Book Review Essay: Para-Ontology and the Truth of Incompleteness, on “The Not-Two: Logic and God in Lacan” by Lorenzo Chiesa

If, for Jacques Lacan, the fundamental non-existence of a sexual relationship—that is, of any actually existing dual and mutual relational parity of man and woman—determines our linguistic subjectivization at a basic level, this fundamental non-rapport also defines the “transcendental” logic of human subjectivity that Lacan pursued by means of mathematical formalization, particularly in his seminars of the early to mid-1970s.   As Lorenzo Chiesa convincingly argues in The Not-Two, psychoanalytic and philosophical reflection on human sexed subjects following Lacan must accordingly take up, on the level of formalism, the implications of the fundamental incompleteness to which this non-rapport points.  This simultaneously logical, ontological, and cosmological inquiry essentially involves the critical elucidation and reconsideration of what Lacan termed “the God hypothesis,” which both arises from and crucially obscures the essential “not-all” of the cosmological whole that itself results, for sexed and linguistic beings, from the fundamental “not-two” of sexuality.

In his wide-ranging and suggestive analysis, Chiesa shows how the implications of this fundamental incompleteness underwrite Lacan’s views about the relationship between psychoanalysis and biology, the complex structural logic of unity or the One signaled by Lacan’s enigmatic “Il y’ a d’l Un” (“there is something of the one”), the problematic structure of the phallic function that essentially mediates the “not-all” of sexuality, and the innovative “formulas of sexuation” which re-inscribe the quantifiers of Fregean logic to figure the complex structures of sexual difference, as produced by the asymmetric relationships and non-relationships of this function.  In addition to providing these lucid and illuminating accounts of Lacan’s own formal innovations drawing out the logic of incompleteness, Chiesa’s analysis also admirably and suggestively points to some of the deepest problems left open for a further development of Lacan’s formalizing project today in both its psychoanalytic and philosophical aspects.  These include: the problem of the relationship of psychoanalysis and biology in the wake of contemporary developments of evolutionary and developmental theory; the problem of anthropogenesis or the biological origin of linguistic signification; and the extended implications of the metalogical or metamathematical “truth of incompleteness” as demonstrated in twentieth-century investigation into the logic of quantification and number. In the course of these discussions, Chiesa engages critically and insightfully with contemporary Lacanians, commentators, and philosophers including Žižek, Johnston, and Meillassoux.   Throughout, the marked rigor and provocativeness of Chiesa’s critical dual engagement with Lacan and post-Lacanian discourses, as well as recent developments of formalism and formalization, situate the book’s analyses at the cutting edge of the contemporary interdisciplinary discussion of the origin and structure of human linguistic subjectivity. The analyses contained here will be an essential part of any continuance of this discussion in the years to come.

In his seminars, Lacan engages centrally and decisively with the positions of traditional philosophers such as Plato, Aristotle, Kant and Hegel, as well as more contemporary logicians such as Frege, on the ontologically fundamental questions of unity, plurality, and the relationship of truth and being.  Despite this engagement, however, he notoriously refrains from developing anything that could be entitled an overall “ontology,” occasionally designating himself an “anti-philosopher” and repeatedly signaling the requirement for a psychoanalytic approach to treat ontological questions only obliquely, from a practically informed clinical perspective at some distance from the generalizing theoretical approach typical of traditional ontology.  In seminar XX (Encore), in particular, Lacan flirts with the possibility of a “para-ontology” as the project of a discourse on the par-être, or “being beside” that is invoked there in close connection with the materiality and structurality of the signifier.  One of the most suggestive aspects of Chiesa’s book is his provocative exploration of the contemporary possibility of the further development of such a “para-ontology,” drawing on Lacan’s own scattered suggestions but fundamentally challenging Lacan’s own marked resistance to positive ontological theory.  As Chiesa argues, the possibility of such a para-ontology can readily be seen as implied in Lacan’s own development of the “transcendental logic” of the truth of incompleteness which he draws from the axiom of the non-existence of the sexual relation.  Such a development would, in particular, extend the implications of this truth in the context of a critical contestation of the traditional “ontotheological” metaphysics of the One.  For such a contestation, the aim will no longer be either to inscribe the necessity of a first signifier or confirm a metaphysical theory of language as grounded in the unity and totality of the universe to which it relates.    Instead, what will be at issue is the extended metaphysical implications of the truth of incompleteness or non-totality itself, repeatedly marked by Lacan on the level of linguistic analysis by the recognition that “there is no meta-language” capable of mastering the whole of signification, or of drawing it into a single overarching unity in relation to the world.

