Deleuze’s zeroness and Peirce’s pure zero regarding the expansion of semiotics’ categorial frame

1 Deleuze’s affiliation to Peirce’s theory of signs and the dissenting Zeroness

The scholarly literature dedicated to Deleuze often assumes that the strength of Deleuzian semiotics of comes from its roots being Bergsonian, even though Deleuze calls upon Peirce in his books about cinema. Ehrat argues that the “‘Bergsonization of Cinema” that the Deleuzian semiotic promotes falls short of Peirce’s requirements because Deleuze’s books on cinema “contains some Peirceish material… but this remains rather inconsequential” (Ehrat 2005: 221). Consequently, Deleuze’s attempt to translate Bergson’s metaphysical thesis into Peirce’s logics fails or, rather, is not achievable due to philosophical restrictions that put Deleuze’s Bergsonism and Peirce far apart (Ehrat 2005: 241–242). According to Bogue (2003: 65–67), one could understand that Deleuze’s semiotic is Bergsonian in its core and Peircean on its surface, so that his “conception of semiotics is not really Peircean but Bergsonian” (Bogue 2003: 67).^[1] Similarly, for Schwab (2000: 110–111), the weight that Deleuze assigns to Bergson’s “image-ontology” forces Peirce’s semiotics to become mostly Bergsonian (Schwab 2000: 110–111). Likewise, O’Neill argues that as “Deleuze’s work on cinema owes more to Bergson than to Peirce, this makes the author’s own elaboration of Peirce’s work a questionable goal” (O’Neill 1998).

This article interrogates the positions that assign to Bergson the major influence over Deleuze’s semiotics of cinema by making a closer assessment of Deleuze’s uses of Peirce’s categorial frame. From this perspective, Deleuze’s semiotics of the cinema shows to be more Peircean than often accounted for. Deleuze succeeded in faithfully – and inventively – developing Peirce’s triadic logic and categories (Firstness, Secondness, and Thirdness). This is the general argument to be sustained, not only from the inspection of Deleuze’s books on cinema (1985 and 1989), but also from raw, scarcely drawn-upon resources reunited in Deleuze’s ninety-two lectures on cinema held between November 1981 and June 1985 (Deleuze 1981–1982; Deleuze 1982–1983; Deleuze 1983–1984).^[2] In these lectures, Deleuze turns to Peirce’s logic and categorial schema to reject the Saussurean affiliation:

[I]f you compare roughly with a contemporary of Peirce who is well known in France: Saussure … he is not French, but he is well known in France … (laughter from everyone) … And he is better known than Pierce’s[sic] entire theory of signs, he proposes as foundation a kind of dual distinction: signifier – signified

One student: Saussure proposes!

Gilles Deleuze: for Saussure, excuses … I said eh … and yes … see that he is already quite important to us … that Peirce definitely cannot be part of this, of a similar lineage.

(Deleuze 1982–1983: lecture 27 part 1)^[3]

Some scholars do not disregard Deleuze’s divergence about Saussurean-based semiotics as it impacts on Deleuze’s works as a counterweight to Bergson’s influence. Dawkins warns that, although Deleuze’s studies about cinema depend on the Bergsonian metaphysics of image, Deleuze decisively calls upon “Charles S. Peirce’s semeiotics to describe … a range of different signs in the cinema” (Dawkins 2005: 8). Similarly, Flaxman states that “Deleuze resolves in the cinema books to draw on … Peirce’s ‘extremely rich classification of signs’ in order to supplement Bergson’s own varieties of images” (Flaxman 2000: 23). If Deleuze believes “that Peirce’s method can be used to expand the movement-image taxonomy that began with Bergson’ fundamental coordinates,” (Deamer 2016: 17), then, he has been faithful to Peirce’s promise, in 1911, of an intended book that would surpass Bergson’s: “I feel confident the book would make a serious impression much deeper and surer than Bergson’s, which I find quite too vague” (Peirce 1958: 428).^[4] Deleuze, in turn, clearly answers Peirce seventy years later in one of his lectures on cinema:

