Again, another document for our reading group - to correspond with AF's Chapter 3: The Principle of Factiality. Please be sure to have read this for our next discussion, this coming week.
Translation of Meillassoux’s
(Credit to Fabio Gironi for the translation. It is also available at hyper tiling blog.)
Let us return now to the question of the one. The problem has thus far consisted, I want to remind you, in determining the existence of a possible procedure for justifying the realist—therefore non-correlationist—character of a mathematical description of the world. This problem for me takes now the following form: can we derive from the principle of factiality the absolute validity of mathematical descriptions of the real? I certainly cannot resolve here this massive problem in which I still am involved. I would simply like to present the first move for this investigation, which leads precisely to the nature of the one, and in particular to the difference within the unity of a thing, and the unity of a sign: what I shall call the difference between the ontic one and a semiotic one [l'un ontique et l'un sémiotique]
What relationship is there, between this question regarding two types of unities and the question of the realist capacity of mathematics?
In order to resolve the problem of the absolute scope of mathematics, I began by trying to identify a minimum requirement of any formal writing—logical or mathematical—that distinguishes it from natural languages [langues naturelles]. I tried to reach a precise and determinative point of difference, capable of distinguishing a symbolic, or formal, language from a natural language. It was necessary to find a characteristic which is minimal enough to be present in all formal languages and precise enough never to be present in a natural language. Indeed, I had an idea about this characteristic, and I had the intuition that it had precisely to do with the ability of thought to access the eternity of contingency. This minimum requirement actually seems to me to consist in a remarkable usage— a systematic and precise usage—of the sign devoid of significance [dépourvu de sense].
The hypothesis that I adopted, roughly, is the following: if the question of the reference [referent?] of mathematics—what does mathematics talk about?—is a piercing question, it is because mathematics consists of a sequence of operations applied to signs which, ultimately, mean nothing. Therefore, in all mathematical writing—at least in its most fundamental forms, such as Set Theory or Category Theory—there would be two kind of signs: the signs signifying operations—which I call operational-signs [signes-opérateurs]—and those signs on which ultimately these operations bear—which I call base-signs [signes-bases] and that have the express function of not signifying anything, and thereby of avoiding any parasitism of operational meaning from another regime of meaning. What we find in formal writings of signs without meaning is, therefore, not a failure of these writings, but what allows for their own singularity and richness.
A simple example is given by the so-called standard axiomatics of Set Theory. To put it very crudely, in this axiomatic one begins from signs—a, b, c— which are usually called ‘sets’. But actually, Set Theory never defines what a set is. These signs, in themselves, don’t mean anything, because they specifically have to—in my terms—provide operational-signs with a base devoid of meaning on which to operate. These signs begin to ‘resemble’ sets, in that they are subject to an operator—the operator of belonging [l’opérateur d’appartenance]. It is this operator, and not the signs of the sets, that carries upon itself the charge of signification: thus, the signs a, b, c will be called sets in that they may be subject to the operator of belonging and allow statements such as ‘a belongs to b’ or ‘b belongs to a’. A set is that which can belong to another set, or what may contain another set: a circular definition which shows that one never really defines a set, i.e., the base-sign, but only the operations that it supports.
My strategy is then as follows: I posit that the minimum requirement for the possibility of mathematical writing (I cannot show here that it is the only condition, but I posit that it is at least one condition) is the possibility to conceive and thematise signs devoid of meaning. Far from being identifiable as a nothing or a nonsense (meant as an absurdity) the sign devoid of meaning is posited as the eminent condition for mathematico-rational thought (and I think we could say the same about logic). The question that I ask is therefore the following: how can we think of the sign devoid of meaning? And the answer that I give to this question consists in showing that the condition for the thinkability of the sign devoid of meaning is the access (whether thematized or not) to the eternal contingency of everything. In short, I attempt to derive from the principle of factiality our ability to produce signs empty of meaning, therefore showing that mathematical discourse moves within a sphere of thought ‘closely associated’ with the absoluteness of contingency. I have so far demonstrated the absolute capability of mathematical descriptions; now I shall, at least, work out the first and necessary step of such an absolutization.