What, then, are the ontological implications of such a logically and meta-logically motivated conception of the fundamental fact of non-totality?  As Chiesa notes (xvi-xvii), Lacan’s own repeated animadversions against metaphysical speculation, especially on the origin of language, here belie his own equally marked tendencies toward a critical and meta-critical development of the implications of the structural “non-all” of language itself, as structuring human subjectivity at a fundamental level.  Despite, or rather in connection with, Lacan’s sporadic identification as an “anti-philosopher” we may thus see in Lacan the “vertiginous fluctuations” of the “(repressed) struggle of an anti-philosophical first philosophy in the making,” (xvi), one which is, as Chiesa points out, repeatedly signaled by Lacan in the form of poignant but undeveloped suggestions and questions left open as basically undecidable without prejudicing the formal question of the One itself.

If, however, Lacan’s own discourse plausibly witnesses the truth of fundamental incompleteness in the way he himself signals, there is no evident reason why these meta-logical or meta-formal questions should not be pursued further, and indeed developed on the level of a paradoxical ontology of the incomplete being of the “not-one” that is itself implied in the fundamental “not-two” of human sexuality.  As Chiesa argues, the recognition of these implications on the level of ontology then poses the important question of how the truth of incompleteness should itself be understood: should we (can we) understand it as the whole “truth about truth”, thus risking re-inscribing at another level a claim of completeness—one which could then only be (at most) “half-said” without fundamental contradiction – or  can we instead see it as pointing to a fundamental illogicality or inconsistency at the level of the totality of being, or of the structurally necessary hypothesis of God as absolute?  (xiv)  As Chiesa points out, the two options evidently left open here evidently correspond to two contrasting ways of interpreting the structural persistence of the “God hypothesis” in the wake of Lacan’s analysis of language.  In particular, the first produces a kind of “religious atheism” that tends, as Chiesa argues, to re-invoke the ultimately untenable position of an absolute (even if only as the “necessity of everything’s non-necessity” that has more recently been defended by Meillassoux).  The second, by contrast, is comparable to the hypothesis of a with the evil deceiver of Descartes’ meditations: here, incompleteness is indeed the “complete truth”, but one only to be maintained on the basis of a “deceivingly inconsistent God,” one whom we will never comprehend, and whose assumption therefore allows us, by means of a decisive critical/practical wager on undecidability, to act as though He did not exist.  On Chiesa’s argument, this second option in particular poses the prospect of re-opening the possibility of freedom by allowing us to occupy the undecidable universe of “inconsistent indifference” (75) that thus appears to be critically and formally indicated.

Operating within this suggestive theoretical framework, Chiesa proceeds, in chapter 1, to analyze the connection between the complex structure of sexual difference as Lacan understands it, on one hand, and the subsistence and varied forms of the “God hypothesis”, on the other (1).  On Chiesa’s analysis, the fundamental non-existence of the sexual relationship means that it is logically impossible ever to enunciate the truth of sexuality as any kind of unity or simple, positive being at all; this leads to the fundamental necessity for speaking beings to represent their own identity and sexuation only by means of the complex and “precarious” set of substitutes that Lacan theorizes as the asymmetric relations of the sexes to the phallic function (2).   This leads to the particular and actually mutually incompatible ways in which jouissance is produced, on both sides of the sexual difference, as a by-product of the actual impossibility of the fusional “one” which is desired in the pursuit of love. (5)  In fact, the varying possibilities thereby defined, across sexual difference, for the acceptance or rejection of phallic jouissance and its fantasmatic projections leads to the necessity to distinguish four types of jouissance, each with its own distinctive way of counting and reckoning the implications of the fundamental “non-totality” of the sexual relationship.