Peirce’s categories, for example, firstness, secondness, thirdness, it seems to me, will allow us to relaunch [Bergson]. And, indeed, this overflows Bergson, it does not contradict him, it is totally a different kind of problem … and that may allow us to push the analysis … further than Bergson gives us the means of. That’s it, so you shall go on. (Deleuze 1981–1982: lecture 13 part 1; emphasis added)^[5]

Deleuze draws decisively upon Peirce’s triadic categories to recreate them on his own account, aiming at classifying signs in movement, the cinematic signs of the cinema. Consequently, in this article we will observe Peirce’s requirement about the formal and categorial frame that conditions his semiotics: “the new concept of a ‘sign’ will be defined exclusively by the form of its logical relationships; and the utmost pains must be taken to understand these relations in a purely formal, or, as we may say, in a purely mathematical way” (EP 2: 389).

This article will show and discuss Deleuze’s approach to Peirce’s “purely mathematical way” of conceiving of a sign by the “form of its logical relationships.”

In fact, in the 1980s, Deleuze catches readers of Cinema 1 (1985) and Cinema 2 (1989) by surprise as he claims for the expansion of the Peircean system of categories (Firstness, Secondness, and Thirdness): “why does Peirce think that everything ends with thirdness … and that there is nothing beyond?” (Deleuze 1989: 33) Deleuze himself answers: “there will be a ‘zeroness’ before Peirce’s firstness” (Deleuze 1989: 31–32). We will see that zeroness corresponds with Peirce’s considerations about the “pure zero,” the “vague zero of indeterminacy,” and the “medad” (a logical relative without correlates). Zeroness either violates or is an authorized and fertile addition to Peirce’s system of categories, which eventually enhances the triadic classification of signs. This puzzling issue, nevertheless, counts on an important exegetical support to be dealt with. Deleuze’s endeavor to extend Peirce’s categories jumps from the past and aligns partially with more recent Peircean scholarship that have been acknowledging that the “mathematical semiotics” of Peirce’s “trichotomic-triadic form” (Toth 2012: 2) involves the zeroth state. In general, the zeroth state is “that which precedes the three categories” (Merrell 1997: 168).^[6]

We believe that the confrontation between Deleuze’s writings on the cinema and the Peircean studies about the zeroth state category – for the most part unaware of Deleuze’s Zeroness – allows a closer inspection of the role that the zeroth state of zeroness plays in Deleuze’s cinematic images. With the aid of Peirce’s scholars, “it is necessary to return to [Deleuze’s] logic and its many ambiguities. To discover the [Peircean] concealed and absent structures and elements of the taxonomy [related to the images of cinema]” (Deamer 2016: xxx).

2 The Peircean scholarly literature about the status of zeroth state and its importance to Peirce’s phenomenological categories

A decade, or so, after Deleuze’s book on cinema from 1983 to 1985, and not as a reaction to Deleuze’s designation of Zeroness on Pierce’s triadic categories, the assumption that the zeroth state as an important issue arose among Peircean scholars. All the positions assumed since then stem from four main passages in Peirce’s writings:

1. Zero is a metaphysical concept – the “pure zero” or “germinal nothing” – that explains the very birth of the universe:

   We start, then, with nothing, pure zero. But this is not the nothing of negation. For not means other than, and other is merely a synonym of the ordinal numeral second. As such it implies a first; while the present pure zero is prior to every first … It is the germinal nothing, in which the whole universe is involved or foreshadowed. (CP 6.217)

1. Peirce admits that there may be “an entirely different series of categories,” not related to formal relations but to the “material” of the phenomenon. The different categories require a new phenomenological branch:

   I have some acquaintance with two different such classifications [of the indecomposable elements of the phaneron], both quite true; and there may be others. Of these two I know of, one is a division according to the form or structure of the elements, the other according to their matter. (CP 1.288)

   … It appears to be a conception of an entirely different series of categories. (CP 1.588)

1. The “zero” or “utter nothingness” or the “vague zero of indeterminacy” is a “limit” that plays a generating role as to the triadic relation:

   It is true that there is another sort of zero which is a limit. Such is the vague zero of indeterminacy. But a limit involves Secondness prominently, and besides that, Thirdness. In fact, the generality of indeterminacy marks its Thirdness. (CP 6.211)

