Accretion Event - a presentation of the expressive compound “A = {h, x, y, z…}, X = {x}, H = {x, u, v, w…}”

By Calum Hazell

1. Note

Formal Shamanic Conduct: Zonal Installations in Supernatural Space (Agger: EyeCorner Press, 2022)¹ stages an indisciplinary practice out of certain anthropologic and set-theoretic resources. Indisciplinary practices stipulate novel parameters for generic executions of conceptual matter because their aim is to recover (through local performances of) the radical democratic and outside-disciplinary status proper to thought as such. Formal Shamanic Conduct is consistent with this objective and advances, in part, a mathematical fiction. No one set-theoretic construct is applied here, but fragments of wellfounded and nonwellfounded convention, language, and patterns of sense-making are integrated in the service of an anthropological mathematics: a practice of mathematics for studying who and what counts as human where and when.

Broadly speaking, Formal Shamanic Conduct simplifies set-theory to an operation of containing human and animal elements in sets, whilst the practice of shamanism is reduced to a mechanism of traversing the thresholds of the sets within which those elements are variously contained. The set-theoretic artefact of containment abbreviates membership relation, and the anthropologic-shamanic one of traversal involves compression of shamanic initiation protocols and affordances amongst the Yanomami, principally due to Kopenawa.² With the exception of the clarification offered at endnote 3, my usage of set-theoretic parlance in what follows is as standard if executed towards non-standard ends.

In the first place, the essay models propositions that “The human is an animal” and that “The animal is a human” with formal proofs as means of excavating and clarifying local containments it is going to be possible, via their simple concatenation at a humanimal zonality, to traverse. For their part, the propositions are karaoke versions of well-known songs about humans and animals. The essay is not interested in endorsing the propositions, but in their explication in the service of a formal shamanic practice.

“The human is an animal” says “The human is a kind of animal,” expressing containment of the human h via containment in a set of all the humans³ H in a set of all the animals A beside all the others x, y, z and so on, that is:

∀h, h ∈ H : H = {h}, ∀ h, h ∈ A, A = {h, x, y, z…} : H ⊊ A.

“The animal is a human” says 1) “This kind of animal is human,” expressing unique containment of this animal x via containment in the set of all such animals X in the set of all the humans H, like so:

∀X ∀ H (∀ x (x ∈ X ⟺ x ∈ H) ⟹ X = H,

and 2) “This animal is a kind of human,” expressing containment of this animal x via containment in the set of all of them X in the set of all the humans H amongst all the other humans u, v, w and so on, so that:

∀x, x ∈ X : X = {x}, ∀_x_, x ∈ H, H = {x, u, v, w…} : X ⊊ H .

Which is to say “The animal is a human” expresses the unique & nonunique humanity for whatever this animal x is going to be due to the nonproper & proper containment of the set of all this animal x X in H, where “&” says perpetually and all at once. This dynamic can be abbreviated in such a way that for every animal, it is the only human for itself.

The proofs are stepwise architectures allowing retrospective extraction of interstitial human and animal situations and circumstances. These are interstitial because between one another and the (preparatory) colloquial and (subsequent) formal sayings of the proposition to hand. “The human is an animal” generates the circumstances H = {h}, A = {h, x, y, z…}, H ⊊ A /A ⊋ H. “The animal is a human” generates the circumstances X = {x}, H = {x}, X = H about its first tendency and X = {x}, H = {x, u, v, w…}, X ⊊ H /H ⊋ X about its second. The humanimal zonality is built out of these, like so:

{(H = {h}), (A = {h, x, y, z…}), (H ⊊ A /A ⊋ H), (X = {x}), (H = {x}), (X =H), (H = {x, u, v, w…}), (X ⊊ H /H ⊋ X)}

Interstitial circumstances are integrated at the humanimal artefact and secured as discrete expressions via their bracketing (( )). The thresholds of the zonality are secured where it is braced ({ }). Given their discrete integration, local circumstances need not constitute a special narrative or linear explanation: The prospects for erecting zonal installations are predicated upon scalable tractabilities of the artefact and the circumstances it hosts. Zonal installations are prepared by stipulating collections of circumstances from out of the humanimal plane. This requires removing the parentheses securing definitions of interest and drawing a circle or circles around them.⁴ A collection composes a novel expressive compound serving as experimental object and nickname for a zonal installation.