There is perhaps some strangeness in the question: how do we produce signs devoid of meaning? Because the signs devoid of meaning are mainly perceived as the manifestation of failure, an inability to produce meaning, rather than as the manifestation of a capacity. What sense is there to ask how we manage to produce the insignificant? And again, what does it all have to do with the question of the one?
A story will allow me to answer these two questions at once.
Imagine, without any concern about verisimilitude, that an archaeologist—working among the ruins of a largely unknown civilization without knowing if it possessed writing— partially unearths, during his research, a tablet on which there is a series of symbols [motifs], like for example:
# # # # # # # #
Suppose that his first reaction is to assume that this line is a frieze, an engraved design on the edges of the tablet. However, a moment later, he modifies his hypothesis and says, with excitement, that it could be a line of writing, like a schoolboy would write the same letter in his notebook in order to learn how to write. Then, going on digging up the tablet, he realizes that it does not contain other lines made of other characters—which would confirm his hypothesis—but a design that convinces him that the first hypothesis was correct: he was indeed dealing with a frieze rather than a line of writing.
The question that can be posed is therefore the following: what shift in vision occurred in our archaeologist, seeing in the very same pattern, respectively, similar symbols as parts of the same frieze, and as occurrences of the same sign? In both cases, the engraved marks were seen in their singular unity and in their collective arrangement: but in what consists the difference for which the symbols have become occurrences of the same sign, a token of the same type?
To explain this difference will lead us to the heart of the question of the one.
When the archaeologist saw the series as a frieze, he saw it as an entity susceptible to aesthetic appreciation in a broad sense: a singular decorative pattern, composed by a determined number of symbols—for example, eight—and whose configuration (shape of the symbols + number of these symbols) could be judged as more or less accomplished, as more or less pleasurable to the eye. In this judgment, the number of these symbols is not indifferent: seven or five symbols could have been less pleasurable than a series of eight—or, on the contrary, more so. In other words, the frieze holds what could be called an effect of monotony, or again what I more generally call an effect of repetition: an effect for which the symbols cease to be the same even though they are supposed to perfectly resemble each other (I say, be similar). To understand this point clearly, we must make a comparison with a melody: it is well known since Bergson that two similar notes (two phonetically indiscernible ‘Dohs’) are understood in different ways if they conclude distinct melodic sequences. Thus a melody—say eight DOHs played in sequence—will produce a final DOH distinct from the initial DOH, because the final one is charged with a melodic past that the initial one does not possess. There is, here, a differential effect of repetition, a melodic effect, which is an effect of monotony regarding time, homologous to that which is presented, regarding space, in the frieze. For I believe, unlike Bergson, that space is as responsible for melodic differences as time is, and that our frieze affects its symbols with a differential effect that makes each of them differ from others even if they are, empirically, rigorously similar.
Why mention this? Because I believe that the monotony effect affects every vision of empirical reality: everything that is seen as one—as an empirical, ontic unit—unfolds itself in a space and in a time that produce differences amongst these things/ones [choses-unes] themselves that, empirically, are not distinguished. Now the enigma, the mystery of the sign is, I believe, that this differential effect inherent in space-time, disappears when we see similar marks not as symbols of the same frieze but as occurrences of the same type. For then we have the right to say that these occurrences are absolutely identical, with no differential effect (neither empirical nor spatio-temporal), i.e., repetitive. It is absolutely the same sign, as a type, that is found in each of its instances: and this type will never vary, regardless of the number of occurrences, so that this time our series (########) could be extended with an ‘etc.’, which denotes the radical indifference of the identity of the type regarding the multiplicity of its occurrences. An ‘etc.’ which would be meaningless for the frieze, since the frieze is a concrete aesthetic reality, and therefore inseparable from the specific and finite number of its symbols.
