## Point

Le point

A term used in the mathematical sciences, and in Lacan’s rethinking of psychoanalysis in terms of the chain of signification, the concept of ‘point’ recurs in various contexts in the *Cahiers pour l’Analyse*.

Two related senses of ‘point’ are operative in the *Cahiers pour l’Analyse*, both of which turn on a concept of ‘point’ as a functional entity without substance in itself. In geometry or topology, the point names an entity that has a location
in space or on a plane but no extension. In the debates concerning the function of infinitesimals, points came to be conceived
as limit entities, that is, sites toward which functions tended, but at which they never arrived. The peculiar nature of the
point was also clear in the debates that followed from Cantorian set theory. In her presentation of set theory, Mary Tiles suggests that one way to get at the gravity of the Cantorian revolution
in mathematics is to consider the question: ‘how many points are there in a line?’.^{1} If a line is infinitely divisible, into smaller and smaller ‘points’, how might we have an arithmetical or analytical account of the ‘number’ of points on a line? This question, which links the geometric ‘point’ to the arithmetical ‘number’, seems to make a mockery of the concept of number itself. Cantor’s work, which effectively answered this question with the
concept of transfinite cardinals, developed that of Richard Dedekind, and in particular the latter’s concept of the ‘cut’, that is, a site in a sequence of real numbers where values of lesser and greater may be quantitatively determined. In this
sense, the geometrical ‘point’ correlates to a genuinely real though non-rational number. Though it lacks any ‘content’ or substance, such a conception of point serves to determine a limit, function, or site.

The second sense of ‘point’ that informs the *Cahiers* comes from the work of Jacques Lacan and his conception of the relation of signifier to signified in the signifying chain. In his seminar on the psychoses, delivered in 1955-56, Lacan described how, although
the signified always slips under the signifier, that is to say, the metonymic movement of the signifier always displaces meaning, there are nevertheless certain ‘attachment points’ where meaning is temporarily fixed. Lacan called these points ‘*points de capiton*’, which has also been variously translated as ‘anchoring points’, ‘quilting points’ or ‘button ties’. The direct reference of the French expression is to an upholstery button, the point in a mattress or other piece of furniture
that holds the object together, preventing the stuffing from spilling out. From this conception of the ‘quilting point’, Lacan develops further the key function of punctuation in the establishment of meaning. Indeed, something of the myriad
senses of ‘point’ is contained in the very concept of punctuation, as a stopping point that fixes a set chain of signifiers.

Later, in his text ‘The Subversion of the Subject and the Dialectic of Desire in the Freudian Unconscious’ (1960), Lacan invokes the *point de capiton* again in his construction of the graph of desire. The key to Lacan’s graph is the intersection of two movements, the vector that constitutes the sliding of signification
(metonymy) and the retroactive movement of the subject across this signifying chain. Using the image of a fish on a hook, Lacan suggests that it is here that the *point de capiton* is in play. Moreover, this *point* has two functions, a diachronic and a synchronic one. Lacan writes:

The diachronic function of the button tie can be found in a sentence, insofar as a sentence closes its signification only with its last term, each term being anticipated in the construction constituted by the other terms and, inversely, sealing their meaning by its retroactive effect.

But the synchronic structure is more hidden, and it is this structure that brings us to the beginning. It is metaphor insofar as the first attribution is constituted in it... [

E, 805].

Lacan’s later concern for knots is foreshadowed here in his emphasis on the ‘button tie’. But the broader concept of point, and its resonance with the mathematical conception, is also evident. Lacking substance, the point is a function, a site, spatial or temporal, that makes meaning and thus discretionary instances, sequences, and objects possible.

### In the *Cahiers pour l’Analyse*

In his ‘La Suture: Éléments pour une logique du signifiant’ (CpA 1.3), Jacques-Alain Miller invokes the peculiar nature of the topological point to describe his own position as a non-analyst addressing an audience of psychoanalysts. He writes:

The opening up of psychoanalysis is not the effect of the liberalism, the whim, the blindness even of he who has set himself as its guardian. For, if not being situated on the inside does not relegate you to the outside, it is because at a certain point, excluded from a two-dimensional topology, the two surfaces join up and the periphery or outer edge crosses over the circumscription.

That I can recognise and occupy that point is what releases you from the dilemma I presented to you, and entitles you to be listening to me to-day. Which will enable you to grasp, Ladies and Gentlemen, to what extent you are implicated in my undertaking and how far its successful outcome concerns you (CpA 1.3:38, trans. 24-25).

