A part-object which is not any part of any object

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Guy Le Gaufey

A part-object which is not any part of any object

l’d like to throw light on the « schema optique » through which Lacan achieved his imaginary dialectic in the first years of the sixties. But l’m not going to present this schema to you as a whole because, in spite of this achievement, the central key — and the main difficulty in my opinion — is tied with a clear understanding of what can be a part-object. According to this purpose, my title is as clear as day : a part-object is not any part of any object. But this light could quickly turn to be sheer mud if we don’t firmly hold the contradictions included in the statement itself.

I’m led to tackle this question that way because I well know that this expression « part-object » is coming from Melanie Klein, and so widely spread in the freudian world today that a lot of people think it is Freud’s invention. It is not, and neither an Abraham’s one, despite the fact that he had created the notion of « partial love of the object » (or « partial incorporation of the object »). Of course, the idea to part a whole and a part of this whole is strong in Abraham’s, and we can surmise that Melanie Klein followed his way of thinking. But she no longer thought of the whole-object relationship as an achievenent of love, as Abraham did, but as the painful result of ambivalence, achieved through the depressive position.

I do know that these part-objects are not exactly objects, but that they were, from their very beginning, more conceived as emotional objects, having a function, rather than a material existence. So that l’m not astonished by the Hinshelwood’s catalogue of such objects, in which the alphabetic order only puts a hint of humour : babies, bad-object, breast, buttocks, child, combined-parent-figure, fæces, father, good-object, milk, mother, mother-with-penis, penis, womb.

I’m not going to stay a long time in this kleinian world which has given us this precious name of part-object, because what « part » means from this list is not clear at all. I can’t understand on what grounds a father, or a mother, or a mother-with-penis is conceived as a « part-object », if not as an imaginary character on a stage, as Melanie Klein wrote in one of her first papers (1929). All of these part-objects are supposed to merge into a synthesis, and even if the depressive position, coming only after 1915, makes the plot a little bit more intricated, the main pattern of synthesis keeps on being available.

Thus, if we agree with the freudian notion of narcissism as the first feeling of unity any human being encounters, we are on the way to understand that kleinian part-objects are always thought as components of the narcissistic unity considered as a whole, even if they come to vanish in so far as entities in this merging. But unfortunately I want to lead you to the exact opposite meaning of the word « part » in « part-object », so I must slip away from any kleinian world on pain of confusion.

In the same way, I’ll keep on avoiding a direct presentation of the lacanian « object a » as such it bas been conceived between, say, 1959 and 1963. In 1959, up to the end of Le désir et son interprétation (by now available in English), this object is still the little other, l’autre imaginaire, and therefore the pattern of narcissism, since « l’autre imaginaire » and the mirror image are said to be equal from 1936.

A lot of discursive operations through these seminars (Le transfert, L’identification, L’angoisse) led Lacan to a complete different understanding of the same name : the object a is then defined as having not any mirror image, as essentialy linked with instincts (oral, anal, regard and voice), and as partial in a very new meaning I want now to emphasize.

Putting aside for a while the instinctual aspect, the absence of mirror image and the « partial » feature both focus on the fact that such an object can’t be counted as « one ». Any kind of unity misses it — and if it can’t be inscribed in any unity, it’s normal it escapes any mirror image as far as mirror image is conceived as the pattern of unity. From now on, we must take care of the fact that any object which can’t be taken into account as one is difficult to be thought as an object. It’s the least we are allowed to expect from any object : to be one object. So it immediatly turns out that this strange object doesn’t own the minimal property of any object. First difficulty. One breast, as a part-object at least, is not one breast, which doesn’t mean in this case a sort of function which would be called « breast ». But, at this point, I must lose any hope to make it clear this way. So I do prefer to come back to the first notion of the mirror stage given by Lacan in the middle of the thirties, and to show off in this first attempt what was then to become this bloody « object a ».

