The above proposition is occasionally useful.
Bertrand Russell & Alfred North Whitehead, Principia Mathematica (below the proof of ‘1 + 1 = 2’)
Tentative. Always tentative. Structural and intentional features are conflated to some degree here to reveal their connections and to show how distinct structures generate distinct intentional features. However, more work is needed to properly bridge the two. Or is it the three?
- ∅
- {∅}
- ∅ ≠ {∅}
- ∅ & { }
- ∅ & { } & {∅}
- ∅ is nothing.
- { } is emptiness.
- {∅} is the emptiness in which nothing appears: absence-in-itself.
- {∅} is the precondition of Being.
- Being is presence-within-absence, or filled emptiness.
- ∅ and {∅} are pre-ontological (Pre(x)), or prior to existence: Pre(x) iff x ∈ {∅, {∅}}.
- Being describes the set generated by the successor function operating on {∅}: S: Ord → Ord, S(n) = n ∪ {n}.
- Therefore, to be is to be in addition to ∅: ∃x iff (x ∈ Im(S) & ¬Pre(x)).
- In Ord, ∅ = 0 and {∅} = 1; however, ∅ and {∅} ∈ Pre(x), so the first member of the set of existent things is 2: S({∅}) = {∅} ∪ {{∅}}, or {∅,{∅}}.
- {∅,{∅}} is presence-in-itself, or Being as such: the first division in absence that makes absence present to itself as that which is in addition to itself.
- 3, or {∅,{∅},{∅,{∅}}}, then, is presence-to-itself, a division in presence that allows for Being to become present to itself. This results from the asymmetry and irreversibility of the Ord sequence. {∅,{∅},{∅,{∅}}} is related to {∅,{∅}} in that {∅,{∅}} is contained in {∅,{∅},{∅,{∅}}}, such that {∅,{∅},{∅,{∅}}} is about {∅,{∅}} but not vice-versa.
- For any existent x (x ∈ Im(S) & ¬Pre(x)) and any y ∈ Ord, x is about y iff (i) y ∈ x, (ii) y ∉ x* for any x* ∈ x at the same Ord level, and (iii) the relation between x and y is irreversible under S; or: R_p(x,y) iff y = S⁻¹(x), where R_p denotes a primary aboutness relation.
- In any case where y itself meets the condition R_p(y,z) iff z = S⁻¹(y), then a secondary aboutness relation holds between x and z, such that R_s(x,y,z) iff z = S⁻¹(y) & z = S⁻¹(S⁻¹(x)). The primary aboutness relation determines the immediate ancestral context of x, while secondary aboutness refers to the chain of ancestral contexts of y, z, …, n contained in x via repeated applications of S⁻¹(x). (Note: the conditions z = S⁻¹(y) and z = S⁻¹(S⁻¹(x)) are equivalent given R_p(x,y); however, it is worth making explicit the connection between S⁻¹(y) and S⁻¹(S⁻¹(x)) in differentiating primary and secondary aboutness)
- 3 generates identity, where Being takes itself as identical with itself. Presence-in-itself is contained in the present-Being of presence-to-itself, so that presence-to-itself is about its own presence-in-itself.
- {∅,{∅},{∅,{∅}}} is identical with the set containing ∅, {∅}, and {∅,{∅}}, and this identity includes an element that is not pre-ontological: {∅,{∅}} (part of what is identical with {∅,{∅},{∅,{∅}}} itself exists).
- 4, or {∅, {∅}, {∅, {∅}}, {∅, {∅},{∅,{∅}}}}, is presence-for-itself, wherein Being’s presence-to-itself takes part of itself as in addition, and so determinable by itself. {∅, {∅}, {∅, {∅}}, {∅, {∅},{∅,{∅}}}} contains presence-in-itself and presence-to-itself in such a way that {∅, {∅}, {∅, {∅}}, {∅, {∅},{∅,{∅}}}} is about {∅, {∅},{∅,{∅}}}, which is itself about {∅,{∅}}.
- 4 generates subjectivity, where Being’s identity as its present-Being takes itself as oriented towards another-Being, as Being-beyond-present-being, or Being-toward.
- {∅, {∅}, {∅, {∅}}, {∅, {∅},{∅,{∅}}}} projects forward from the initial identification of {∅, {∅},{∅,{∅}}} with {∅, {∅}} as something in addition to this identity that is then identical with the set that contains the members of initial identity: ∅, {∅}, {∅,{∅}}, and {∅, {∅},{∅,{∅}}}.
- Time is the result of additions to 4, where Being-toward is realized in additions that have aboutness relations with their ancestors, such that Being’s prior determination (y = S⁻¹(x)) is a member of its present-Being (S(x)), and its prior determination has a primary aboutness relation to y = S⁻¹(S⁻¹(x)), and so forth, terminating in the initial identity of {∅,{∅},{∅,{∅}}} with {∅,{∅}}.
- A time slice refers to any individual step in the sequence (which contains all prior steps), while time proper is the whole sequence understood in its (always-already incomplete) totality. Time is incomplete because it is open by the non-closure of S, with each addition containing all prior members of Ord and so all prior primary aboutness relations, rendering Being-across-Time as always-in-addition-to. Each time slice is a unique addition; time is the whole of all unique additions without end.
Number is the ruler of forms and ideas, and the cause of gods and daemons.
Pythagoras, quoted by Iamblichus of Chalcus in The Life of Pythagoras
Image: Black Square by Kazimir Malevich (1915)
Fish In the Afternoon 