The following installation is interested in the collection “A = {h, x, y, z…}, X = {x}, H = {x, u, v, w…}”. It maintains fidelity to the humanimal artefact by adopting this little string as a nickname and to the malleability of humanimal circumstances by what it does to them. The compound unfolds throughout the installation to which it gives its name. The unfolding of a compound makes an experiential pathway along which a zonal operator would be ramified or undergo. Adjustments, prostheses, erasures delivered on the human and animal circumstances at a given installation are intentional or nonintentional impressions made on a pathway: stumpings of thought, ornaments of navigation, technologies of prayer.

2. “A = {h, x, y, z…}, X = {x}, H = {x, u, v, w…}”

I have prepared this installation by making adjustments to each of the definitions out of which the region A = {h, x, y, z…}, X = {x}, H = {x, u, v, w…} – that is to say, its nickname – is composed. I have substituted for the elements x in A and H those c and d respectively, and for h in A an element b, because according to the event this installation has been designed to stage, there should be no incidents of proper containment amongst A, H, and X.⁵ Similarly, I have renamed x in X “x₀” to avoid recursive/tautonymous structuration of X as regards its unique inhabitant. These adjustments are presented at the stations I through III above. It will also be observed that, as clarified diagrammatically via II and V, “X = {x₀}” is to say “{{b, c, y, z, d, u, v, w…}}” due to x₀ = {b, c, y, z, d, u, v, w…}. So “x₀” names a collection {b, c, y, z, d, u, v, w…} intersecting with (⋂) A and H about {b, c, y, z} and {d, u, v, w} respectively. Intersections of A and H with a being = x₀ are detailed at the stations IV through VI and illustrated with recourse to the items and as regards x₀ ⋂ A and x₀ ⋂ H respectively.

With these adjustments to hand, the expressive compound “A = {h, x, y, z…}, X = {x}, H = {x, u, v, w…}” captures an accretion event in medias res: It stages an encounter for the zonal practitioner with a being = x₀ who appears to be assembling themself at a world = X out of animal {b, c, y, z} and human {d, u, v, w} elements. In the terms of this installation, then, the members of the sets A and H do not refer to particular kinds of animals and humans, but to particular properties, qualities, tendencies, abilities. Perhaps at this installation, these elements b, c, y, z, d, u, v, w are talking in no particular order about facial hardware, wingspan, deriving joy from the good fortune of others, vocalisation, floppy ears, occupation, autotomy, lacking eyes. And they are the ingredients out of which a being = x₀ is gradually becoming, resolving, alighting upon themself at a world = X before the zonal practitioner. Let A contain all animal elements and H all human elements where ‘elements’ are properties, qualities, tendencies, abilities. The animal elements called b, c, y, z and the human ones called d, u, v, w are the properties, qualities, tendencies, abilities with which a being = x₀ has been determined to assemble themself. “x₀ = {b, c, y, z, d, u, v, w…}” is to say as things stand, or at the juncture at which the encounter is staged, “x₀ ⋂ A = {b, c, y, z}” and “x₀ ⋂ H = {d, u, v, w},” where as things stand is to say “… at x₀, A, and H respectively.”

The ellipses attached as suffixes to the definitions for A and H indicate these sets contain more elements than those b, c, y, z and d, u, v, w respectively, rendering their cardinality indeterminate. These additional elements, whose indeterminacy is both a qualitative and quantitative concern, are abbreviated with recourse to a proliferation of items populating the sets of all animal and all human elements above. The ellipsis attached as a suffix to the definition for a being = x₀ indicates b, c, y, z, d, u, v, w do not account for every property, quality, tendency, ability with which they are determined to assemble themself in a world = X. Instead, they would like to gradually become less disjoint, to intersect more and more with A and H⁶ via determination of some animal and human elements in the sets containing all of them respectively. In my diagram, the items and operate as formal placeholders for the prospective determination of some additional human and animal elements about which a being = x₀ would also intersect with the sets containing all of them respectively. There is a being = x₀, then, who is intersecting and almost disjoint (and prospectively less so) with the sets of all animal elements A and all human ones H about some elements determined by the names b, c, y, z, d, u, v, w they are called. The ellipsis attached as a suffix to the definition for a being = x₀ indicates the site from which these names and other names have been and are being derived. The names b, c, y, z, d, u, v, w are animal and human resolutions for problems a being = x₀ poses of themself of the kind: “What do I like to eat?” “What jewellery do I wear?” “How do I dance?” “Am I confident?” “Do I have tusks?” “Can I breathe in the water?” “What is my fragrance?” “Are my teeth very sharp then?” These problems are structured with recourse to the items and in the diagram above, where dotted circles indicate sites (within the parameters of a being = x₀) from which they are posed, dotted lines compose pathways along which they are extended, and dashed circles indicate sites (within the parameters of A and H) at which they are animally or humanly received and resolved.