The unitary class

You should be wrong if you thought of the mirror stage as a well-known story which tells us that a baby between nine and twelve months facing a mirror recognizes the image of this body as his own, and shows a brief exultation which alerts us of the fact. This is true, but only for the child’s psychologist. And if you refer to a text of Lacan dated 1936 (« Au-delà du principe de réalité », the only one contemrorary with the first version of the mirror stage as presented in Marienbad), you will realize that the introduction of this mirror stage immediately had an epistemological value for Lacan himself (and only for him, I guess, in that time).

That’s the point we are starting from. For a huge question remains in the psychological version : why on earth the mirror image holds such a property of unity ? Where does this new unity come from ? ls it the difference between content and form ? Lacan gives us some information that he was not insensitive to this gestaltist argument, since in this text he criticizes the blind associassionism of what he then calls « the freudians of the second generation », saving Freud himself of such an assault in recognizing that he has given us the so precious notion of « imago » — and that of course he has introduced the great idea of libido linked with narcissism. But the question about the origin of unity is now only sharper than ever : is it coming from the baby becoming able to discem this difference between content and form, thanks to his neuronic developments, or is this unity coming from the mirror image itself, bringing a totaly new information into the baby’s mind ? Lacan’s preference is readable only in a difference between two adjectives : this form of the body « is given to the baby only as a Gestalt, that is to say in an exteriority in which this form is more constituend than constituted […]1 ». The agent is not on the baby’s side : he receives this unity as a sort of gift, even if this difference is but a preference, and not an exclusion. The participation of the so-called « baby » does exist, but it doesn’t come first.

I’ve said enough of it to introduce now a new deal, which is nothing else than a basic logical given : the unitary class, the class which has only a single member and is therefore called a singleton.

When you begin to study logic, even like an amateur (as I am), you immediately encounter the notion of class in general (or set, I don’t want to do any technical difference between these two words for the time being). And the definition of such a central concept is the easiest thing you can imagine : for instance, you open your Kleene’s (Mathematical Logic, 1967, p. 135) and you read : « A set is constituted by objects thought of together. » Or Robin’s (Mathematical Logic : A First Course, 1969, p. 171) : « Roughly speaking, a set is a collection of objects and is thought to have an independant existence of its own […] ». Or still Halmos, more direct if possible in his Naïve Set Theory (p. l) : « A pack of wolves, a bunch of grapes, or a flock of pigeons are all examples of sets of things. »

David Lewis, who reports these quotations in his last book Parts of Classes humouristly comments : « So far, so good ». And he decisively goes on :

But after a time, the unfortunate student is told that some classes — the singletons — have only a single member. Here is a just cause for student protest, if ever there was one. This time, he has no « many ». He has no elements or objects — I stress the plural — to be « combined » or « collected » or « gathered together » into one, or to be « thought of together as one ». Rather, he has just one single thing, the element, and he has another single thing, the singleton, and nothing he was told gives him the slightest guidance about what that one thing has to do with the other. Nor did any of those familiar examples concern single-membered sets. His introductory lesson just does not apply.

This wouldn’t be worrying if such unitary classes were very rare, sorts of peculiarities reserved for tourists only ; but they are in fact the perpend, the real brick which all classes are made of. Here, we need to recall some basic law of classes logic : a class can always be divided into its sub-classes, up to the sub-classes composed of one element only — our « singletons ». So that singletons are everywhere. But, in the same time, we have not to forget that nobody is never allowed to break such a singleton to lay one hand on this element which « belongs to » this singleton. Each singleton is a dead end in any decomposition of any class. And more : it’s strictly required that our student be able to make a perfect difference at any time between an element and the singleton composed of this same element — on pain of severe confusion, I mean : the immediate inability to make any step more in logic. It’s thus a very serious law, impossible to slip away by any means.

