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.
(Credit to Fabio Gironi for the translation. It is also available at hyper tiling blog.)
Two Ones
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.