The function of the point is also evoked throughout this article in the concept of suture, which uses a medical analogy close to the *point de capiton* or fish hook found in Lacan’s teaching. Serge Leclaire picks up on this aspect of suture in his response to Miller, ‘L’Analyste à sa place?’ (CpA 1.4), criticizing Miller’s claim that all discourse sutures, even the discourse of the analyst, when he writes: ‘[T]o arrive at such a discourse, it is necessary, if I may say, to have a secure grasp of the point that makes the articulation
of a logical discourse possible, that is, the point that Miller presents to us as the weak point [*le point faible*], and simultaneously the crucial point of every discourse, namely the suturing point [*le point de suture*]’ (CpA 1.4:50).

As the title suggests, the point is a key concept in Jean-Claude Milner’s ‘Le Point du signifiant’ (CpA 3.5) in Volume 3. Milner’s goal in this article is to read the ‘logic of the signifier’ developed by Miller in ‘La Suture’ (CpA 1.3) in Plato’s *Sophist*. In effect, Milner argues that Plato seeks to mobilize the concept of non-being as a way to entrap and ultimately to mark
or find the name [*onoma*] of the Sophist, sophistry being aligned with the falsity or ‘non-being’ to be excluded from philosophy. And yet, being ignorant of the zero, Plato is unable to see the warping effects of non-being on discourse itself, that is, his own dialogue fundamentally misrecognizes
these discrete ‘points’ that make the discourse, and the ultimate bestowal of a proper name for the Sophist, possible. Here the act of bestowing a proper name accomplishes a function akin to Lacan’s *point de capiton*, fixing a sliding signification in a point of meaning. What is significant here, however, is that the signifier that is ‘fixed’ is precisely negativity or non-being itself, covered over by the proper name. Milner concludes that to make manifest the
play of ‘the diffracted reflection of the signifier’ in Plato’s text, ‘one must imagine Plato directing a blind eye towards a point whose unicity, position and validity can only subsist as strangers
to the gaze itself, just shy of misrecognition’ (CpA 3.5:82). He cites André Breton: ‘In order to situate the point that renders the object alive’, it is necessary ‘to place the candle well’. It is the Lacanian logic of the signifier that allows this point to be localized.

The crucial function of the point, in its repeated punctuality, is also central to Milner’s analysis of Louis Aragon in Volume 7 (CpA 7.2). For Milner, Aragon’s writing evokes the essential dilemmas of a literature of realist representation by showing how any attempt at representation fails to avoid the element of artifice constitutive of representation itself. Aragon, Milner suggests, has made of this paradox the principle of a literary machine that functions indefinitely on all its various levels, such that in the entangled play of doublings and displacements, in the movement from individuated insignia to the anonymous I, from the written to the writing, ‘we cannot grasp any term without being obliged to follow the structure which is the past and future of this point, and its unceasing multiplication’ (CpA 7.2:55).

For Milner, the point is essentially tied to negativity, a phenomenon that is also evoked in an arcane French grammatical
rule that posits negation in the form *ne...point* rather than *ne...pas*. François Regnault cites this formulation himself in his contribution to Volume 6, ‘La Pensée du prince’ (CpA 6.2) in a letter from the Princess Elisabeth of Bohemia to Descartes criticizing Machiavelli (CpA 6.2:51). More pertinently, however, Regnault uses the concept of point in this article to develop a series of claims concerning
the epistemological break and its role in the advent of a materialist science of politics. In Descartes, Regnault argues, there is only ‘one point of view’, namely that of philosophy or metaphysics, to which all others should be subordinated. In Machiavelli (as in Archimedes),
by contrast, the point is material and gives the political theorist leverage on what happens in history. Regnault insists
that, ‘beyond the points designated here, there are no other points, and more particularly, there is no tertium punctum, no Sirius
from which to consider Descartes and Machiavelli’ (CpA 6.2:40). Where Machiavelli takes the first steps towards a science of history, Descartes does not so much refuse such a science
as ‘deny’ it in advance, by ‘going beyond it’, i.e. by privileging metaphysics over history. Clear and distinct ideas have no purchase on politics, and thus ‘Cartesian politics is a politics like any other; not a science, but a [mere] strategy’ (CpA 6.2:40).

