in a conditional relation, subtraction segregates a virtual structure () apart from actual structure {}. a virtual structure is only held up by the dots in it.

a full subtraction removes all actual structure to leave behind only virtual structure:

1' - 1' = · {·} - · {·} = · (·)

2' - 2' = · {·} { · {·} } - · {·} { · {·} } = · (·) ( · (·) )

3' - 3' = · {·} { · {·} } { · {·} { · {·} } } - · {·} { · {·} } { · {·} { · {·} } } = · (·) ( · (·) ) ( · (·) ( · (·) ) )

a partial subtraction leaves behind a mixture of actuality and virtuality

2' - 1' = · {·} { · {·} } - · {·} = · {·} ( · {·} )

2' - 1' - 1' = 2' - 2' = · (·) ( · (·) )

3' - 1' = · {·} { · {·} } { · {·} { · {·} } } - · {·} = · {·} { · {·} } ( · {·} { · {·} } )

3' - 1' - 1' = 3' - 2' = · {·} ( · {·} ) ( · {·} ( · {·} ) )

3' - 1' - 1' - 1' = 3' - 3' = · (·) ( · (·) ) ( · (·) ( · (·) ) )

virtuality can be erased by a “double parenthesis”

((1' - 1')) = (· (·)) = · = 0'

((2' - 1')) = (· {·} ( · {·} )) = · {·} = 1'

((2' - 1' - 1')) = ((2' - 2')) = 0'

((3' - 1')) = (· {·} { · {·} } ( · {·} { · {·} } )) = · {·} { · {·} } = 2'

((3' - 1' - 1')) = ((3' - 2')) = 1'

((3' - 1' - 1' - 1')) = ((3' - 3')) = 0'

in these subtractions, we only “virtualized” the outer parts of the structure. we can also subtract an inner part:

2' - (2' - 1') = · {·} { · {·} } - ·{·} ( · {·}) = · (·) { · (·) }

3' - (3' - 1') = · {·} { · {·} } { · {·} { · {·} } } - · {·} { · {·} } ( · {·} { · {·} } ) = · (·) ( · (·) ) { · (·) ( · (·) ) }

***

based on this observation, we can now define virtual numbers that are

0" = 0' = ·

1" = · (·)

2" = · (·) ( · (·) )

3" = · (·) ( · (·) ) ( · (·) ( · (·) ) )

which satisfy these properties:

n" = n'-n'

((n")) = ·

m' - n" = m'

(n+1)" = n" (n")

and we will also define marginal numbers that are

1^ = · {·}

2^ = · (·) { · (·) }

3^ = · (·) ( · (·) ) { · (·) ( · (·) ) }

which satisfy these properties

1^ = 1'

n'-1' = (n-1)' ( (n-1)' )

n^ = n'-(n'-1') = (n-1)' { (n-1)' } - (n-1)' ( (n-1)' ) = (n-1)" { (n-1)" }

n^-1' = n" = n'-n'

(n+1)^ = n" { n" }

and we’ll also have marginals of a number:

1^^3 = · {·} ( · {·} ) ( · {·} ( · {·} ) ) = 1^ (1^) (1^ (1^)) = 1^^2 (1^^2)

2^^3 = · (·) { · (·) } ( · (·) { · (·) } ) = 2^ (2^)

3^^3 = · (·) ( · (·) ) { · (·) ( · (·) ) } = 3^

m^^m = m^

m^^n = m^(n-1) ( m^(n-1) ) (if n>m)

(we’ll use the notation m^n for nth degree marginal of m, which is something else)

thus, the increment rules for our various numbers are:

n+1 = n union {n} (next natural/structural number)

(n+1)' = n' n (next conditional number)

(n+1)" = n" (n") (next virtual number)

(n+1)^ = n" {n"} (next marginal number)

and the rules that relate them to each other are:

