Sans objet

conceptualized.

Let us now set aside the zero lack in order to consider only that which is
produced by the alternation of its evocation and its revocation, the zero
number.

The zero understood as a number, which assigns to the subsuming concept the
lack of an object, is as such a thing - the first non-real thing in thought=
.

If of the number zero we construct the concept, it subsumes as its sole
object the number zero. The number which assigns it is therefore 1.

Frege's system works by the circulation of an element, at each of the place=
s
it fixes: from the number zero to its concept, from this concept to its
object and to its number - a circulation which produces the 1.
[5]<http://www.lacan.com/symptom8_articles/miller8.html#_ftn5>This
system is thus so constituted with the o counting as 1. The counting
of
the 0 as 1 (whereas the concept of, the zero subsumes nothing in the real
but a blank) is the general support of the series of numbers.

It is this which is demonstrated by Frege's analysis of the operation of th=
e
successor, which consists of obtaining the number which follows n by adding
to it a unit: n' the successor of n, is equal to n + 1, that is, ... n... (=
n
+ 1) =3D n'... Frege opens out the n + 1 in order to discover what is invol=
ved
in the passage from n to its successor.

You will grasp the paradox of this engendering as soon as I produce the mos=
t
general formula for the successor which Frege arrives at: the Number
assigned to the concept "member of the series of natural numbers ending wit=
h
n" follows in the series of natural numbers directly after n'.

Let us take a number. The number three. It will serve to constitute the
concept "member of the series of natural numbers ending with three". We fin=
d
that the number assigned to this concept is four. Here then is the 1 of n +
1. Where does it come from? Assigned to its redoubled concept, the number 3
functions as the unifying name of a set: as reserve. In the concept of'
member of the series of natural numbers ending with 3", it is the term (in
the sense both of element and of final element).

In the order of the real. the 3 subsumes 3 objects. In the order of number,
which is that of discourse bound by truth, it is numbers which are counted:
before the 3, there are 3 numbers - it is therefore the fourth.

In the order of number, there if an addition the 0 and the 0 counts for 1.
The displacement of a number, from the function of reserve to that of term,
implies the summation of the 0. Whence the successor. That which in the rea=
l
is pure and simple absence finds itself through the fact of number (through
the instance of truth) noted o and counted for 1.

Which is why we say the object not-identical with itself invoked-rejected b=
y
truth, instituted-annulled by discourse (subsumption as such) - in a word,
sutured.

The emergence of the lack as 0, and of 0 as 1 determines the appearance of
the successor. Let there be n; the lack is fixed as which is fixed as 1: n =
+
1; which is added in order to give n' - which absorbs the 1.

Certainly, if the Lot n + 1 is nothing other than the counting the zero, th=
e
function of addition of the sign + is superfat1ory, and we must restore to
the horizontal representation of the engendering its verticality: the 1 is
to be taken as the primary symbol of the emergence of lack in the field of
truth, and the sign + indicates the crossing, the transgression through
which the 0 lack comes to be represented as 1, producing, through this
difference of n to n' which you have seen to be an effect of meaning the
name of a number.

Logical representation collapses this three-level construction. The
operation I have effected opens it out. If you consider the opposition of
these two axes, you will understand what is at stake in logical suturing,
and the difference of the logic which I am putting forward to logician's
logic.

That zero is a number: such is the proposition which assures logical
dimension of its closure.

Our purpose has been to recognize in the zero number the suturing stand-in
for the lack.

Remember here the hesitation perpetuated in the work of Bertand Russell
concerning its localization (interior? or exterior to the series of
numbers?).

The generating repetition of the series of numbers is sustained by this,
that the zero lack passes, first along a vertical axis, across the bar whic=
h
limits the field of truth in order to be represented there as one,
subsequently cancelling out as meaning in each of the names of the numbers
which are caught up in the metonymic chain of successional progression.

Just as the zero as lack of the contradictory object must be distinguished
from that which sutures this absence in the series of numbers, so the 1. as
the proper name of a number, is to be distinguished from that which comes t=
o
fix in a trait the zero of the not-identical with itself sutured by the
identity with itself, which is the law of discourse in the field of truth.
The central paradox to be grasped (which as you will see in a moment is the
paradox of the signifier in the sense of Lacan) is that the trait of the
identical represents the non-identical, whence is deduced the impossibility
of its redoubling,
[6]<http://www.lacan.com/symptom8_articles/miller8.html#_ftn6>and from
that impossibility the structure of repetition, as the process of
differentiation of the identical.

Now, if the series of numbers, metonymy of the zero, begins with its
metaphor, if the o member of the series as number is only the
standing-in-place suturing the absence (of the absolute zero) which moves
beneath the chain according to the alternation of a representation and an
exclusion - then what is there to stop us from seeing in the restored
relation of the zero to the series of numbers the most elementary
articulation of the subject's relation to the signifying chain?

The impossible object, which the discourse of logic summons as the
not-identical with itself and then rejects as the pure negative, which it
summons and rejects in order to constitute itself as that which it is, whic=
h
it summons and rejects wanting to know nothing of it, we name this object,
in so far as it functions as the excess which operates in the series of
numbers, the subject.

Its exclusion from the discourse which internally it intimates is suture.

If we now determine the trail as the signifier, and ascribe to the number
the position of signified, the relation of lack to the trait should be
considered as the logic of the signifier.

*Relation of Subject and Signifier*

In effect, what in Lacanian algebra is called the relation of the subject t=
o
the field of the Other (as the locus of truth) can be identified with the
relation which the zero entertains with the identity of the unique as the
support of truth. This relation, in so far as it is matrical, cannot be
integrated into any definition of objectivity - this being the doctrine of
Lacan. The engendering of the zero. from this not-identical with itself
under which no thing of the world falls, illustrates this to you.

