Jacques-Alain
Miller
http://www.lacan.com/symptom8_articles/miller8.html
[…]
Concept of
the Logic of the Signifier
What I am aiming to restore, piecing together indications dispersed through the work of Jacques Lacan, is to be designated the logic of the signifier - it is a general logic in that its functioning is formal in relation to all fields of knowledge including that of psychoanalysis which, in acquiring a specificity there, it governs; it is a minimal logic in that within it are given those pieces only which arc necessary to assure it a progression reduced to a linear movement, uniformally generated at each point of its necessary sequence. That this logic should be called the logic of the signifier avoids the partiality of the conception which would limit its validity to the field in which it was first produced as a category; to correct its linguistic declension is to prepare the way for its importation into other discourses, an importation which we will not fail to carry out once we have grasped its essentials here.
The chief advantage to be gained from this process of minimisation is the greatest economy of conceptual expenditure, which is then in danger of obscuring to you that the conjunctions which it effects between certain functions are so essential that to neglect them is to compromise analytic reasoning proper.
By considering the relationship between this logic and that which I will call logician's logic, we see that its particularity lies in the fact that the first treats of the emergence of the second. and should be conceived of as the logic of the origin of logic - which is to say, chat it docs not follow its laws, but that, prescribing their jurisdiction, itself falls outside that jurisdiction.
This dimension of the archeological can be grasped most succinctly through a movement back from the field of logic itself, where its miscognition. at its most radical because closest to is recognition is effected.
That this step repeats something of that which Derrida has shown to be exemplary to phenomenology [1] will conceal to none but the most hasty this crucial difference, that here miscognition finds its point of departure in the production of meaning. We can say that it is constituted not as a forgetting, but as a repression.
To designate it I choose the name of suture. Suture names the relation of the subject to the chain of its discourse; we shall see that it figures there as the clement which is lacking, in the form of a stand-in. For, while there lacking, it is not purely and simply absent. Suture, by extension - the general relation of lack to the structure - of which it is an element, inasmuch as it implies the position of a taking-the-place-of.
It is the objective of this paper to articulate the concept of suture which, if it is not named explicitly as such by Jacques Lacan, is constantly present in his system.
Let it be absolutely clear that it is not as philosopher or philosopher's apprentice that I am speaking here - if the philosopher is as characterized by Heinrich Heine in a sentence quoted by Freud, "with his nightcaps and the tatters of his dressing-gown, patching up the gaps in the structure of the universe". But take care not to think that the function of suturation is peculiar to the philosopher: what is specific to the philosopher is the determination of the field in which he operates as a "universal structure". It is important that you realize that the logician, like the linguist. also sutures at his particular level. And, quite as much. anyone who says "I".
In order to grasp suture we must cut across what a discourse makes explicit of itself, and distinguish from its meaning, its letter. This paper is concerned with a letter - a dead letter. It should come as no surprise if the meaning then dies.
The main thread of this analysis will be Gottlob Frege's argument in Grundlagen der Arithmetik, [2] crucial here because it puts into question those terms which in Peano's axiomatic, adequate for a construction of a theory of natural numbers, are taken as primary - that is, the zero, the number, the successor. [3] This calling into question of the theory, by disintricating, from the axiomatic where the theory is consolidated, the suturing, delivers up this last.
The Zero and the One
Here then is the question posed in its most general form;
What I am aiming to restore, piecing together indications dispersed through the work of Jacques Lacan, is to be designated the logic of the signifier - it is a general logic in that its functioning is formal in relation to all fields of knowledge including that of psychoanalysis which, in acquiring a specificity there, it governs; it is a minimal logic in that within it are given those pieces only which arc necessary to assure it a progression reduced to a linear movement, uniformally generated at each point of its necessary sequence. That this logic should be called the logic of the signifier avoids the partiality of the conception which would limit its validity to the field in which it was first produced as a category; to correct its linguistic declension is to prepare the way for its importation into other discourses, an importation which we will not fail to carry out once we have grasped its essentials here.
The chief advantage to be gained from this process of minimisation is the greatest economy of conceptual expenditure, which is then in danger of obscuring to you that the conjunctions which it effects between certain functions are so essential that to neglect them is to compromise analytic reasoning proper.
By considering the relationship between this logic and that which I will call logician's logic, we see that its particularity lies in the fact that the first treats of the emergence of the second. and should be conceived of as the logic of the origin of logic - which is to say, chat it docs not follow its laws, but that, prescribing their jurisdiction, itself falls outside that jurisdiction.