{n'} = n (structure a conditioning)

n'-n' = n" (virtualise a conditioning)

n'-(n'-1') = n^ (marginalize a conditioning)

***

we can now repeat alain badiou’s analysis of mallarme’s poem in his “theory of the subject”:

Stilled beneath the oppressive cloud

that basalt and lava base

likewise the echoes that have bowed

before a trumpet lacking graceo what sepulchral wreck (the spray

knows, but it simply drivels there)

ultimate jetsam cast away

abolishes the mast stripped bareor else concealed that, furious

failing some great catastrophe

all the vain chasm gaping widein the so white and trailing tress

would have drowned avariciously

a siren’s childlike side

badiou calls these as two metonymical chains, each of which progress rightward by “vanishing terms”, and the first chain, after the “horn”, ends at the “or else..” that “annuls the vanishing itself”.

we will designate a vanishing term by a “marginal number” and its incrementing n^-1′ {n”} = n” {n”} = (n+1)^ where a marginal n^ loses its 1′ and confronts its virtuality in brackets {n”} to reach the next marginal (n+1)^.

we will designate annulment of a final term by a virtualisation ((n^-1′)) = ((n”)) = · where a marginal number n^ loses its 1′ to become the virtual number n”, and then it is “double parenthesized” to reach to the initial unicity of 0′ = · for a new beginning.

so it becomes:

foam ==> 0' = ·

ship ==> 0' 0 = · {} = ·{·} = 1' = 1^

wrecked ==> 1^-1' = ·{·} - ·{·} = ·(·) = 1"

mast ==> 1" {1"} = ·(·){·(·)} = 2^

stripped and abolished ==> 2^-1' = ·(·)(·(·)) = 2"

horn ==> 2"{2"} = ·(·)(·(·)){·(·)(·(·))} = 3^

ineffectual ==> 3^-1' = ·(·)(·(·))(·(·)(·(·))) = 3"

or else... ==> ((3")) = 0' = ·

siren ==> 0' 0 = · {} = ·{·} = 1' = 1^

drowned ==> 1^-1' = ·{·} - ·{·} = ·(·) = 1"

hair ==> 1" {1"} = ·(·){·(·)} = 2^

the full equation of the poem counts through the marginal numbers 1^ 2^ 3^ 1^ 2^:

0' 0 = 1^

0' 0 -1' {1"} = 2^

0' 0 -1' {1"} -1' {2"} = 3^

((0' 0 -1' {1"} -1' {2"} -1')) 0 = 1^

((0' 0 -1' {1"} -1' {2"} -1')) 0 -1' {1"} = 2^

at the same time, it counts through the virtual numbers 0″ 1″ 2″ 3″ 0″ 1″:

0' = · = 0"

0' 0 -1' = 1"

0' 0 -1' {1"} -1' = 2"

0' 0 -1' {1"} -1' {2"} -1' = 3"

((0' 0 -1' {1"} -1' {2"} -1')) = · = 0"

((0' 0 -1' {1"} -1' {2"} -1')) 0 -1' = 1"

the key point is that the 0″ at the beginning is not the same as the 0″ reached after 3″. the double parenthesis that “crushes down” the virtuality of 3″ condenses it in a new unicity that is different from the first one. but still, there is no guarantee that this second chain will be the “correct one” (whatever that means).

badiou calls this equation of counting, the “structural dialectic”. he calls “historical dialectic” what concerns the “event”, which in our case is the dot: 0″ = 0′ = · a unicity “less than zero” that marks the beginning of both conditionality and virtuality.

{0"} = {0'} = {·} = {} = 0

***

let’s compare and combine various singular existences.