So the ellipsis attached as a suffix to the definition for a being = x₀ indicates the mechanics of determination of animal and human elements in the sets containing all of them from without them. There is a being called “x₀” because they are placed just at their origins, so that the sets of all animal and all human elements are known as “A“ and ”H“ relative to an origin with that name. There is a being = x₀ who orchestrates the accretion event captured (with its various adjustments to hand) at the expressive sequence “A = {h, x, y, z…}, X = {x}, H = {x, u, v, w…}” insofar as they stipulate the terms of disjointness enjoyed between 1. themself and the sets of all animal and all human elements, 2. the sets of all animal and all human elements, 3. the sets of all animal and all human elements and a world = X they inhabit alone:

1. As per endnote 6, the sayings “| x₀ ⋂ A | < ∞” and “| x₀ ⋂ H | < ∞” regulate the intersection a being = x₀ enjoys and can possibly enjoy with the sets of all animal and all human elements. Which is to say both as things stand (when x₀ ⋂ A = {b, c, y, z} and x₀ ⋂ H = {d, u, v, w}) and prospectively (when x₀ = {b, c, y, z, d, u, v, w…}). There are no prospects for the proper containment of A and/or H as subsets in a being = x₀ because a being = x₀ is discerning as to the animal and human elements they are determined to stipulate and with which they are determined to assemble themself in a world = X.⁷ So there are no prospects for the ellipsis attached as a suffix to the definition for a being = x₀ and those attached as suffixes to the definitions for the sets of all animal elements and all human ones to ever meet: a being = x₀ will always enjoy some finite intersection with/disjointness from A and H whatever animal and human elements it would determine in the sets containing all of them respectively.

2. A ⋂ H = {} is a fact stipulated out of a being = x₀ who inhabits a world = X uniquely about the sets of all animal and all human elements where elements are understood as properties, qualities, tendencies, abilities. A ⋂ H = {} is to say that there are no prospects for the determination of a human element in the set of all animal elements nor an animal element in the set of all human ones. So just as there are no prospects for the ellipsis attached as a suffix to the definition for a being = x₀ and those attached as suffixes to the definitions for the sets of all animal elements and all human ones to ever meet, nor are there for the ellipses attached as suffixes to the definitions for the sets of all animal elements and all human ones to ever meet: A and H will always enjoy disjointness from one another whatever animal and human elements are determined in them from without them respectively. Moreover, just as there are no prospects for the proper containment of A and/or H as subsets in a being = x₀, so is A ⋂ H = {} ultimately to say there are no prospects for the proper containment of a being = x₀ in either A or H because a being = x₀ contains as elements those collections of elements {b, c, y, z} and {d, u, v, w} about which A and H are disjoint.

3. The disjointness of a world = X from the sets of all animal elements and all human ones is illustrated via the expressions “X ⋂ A = {}” and “X ⋂ H = {}” describing the vertical items ![[Pasted image 20221219193013.png]] above. There is a world = X inhabited uniquely by a being = x₀ who has been determined to contain in themself elements = {b, c, y, z, d, u, v, w…}. There is not a world = X containing these elements, but containing only {{b, c, y, z, d, u, v, w…}}, that is, a being = x₀.⁸ So a being = x₀ does not share its problems with, or allow them to touch, a world = X, but assembles of themselves an architecture for them that they would resist the disjointness enjoyed by a world = X and the sites of their animal or human resolution.

A is a library or archive of all the animal elements and H of all the human ones over which there is a being = x₀ in a state of survey,⁹ or who is curating them. Which is to say the sets of all animal elements and all human ones, where elements are understood as properties, qualities, tendencies, abilities, are curated of a being = x₀ who is determined to discern themself from out of them. There are no elements determined by the names they are called but those curated and surveyed, such as {b, c, y, z} and {d, u, v, w} respectively, of a being = x₀. Thus, as to the question “Whence are the elements at A and H in addition to those {b, c, y, z} and {d, u, v, w}¹⁰ respectively derived?” the zonal practitioner will respond that “They are derived from a being = x₀ in A and H insofar as they are disjoint from them. They are derived from the sets of all the animal elements and all the human ones from without them.” A and H are animal and human continua – horizons of all animal elements and all human ones – observing and observed of their constitutive disjointness at an origin = x₀ in a world = X. There is a being = x₀ for whom the sets of all animal elements and all the human ones compose, across their styles and degrees of disjointness, a templum = X. A and H are, in the style of Varro, ‘truthful trees… that within them the regions are set where [a being = x₀ is] to gaze.’¹¹ So that there is a being = x₀ who is envisioning a world = X as does the auspex their templum, who is painting and navigating a world = X with some instruments that are not of it but, quite discerningly out of A and H, in of them.