The ‘point’ is a recurring theme in Volume 9, devoted to a ‘genealogy of the sciences’. In his ‘Réponse au Cercle d’Épistémologie’ (CpA 9.2), Michel Foucault addresses the function of a point in establishing an essential discontinuity in a discourse, or the history of a discourse. In Foucault’s view, what the contemporary historian now ‘undertakes to discover is the limits of a process, the point of change of a curve, the reversal of a regulatory movement, the bounds of an oscillation, the threshold of a function, the emergence of a mechanism, the moment a circular causality is upset’. Discontinuity is no longer an obstacle, but a practice, involving the ‘regulated use of discontinuity for the analysis of temporal series’ (CpA 9.2:11, trans. 300).

The point is a crucial concept for Jacques-Alain Miller’s case for structural causality in ‘Action de la structure’ (CpA 9.6) where it is closely tied to the concepts of suture, place and lack. Miller argues that every structure includes a ‘lure’ or ‘decoy’ [*leurre*] which takes the place of the lack [*tenant lieu de manque*], but which is at the same time ‘the weakest link of the given sequence’, a ‘vacillating point’ which only partially belongs to the plane of actuality. ‘The whole virtual plane (of structuring space) is concentrated’ in this vacillating point. The place of this function ‘can be named the utopic point of the structure, its improper point, or its point at infinity’ (CpA 9.6:97). These are the points at which the ‘transcendental’ space of structuration intersects with experiential, structured space. To perceive the reality of these vacillating points, it is necessary to ‘regulate our gaze’ and to accomplish a ‘conversion of perspective’. If this is to be more than imaginary and subject to misrecognition, it can only be done by ‘transform[ing] some state of a structure’, and setting out from the utopic points that are specific to each of the levels at which lack mobilises structure.

In the first of his two contributions to the *Cahiers*, ‘La Subversion infinitésimale’ (CpA 9.8), Alain Badiou develops at length a concept of ‘infinity points’ that in many respects ties together the mathematical and Lacanian concerns of the journal. Badiou’s goal in this piece is
to account for the properly subversive nature of the infinite in the mathematical sciences, a concept which puts paid to ideological notions of quality and continuity that continue to
obscure this scientific ‘subversion.’ At the outset, Badiou establishes the domain of natural numbers as composed of an endless series of places that successive
members of this domain come to occupy. If then we are to posit a place that is unoccupiable by any member of the domain, it
will have to be marked by an inscription that is ‘supplementary’ to the ordinary series of supplementary inscriptions which count out this domain. What Badiou calls an ‘infinity-point of the domain’ is a number which can be ‘forced’ to occupy this ‘unoccupiable empty space’ (CpA 9.8:119). Such an occupation will have to meet two conditions. On the one hand, if such an ‘impossible’ number is to be conceived as possible, it will have to figure as impossible for the original domain, and thus be constructed
on the basis of the ‘initial procedures’ operative in the domain. On the other hand, as impossible it must of course be constructed as ‘exterior’ to the domain itself, precisely as a ‘supplement’ to the domain as a whole.

The bulk of Badiou’s article is a technical explication of this procedure, which evokes a range of themes from the *Cahiers*, from Miller’s introduction of himself as occupying a point neither inside nor outside the field of psychoanalysis, to Milner’s
emphasis on the essentially vacillating nature of the ‘point of the signifier’. Badiou formulates his main ‘epistemological thesis’ as follows: ‘in the history of mathematics, the marking of an infinity-point constitutes the transformation wherein are knotted together
those (ideological) obstacles most difficult to reduce’ (CpA 9.8:128). In the case of the calculus of real numbers, such an obstructive infinity-point was marked by the concept of an infinitely
small numerical element or point, the inscription of a number at the place marked as ‘smaller than all other numbers’. It is precisely this ‘punctualisation’ of the infinite that classical philosophy, here exemplified by Hegel, was determined to reject. More generally, as Hegel’s contemporary Galois anticipated, it is ‘by establishing oneself in the constitutive silence, in the unsaid of a domainial conjuncture, [that] one maintains the chance
of producing a decisive reconfiguration’ of this domain (128).

### Select bibliography

- Lacan, Jacques.
*Écrits*. Paris: Seuil, 1966.*Écrits*, trans. Bruce Fink, in collaboration with Héloïse Fink and Russell Grigg. New York: W.W. Norton, 2006. - Lacan, Jacques.
*Le Séminaire, livre III: Les Psychoses*, ed. Jacques-Alain Miller. Paris: Seuil, 1981.*Seminar III: The Psychoses*, ed. Jacques-Alain Miller, trans. Russell Grigg. New York: W.W. Norton, 1993. - Tiles, Mary.
*The Philosophy of Set Theory: An Historical Introduction to Cantor’s Paradise*. New York: Dover, 2004.

## Notes

1. Tiles, *The Philosophy of Set Theory*, 1. ↵