1 is the structural empty existence. 1′ is the conditional “conscious” existence. 1″ is the virtual “subtracted” existence, and 1^ is the marginal existence, which is equal to 1′ for it is the first one.

when conditional 1′ faces structural 1, it captures the structure to become 2′:

1' 1 = ·{·} {{}} = ·{·}{·{·}} = 2'

when virtual 1″ is presented by its virtuality (1″), it becomes the next virtual 2″:

1" (1") = ·(·) (·(·)) = 2"

when virtual 1″ is presented by itself in brackets {1″}, it becomes a marginal 2^:

1" {1"} = ·(·) {·(·)} = 2^

so, when virtual 1″ faces structural 1, what happens?

1" 1 = ·(·) {{}} = ?

it cannot propagate its dots as in the conditional 1′, but stays side by side with this structure, closing itself to any further incrementing.

in this case, further incrementing only becomes possible by a “crushing down” of this prior virtuality by a double parenthesis, which also will “open” the structure by dotting it:

((1" 1)) = ((·(·) {{}})) = (({{}})) = ·{·} = 1'

so, a structural number without dots is always found side by side with a virtual number that is only held up by its dots. they form an unincrementable deadlock, which can only be surpassed by a double parenthesis. we can call these as locked/self-locking/self-blocking/deadlock etc. numbers:

[1] = 1" 1 = ·(·) {{}}

[2] = 2" 2 = ·(·)(·(·)) {{},{{}}}

[3] = 3" 3 = ·(·)(·(·))(·(·)(·(·))) {{},{{}},{{},{{}}}}

which satisfy these properties

[n] = n" n

[[n]] = [n" n] = n'

where the inner [] refers to the interlocking and the outer [] refers to “unlocking” (equivalent to annuling by double parenthesis).

a structure cannot hold it by itself, it can only hold up by locking a prior virtual number. it is not only conditionalities that count from the dot. *everything* stems from the unicity of the dot. natural numbers are only possible besides the virtual numbers.

nature emerges from the self-locking of virtuality by structure.

and the surpassing of this self-locking reaches to conditionality.

the only exception to this self-locking is the virtual 0″ that cannot be locked or unlocked. in zero, virtuality captures nature where locking/unlocking is undefined:

0" 0 = [0] = · {} = ·{·} = 1'

[[0]] = [0]

locking of a unicity amounts to its singular conditioning.

moreover, locking/conditioning of unicity 0″ is identical to an unlocking of the lock 1″ 1:

0" 0 = [0] = 1' = [[0]] = [[1]] = [1" 1]

[ · {} ] = [ ·(·) {{}} ] = 1'

thus, there is an special equivalence between zero and one, when their virtuality is locked by structure and then unlocked. because, both operations yield to the independent existential conditionality 1′.

consider various ways of founding a singular conditionality:

n'

n'-n' = n"

[n"] = 0"

0" 0 = [0] = 1'

in the case of [0], we first subtract all conditionality from itself to make them virtual. we then annul this virtuality to get a pure unicity. thus, we can combine it with an empty set to create an existence.

1'-1' = 1"

1" 1 = [1]

[[1]] = 1'

in this case, we have a virtual 1″ that is left behind by a singular subtraction. so, it cannot combine with an empty set, and it has to combine with a 1, and even then, it becomes locked, and does not naturally lead to a conditionality. only after this locking is unlocked that we reach at a singular existence.

n'

n'-(n'-1') = n^

((n^)) = 1'

another way to reach a conditionality is to annul a marginality. we first subtract all-but-one conditionality to reach a marginality, and then annul it to get 1′.

***

what does it mean for “yetki/authority” and “beden/body” being cross-determined by their respective moments “irade/will” and “sistem/system”? (reference) here, body is a virtual number n” derived by subtracting conditionality from itself n’-n’. authority is a structure n that locks, that is the self-locking of this body n”. will and system emerges from the unlocking. then, body forms the system that captures the authority, and authority forms the will that captures the body.

[2] = 2" 2 = ·(·)(·(·)) {{},{{}}}

[[2]] = [2" 2] = [ ·(·)(·(·)) {{},{{}}} ] = 2'

now we can rewrite conditional numbers in terms of “body” and “authority” locked+unlocked as a cross-determination of “system” and “will”.

we will call locking + unlocking = encapsulation.