These include not only the masculine form which, in attempting to totalize “uncovers its very nontotalizability” and the feminine phallic forms, which replaces the requirement of a totalizing One with the singular “one by one” of an always unlimited and never totalized count, but also an asexual form which results from a “fantasy of masculine phallic jouissance”, projected onto woman as totalized, and a nonsexual feminine jouissance of nontotalization as such.  (p. 4).  After developing the formal and logical structure of these positions and relationships as articulating the complex logic of the phallic function itself, Chiesa considers its implications for the structure of the symbolic order as non-totalizable (or as “barred”) and for the consequent availability (or lack thereof) of the Other as mediated by a symbolically mediated desire that is itself correspondingly and essentially non-total. (14-15).   In essential connection with the asymmetry of the phallic function and the (differently mediated and dissimulated) non-totality of the symbolic order, here “God is but a name for the paradoxical status of the symbolic order … suspended as it is between its making One and its being not-One” (16), or between the constitutive “effect” of the symbolic order in producing oneness and totality on the level of the imaginary, and the real maintenance of essential nontotality on which this imaginary production relies.

In chapter 2, Chiesa turns to the fascinating and deeply relevant issue of the relationship of Lacanian psychoanalysis to biology, and the contemporary implications of Lacan’s delicate combination of invocation and critique of the life sciences in light of contemporary discussions of the biological basis of human subjectivity.  On the one hand, as Chiesa underscores (26), Lacan understands the domain of the natural broadly in terms of the “real” of the impasse of logical symbolization; there is thus, for Lacan, no straightforward or directly correspondent symbolization of the biological basis of human sexuality, and biological nature is rather broadly to be conceived as the realm of “indifference” that gives rise to the difference of human sexuality only by means of a failed resolution of this essential deadlock.  This implies a deep-seated critique of the essentialist and teleological assumptions of biological explanation, a critique which, as Chiesa argues, can and should be extended also to bear on the projects of contemporary “evolutionary developmental” explanations and the “new Darwinian synthesis” (31).  On the other hand, however, Lacan’s account of the etiology of human sexuality nevertheless quite evidently requires a deep engagement with the question of the “natural” origins of symbolic subjectivity, and raises thereby the question of how these origins can be thought while maintaining the pure “indifference” of nature also implied by Lacan’s critique.   As Chiesa argues, an account of the origin of the symbolic can then only succeed by situating itself within the consequences of the contingent material “real” of structure itself, and in particular those of the incompleteness which is structurally implied in its logic as its “illogical limit”  (p. 44).

How, though, is the pure indifference of nature as such to be positively thought, in such a way as also to account for the possible emergence of sexed human subjectivity from it?  Here, Chiesa discerns in Lacan’s texts two somewhat opposed strands of argumentation.  On the first of these, natural biological life already includes the fundamental difference of sexuality; here, difference is already a “real lack”  in nature that can only be overcome (or precariously substituted for) by the advent of human symbolic reality (60).  As Chiesa argues, the invocation of such a fundamental biological difference, however, actually oversteps the boundaries of Lacan’s materialism in that, by invoking a difference which would thereby by other than the One, it essentially also re-invokes an essential principle of One or unity which has no actual warrant therein.  For this reason, the second, distinct strand of Lacan’s argument, on which nature is to be thought rather as fundamental in-difference and indeed as thoroughly “indifferent to difference”  (p. 62) is to be preferred.  On this position, though “everything emerges” from signification, “nothing really changes with it”; the advent of symbolic reality does not fundamentally transform the underlying real of nature, but actually only allows it to be thought explicitly as non-differential.  (62-64).