Once here, Lewis sets off on a very interesting hypothesis he calls : the « lasso hypothesis ». Trying to help our poor student through bringing to him some clear difference between an element and its singleton, Lewis notices that we learnt in school to picture a class by drawing pictures of its members, and then drawing a picture of a lasso around them. He then keeps on writing :

What if this were the exact truth of the matter ? Maybe the singleton of something x is not, after all, an atom ; but rather consists of x plus a lasso.

At first, it seems comfortable to think this way, since we can directly get the difference between an element and its singleton : this so-called « lasso ». But we really need to move forward very carefully in these basic matters, for embarrassing questions arise once again : if the singleton of x consists of x plus a lasso, and the singleton of y consists of y plus a lasso, can it ever be that the same lasso is used twice over ? Or can it ever be that one single lasso will do for making all the singletons ? The reply is very simple, because Lewis immediately gives an unanswerable logic demonstration which shows that there must be a different lasso each time. So that each thing has its own lasso. I don’t want to stress here this result, but it is very significant on metaphysical grounds (Duns Scot, for example, is not so far with his strange concept of hæcceité). But the most important question still remains, because in this logic world which we move in by now, there is room for elements and for classes, strictly nothing else. And this lasso is, at first glance at least, neither an element nor a class. Of course, everyone could be tempted to consider it as en equivalent of the null class, and thus to take this lasso as a class, even if a very special one, still harder to realize, still far-off any intuition than our singleton : the null class. Unfortunately, in doing so, our question crops up again : what on earth such a lasso, now considered in itself, would be made of, if it has to be thought of the most basic constituend of any relation ?

The mystery of singletons — which was our first problem according to Lewis — is now reduced to this mystery of lassos, but even if this latter turns out to be as obscure as the former, its worth consists of a better localization of the black hole. Yet, it is not very surprising if Lewis, in his deadpan style, comes to call the conclusion of this chapter : « Credo ». What else in front of a sheer mystery ?

I have to say, gritting my teeth, that somehow, I know not how, we do understand what it means to speak of singleton. […] And somehow that ordinary things have singletons, and singletons have singletons […] We know even that singletons comprise the predominant part of Reality2.

In this mystery, I would like to do an hypothesis of mine : the evidence of the existence of singleton is so strong and dark because, for everyone, it coincides with the identification to the mirror image, in so far as we thought of the body in front of the mirror as a logic element, and the mirror image as the very spring of all singletons. And I immediately add that I want no one here to shout about this as « psychologism ». After all, when our logic handbooks give us their definition of a class from any « many », they use too a very common feeling, coming from our first childhood and according to which we do know what it means to gather together a « many » into one manifoldness, at least. We do know for a long time the difference between a heap of sand and a sandpie or a sandcastle. The passage from many to one is presented as secure only because of this psychological background. I don’t claim anything more here than the right to throw light on the background of singletons mystery. I ignore if such a precision could matter for logicians ; I believe not. But I want to develop what this viewpoint can teach us about the so-called « element », since my hypothesis implies that the body in front of the mirror has to be conceived with the properties of the logic element such as already stressed. And this could be of some importance to us.

The body without any image

The common feeling is that the mirror image is the manifold of a unity which already exists with the body in front of the mirror. The body would have its own unity, and it would only be unable to catch it without the help of a mirror. All that would be nothing but a question of knowing. I can’t agree with this view, among other things because the alleged primarity of the body unity is not something we can easil, y trust.

If we refer to the session of the 7th of June 1961 in the seminar Le transfert translated by Cormac, we can read something about :

[…] the contrast between this thing which can be sketched of something which is projected in front of the baby, which attracts him, with which he persists in playing, and this incomplete thing which is manifested in his own gestures.

To introduce now this strange thing which could be a body « without any image », I want to talk about this « incomplete thing », or rather this « quelque chose d’incomplet » which stands in front of the mirror, at this place we call « the baby ». The french turn of phrase indeed means « the incomplete thing », as translated, but it means as well that « something » in it, a sort of feature, turns out to be « incomplete » ; it’s as much a feeling as a reality. We must pay attention to this little detail, because « something of » (quelque chose de) is, grammaticaly speaking, a partitive, and we are running after what could be a part object which would be actually partial. An « incomplete thing » too easily suggests once again a sort of pie where a part would be missing, and I still want to dismiss this representation of incompleteness.

