found a meaningful connection with such a geometry but the numbers suggest that this basic system is simple and in some sense complete.

Mu: I'm hearing feedback.

[Get the mail. A letter from Brother David Ellison-Bey to Brother Andrius Kulikauskas. "Islam. Keep it real. Uplift fallen humanity. Peace."]

[Brother David about Brother Tim's vegetarian bakery.]

[Brother David with the flower.]

I now step outside of myself, so to speak, to consider why these three structures and why these three languages describe how we experience life.

I think the answer why is given by four more frameworks which I have encountered in Scriptures but also find evidence for through empirical investigation. Each of these frameworks consists of eight perspectives but by considering their different purposes I was able to distinguish them as four distinct frameworks. They are all purposeful in that they describe not simply how we experience life but rather how our life within system relates with life beyond system. These are four frameworks of reservations which express how and why we might stay within system and yet also how and why even so we might go beyond system. There are reservations of the body in terms of needs, of the mind in terms of doubts, of the heart in terms of expectations, and of the will in terms of values. Each framework provides six ways of functioning within a system. But each also provides a seventh way by which we can, within system, ignore ourselves and focus on others, and also an eighth way, by which we could be without system. Our mind can encompass six ways but with eight ways we find ourselves as if inside a house of intuition with eight familiar rooms which yet our mind cannot encompass all at once.

For example, we have six needs and six operating principles for addressing them - for survival, we cling to what we have; for security, we get more than what we need; for society, we avoid extremes; for self-esteem, which is survival of the psyche - we choose the good over the bad, for opportunity, we choose the better over the worse, and for self-fulfillment, we choose the best over the rest. These operating principles are rather mathematical and I'm curious where we might encounter them in mathematics. They are quite mechanical but there is freedom in a seventh option, which is that I can ignore my own needs by taking up the needs of another. And there is an eight option which is that I may be perfect and have no needs at all.

What is the purpose of such reservations? They seem to resist a spirit that wishes, a spirit which is beyond system and yet goes beyond itself into system, which is to say, into us, and may thus live through us. I imagine that such a spirit starts out with no perspective but proceeds to take up a perspective, and then a perspective upon a perspective, and finally a perspective upon a perspective upon a perspective, by which it can see itself go beyond itself. At each stage, the spirit wishes but we have reservations. The spirit wishes for nothing, is self-sufficient, whereas our bodies have needs. The spirit wishes for something, is certain, whereas our minds have doubts. The spirit wishes for anything, is at peace, whereas our hearts have expectations. The spirit wishes for everything, is loving, whereas our wills have values. I think of this spirit as God. The frameworks of reservations relate God within us, within system, with God beyond us, beyond system.

In future episodes I will describe how we respond to our needs with operating principles, we respond to our doubts with counterquestions, we respond to our expectations with directions to and from the good, and we respond to our values with investigatory questions. In each case, we can choose whether to live in terms of our own selves, within system, or to live in terms of God beyond system. For example, in the gospel of John, there are eight statements by Jesus of the form "I am...", which apply the operating principles with regard to God. Jesus says, "I am the resurrection and the life", and I take that to mean that we can cling to God's glory rather than cling to our own lives. Understandably, we have reservations about making such a choice with our lives. Yet, from a practical point of view, such choices are what our lives are all about. I will be producing a series of videos to detail how these four frameworks of reservations arise in practical applications of Math 4 Wisdom. For example, I will show how the counterquestions are useful for freeing our minds, and thus for peacemaking, for engaging the violent, for letting go of our own point of view and looking at everything from their point of view. I will also be looking for where these structures arise in mathematics. For example, our expectations and emotional responses give rise to a language of moods which plays with the boundary of our self by means of six transformations: reflection, shear, rotation, dilation, squeeze and translation. Some of these are Mobius transformations. All six are related to the Lie group or Lie algebra of SU(2). I am curious if gauge theories may express the four reservations in terms of zero dimensions, one dimension, two dimension and three dimensions of freedom for perspectives, and this brings to mind the Standard Model U(1) x SU(2) x SU(3) if we could supplement it with a zero-dimensional gauge theory, perhaps for gravity, however contradictory that may be.

Personally, a particularly significant working hypothesis of mine since the year 2000 is that wishes take up reservations in six empathies which are expressed structurally as the divisions, conceptions, circumstances, argumentation, verbalization and narration. This makes me curious to understand the Poincare group of Minkowski spacetime isometries as it similarly relates four dimensions and six pairs of dimensions.

I further expect that there is a bigger picture in which the four reservations and six empathies are ten structures within a system of twenty four structures that describe the flow of experiences of an individual first person – Me. Empirically, as data for this investigation, I am studying and modeling every aspect of the meaningful experiences in my life by which I have grown as a person. These are experiences where I have let go of my old self and taken up a new self, which is to say, I have let go of life as usual, and however briefly have lived eternal life, here and now.

[More feedback - Senele - sweathearts - SU(2) - take it higher.]

Theoretically, at this point, if I want to point to evidence of twenty four structures, then I need to show a broader context of how to model the worlds of the zeroth person, God, the first person, I, the second person, You, and the third person, Other.

[Organizing the ways I figured things out.]