   That perfect cosmology must therefore show that the whole history of the three universes, as it has been and is to be, would follow from a premiss which would not suppose them to exist at all … But that premiss must represent a state of things in which the three universes were completely nil. Consequently, whether in time or not, the three universes must actually be absolutely necessary results of a state of utter nothingness. (CP 6.490)

1. Zero takes the form of a relative without a correlate (medad) in the logic of relatives:

   [A] medad must mean an indecomposable idea altogether severed logically from every other; a monad will mean an element which, except that it is thought as applying to some subject, has no other characters than those which are complete in it without any reference to anything else; a dyad will be an elementary idea of something that would possess such characters as it does possess relatively to something else but regardless of any third object of any category; a triad would be an elementary idea of something which should be such as it were relatively to two others in different ways, but regardless of any fourth; and so on. (CP 1.292)

Henceforth, we will present an overview of the contribution of scholars that have been drawing upon the previous Peircean passages. Their position will be arranged according to some recurring issues in Peircean and, scarcely, in Deleuze studies, respectively (a) system of categories, the (b) Peircean phenomenology (phaneroscopy), the (c) generative categorial role, the (d) semiotic effectiveness, the (e) doctrine of continuity, and the (f) logic of relatives. The listed items correspond with the following topics and subtopics:

1. zeroth state holds/does not hold a categorial status along with Firstness, Secondness, and Thirdness in the semiotic triad;^[7]

   1. zeroth state is not a category at all since Thirdness covers its function;

      Fernandes states that “The problem lies in the fact that Deleuze ‘inserts’ one more category into Peirce’s crystalline philosophical structure” (Fernandes 2019: 61).^[8] According to Fernandes, although a link to Peirce’s metaphysical nothingness^[9] (“pure zero”) might be easily traced to Zeroness, it is not a good solution because nothingness “cannot correspond to what Deleuze calls zeroness. Nothingness, on Peircian grounds, is just a ‘mere capability of getting thought’; but zeroness is already and de facto image, a selection from material world” (Fernandes 2019: 57). Moreover, for Fernandes, Deleuze’s approach to Peirce’s semiotics is partial because he did not study it along with Peirce’s metaphysics of continuity (synechism). In fact, continuity between matter and mind as the reality of Thirdness includes what Deleuze denominates as Zeroness: “If Deleuze had gone beyond semiotics and had studied Peirce’s metaphysics, maybe he would have thought that continuity … is the condition of images” (Fernandes 2019: 65), not Zeroness.

      Similarly, Marks – a Deleuzian scholar – relying on Peirce’s “pure zero,” maintains that “a kind of ‘before of Firstness’” reveals that the semiotic triadic relation is problematic and incomplete, as Deleuze understands it, mostly because “Peirce’s degree zero [breaks] the dominance of an ossified Thirdness” (Marks 2000: 200). Nevertheless, the residual “Zero in the Peircean categories” (Marks 2000: 197) is eventually absorbed in Thirdness: “that exists before a sign has been marshaled for particular uses, returns (in some sense) at the level of Thirdness” (Marks 2000: 196). Consequently, the zeroth state does not reach a categorial status, although Deleuze’s Zeroness brings a positive impact over Peirce’s systems of categories.

   2. zeroth state plays a role in the semiotic triad, but it does not sustain the status of a category; it is a pre-semiotic nothing instead

      For Merrell,^[10] even though nothingness does not hold the status of a category in the Peircean system, as Firstness, Secondness and Thirdness do, it comes first as the precursor, coordinator, and propeller of all the relations that the categorial system allows, for “‘nothingness’ … is that which precedes the three categories” (Merrell 1997: 168) and is related to a Zen state (Merrell 1997: 183). Moreover, zero stands for the “presence-absence” of “A pure state of pre-Firstness is {0}” (Merrell 1997: 183).

      Similarly, for Brier, even though the pure zero does not play a categorial role in Peirce’s evolutionary metaphysics, “Peirce writes that the three worlds – Firstness …, Secondness …, and Thirdness … – must evolve from this no-thing in an evolutionary metaphysics” (Brier 2014: 208). This clause, likewise Merrell’s position, moves Peirce’s metaphysics closer to the void and emptiness in Buddhist or Vedic thinking, and Christian mysticism (Brier 2014: 210).

