We can distinguish “conditional counting” from the usual “structural counting” by “opening” and “dotting” the usual number representations given in set theory:

structural counting:

0 = {}

1 = {0} = {{}}

2 = {0,1} = {0,{0}} = {{},{{}}}

3 = {0,1,2} = {0,{0},{0,{0}}} = {{},{{}},{{},{{}}}}

...

conditional counting:

0' = ·

1' = · {·}

2' = · {·} { · {·} }

3' = · {·} { · {·} } { · {·} { · {·} } }

...

the conditional number n’ indicates a conditionality among n binary variables or “events” that either happen or do not happen, i.e. they will be either counted or not counted, and this uncertainty is included in the representation of a conditional number in the form of “dots”.

in 0′, there is only a single dot

0' = ·

it’s a pure eventuality that can be a starting point for conditional counting. uniqueness of a quality comes from its power to begin a new conditional counting, and 0′ indicates this new beginning. 0′ is unicity.

if a unicity is “closed”, this beginning is destroyed and structured in an emptiness.

{0'} = {·} = {} = 0

to count from 0′ to 1′, we confront it with 0. we confront uniqueness with emptiness to reach a single eventuality.

0' 0 = · {} = · { · } = 1'

when 0′ meets 0, the dot of unicity propagates in the emptiness to found “being”. through the emptiness, unicity divides into “to be or not to be”.

while 0′ is a pure uniqueness, 1′ is an independent singular existence. among its two dots, only the inner dot “counts” it as happening/existing, and the outer dot “passes over” it as not happening/not existing. this is the division of the uncertainty expressed by 1′.

ambiguity of an independent and hidden singular existence is given by 1′, such as in this image

the outer dot of “not to be” is necessary for 1′ to stay conditional. if its outer dot is destroyed by a structural closure, a proof or guarantee of existence, it becomes an empty counting of an indifferent 1.

{1'} = {· {·}} = {{}} = 1

thus the basic freedom of simple existence is a freedom to “pass over”, a freedom not to count, not to be. otherwise, the problem of existence is foreclosed, and 1′ of independent existence becomes 1 of indifferent capital.

however, an independent singular existence 1′ needs to face its indifferent oneness to reach the dilemma of 2′.

1' 1 = · {·} {{}} = · {·} { · {·} } = 2'

the 1′ faces 1 with the following question: “i don’t know whether i am or not. however, i also don’t know if it matters or not, for i am only one, with respect to…” and this self-conditioning is always made with respect to a second. self-conditioning of 1′ is the essence of the second, or 2′.

2′ is the conditionality of two existences.

2' = · {·} { · {·} }

the four dots of 2′ correspond to 0,0 / 1,0 / 0,1 / 1,1.

the uncertainty in 2′ involves the interdenial between these two events, a dilemma. the dilemma of 2′ can be illustrated by the visual illusion known as Rubin’s vase:

if faces are counted first, it is decided 1,0 that only faces exist. if the vase is counted first, it is decided 1,0 that only vase exists.

in an asymmetric dilemma, one of the existences presents itself more strongly, to be quickly recognized and begin the counting, while the second existence comes later and cancels the first existence with a countering denial stronger than the existence counted at first. it is in fact a count from 1′ to 2′.

an example is the image below

if you look closely, you immediately recognize the woman, and you are almost sure of this image, thus we have a 1′ with an insignificant outer dot, an almost-sure singular existence. then, when you scan its figure as a whole, a second stronger existence, a skull is recognized, and the woman’s existence becomes conditional to the skull. in this process, the first recognition, 1′ of the woman is denied by the 1′ of the skull, then, it becomes a 1 of a woman, an empty existence to be “re-counted” conditional to the skull in a new 2′.

the counting from 1′ to 2′ has to face an empty existence of a 1. thus, the second existence must first be emptied of its previous conditionalities to be “re-counted” in a 2′

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

after a 1′ has been emptied to become a 1 and re-counted in the 2′, after it received the dots of its denier, it is now possible to initiate a re-counting of its own, which will then bounce back, and so on. after the skull has been recognized, the image can be continually re-counted beginning alternately from one of its two elements, finally reaching a reciprocal determination of the 2′.

when an 2′ is closed, its conditionality is destroyed and the dilemma of 2′ becomes an indifferent structural count 2 of capital.