There is something absolutely non-differential and therefore non-spatio-temporal, something eternal, in the sign as such, and I say in the sign as in the sign devoid of meaning. Here the meaning of our story will emerge: it generally addresses the immaterial character of language through the question of the eventual ideality of meaning [l’idéalité éventuelle du sens]—of its resistance, for example, to historicity and context—and of its possible identity in the minds of two readers of a same text in two different epochs. But here, this is the point that I wanted to highlight, our archaeologist had, I think, the experience of an eternity, of a pure identicality—an eternity in kind and not in meaning [du type et non du sens], that resists the differential effect of the empirical marks—and this experience of eternality is an experience of the sign and not of meaning: it is these signs devoid of meaning that prompted this experience, rather than meaningful signs. And these signs have indeed proved to be devoid of meaning, as they were not really signs [signes], but symbols [motifs]: our archaeologist has therefore experienced a vision capable of seizing, within an empirical mark, something eternal—a mode of unification of the marks not subject to the effect of spatio-temporal repetition—starting from a single semiotic unity, a unity for which each mark has become a one-occurrence of an identical type and, as such, indefinitely reproducible.
It is the eternal unity of the sign-type [signe-type] that allows access to the thinkability of its unlimited iteration of the same: to the ‘etc.’ that follows the series of occurrences, and that did not exist for the aesthetic vision of the frieze. In other words, the eternal is directly present as that which differentiates the sign from the mark [différencie le signe de la marque], and therefore as the meaning, (since?) neither the reference nor the essence are—a fortiori—present. [Autrement dit, l’éternel est présent à même ce qui différencie le signe de la marque, et alors que le sens, ni a fortiori la référence ou l’essence ne sont présents].
So, to have access to the sign devoid of meaning as such requires access to something eternal within the occurrence, that is its kind. Hence the question: what is the nature of this eternity?
My thesis is as follows: the eternity engaged in the grasping of the semiotic unit has its source in the grasping of the contingency of the occurrence of the sign.
Let me explain this point, to conclude. When I perceive some thing, or an empirical mark, I perceive this mark with its empirical determinations, and I perceive it as a fact. But the perception of the mark, and of its ontic unity, makes its empirical determinations come into focus first, and then, in second place, its facticity: I perceive a mark, and moreover a factual one. On the other hand, if I see the facticity of this mark as such—if I bring it to the forefront—then I know that this mark is identical in the whole of reality, and it does not vary in space nor in time. Then I will operate a unification of the mark that is of another kind than the ontico-empirical one for which I shall precisely see the eternal contingency present in that mark. I unify the mark around its contingency, and not around its empirical determinations. I can then see, in a multitude of similar marks, a kind of eternal unity, and as such not subject to the differential effect of repetition.
Now, going back to the vision of the mark-one [marque-une] as the occurrence-one [occurrence-une] of a sign-type [signe-type]. What do I do precisely, when I see a sign as a sign: when I stop considering a mark as a thing, in order to consider it as a sign? Well, I am making of this mark an essentially arbitrary entity, i.e., contingent in its being a sign. That is, I can not thematize the idea of a sign—cannot think the sign as a sign—without letting the contingency of its determinations come to the fore. What does this mean? As a thing, the mark can be thought as the necessary effect of a certain number of causes: possibly related to erosion, to a shock, to a constrained human action, etc. Even if this necessitarianism is illusory, it shows that the mark-thing [marque-chose] doesn’t require that its contingency be thematised to be grasped. Therefore even if I am a Spinozist, the same mark, now become sign, must be necessarily posited as arbitrary, since a sign has the characteristic of not having in itself any necessary determinations. Certainly there are structural constraints in a language (the signs for distinct things must be separate), but the characteristic of a sign, or of a system of signs must be capable of being encoded—transcribed—into another, structurally identical, system of signs. A sign therefore exhibits its contingency ‘on its front line’, so to speak—at least when I grasp it as a sign, one that I thematize as such.
Now, when I deal with a sign devoid of meaning, I am dealing with a sign which does not refer to a sense, a reference, but only to itself as a sign: to think a sign devoid of meaning is necessarily to thematize the sign as a sign, hence to think its own arbitrariness—by letting its eternal contingency come to the fore—to unify it around its contingency, and finally to let it proliferate in accordance with a succession of occurrences released from the differential effect of repetition.
Therefore, it seems to me that there is a possibility to derive the possibility of mathematical discourse—i.e., a discourse structured around the sign devoid of meaning—starting from the principle of factiality, by ontologically basing the difference between the ontic one and the semiotic one. Here there is the first step, I believe, towards a possible absolutization of the mathematical descriptions of the real.