At this point, our reflections about the logic element in front of its singleton have taught us that it owns a bizarre kind of unity. Indeed, even if we are led to consider it as taking up some space, and therefore is indefinitely divisible, we must think of it as having no part. Mind you that it is exactly why, in the first steps of set theory, our student is firmly invited to say that this element « belongs to » its singleton, but is not by the same way « included in » this same singleton. A is included in B if and only if each part of A belongs to B and in so far as the totality of the parts of A belong to B too. So that only classes can be included in because they do own parts. The element can belong to, but can’t never be included in, simply because we are not allowed at any time to consider it as composed of parts.

The element will be an element if and only if I consider it as a totality without any kind of unity : a mere multiplicity such as I will be unable to separate any part of it since to do this I should have the possibility to print on it a sort of stamp which would immediately cut out a part, and that part would then precisely bear the kind of unity we are not allowed to use at this time of the game. Of course, it is not very easy to realize what can be this sort of being — a mere multiplicity — in which we would be unable at all to cut out any piece, since in such a multiplicity we don’t have any lasso.

I’m interested in this strange quality developped by a mere multiplicity because I think that it is the fundamental ground of what Lacan named (december 1961) « le trait unaire » in his attempt to translate very carefully what Freud had pointed out as « ein einziger Zug ». « Unaire » means a feature such as there won’t be any part of it as such. Naturally, it is not at all an empirical observation, and we would be misleading if we went that way : on the other hand, we must understand it as a definition, knowing that a definition needn’t to be right, but only to be relevant. But I want to be clearer on this point and, to this end, I need to give you a brief comment on the introduction by Lacan of his utterance : Yadlun (There’s one).

When he introduced this sort of onomatopoeia, like a refusal of any grammatical articulation in this pure emergence of unity into discurse, he didn’t connect it with his mirror stage. He only invented a new word, a neologism, the adjective « unien » (depicted as the anagram of « ennui ») to mean the exact opposite of « unaire ». « Unien » means exactly the kind of unity which belongs to a class (or a set, as we saw with Lewis' lasso), whereas « unaire » refers to the unity of the element. So that he comes to this kind of statement, for instance in his session of 19th April1972 :

There can’t be any one out of the figure of a bag, a bag with a hole in it. Nothing is one without getting out, or getting in the bag. This is the basis of unity, intuitively speaking.

The point here is about the mouvement itself. We can present the same thing in another way. Let’s suppose we have two collections of elements (classically : forks and knives), and strictly unable to count them separately. We are going to decide that, each time we’ll take (or mark) an element on our right, we’ll take (or mark) another one on our left. We are going to practice the tac-tac method : each time it will turn out l’m able to practice my tac-tac, nothing is to be expected. We are only invited to keep on doing the same way. But let’s suppose now that, suddenly, I will be able to mark the element on my right, whereas there is no more element on my left. I could then cry : Yadlun. This unity is neither to be searched on the side of the blank on my left, nor on the side of the element on my right : it is precisely located in the fact that this element is confronted for the first time with the lack of its colleague. At this point, it turns out that the set on my left is achieved, it has reached its « union » quality, whereas the set on my right appears to have one element more than the other.

I now seek to present you the mirror stage and its imaginary identificatien on this pattern of an element confronted with its own singleton, even if such an event is close to be a joke (but each imaqinary attempt to point out the imaginary unity as such is necessarily a sort of joke). One of the consequences of such a view is to held that « something » in this body in front of the mirror is not to be taken into account as reflected in the mirror image. This body has « something » incomplete, but all the rest of it is complete ; don’t you go thinking that l’d want to suggest you that nothing of this body would be reflected in the mirror. It’s a property reserved only for vampires. This body is certainly reflected in a very common sense in the mirror image ; but in so far as we regard this body as encountering, discovering its first feelings of unity, we can’t adorn it with the unity that we, as observers, are able to perceive in it. In such a position, we would make a serious mistake since this body we would be looking at would have the exact value of the mirror image for us. We would have done nothing but to repeat the same problem, shifting the emphasis from the relation between the body and its mirror image, to the relation between us and this body. This slide is devoid of interest : we have to realize the relation between the so-called body and its mirror image, once and for all, on the pattern of the strange relationship between a logic element and its singleton.