{2'} = {· {·} { · {·} }} = {{},{{}}} = 2

here, the new comma inside the 2 indicates the indifference between the two elements that resulted from the structural closure of 2′. the second element’s existence simply comes “secondly”, it does not bear any conditionality to the first one. it only has the structure of a conditionality, but not the conditionality itself. it is just the empty skeleton of a two-sided conditionality, frozen in an indifferent marking.

to count from 2′ to 3′, we have to confront 2′ with its empty structure 2.

2' 2 = · {·} { · {·} } {{},{{}}} = · {·} { · {·} } { · {·} { · {·} } } = 3'

in this counting, the four dots of the dilemma 2′ propagate towards the structure of 2. filling of this structure gives the three-sided uncertainty of 3′.

we can also count to 3′ by beginning from the unicity of 0′

0' 0 1 2 = 1' 1 2 = 2' 2 = 3'

`0' 0 1 2 = · {} {{}} {{},{{}}} =`

1' 1 2 = · {·} {{}} {{},{{}}} =

2' 2 = · {·} { · {·} } {{}} {{},{{}}} =

3' = · {·} { · {·} } { · {·} { · {·} } }

from the unicity 0′, we first reach an independence of 1′ and after that, a dilemma of 2′. the uncertainty of 3′ only appears afterwards, when the dilemma 2′ faces the simple 2. the conditionality of a “third element” is identical to this confrontation.

the confronting 2′ 2 = 3′ can be considered as follows: “we were to choose among these two, but we see that they are no different.” we see this fact only conditional to a third, and this seeing is identical with this conditionality with respect to a third. and in the end, what we get is an uncertainty among three elements, given by 3′.

the illusions of 2′ above are also examples of 3′. here, the third element is the picture itself, the intention of the painter, who purposedly put this dilemma inside the picture. the realization of this third element cancels the initial dilemma, and creates a new conditionality with respect to this third element: “what did he mean by a woman, what did he mean by a skull, why did he combine them in this way” etc.

if a 3′ is closed, it becomes the empty structure 3:

{3'} = {· {·} { · {·} } { · {·} { · {·} } }} = {{},{{}},{{},{{}}}} = 3

we can then condition it to a fourth element by the confrontation 3′ 3 = 4. an example “fourth element” would be myself telling about the 3′ example above. in this case, we bracket the woman, skull and their painter, and we begin to question some “hidden cause” behind my sentences: “why did he use such an example? do the woman or the skull have any significance for his whole issue of ‘qualia’? is there any correlation between the painter’s intentions and the usage of this example in this article?” as a result, we have come to a four-sided uncertainty of 4′.

the next step could be closing this as {4′} = 4 and propagating its dots toward a fifth element, you the reader and your conditionality with respect to these four elements combined together: 4′ 4 = 5′.

to conclude, uncertainty is conditional counting, and qualia is the uncertainty of this conditional counting, always confronting a given conditionality with its neutral structure and making it reach towards new elements that expand the space of uncertainty.

(note: here, we have only looked at “full conditionalities” that condition the new element to all of the previous elements. there are also “partial conditionalities” that condition the new element to only some of them, or maybe none of them.)

(side note: a conditional number counts through the state of a situation. in being and event, badiou distinguishes between a situation and a state of a situation. a situation is a multiple or a set S that consists of elements that belong to it. the state of this situation is the power-set p(S) that consists of all sub-multiples or subsets of S. if S has n elements, it has 2^{n} subsets and every subset of S can be designated by an n-sized bit string. in a conditional number n’, there are 2^{n} dots, and each of these dots correspond to an n-sized bit string. therefore, in badiouan terms, the conditional number n’ counts through the 2^{n} elements of a state of a situation with n elements.)

***

Continuation: Virtuality is what is left behind by conditional subtraction

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