Chiesa then considers in these terms the ontological assumptions underlying Zizek’s development of Lacan’s project, particularly in its invocation of Schellingian and Hegelian ontological structures of negativity, instability, and “radical gap” as fundamental to the natural emergence of signification.  As Chiesa convincingly argues, Žižek’s invocation of such a “transcendental” horizon of pure and negative differentiality at the origin of possible signification is, at one level, to be lauded in its steadfast refusal to think the basis of signification as resting in any kind of positive substance or discrete natural basis of the anthropological.  Here, substance is sustained only by the pure difference “in the real” which appears retroactively through symbolic differentiality but “cannot be reduced to it” (66).  However, in invoking such a principle of pure difference as origin, Žižek also risks treating this primary “difference in the real” itself as a kind of vitalistic or even quasi-subjective agency of nature, a kind of basic “negative energy” which accordingly determines the intrinsic “curvature” of the real independently of symbolic reality itself.  Against this tendency toward what amounts to a kind of “animism of the not-all” (66) in Žižek’s invocation of the primacy of difference, Chiesa argues that pure difference must also be thought as a more basic in-difference: that is, as a more basic presymbolic Real that “contingently becomes difference yet also remains indifferent to difference.”  (66)  Here, in other words, the basis of substance in the Real is not to be thought simply in terms of the primary negativity (the “gap” or “hole”) which can be read back into it, given the linguistic differentiality of the symbolic, but also, just as deeply, as a more basic indifference to both the symbolic and the non-symbolic, from which the symbolic may indeed contingently emerge, but which, as thus neutral, remains indifferent to this emergence.  (67)

Chiesa next turns to Johnston’s “transcendental materialism,” which attempts under the banner of an ontological appeal to a “weak nature” rent by essential gaps, flaws, and imperfections, to account for the natural conditions of possibility for the “transcendence-in-immanence” of subjectivity and language.  Such an account is needed, on Johnston’s argument, to avoid appealing to any of various related forms of idealism, spiritualism, and theism in accounting in materialist terms for the apparent “break with nature” that language and sexed human subjectivity represent.  Chiesa agrees that such a “genetic/materialist” investigation into the genesis of language from the Real of nature is necessary, but challenges the way in which Johnston locates the essential “barring,” “inconsistency,” or differentiality at the structural core of the symbolic already in the “weakness” of nature itself.  For Johnston, prior to the historical origin of symbolic differentiality, nature is already essentially “holey,” (71), barred, and “not-One”.  In positing a “weak” nature in this sense, Chiesa argues, Johnston does not sufficiently distinguish between the “barred” real of nature and the “barred” symbolic; in other words, in thinking the natural as always-already differential, Johnston does not succeed in thinking it as essentially indifferent to the symbolic and the (actually entirely contingent) possibility of its emergence.  Chiesa’s alternative suggestion is, without refusing the genetic investigation itself, to understand the emergence of linguistic subjectivity as both involving and not involving a fundamental “break” with nature (70).  From this perspective, while the exceptionality of human subjectivity is itself to be interrogated as Real, this exceptionality is itself to be understood not only as grounded in the structure of the natural world itself but also as having a broader formal-logical basis in the global truth of the incompleteness of the world, and the constitutive indifference (including the indifference of “nature” to “culture”) to which it leads.

This indifference, Chiesa suggests, might be seen as subsuming not only actual structural conditions retrospectively recognizable as contingently producing the emergence of the symbolic, but also the stark contingency of this emergence itself: the fact that nothing necessitates it, and that all of our linguistic activities and interactions remain sustained by nothing more than the bare contingency of our biology (73).  In this way, Chiesa closes the chapter by suggesting, what Lacanian theory and the development of a “para-ontology” along the lines recommended here would have ultimately to pose is the finally undecidable alternative between, on one hand, a final being of the “not-one”—an utter and uncircumscribable absence of unity in the Real—and, on the other, the “materialist theology” of a not-One that nevertheless (paradoxically) is One as a basic ontological principle of gap, lack or difference.  What is thus most important, according to Chiesa, is not necessarily to decide this alternative between (an extended and modified development of) Johnston’s transcendental materialism and Zizek’s theologically tinged ontology of the drive, but rather to comprehend the undecidability of the alternative thereby produced as a development of the essential ambivalence of Lacan’s “god hypothesis” and as suggesting ultimately a kind of “materialist agnosticism” grounded in the “in-difference between difference and indifference” itself (74).