1' = [ ·(·) {{}} ] = [ · {} ] = ·{·}

2' = [ ·(·)(·(·)) {{},{{}}} ] = ·{·}{·{·}}

3' = [ ·(·)(·(·))(·(·)(·(·))) {{},{{}},{{},{{}}}} ] = ·{·}{·{·}}{·{·}{·{·}}}

we conclude that the dual self-locking is foundational and “encapsulation” is only an operator to make it into a cross-determining self-locking. we can call these as the encapsulated numbers. we will show [[]] using〚〛.

〚0〛= [ · {} ] = 1'

〚1〛= [ ·(·) {{}} ] = 1'

〚2〛= [ ·(·)(·(·)) {{},{{}}} ] = 2'

〚3〛= [ ·(·)(·(·))(·(·)(·(·))) {{},{{}},{{},{{}}}} ] = 3'

(more on〚0〛and [0] shall come later)

so,〚0〛and〚1〛must be distinguished as “different forms” of independent singular existence 1′. different encapsulations that correspond to the same conditional number. conditional numbers n’ are not elementary. they are encapsulated as a structure + virtuality in〚n〛= [n” n], a cross-determination of “body” and “authority” forming into and captured by “will” and “system”.

***

what happens if an encapsulation is not balanced?

say if we have

[3" 1] = [ ·(·)(·(·))(·(·)(·(·))) {{}} ] = ?

[3" 2] = [ ·(·)(·(·))(·(·)(·(·))) {{},{{}}} ] = ?

the first should give the marginal 3^, and the second should give the “second degree” marginal 3^2:

[3" 1] = ·(·)(·(·)){·(·)(·(·))} = 3^

[3" 2] = ·(·){·(·)}{·(·){·(·)}} = 3^2

[n" m] = n^m (if m < n)

we can also have n<m such as

[1" 2] = [ ·(·) {{},{{}}} ] = ?

[2" 3] = [ ·(·)(·(·)) {{},{{}},{{},{{}}}} ] = ?

[1" 3] = [ ·(·) {{},{{}},{{},{{}}}} ] = ?

these might lead to “negative” numbers that has more structure than virtuality. do they exist? we don’t know, but let’s call them 1^2, 2^3, 1^3 anyway.

***

how can we write conditional operations in terms of encapsulation?

conditional incrementing:

(n+1)' = n' n

(n+1)' = [ (n+1)" n+1 ] = [ n"(n") n u {n}]

n' n = [n" n] n

==> [n" n] n = [ n"(n") n u {n}]

what does this mean? this equation tells what is happening “inside” a conditional number when it is being incremented. as we recall, n’ confronts n to increment. and we know that n’ contains n” and n as encapsulated. what happens is that n enters inside the encapsulation by dividing into (n”) and {n}. the first one extends the virtuality n” and second one unites with the inner structure n. an encapsulated “conditionality” is always doubly incremented in its two dimensions: “virtual” + “structural”.

example:

〚0〛0 = [ · {} ] {} = [ ·(·) {{}} ] =〚1〛

〚1〛1 = [ ·(·) {{}} ] {{}} = [ ·(·)(·(·)) {{},{{}}} ] =〚2〛

〚2〛2 = [ ·(·)(·(·))(·(·)(·(·))) {{},{{}},{{},{{}}}} ] =〚3〛

***

we now know that a conditioning n’ is better be shown as [n” n] an encapsulation of a virtual number and a structural number. we also know that an mth degree marginal n^m can be shown as [n” m], and it also follows that n^n = [n” n] = n’. we can also increment these encapsulated numbers. but we cannot yet subtract from them. and we also don’t know about numbers like 1^2.

the simplest subtraction is the one that yields full virtuality.

how to get full virtuality from a marginal?

n^-1' = n"