This dimension of the archeological can be grasped most succinctly through a movement back from the field of logic itself, where its miscognition. at its most radical because closest to is recognition is effected.
That this step repeats something of that which Derrida has shown to be exemplary to phenomenology [1] will conceal to none but the most hasty this crucial difference, that here miscognition finds its point of departure in the production of meaning. We can say that it is constituted not as a forgetting, but as a repression.
To designate it I choose the name of suture. Suture names the relation of the subject to the chain of its discourse; we shall see that it figures there as the clement which is lacking, in the form of a stand-in. For, while there lacking, it is not purely and simply absent. Suture, by extension - the general relation of lack to the structure - of which it is an element, inasmuch as it implies the position of a taking-the-place-of.
It is the objective of this paper to articulate the concept of suture which, if it is not named explicitly as such by Jacques Lacan, is constantly present in his system.
Let it be absolutely clear that it is not as philosopher or philosopher's apprentice that I am speaking here - if the philosopher is as characterized by Heinrich Heine in a sentence quoted by Freud, "with his nightcaps and the tatters of his dressing-gown, patching up the gaps in the structure of the universe". But take care not to think that the function of suturation is peculiar to the philosopher: what is specific to the philosopher is the determination of the field in which he operates as a "universal structure". It is important that you realize that the logician, like the linguist. also sutures at his particular level. And, quite as much. anyone who says "I".
In order to grasp suture we must cut across what a discourse makes explicit of itself, and distinguish from its meaning, its letter. This paper is concerned with a letter - a dead letter. It should come as no surprise if the meaning then dies.
The main thread of this analysis will be Gottlob Frege's argument in Grundlagen der Arithmetik, [2] crucial here because it puts into question those terms which in Peano's axiomatic, adequate for a construction of a theory of natural numbers, are taken as primary - that is, the zero, the number, the successor. [3] This calling into question of the theory, by disintricating, from the axiomatic where the theory is consolidated, the suturing, delivers up this last.
The Zero and the One
Here then is the question posed in its most general form;
what is it that functions in the series of whole natural
numbers to which we can assign their progression?
numbers to which we can assign their progression?
And the
answer, which I shall give at once before establishing it:
in the process of the constitution of the series,
in the genesis of progression,
the function of the subjet, miscognized is operative.
in the genesis of progression,
the function of the subjet, miscognized is operative.
This
proposition will certainly appear as a paradox to anyone who knows that the
logical discourse of Frege opens with the exclusion of that which is held by
empiricist theory to be essential for the passage of the thing to the unit, and
of the set of units to the unit of number: that is, the function of the
subject, as support of the operations of abstraction and unification.
For the unity which is thus assured both for the individual and the set, it only holds in so far as the number functions as its name. Whence originates the ideology which makes of the subject the producer of fictions, short of recognizing it as the product of its product - an ideology in which logical and psychological discourse are wedded, with political discourse occupying the key position, which can be seen admitted in Occam, concealed in Locke, and miscognized thereafter.
A subject therefore, defined by attributes whose other side is political, disposing as of powers, of a faculty of memory necessary to close the set without the loss of any of the interchangeable elements, and a faculty of repetition which operates inductively. There is no doubt that it is this subject which Frege, setting himself from the start against the empiricist foundation of arithmetic. excludes from the field in which the concept of the number is to appear.
But if it is held that the subject is not reducible, in its most essential function, to the psychological, then its exclusion from the field of number is assimilable to repetition. Which is what I have to demonstrate.
You will be aware that Frege's discourse starts from the fundamental system comprising the three concepts of the concept, the object and the number, and two relations, that of the concept to the object, which is called subsumption and that of the concept to the number which I will call assignation. A number is assigned to a concept which subsumes objects.
What is specifically logical about this system is that each concept is only defined and exists solely through the relation which it maintains as subsumer with that which it subsumes. Similarly, an object only has existence in so far as it falls under a concept, there being no other determination involved in its logical existence, so that the object takes its meaning from its difference to the thing integrated, by its spatio-temporal localization, to the real.
Whence you can see the disappearance of the thing which must be effected in order for it to appear as object - which is the thing in so far as it is one,
It is dear that the concept which operates in the system, formed solely through the determination of subsumption, is a redoubled concept: the concept of identity to a concept.
This redoubling. induced in the concept by identity, engenders the logical dimension, because in effecting the disappearance of the thing it gives rise to the emergence of the numerable.