   3. Zeroness is a category, and it occupies the pre-level regarding Firstness, Secondness, and Thirdness

      Bense in the 1970s, before Deleuze, stated that Zeroness is an “ontological category” with regard to Firstness, Secondness, and Thirdness, which are, in turn, “semiotic categories” (Bense 1975: 65).^[11] Following Bense, Toth assumes that a “pre-semiotics” based on a “categorial nothingness” can extend Peirce’s phenomenological categories (Toth 2011a: 1). Accordingly, “the [ontological] category of zeroness” (Toth 2008a: 5; emphasis added) shall be represented by a “nod” (Toth 2009b: 1) in the core of the Peircean categorial model to underlie a phenomenological category (Toth 2008a: 1).^[12]

      Ji in turn associates Zeroness, to which he ascribes a full categorial status, with Peirce’s pure zero. He aims at extending the Peircean categories to interpret the language of the living cell (Ji 2017: 278–280). Zeroness is the category of “Transcendentality/Ineffability” (Ji 2017: 277) that puts together quantum physics and Zen; it is at the core of Firstness, Secondness, and Thirdness, and it underlies a Peircean-inspired biosemiotics.

      Deamer understands that Zeroness is a category, but it occupies a pre-semiotic level, since it dissolves “into the other images” (Deamer 2016: 33).

1. some phenomenon instantiates zeroth state and makes it experienced

   1. Zeroness corresponds with perception, but it sustains a vanishing, embryonic status

      According to Deamer, Deleuze’s Zeroness as a phenomenological category corresponds with perception, but it plays a vanishing role in the semiotics of the cinema: “The perception-image is designated as having ‘zeroness,’ as disappearing or being ‘identical’ to the other three images” (Deamer 2016: 30), so that we can “bracket off perception-images” (Deamer 2016: 33).

   2. perception is a phenomenon, but Zeroness does not underlie it

      For Fernandes, there is no need for Zeroness to account for perception: “perception-image is just a concept created [by Deleuze] to designate the emergence of perception (one of the modes of thought) in the own matter, that is, matter perceiving itself, perceptive matter, that is, mind” (Fernandes 2019: 64–65). Zeroness as phenomenological category is obsolete because it is Thirdness that motivates perception instead.

1. zeroth state plays a generative role regarding Firstness, Secondness, and Thirdness as the empty set axiomatic-semiotic function

   1. nothingness is the principle that passes its generating role on to the triadic set of relations through an empty set

      Merrell believes that there is a “node,” or “fourth point, or “zero” by means of which the categories “become interrelatedly, interdependently conjoined by the virtual ‘emptiness’” (Merrell 2001: 33, 2012b: 78–79, 82–83). Therefore, the nothingness of Peirce’s “germinal nothing” has a generative role in the categories, but it is not itself a category. Consequently, “0 as ‘nothingness’ or ‘emptiness,’” (Merrell 1997: 168) is the origin “that cannot be recovered … since it precedes all semiosic activity” (Merrell 2012b: 126n.). The zero of nothingness even antecedes the empty set, which is the “‘pivotal point’ of the sign” that underpins any semiotic relations (R-O-I) and makes them rotate in a set-theoretically basis (Merrell 2012b: 126). The function of the zero or fourth point is demonstraded insofar as it is the “node” in the form of any logical relationship:

      A First is {0}, after some possibility has emerged. A Second is {0; 1; 2}, the result of the sign’s having come into contact with something else, and is now an ‘object’ that can be acted upon by some agent. And a Third is {0; 1; 2; 3}, sign interaction during the course of which a mediator and moderator brings about ongoing interaction between a First and a Second and between them and that mediating Third itself. (Merrell 1997: 183)

   2. the zero is the empty set and performs Zeroness and generates “pre-semiotic sign classes”

      The “pre-semiotic relation” might take the following form: “PSR = (3.a 2.b 1.c 0.d)” (Toth 2011a: 2); whereas “(3.a 2.b 1.c)” (Toth 2008a: 1) represents the semiotic relation (SR). There are thereby “pre-semiotic sign classes” formed by the permutations of PSR and SR, because “the category of zeroness … guarantees the quality of these sign classes … and integrates SR into PSR” (Toth 2008a: 5). Furthermore, Toth demonstrates by means of a matrixial calculus that the insertion of Zeroness among the three Peircean semiotic categories allows a non-static presentation of Peirce’s ten semiotic classes, so that they appear as an auto-generative product of “dynamic morphisms [relationships between elements]” (Toth 2009b: 2). For instance, the first Peircean sign class (3.1 2.1 1.1) can be noted dynamically by the formula “[[[0, 1, 2, 3], [0, 1]], [[0, 1, 2], [0, 1], [[0, 1], [0, 1]]]]” (Toth 2009b: 3).