Thus I come back to this « quelque chose d’incomplet » we had seen before, and that Lacan places on the side of the body proper. In a way, this body is no more no less « complete » than the logic element itself. Strictly speaking, nothing is missing in it, and yet it is going to encounter « something » in its mirror image that it neither knew nor held before : its unity. But please, remember here a property of any logic element as such David Iewis has taught us : any logic element has its own lasso, its own capacity to belong to its own singleton, if and only if it does exist. So, shouldn’t we know where stands the lasso, in the time the element is only an element ? But an element is not supposed to have pockets to screen its lasso when it doesn’t need it. So, are we going to consider that an element is an element only when it is short of its own lasso ? But if we did so, we might have to face the fact that an element is never supposed missing anything. And worse : in what kind of paradise such a supply of lassos could silently languish after their beloved element ? It’s clear that our so precious notion of lasso leads us to a mere contradiction.

The imaginary dialectic

That’s exactly what I was looking for from the very beginning, because if we are not held back (as we are each time we must face a contradiction), we risk to miss the point of this last session ending the seminar Le Transfert. During this session, Lacan tries to present his imaginary dialectic through the positionning of what is present and what is missing on both sides. To this end, he draws a sort of graffiti showing what is to be counted as libidinaly cathected on each side of the mirror :

To make it a little clearer, we are going to call the left side W (the wordly side), and the right one M (mirror side) ; on each side, the horizontal line separates a « head » and a « bottom ». The bottom of W, supposed to mean the narcissistic cathexis of the body proper, the body as an organism, as composed of parts, is sent to the bottom of M, which is regarded as the image proper, exactly : what is visible in the mirror. The head of M is not included in the image ; it is neither visible from W nor from anywhere. But a sort of equivalence is so reached between the head of W and the bottom of M, between the remainder of the libidinal cathexis, and all the libido cathected in the mirror image. The crucial identification to the mirror image is leaned on this : it is possible for the baby to take this image as his own, not only because he discovers a parallel between the visible motions of the image and his cenesthethic feelings, but because he has not libidinaly deserted W. Taking the most beautiful image to stand for our bottom of M, namely Venus Aphrodite, Lacan comments this equivalence this way :

If it (the phallus) is there before us in this dazzling body of Venus, it is precisely in so far as it is not there.

I hope you are not going to be that distracted by this new and so powerful word : « phallus ». At this place, it doesn’t mean nothing about sexual ratio or anything of that kind. She-babies encounter their mirror image as well as he-babies. That’s why I keep on prefering to speak about lassos that about phalluses : the plural sounds here very strange, but l’m lead to think that the phallus taken as universal is a very dangerous matter. Anyway, this equivalence once given, what could on earth be the destiny of this head of W, this libidinal cathexis which seems to weigh as much as the totality of that of the image, but which is straightoff set as definitely out of any direct visibility.

It is striking that a couple of weeks before Lacan presents this graffiti, he made for the first time the remark according to which, when the baby faces the mirror image and is jubilant, he or she turns round to catch the regard of the adult who’s carrying him in this arms. As far as I know, Lacan had not done the least mention of this single fact before this 7th of June 1961. That day, Lacan asked : « From the 0ther, what can come ? » I now want to take a close look at his answer :

We advance and we say : there can only come the sign, the image of o, i (o).