[n" 1] - [1" 1] = n"

[2" 1] - [1" 1] = 2"

[ ·(·)(·(·)) {{}} ] - [ ·(·) {{}} ] = ·(·)(·(·))

the structures cancel each other along with the encapsulation, and only virtuality is left.

how to get full virtuality from any conditional:

n'-n' = n"

[n" n] - [n" n] = n"

similarly, structures cancelled each other.

how about

(n+1)'-1' = [(n+1)" n+1] - [1" 1] = [n" n] n - [1" 1]

(n+1)'-1' = n'(n') = [n" n]([n" n])

thus,

[n" n] n - [1" 1] = [n" n]([n" n])

here, subtraction of 1′ converts n={n’} into (n’)

also,

(n+1)^ = ?

(n+1)^ = [(n+1)" 1] = [n"(n") 1]

(n+1)^ = (n+1)'-((n+1)'-1') = [(n+1)" (n+1)] - [n" n] ( [n" n] )

[(n+1)" (n+1)] = [n" n] n = [n" n] {n'} = [n" n] { [n" n] }

[n" n] { [n" n] } - [n" n] ( [n" n] ) = [n"(n") 1]

here, the structures cancel each other, and 1 remains from {} – (). after the structure being determined, the virtuality simply extends itself by (n”).

to sum up,

[n" 1] - [1" 1] = n"

[n" n] - [n" n] = n"

[n" n] n - [1" 1] = [n" n]([n" n])

[n" n]{[n" n]} - [n" n]([n" n]) = [n"(n") 1]

how do we write:

[n" n]{[n" n]} = ?

[n" n]([n" n]) = ?

here’s the problem:

[1" 1]{[1" 1]} = ·{·}{·{·}} = [ ·(·)(·(·)) {{},{{}}} ] = [2" 2]

[1" 1]([1" 1]) = ·{·}(·{·}) = [·(·) {{}}] ( [·(·) {{}}] ) = [ ·(·)(·(·)) ??? ]

do we need to keep the structure divided? do we need to distribute conditionality among virtualities by keeping their encapsulations separate? do we need separate encapsulations for each cell of virtuality that do not fit to the structure? it will cause an exponential multiplication of encapsulations.

the problem is the ordering. if first elements are subtracted, we get the marginals such as 3^2 3^1 that can be encapsulated as [3″ 2] [3″ 1] etc. but if the last elements or middle elements are subtracted, we have a gap that cuts after some structure. if there are () inside {}, no problem. a virtuality is being marginally structured. but if there are {} inside (), then virtuality breaks the structure into pieces, yielding exponentially many encapsulations to express a single conditionality. we can call these as broken conditionalities.

we defined n’=[n” n], but from now on, we drop this special definition of n’ and we use n’ to refer to any number that expresses a conditionality by a dotted combination of {} and ().

(n+1)' is broken if n' is broken

(n+1)' is broken if (n+1)' = n'(n') and n'!=n"

an unbroken conditionality is always a marginal n^m = [n” m] (where n>=m for the time being) in which there are n-m initial unstructured virtualities and m structured virtualities. this is also the definition of an encapsulation. we can call these n-m latent elements and m manifest elements of an encapsulation. if l=n-m, it becomes [(l+m)” m]. thus, an encapsulation is a partial structuring of a virtuality.

***

we can now consider the case for n^m = [n” m] when n<m. it means that there are more elements in the structure than the virtuality. an element of virtuality must correspond to more than one element of its structure, such as

0^1 = [· {{}}]

there are no elements, one of which is manifest.