1. zeroth state as a phenomenological category is/isn’t semiotically effective, so that it produces/does not produce a sign class of its own

   1. the zero of nothingness generates all signs, but there are no signs of nothingness, only the “pre-Sign”

      According to Merrell, the zero of nothingness is responsible for the relations to become meaningful as signs, since “the tripod’s central ‘dot’ contains the wherewithal for engenderment of an indefinite number of signs” (Merrell 2012a: 10). It provides the “semiotic tripod” with its “time-bound nature” (Merrell 1997: 12–13, 25–26, 32–33). Nonetheless, there is no sign of nothingness, since the zeroth state of semiosis lies in the undifferentiated backstage of signhood. Nothingness is like a “silent Tao” that puts signs in movement, but it is not out there signaling itself, “it is a sea of possibilities … it cannot effectively enter into the animation and bustle of flesh-and-bone signs, it cannot interact with other signs, nevertheless ‘0’, the ‘pivotal point’ of the sign, is essential to the equation” (Merrell 2012b: 120, 126). There is only an overarching “pre-Sign” class belonging to a “pre-Semiotic” or “pre-Signness” state of “all possible possibilities,” which precedes the sign-making in the triadic relationship (Merrell 2012a: 39).

   2. Zeroness as the empty set adds a fourth trichotomy to Peirce’s categorial frame, consequently, there are “zero-signs”

      According to Toth, the integration of SR (semiotic relations) in PSR (pre-semiotic relations) adds a forth trichotomy of “zero sign” formed by the relationship between SR and Ø (empty set or Zeroness) in Peirce’s three trichotomies: “Since zero-signs, like the three basic fundamental categories [SR], are trichotomically subdivided, ℙSR presupposes a tetradic-trichotomic sign relation, which we call the extended Peircean sign relation that ZR+ = (R, O, I, ∅) describes” (Toth 2009c: 1; see also Toth 2009a: 5; and Toth 2009b: 1–2). Zeroness, hence, integrates a pre-semiotic “tetradic relation” (“3.a 2.b 1.c 0.d”) that occupies the space of the quantitative semiotic relations, whose form is “3.a 2.b 1.c” (Toth 2008a:1).

      In accordance with Toth, for Semetsky, a Deleuzian scholar, Deleuze’s zero performs a generic and undetermined sign that can be related to Peirce’s idea of the “‘blank zero’.” Moreover, zero is a sign that semiotically stands for Peirce’s “‘primeval chaos’,” therefore, “the quasi-purpose of this sign is to produce sense and initiate the process of creating order out of chaos” (Semetsky 2013: 33).

   3. Zeroness extends Peirce’s categories to the living world, so there is an experienced “nilsing”

      Ji reshapes Peirce’s triadic categories to develop the “Quark Model of the Peircean Sign” (Ji 2017: 276) for biomedical uses by including Zeroness as a category that carries out the “Nilsign” or “sign-less” class, which emits signs without representamen: “Peirce already thought about what I call Zeroness in his concept of Pure Zero, but he apparently did not consider the sign associated with his Pure Zero comparable to my ‘signless’ or ‘nilsign’” (Ji 2017: 277).

1. zeroth state and Peirce’s continuity

   1. Thirdness establishes the continuity between matter and mind (Peirce’s synechism)

      For Fernandes, the Deleuzian Zeroness is what Peirce understands as the continuity between matter and mind. Thus, continuity is already represented by Thirdness and can do without Zeroness: “Thus, we conclude that Peirce’s concept of continuity, that expresses the mental character of universe, is much more coherent than Deleuze’s Zeroness. Perception is the perception of matter by itself and a matter perceiving matter is mind, as Peirce understands it” (Fernandes 2019: 65). Consequently, the zeroth state of perception stands for continuity (Thirdness).