This phrase is obscure enough to give way to numerous misunderstandings. I must advance too and say : this sign comes to assent to the equation in which the head of W equals the bottom of M. To introduce, or to agree with an identity, you have to get first a differentiation between the two menbers of your identity. This assent too both recognises the difference between the head of W and the bottom of M, and straightoff, the identification of them as well.

Working on this, I was led to consider another feature belonging to this kind of scene I had to encounter on my own, and to which Lacan doesn’t seem te have paid any attention. When you carry a baby in your arms in front of a minor, according to the very usuel laws of optics, your own image is reflected in the mirror, for the baby as well as for you. It wouldn’t be the case only with a very narrow mirror, and the baby standing really far from it, and then I think all of it wouldn’t work. So that, when the baby turns back and encounters your regard, he is looking as well at the face a few seconds before he could see in its specular aspects (that is : inverted). Therefore, he is doing on the other’s face the operation he is not able to undertake with himself only : he can’t have a look to his own face to compare it with the image of it. The regard of the other comes as an equivalent of this out of reach head of W. If you agree with this — which is not in Lacan’s — you will be able to understand the opposition Lacan introduces, immediately after the quotation I gave, between the « echte Ich » and the ideal ego, that is i (o).

The « echte Ich », the « authentic ego », is the one supposed to turn back. This one is then located at a very precise crossing : hardly has he come to identify himself as a visible person in the mirror that, almost at the same time, he has to be recognized as a sighted person as well. And this recognition is described by Lacan as a tie between this « echte Ich » and a signifier, that is something we already know as having no part.

This crossing is a crossing between two of the three lacanian dimensions, and this aspect of the body which escapes the imaginary dimension of the identification with the mirror image, is precisely what comes to be recognized by something as indivisable as a regard. From this sort of point, the difference and the equivalence of the head of W and the bottom of M is given : this point is named by Lacan « I », and we here can add that this difference/equivalence is, properly speaking, the phallus. This latter is never « something », but the basis of any ratio between anythig you can imagine.

The lasso-phallus has therefore to be conceived as a passage from a mere rnultiplicity to a whole composed of parts, and that teaches us to make a difference between two values of the body, those which appear on the W side with the head and the bottom.

The bottom is to be taken as the imaginary body, the body such as we can see it in the mirror, included this mirror which is the other’s body. But the head is to be taken as the real body — not because it would be more actual, or more instinctual, or more « physical » —, but because it necessarily escapes the imaginary unity as a result of its true nature, at least if we were right to describe it as a mere multiplicity.

This inability to enter imaginary unity is the reason of the equivalence aforementioned. Exactly like when someone comes and tells you : your money or your life, stand and deliver ! These two things — money and life — certainly not equal previously, are now on the scales. The same in front of the mirror, when time has come for the discovering of the specific unity : your image or your presence.

All the diversity, all the world of visible things as well as of significations, all this world indefinitely decomposable into parts, is laid down as equal to this mere multiplicity in which none part can be counted as one, which simply means that this multiplicity is to be taken as « unaire » (on the pattern of the german mathematician Richard Dedekind when he defined a real number — which is no doubt a multiplicity — as a cut between all the real numbers before it and all the real numbers after it. This cut is a veru good example of what could be a « trait unaire » according to Lacan).

To conclude, I’d like to give you another kinds of examples through which we could have the idea of the body proper as « unaire ». The first — but maybe the darker — is around birth : when the new being seems so weird that we are not well-prepared enough to consider it so quickly as a compound being. His coming from approximatively nowhere points it out as « unaire », for a short time. But this momentary feeling is certainly stronger at the other edge of the stage, when appears, the time of a glance, the rigor mortis, the rigidness of a corpse : this kind of unity is weird enough, too, to give us a little idea of what is beyond all ideas : just between the stiffness of a stick — and the indefinite multiplicity of this dust we are said to return to, and which escorts us all along this world full of living and charming lassos.

1. J. Lacan, Écrits, Le Seuil, Paris, 166, p. 95.

2. D. Lewis, Part of Classes, op. cit., p. 59.


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