A rather similar 24-fold system arises in surveying the ways of figuring things out in any scientific discipline. Notably, in mathematics we can document the patterns by which we solve problems as collected by George Polya, Paul Zeitz and others. I have gathered almost 200 examples of mathematical problem solving and sorted and organized them in a system of 24 ways of figuring things out mathematically. We start by acknowledging that we are always free to work independently and start from scratch. Four algebraic ways focus on a single problem, identifying the center, leveraging balance, forming sets of solutions (as with roots of polynomials), and lists of solutions (as with basis elements for a vector space)). Four analytic ways consider a sequence of problems, noting maximal or minimal situations, then least upper bounds and greatest lower bounds, and ultimately, limits. Algebraic and analytic approaches are intertwined by a three-cycle which has us extend the domain of our solutions, apply continuity to discover critical points, and self-superimpose sequences to formulate new relationships. These presystemic ways all come together in the most central way of figuring things out, which is to set up a symmetry group that expresses the possible states of a mathematical observer. At this point we can work in a system and its metasystem, which are related in four ways: revealing truth through contradiction, working with simplifying models, thinking backwards in terms of implications, and leveraging the ambiguity in variables. Indeed, I can devote a single episode to the 24 ways of conceiving variables. Pairs of these levels yield the six ways of visualizing mathematical structures, which arise by restructuring sequences, hierarchies and networks. Finally, in working within an explicit system, we can appreciate the significance of context. It is time to ask you a simple question, what is 10 + 4 ? (You can pause if you want to think about it.) The answer is 2 o'clock because 10 o'clock plus 4 o'clock is 2 o'clock! Cuckoo, cuckoo! This joke shows that we can never make a system explicitly unambiguous.

I am excited to think that we can work together to collect more examples and sharpen this system of ways of figuring things out. We can do this not only for mathematics but for other scientific disciplines as well. I have collected a database of more than 1,000 examples and together we can collect thousands more. I have sketched out the 24 ways of figuring things out in physics, in biology, in neuroscience, and in chess. I have systematized the business innovation games known as Gamestorming, and argued how such game playing could be the foundation for universal grammar. I have also sketched out the ways of figuring things out used by personalities such as Jesus, the Gaon of Vilna and my very self. Basically the same epistemological system arises for any observer You who is mastering their domain of interest or making what they can of their own life. I wonder how these different epistemological systems unfold, along with their observers. So let us work together on that and learn the language of wondrous wisdom.

[Organizing how I imagine God.]

I do think a lot about God because of my ambition to know everything. In 2016, I surveyed and structured the ways that I imagine God. Mathematically, I think of God as a state of contradiction, beyond system, in which all statements are true. What could arise from such a state? How could it give rise to a state of noncontradiction, a self standing logical system? I think of God as motivated by the question, Is God necessary? Would God exist even if God did not? This brings to mind a proof by contradiction, which divides everything into two tracks: if God, then God, trivially, as in the spiritual world. If not God, and yet even so there emerges God - such is the physical world, if God necessarily exists. This line of imagining yields a God who understands, a God who comes to understand, and a God who is understood, by which they are the same God. Mathematically, this might be modeled by the mythical finite field with one element, which can be interpreted as zero or one or infinity. This further brings to mind the anharmonic group, and its action on zero, one and infinity. Conceptually, the three divine vantage points contribute a threefold structure, an eightfold structure and a tenfold structure, and an additional three-cycle combines for a system of 24 ways of imagining God which I call God's dance.

My hypothesis is that there is a 24 fold worldview for each of the four persons - God, I, You, Other - which unfold in that order. God's dance models the ways of the spirit. My flow of experiences manifests the structures revealed when spirit recedes. Your epistemologies, your houses of knowledge are representations of structures in terms of investigations. Other’s reality consists of 24 kinds of love for unity, namely, the unity of the representations of the structure of spirit.

[More feedback - childhood - take it higher - up with a sow thistle - tree house.]

All that I have sketched so far reflects the limits of the imagination. But there is more!

We can go beyond the imagination by considering the unimagineable. ...

At the end of his life, Jesus said to his disciples, "I have much more to tell you but you cannot bear it now. But when the Spirit of truth comes, he will guide you into all truth..."

What could be this bad news that is hard to bear? I think it is the most wondrous wisdom that I have come upon.

God doesn't have to be good.

Life is the fact that God is good but eternal life is understanding that God doesn't have to be good, life isn't fair. God is beyond conditions and good is God within conditions.

I came to this from investigating the Gospel of John, and from the language of wondrous wisdom and from personal experience. Experience of a good person -

This is elaborated by the table of life, which defines what is life and what is eternal life. [...]

How to fit everything together, all of wondrous wisdom [...]

Finally, I will briefly sketch what I would argue is beyond those limits based on my exploration of those limits. For we can allow for what we cannot imagine, as we know from mathematics but also our choices in every day life. The imagineable is given by a perspective on a perspective but we can also consider the unimagineable as given by a perspective on a perspective on a perspective, and likewise the definite given by a perspective, and the indefinite given by no perspective at all. These are four stages by which the spirit goes beyond itself. It starts as indefinite, then proceeds to the definite, and then the imagineable, and finally, the unimagineable. This is how God, as such, the indefinite, becomes somebody's God, my God or your God, the unimagineable. This is the transcendence by which God manifests himself, in that those things are which show themselves to be. I think an example in math is the Morita equivalence by which rings are equivalent if their categories of modules are equivalent.

The unimagineable God is those very persons - God, I, You, Other - who we take to inhabit the four respective worldviews that we imagine. We accept and leverage such persons who we don't directly imagine but rather who we choose to imagine through.

Furthermore, as I try to chase down the assumptions that I have been making in exploring the imagineable, there is a most fundamental notion of perspective and what it means to define it, to understand it, what it means for anything to be definite.

Four columns of going beyond oneself - indefinite, definite, imagineable, unimagineable - God becomes somebody's God

Four level of self-understanding - 4, 3, 2, 1

Wisdom of lost child

[Chimney (or well) - relation between God and person]