   2. there is a continuum of time that explains sign-making, but the nothingness that generates the semiotic profusion is itself timeless

      “What Peirce calls nothingness [is] an orb without time that brings the three categories together” (Merrell 2012b: 120). The germinal nothing, zero, holds a special relationship with time (Merrell 1997: 32–33) as nothingness participates in the continuum of time, insofar as the empty set stands for it as to generate the time-bound characteristic of the semiosis. Reciprocally, the possibility of the empty set, ∅, involves the previous existence of its precursor, nothingness (Merrell 2012b: 346).

1. zeroth state is/is not a Peirce’s medad

   1. zeroth state has no function because the medad is inscrutable

      According to Marty, one can object that “The case of the nonvalent elements that Peirce calls ‘medads’ is apart and can be neglected” (Marty 1990: 108). As long as the medad entertains no relationship at all, it cannot be experienced, so that it remains ineffective as a sensory phenomenon.

   2. the empty set is a “0-valued relation” and belongs in the triadic sign relation

      Toth assumes that if Peirce introduced the medad, and if the medad takes part in the relation with the monad, dyad, and triad, then a “0-valued relation” also belongs to the triadic sign relation. Thus, there is a categorial Zeroness on behalf of the medad (Toth 2009b: 1). Accordingly, the Peircean categorial model shall be rearranged to include the medad. It eventually turns out to be a tetradic model: “the early Peircean sign model is not compatible with the triadic sign relation, but requires a tetradic sign relation as the relation (Medad, Monad, Dyad, Triad) does” (Toth 2009a: 6).

      The showcase of the Peircean scholarship previously presented disallows what Bogue so categorially declares: “when Deleuze argues that the perception-image is a ‘Zeroness,’ it is evident that he is no longer thinking in Peircean terms” (Bogue 2003: 67). On the contrary, the Peircean literature on the zeroth state encourages such a declaration as “Deleuze uses the Bergsonian formula, and retains only the Peircian structure of the trichotomies through the zeroness of perception and the perception-image” (Deamer 2016: 32).

      Hereafter, we will approach Deleuze’s semiotics to understand the the inflexions that it requires from Peirce’s triadic logic and categories regarding the classification of the images of the cinema in order to tackle the relationship that Deleuze established between Zeroness and perception-image.

3 Deleuze’s semiotics of the cinema and the expansion of triadic logic

For the sake of a comparative approach, the following exposition of Deleuze’s Zeroness will follow the pattern adopted to summarize the Peircean scholarship on the zeroth state in the previous section. Therefore, the following items will be arranged according to (a) system of categories, the (b) Peircean phenomenology (phaneroscopy), the (c) generative categorial role, the (d) semiotic effectiveness, the (e) doctrine of continuity, and the (f) logic of relatives.

4 Final remarks

Deleuze contributes to Peirce’s categorial frame by primarily adding Zeroness to it. Nevertheless, he was not informed about Peirce’s reflections on the “pure zero” or “germinal nothing.” There is evidence that bear witness for Deleuze not having access to Peirce’s metaphysical and mathematical writings: “the only reading that Deleuze mentions in his books and lectures about Peirce’s semiotics is Gérard Deledalle’s book, Écrits sur le signe, which is an annotated collection of comments on some of Peirce’s writings on signs” (Girel 2014). If Deleuze knew further about Peirce’s work than the system categories, the phenomenology, the triadic logic and the classification of signs, he would be aware that “A mathematician and metaphysician as prolific as Peirce … could not have failed to think about the zeroth state as it is involved in the three terms of the triadic relation” (Cardoso Jr. 2018a: 187).

If we consider Deleuze’s seconded-handed reading of Peirce’s work, his Zeroness from the 1980s unexpectedly echoes with Peirce’s writings to which Deleuze did not have access. Additionally, the Peircean scholarly literature, only in recent years, most of the times independently from Deleuze’s thesis on Peirce, has become aware of the importance of Peirce’s concept of zero for the triadic logic and categories. In fact, Deleuze’s non-Peircean account of the function of the zeroth state anticipates some of the recent Peircean scholarship’s efforts. If we compare the items one by one (from a to f) that structures this essay, one can observe that the most recent voices match, partially agree, contrast or even sharply conflict with Deleuze’s views on the Peircean zeroth state concerning: (a) the system of categories, (b) the phenomenology of the zeroth state, (c) the generative categorial role of the zeroth state, (d) the semiotic effectiveness of the zeroth state, (e) the zeroth state related to the doctrine of continuity, and (f) the zeroth state and the logic of relatives. Deleuze’s accomplishment, however, overcomes the scholarly literature since he extended Peirce’s zeroth state in diverse and innovative ways.