For example, if I group what falls under the concept "child of Agamemnon and Cassandra", I summon in order to subsume them Pelops and Teledamus. To this set I can only assign a number if I put into play the concept "identical to the concept: child of Agamemnon and Cassandra". Through the effect of the fiction of this concept, the children now intervene in so far as each one is, so to speak, applied to itself - which transforms it into a unit, and gives to it the status of an object which is numerable as such. It is this one of the singular unit, this one of identity of the subsumed, which is common to all numbers in so far as they are first constituted as units.
From this can be deduced the definition of the assignation of number: according to Frege "the number assigned to the concept F is the extension of the concept identical to the concept F". Frege's ternary system has as its effect that all that is left to the thing is the support of its identity with itself, by which it is the object of the operative concept, and hence numerable.
The process that I have just set out authorizes me to conclude the following proposition, whose relevance will emerge later, - the unit which could be called unifying of the concept in so far as it is assigned by the number is subordinate to the unit as distinctive in so far as it supports the number.
As for the position of the distinctive unit, its foundation is to be situated in the function of identity which, conferring on each thing of the world the property of being one, effects its transformation into an object of the (logical) concept.
At this point in the construction, you will sense all the importance of the definition of identity which I am going to present.
This definition which must give its true meaning to the concept of number, must borrow nothing from it [4] - precisely in order to engender numeration.
This definition, which is pivotal to his system, Frege takes from Leibniz. It is contained in this statement: eadem sunt quorum unum potest substitui alteri salva veritate. Those things are identical of which one can be substituted for the other salva veritatewithout loss of truth. Doubtless you can estimate the crucial importance of what is effected by this statement: the emergence of the function of truth. Yet what it assumes is more important than what it expresses. That is, identity-with-itself. That a thing cannot be substituted for itself, then where does this leave truth? Absolute is its subversion.
If we follow Leibniz's argument, the failing of truth whose possibility is opened up for an instant, its loss through the substitution for one thing of another, would be followed by its immediate reconstitution in a new relation: truth is recovered because the substituted thing, in that it is identical with itself, can be the object of a judgement and enter into the order of discourse: identical with itself, it can be articulated.
But that a thing should not be identical with itself subverts the field of truth, ruins it and abolishes it.
You will grasp to what extent the preservation of truth is implicated in this identity with itself which connotes the passage from the thing to the object. Identity-with-itself is essential if truth is to be saved.
Truth is. Each thing is identical with itself.
Let us now put into operation Frege's schema, that is, go through the three-stage itinerary which he prescribes to us. Let there be a thing X of the world. Let there be the empirical concept of this X. The concept which finds a place in the schema is not this empirical concept but that which redoubles it, being "identical with the concept of X". The object which falls under this concept is X itself, as a unit. In this the number, which is the third term of the sequence, to be assigned to the concept of X will be the number 1. Which means that this function of the number 1 is repetitive for all things of the world. It is in this sense that this 1 is only the unit which constitutes the number as such, and not the 1 in its personal identity as number with its own particular place and a proper name in the series of numbers.
Furthermore, its construction demands that, in order to transform it, we call upon a thing of the world - which, according to Frege, cannot be: the logical must be sustained through nothing but itself.
In order for the number to pass from the repetition of the 1 of the identical to that of its ordered succession, in order for the logical dimension to gain its autonomy definitively, without any reference to the real, the zero has to appear.
Which appearance is obtained because truth is, Zero is the assigned to the concept "not identical with itself". In effect, let there be the concept "not identical with itself". This concept, by virtue of being a concept, has an extension, subsumes an object. Which object? None. Since truth is, no object falls into the place of the subsumed of this concept, and the number which qualifies its extension is zero.
In this engendering of the zero, I have stressed that it is supported by the proposition that truth is. If no object falls under the concept of non-identical-with-itself, it is because truth must be saved. If there are no things which are not identical with themselves, it is because non-identity with itself is contradictory to the very dimension of truth. To its concept, we assign the zero. It is this decisive proposition that the concept of not-identical-with-itself is assigned by the number zero which sutures logical discourse.
For, and here I am working across Frege's text, in the autonomous construction of the logical through itself, it has been necessary, in order to exclude any reference to the real, to evoke on the level of the concept an object not-identical-with-itself, to be subsequently rejected from the dimension of truth.
The zero which is inscribed in the place of the number consummates the exclusion of this object. As for this place, marked out by subsumption, in which the object is lacking, there nothing can be written, and if a 0 must be traced, it is merely in order to figure a blank, to render visible the lack.
From the zero lack to the zero number, the non-conceptualisable is 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 places 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]
This system is thus so constituted with the 0 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 the 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) = n'... Frege opens out the n + 1 in order to discover what is involved in the passage from n to its successor.