Notes

1. Formal Shamanic Conduct can be purchased via https://www.eyecorner.press/books-recent/formal-shamanic-conduct. Calum can be contacted via calum.hazell.2017@live.rhul.ac.uk.
2. See Davi Kopenawa and Bruce Albert, The Falling Sky: Words of a Yanomami Shaman, trans. Nicholas Elliott and Alison Dundy, (Cambridge, MA, and London: The Belknap Press of University of Harvard Press, 2013).
3. In violation of a set-theoretic stylistic convention according to which lower and uppercase renderings of the same characters pick out the same object, Formal Shamanic Conduct adopts a convention such that items c are nontrivially beings contained and items C nontrivially containers for all such beings.
4. The number of circles drawn simply reflects the proximity or distance of the definitions of interest to or from one another at the stock rendering of the artefact indicated above.
5. I.e, X and H ⊊ A ∵ A = {h, x, y, z…}, and X ⊊ H ∵ H = {x, u, v, w…}.
6. I.e., x₀ ⋂ A = {b, c, y, z} is to say “| x₀ ⋂ A | < ∞” and x₀ ⋂ H = {d, u, v, w} “| x₀ ⋂ H | < ∞” where the items || talk about the cardinality of an intersection and those < ∞ about its finitude. Taken together, the notations of the kind “| x₀ ⋂ S | < ∞” inform us that there is a being = x₀ who is almost disjoint with such and such set S on account of their finite intersection, and those of the kind x₀ ⋂ S = {m₁, m₂, m₃, m₄} inform us as to the members (m_n) of the intersection they enjoy. Thus, intersections of a being = x₀ with sets A and H are both and variously = 4 considering {b, c, y, z} and {d, u, v, w}.
7. Indeed, to be discerning in a world = X as to their determination of particular animal and human elements is a definition for a being = x₀.
8. I.e., a world = X is understood as having a cardinality = 1 (since X = {{b, c, y, z, d, u, v, w…}}), whereas there is a being = x₀ who it contains as its unique inhabitant understood as having an indeterminate cardinality < ∞ and > 8. < ∞ since a being = x₀ determines themself discerningly, and > 8 since (as to which the zonal practitioner is witness) a being = x₀ is discerning themself.
9. E.g. see (Gilles Deleuze and Félix Guattari, What is Philosophy?, trans. Hugh Tomlinson and Graham Burchell, (New York: Columbia University Press, 1994), 20-1) the deleuzoguattarian concept of concept in relation to its components:

A concept is a heterogenesis, that is to say, an ordering of its components by zones of neighborhood… The concept is in a state of survey in relation to its components, endlessly traversing them according to an order without distance. It is immediately co-present to all its components or variations, at no distance from them, passing back and forth through them: it is a refrain, an opus with its number.

1. I.e, as per the ellipses attached as suffixes to the definitions for the sets containing all animal elements (where A = {b, c, y, z…}) and all human ones (where H = {d, u, v, w…}).
2. Reflecting upon cultic augural practices of pre-Imperial Rome, Varro informs us (see Marcus Terentius Varro, De Lingua Latina, trans. Roland G. Kent, (Massachusetts: Harvard University Press, 1938), 275-7) that the geographic/geometric complex of the templum was constructed by an auspex through the recital of a sacred formula precipitating the suspension of a nominated parcel of land from the surrounding landscape. In De Lingua Latina, this formula, whose precise wording varied in accordance with the specific motivation of the auspicium, is recorded as follows:

Temples and wild lands be mine in this manner, up to where I have named them with my tongue in proper fashion.

Of whatever kind that truthful tree is, which I consider that I have mentioned, temple and wild land be mine to that point on the left.

Of whatever kind that truthful tree is, which I consider that I have mentioned, temple and wild land be mine to that point on the right.

Between these points, temples and wild lands be mine for direction, for viewing, and for interpreting, and just as I have felt assured that I have mentioned them in proper fashion.

‘In making this temple,’ Varro summarises, ‘it is evident that the trees are set as boundaries, and that within them the regions are set where the eyes are to view, that is we are to gaze…’

© 2022 Calum Hazell

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