1^2 = [·(·) {{},{{}}}]

there is an element, two of which is manifest.

for a marginal, we know that

n^m - m' = n"

n^m - [m" m] = n"

so it should be that:

n^m = n" + [m" m]

where + refers to “unsubtraction”.

so, in our case, the “negative” marginal is an unsubtraction of 1′ onto 0″:

0^1 = 0" + [1" 1]

[· {{}}] = · + [·(·) {{}}]

0^1 = · + ·{·}

if virtuality is what is left behind from subtraction, unsubtraction means the structuring of virtuality. pure unsubstraction is an excess of structuring that do not yet have a corresponding virtuality. in the case of 0^m, it is an pure encapsulation that encapsulates nothing, just a dot, a unicity. an empty packet.

0^m = [· m]

it is not an empty set 0 but an empty packet 0^m because it is has a structure of m elements that has only a pure unicity for its virtuality or “content”. it is an encapsulation, a structuring that is “undecided”, a packet without “content”.

so how do we fill a packet? make 0^1 into 1^1?

***

we can re-define incrementing for marginal encapsulations:

[n" n] n ==> [n" n] [n" n]

[(n+1)" n+1] = [n" n] [n" n]

[(n+1)" m+1] = [n" m] [n" m]

***

we need to separate incrementing into virtual and structural increments

[n" m] [~ m] = [n" m+1]

[n" m] [n" ~] = [(n+1)" m]

[n" m] [n" m] = [n" m] [~ m] [n" ~] = [(n+1)" m+1]

thus we first structure a packet, then fill it with content as follows

[0" 0] [~ m] = [0" m]

[0" m] [n" ~] = [n" m]

where the content n” may be more complex than its structure m. the excess of content will be put in a “payload”, which we have not yet defined.

***

what is a payload? what is a header? what is a packet?

packet is an encapsulation, a partially structured virtuality.

header is the structured part of a packet’s virtuality.

payload is the unstructured part of a packet’s virtuality, apart from its size.

how can a payload encapsulate another packet? how do these packets relate?

***

how are encapsulations broken and gathered?

example:

[ [l" l]([l" l]) 1 ]

here is a l-sized payload that is encapsulated by a 1-sized header.

[ [l" l]([l" l])([l" l]([l" l])) 2 ]

the header’s become 2. let’s introduce payload numbers, payload number l belonging to header m (remember that 〚l〛=[l” l]):

l*1 = 〚l〛 ( 〚l〛 )

l*m = l*(m-1) ( l*(m-1) )

l*0 = 〚l〛

l*1 = 〚l〛(〚l〛)

l*2 = 〚l〛(〚l〛) (〚l〛(〚l〛) )

l*3 = 〚l〛(〚l〛) (〚l〛(〚l〛) ) (〚l〛(〚l〛) (〚l〛(〚l〛) ) )

these allow us to form packets with payloads:

[l*m m]

but we need packets with sub-payloads:

a*b*1 = [a" b] ( [a" b] )

a*b*m = a*b*(m-1) ( a*b*(m-1) )

[n" m] = [a*b*m m]

[a" b] = [c*d*b b]

...

now we have something new. we have written a virtual number n” as a “payload number” a*b*m that is not really virtual in the sense of our definition. our definition only allowed for numbers such as these

0" = 0' = ·

1" = · (·)

2" = · (·) ( · (·) )

3" = · (·) ( · (·) ) ( · (·) ( · (·) ) )

but now, we also allowed “payload numbers” that contain an encapsulation instead of dots. we extended the definition of an encapsulation to include payloads, to be either [n” m] or [a*b*m m]. in the first case, there is an excess of information/virtuality over structure, whereas in the second, this excess (or part of it) is covered by a sub-structure, and rest of it can also be covered by a sub-sub-structure etc.

we can imagine a situation where:

[n" m] => [a*b*m m]

[a" b] => [c*d*b b]

[c" d] => [e*f*d d]

[e" f] => [g*h*f f]

[g" h] => [i*h h]

[i" i]

it goes until the whole virtuality is covered by structure, where n-m=a, a-b=c, c-d=e, e-f=g, g-h=i. in this case, all virtuality is covered at the end. there can also be cases where some of the virtuality is left uncovered.

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