You will grasp the paradox of this engendering as soon as I produce the most general formula for the successor which Frege arrives at: "the Number assigned to the concept member of the series of natural numbers ending with 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 find 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 real is pure and simple absence finds itself through the fact of number (through the instance of truth) noted 0 and counted for 1.
Which is why we say the object not-identical with itself invoked-rejected by 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, the function of addition of the sign + is superfatory, 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 which 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 to 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] 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, which 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 to 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 the exclusion of the subject 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 of 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 in 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, so much so that they could even be called its reflections, by the repetition of 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 of 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 of 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]
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 subject 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 it 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 that 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 subject 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] Edmund Husserl, L'origine de la géometrie, translation and introduction by Jacques Derrida, PUF, 1962.
[2] German text with English translation published under the title The Foundations of Arithmetic, Basil Blackwell, 1953.
[3] 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] Which is why we must say identity and not equality.
[5] I leave aside the commentary of paragraph 76 which gives the abstract definition of contiguity.
[6] And, at another level, the impossibility of meta-language (cf by Jacques Lacan, Cahiers pour 1'analyse, No I, 1966).
[7] Dedekind, quoted by Cavailles (Philosophie mathémathique, p 124, Hermann, 1962).
For the unity which is thus assured both for the individual and the set, it only holds in so far as the number functions as its name. Whence originates the ideology which makes of the subject the producer of fictions, short of recognizing it as the product of its product - an ideology in which logical and psychological discourse are wedded, with political discourse occupying the key position, which can be seen admitted in Occam, concealed in Locke, and miscognized thereafter.
A subject therefore, defined by attributes whose other side is political, disposing as of powers, of a faculty of memory necessary to close the set without the loss of any of the interchangeable elements, and a faculty of repetition which operates inductively. There is no doubt that it is this subject which Frege, setting himself from the start against the empiricist foundation of arithmetic. excludes from the field in which the concept of the number is to appear.
But if it is held that the subject is not reducible, in its most essential function, to the psychological, then its exclusion from the field of number is assimilable to repetition. Which is what I have to demonstrate.
You will be aware that Frege's discourse starts from the fundamental system comprising the three concepts of the concept, the object and the number, and two relations, that of the concept to the object, which is called subsumption and that of the concept to the number which I will call assignation. A number is assigned to a concept which subsumes objects.
What is specifically logical about this system is that each concept is only defined and exists solely through the relation which it maintains as subsumer with that which it subsumes. Similarly, an object only has existence in so far as it falls under a concept, there being no other determination involved in its logical existence, so that the object takes its meaning from its difference to the thing integrated, by its spatio-temporal localization, to the real.
Whence you can see the disappearance of the thing which must be effected in order for it to appear as object - which is the thing in so far as it is one,
It is dear that the concept which operates in the system, formed solely through the determination of subsumption, is a redoubled concept: the concept of identity to a concept.
This redoubling. induced in the concept by identity, engenders the logical dimension, because in effecting the disappearance of the thing it gives rise to the emergence of the numerable.
For example, if I group what falls under the concept "child of Agamemnon and Cassandra", I summon in order to subsume them Pelops and Teledamus. To this set I can only assign a number if I put into play the concept "identical to the concept: child of Agamemnon and Cassandra". Through the effect of the fiction of this concept, the children now intervene in so far as each one is, so to speak, applied to itself - which transforms it into a unit, and gives to it the status of an object which is numerable as such. It is this one of the singular unit, this one of identity of the subsumed, which is common to all numbers in so far as they are first constituted as units.
From this can be deduced the definition of the assignation of number: according to Frege "the number assigned to the concept F is the extension of the concept identical to the concept F". Frege's ternary system has as its effect that all that is left to the thing is the support of its identity with itself, by which it is the object of the operative concept, and hence numerable.
The process that I have just set out authorizes me to conclude the following proposition, whose relevance will emerge later, - the unit which could be called unifying of the concept in so far as it is assigned by the number is subordinate to the unit as distinctive in so far as it supports the number.
As for the position of the distinctive unit, its foundation is to be situated in the function of identity which, conferring on each thing of the world the property of being one, effects its transformation into an object of the (logical) concept.
At this point in the construction, you will sense all the importance of the definition of identity which I am going to present.
This definition which must give its true meaning to the concept of number, must borrow nothing from it [4] - precisely in order to engender numeration.