This article paves the way for further investigations that might promote reflections on the mathematical semiotics of the zeroth state hereby discussed. The Deleuzian signs of the cinema could be triadically encoded from the six images and intracategorial triads presented as Tables 1 and 2 indicate. It is evident that the Deleuzian new realm of the signs of cinema becomes possible because of the accuracy and strength of Peirce’s triadic logic. In fact, Deleuze deflects Peirce’s categorial frame to expand it in a dynamic and cosmological sense:

The pattern of adicities [monadic, dyadic, triadic] is striking. It could easily be extended for any finite number of trichotomies, and we would expect that it should be so extended for any additional trichotomies germane to signs analysis. We would expect such an extension, just as we would expect that an inverse square law, found to apply to the solar system, will apply to other planetary systems. (Short 2007: 237)

Second, in the mathematical semiotics of the zeroth state remains a line of flight that touches a mathematical-ontological issue concerning the model that consists of the effectiveness of the zeroth state. Bense’s axiomatic semiotics (1981), Merrell’s set-theoretical approach (1997), Toth’s many matrixial operations (2008b, 2009d, 2010, 2011b), and Marty semantic model (2014) focus on Peirce’s semiotics and choose an axiomatic model to cover Peirce’s triadic categories.

Merrell, for instance, goes as far as to assign the incompleteness of the Peircean trichotomies to the fundamentally unstable axiomatic model that pervades semiosis by assimilating Peirce’s nothing to the Lowenheim-Skolem theorem. For Skolem, the axiomatic set theory cannot avoid being relativistic, for no axiomatization whatsoever guarantees that an interpretation or model is accomplished once and for all. In fact, every axiomatized model is surrounded by uncountable alternative models (Skolem 1967 [1922]: 296).^[17] According to the explanation that Merrell provides, as a consequence of Lowenheim-Skolem theorem, the set-theoretically approach to the Peirce’s mathematical semiotics hosts a fundamental uncertainty with regard to the empty set, Ø, because “0 as ‘nothingness’ or ‘emptiness’,” (Merrell 1997: 168) enables the axiom that performs the categories: Ø < 1:{ Ø } < 2:{ Ø,{ Ø }} < 3:{ Ø,{ Ø },{ Ø,{ Ø }}}. The pure zero of nothing, being an open set, generates the set of categories as one derivative counting model in a myriad of possible ones.

In recent years, vis a vis, the Deleuzian scholarly studies have become increasingly aware of the importance of the zeroth state for Deleuze’s philosophy. They have been studying zero on behalf of Deleuze’s mathematical model, which is applied in a wide range of philosophical subjects, from ontology to education. As the Deleuzian mathematical model draws heavily upon differential calculus and geometry, zero expresses the differential relation of intensive – tending to zero – magnitudes (Duffy 2006; Lawlor 2019; Longo 2016; Rabouin 2012). Still, it seems that the Deleuzian scholars have not yet associated Deleuze’s differential zero to his Zeroness. If, for Deleuze, the generation of perceptions in the sensory body is to be understood as a differential relation between intuitions and sensations that appears in space and time (Cardoso Jr. 2018b), then the function of zero in the Deleuzian mathematical-ontological model may be reasonably extended to Zeroness, since it is the category that explains the emergence of signs related to perception-images.

In short, Deleuze offers a differential model to think Peirce’s categorial frame; whereas Peirce’s scholarship seems to follow the way that Deleuze harshly disapproved: “Mathematical writing systems were axiomatized, in other words, restratified, resemiotized, and material flows were rephysicalized. It is a political affair as much as a scientific one: science must not go crazy. Hilbert and de Broglie were as much politicians as they were scientists: they reestablished order” (Deleuze and Guattari 2005: 143).