What constitutes this relation as the matrix of the chain must be isolated
in the implication which makes the determinant of ( exclusion of the subjec=
t
outside the field of the Other its representation in that field in the form
of the one of the unique. one of distinctive unity, which is called "unary"
by Lacan. In algebra, this exclusion is marked by the bar which strikes the
S of the subject in from of the capital A, and which is displaced by the
identity of the subject onto the A, according to the fundamental exchange o=
f
the logic of the signifier, a displacement whose effect is the emergence of
signification signified to the subject. Untouched by the exchange of the
bar, this exteriority of the subject to the Other is maintained, which
institutes the unconscious.

For: - if it is clear that the tripartition which divides (1) the
signified-to-the-subject, (2) the signifying chain whose radical alterity i=
n
relation to the subject cuts off the subject from its field, and finally (3=
)
the external field of this reject, cannot be covered by the linguistic
dichotomy of signified and signifier; - if the consciousness of the subject
is to be situated on the level of the effects of signification, governed, s=
o
much so that they could even be called its reflections, by the repetition o=
f
the signifier: - if repetition itself is produced by the vanishing of the
subject and its passage as lack - then only the unconscious can name the
progression which constitutes the chain in the order of thought.

On the level of this constitution, the definition of the subject comes down
to the possibility o/ one signifier more.

Is it not ultimately to this function of excess that can be referred the
power of thematisation, which Dedekind assigns to the subject in order to
give to set theory its theorem of existence? The possibility of existence o=
f
an enumerable infinity can be explained by this, that "from the moment that
one proposition is true, 1 can always produce a second, that is, that the
first is true and so on to infinity".
[7]<http://www.lacan.com/symptom8_articles/miller8.html#_ftn7>

In order to ensure that this recourse to the subject as the founder of
iteration is not a recourse to psychology, we simply substitute for
thematisation the representation of the subject (as signifier) which
excludes consciousness because it is not effected for someone, but, in the
chain, in the field of truth, for the signifier which precedes it. When
Lacan faces the definition of the sign as that which represents something
for someone, with that of the signifier as that which represents the subjec=
t
for another signifier. he is stressing that in so far as the signifying
chain is concerned, it is on the level of its effects and not of its cause
that consciousness is to be situated. The insertion of the subject into the
chain is representation, necessarily correlative to an exclusion which is a
vanishing.

If now we were to try and develop in time the relation which engenders and
supports the signifying chain, we would have to take into account the fact
that temporal succession is under the dependency of the linearity of the
chain. The time of engendering can only be circular - which is why both
these propositions are true at one and the same time. that subject is
anterior to signifier and that signifier is anterior to subject - but only
appears as such after the introduction of the signifier. The retroaction
consists essentially of this: the birth of linear time. We must hold
together the definitions which make the subject the effect of the signifier
and the signifier the representative of the subject: it is a circular.
though non-reciprocal, relation.

By crossing logical discourse at its point of least resistance, that of its
suture, you can see articulated the structure of the subject: as a
"flickering in eclipses", like the movement which opens and closes the
number, and delivers up the lack in the form of the 1 in order to abolish i=
t
in the successor.

As for the +. you have understood the unprecedented function which it takes
on in the logic of the signifier (a sign, no longer of addition, but of tha=
t
summation of the subject in the field of the Other, which calls for its
annulment). It remains to disarticulate it in order to separate the unary
trait of emergence, and the bar of the reject: thereby making manifest the
division of the subject which is the other name for its alienation.

It will be deduced from this that the signifying chain is structure of the
structure.

If structural causality (causality in the structure in so far as the subjec=
t
is implicated in it) is not an empty expression, it is from the minimal
logic which I have developed here that it will find its status.

We leave for another time the construction of its concept.

*Notes:*

[1] <http://www.lacan.com/symptom8_articles/miller8.html#_ftnref1> Edmund
Husserl, *L'origine de la g=E9ometrie*, translation and introduction by
Jacques Derrida, PUF, 1962.

[2] <http://www.lacan.com/symptom8_articles/miller8.html#_ftnref2> German
text with English translation published under the title *The Foundations of
Arithmetic*, Basil Blackwell, 1953.

[3] <http://www.lacan.com/symptom8_articles/miller8.html#_ftnref3> Our
reading will not concern itself with any of Frege'g various inflections of
his basic purpose, and will therefore keep outside the thematisation of the
difference of meaning and reference, as well as of the later definition of
the concept in terms of predication, from which is deduced its
non-saturation.

[4] <http://www.lacan.com/symptom8_articles/miller8.html#_ftnref4> Which is
why we must say identity and not equality.

[5] <http://www.lacan.com/symptom8_articles/miller8.html#_ftnref5> I leave
aside the commentary of paragraph 76 which gives the abstract definition of
contiguity.

[6] <http://www.lacan.com/symptom8_articles/miller8.html#_ftnref6> And, at
another level, the impossibility of meta-language (cf by Jacques
Lacan, *Cahiers
pour 1'analyse*, No I, 1966).

[7] <http://www.lacan.com/symptom8_articles/miller8.html#_ftnref7> Dedekind=
,
quoted by Cavailles (*Philosophie math=E9mathique*, p 124, Hermann, 1962).


------------------------------

This text was published in French in *Cahiers pour l'analyse* 1, Winter
1966, subsequently its English version translated by Jacqueline Rose
appeared in *Screen* 18, Winter 1978.

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