This definition, which is pivotal to his system, Frege takes from Leibniz. It is contained in this statement: eadem sunt quorum unum potest substitui alteri salva veritate. Those things are identical of which one can be substituted for the other salva veritatewithout loss of truth. Doubtless you can estimate the crucial importance of what is effected by this statement: the emergence of the function of truth. Yet what it assumes is more important than what it expresses. That is, identity-with-itself. That a thing cannot be substituted for itself, then where does this leave truth? Absolute is its subversion.
If we follow Leibniz's argument, the failing of truth whose possibility is opened up for an instant, its loss through the substitution for one thing of another, would be followed by its immediate reconstitution in a new relation: truth is recovered because the substituted thing, in that it is identical with itself, can be the object of a judgement and enter into the order of discourse: identical with itself, it can be articulated.
But that a thing should not be identical with itself subverts the field of truth, ruins it and abolishes it.
You will grasp to what extent the preservation of truth is implicated in this identity with itself which connotes the passage from the thing to the object. Identity-with-itself is essential if truth is to be saved.
Truth is. Each thing is identical with itself.
Let us now put into operation Frege's schema, that is, go through the three-stage itinerary which he prescribes to us. Let there be a thing X of the world. Let there be the empirical concept of this X. The concept which finds a place in the schema is not this empirical concept but that which redoubles it, being "identical with the concept of X". The object which falls under this concept is X itself, as a unit. In this the number, which is the third term of the sequence, to be assigned to the concept of X will be the number 1. Which means that this function of the number 1 is repetitive for all things of the world. It is in this sense that this 1 is only the unit which constitutes the number as such, and not the 1 in its personal identity as number with its own particular place and a proper name in the series of numbers.
Furthermore, its construction demands that, in order to transform it, we call upon a thing of the world - which, according to Frege, cannot be: the logical must be sustained through nothing but itself.
In order for the number to pass from the repetition of the 1 of the identical to that of its ordered succession, in order for the logical dimension to gain its autonomy definitively, without any reference to the real, the zero has to appear.
Which appearance is obtained because truth is, Zero is the assigned to the concept "not identical with itself". In effect, let there be the concept "not identical with itself". This concept, by virtue of being a concept, has an extension, subsumes an object. Which object? None. Since truth is, no object falls into the place of the subsumed of this concept, and the number which qualifies its extension is zero.
In this engendering of the zero, I have stressed that it is supported by the proposition that truth is. If no object falls under the concept of non-identical-with-itself, it is because truth must be saved. If there are no things which are not identical with themselves, it is because non-identity with itself is contradictory to the very dimension of truth. To its concept, we assign the zero. It is this decisive proposition that the concept of not-identical-with-itself is assigned by the number zero which sutures logical discourse.
For, and here I am working across Frege's text, in the autonomous construction of the logical through itself, it has been necessary, in order to exclude any reference to the real, to evoke on the level of the concept an object not-identical-with-itself, to be subsequently rejected from the dimension of truth.
The zero which is inscribed in the place of the number consummates the exclusion of this object. As for this place, marked out by subsumption, in which the object is lacking, there nothing can be written, and if a 0 must be traced, it is merely in order to figure a blank, to render visible the lack.
From the zero lack to the zero number, the non-conceptualisable is 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 places 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]
This system is thus so constituted with the 0 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 the 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) = n'... Frege opens out the n + 1 in order to discover what is involved in the passage from n to its successor.
You will grasp the paradox of this engendering as soon as I produce the most general formula for the successor which Frege arrives at: "the Number assigned to the concept member of the series of natural numbers ending with 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 find 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 real is pure and simple absence finds itself through the fact of number (through the instance of truth) noted 0 and counted for 1.
Which is why we say the object not-identical with itself invoked-rejected by 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, the function of addition of the sign + is superfatory, 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 which 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 to 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] 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, which 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 to 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 the exclusion of the subject 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 of 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 in 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, so much so that they could even be called its reflections, by the repetition of 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 of 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 of 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]
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 subject 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 it 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 that 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 subject 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] Edmund Husserl, L'origine de la géometrie, translation and introduction by Jacques Derrida, PUF, 1962.
[2] German text with English translation published under the title The Foundations of Arithmetic, Basil Blackwell, 1953.
[3] 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] Which is why we must say identity and not equality.
[5] I leave aside the commentary of paragraph 76 which gives the abstract definition of contiguity.
[6] And, at another level, the impossibility of meta-language (cf by Jacques Lacan, Cahiers pour 1'analyse, No I, 1966).
[7] Dedekind, quoted by Cavailles (Philosophie mathémathique, 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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