Chapter 3 turns to the investigation of Lacan’s views on logic and its essential connection with the transcendental structure of human linguistic subjectivity.  As Chiesa argues, Lacan sees the fundamental fact of sexual non-relationship as “standing for” the constitutive limits of logic and as sustaining its essential structure.  In this way, for Lacan, “the truth of phallic incompleteness lying at the core of Homo sapiens’ transcendental structure equates with the incompleteness of our logical thought as a species-specific linguistic trait capable of founding knowledge.”  (81). In close connection with this, it is especially important for psychoanalysis, as itself (in Lacan’s words) a complex “junction between knowledge and truth,” to pursue and interpret the formal results in the history of the development of logic which have verified and inscribed its essential deadlocks with respect to completeness, consistency, and the (im)possibility of attaining a total and consistent logical knowledge of the “truth about truth.”  (82)  It is in this respect that Frege’s development of the logic of universal and existential quantification poses, in a basic way, the possibility of rendering explicit the structural sources of incompleteness and undecidability, and thereby verifying through its own constitutive impasses – including eminently the failure of the logicist project of grounding number and arithmetic in logic – the essential link between the truth of incompleteness and the structure of logic itself.  For Lacan, Frege’s apparatus of quantification captures in particular, through its essential inscription of “empty” variable places, the underlying structure of the “empty place” or locus of the letter, which makes all symbolic writing possible underwrites predication as essentially involving the unsaturation or “holing” of the sentential function: with respect to ontology, this ensures that Frege’s logic fundamentally supersedes the Aristotelian logic which by contrast presupposes a substantial “copula” linking subject and verb (96-97).

Most significant for Lacan’s exploration of the foundations of logic as grounded in sexuation, however, are the possibilities that Frege’s quantificational symbolism offer for formalizing the “not-two” structure of the phallic function itself.  In chapter 4, Chiesa turns to a clear and detailed analysis of Lacan’s unorthodox modification of this symbolism to produce the famous “formulas of sexuation” as a logical writing of the structurally decisive impossibility of the writing of the (non-existent) sexual relationship itself.  As Chiesa emphasizes, these formulas are to be understood as formalizing the ways in which an initially undetermined living animal becomes essentially inscribed within the phallic function and thereby attains sexed subjectivity; as such, the inscription of the formulas represents nothing other than Lacan’s formalized elaboration of his own conception of the passage through the Oedipus complex and its possible resolution (105).  Here, in other terms, each of the formulas represents a possible attempt at resolution of the fundamental deadlock of the non-existence of the sexual relationship at the basis of the phallic function, thereby articulating basic positions about generality and particularity and the possibility of its articulation.  For example, as Chiesa argues by devoting close attention to its development from earlier seminars to its final presentation in seminars XIX and XX, the formula for masculine sexuation effectively presents it as structured by the contradiction between the totality of men as structured by the phallic function and the existence of a singular man who is not.  This contradiction itself provides the basis for what is effectively the structure of male sexuation with respect to the generality of its possible writing, so that “it is insofar as there is every man that it [the phallic function] can be written, and insofar as there exists one whole man that it cannot be written, as a result of which … particular men do ultimately exist.”  (110).  Conversely, the feminine formulas figure, on Chiesa’s reading, a feminine sexuality that is decisively “not-all” phallic, but which nevertheless exists as a kind of real “beyond” the phallus, suspending the position of feminine sexuality essentially between inclusion and exclusion in the symbolic order.  This is now not the position (as with the masculine one) of an essential contradiction between generality and particularity, but rather, as Chiesa elucidates, that of an undecidable symbolic existence that essentially structures the (non-)totality of the symbolic order itself (117).

As Chiesa argues in closing the chapter, the duality thereby introduced between the inconsistent position of male sexuation and the undecidable one of female sexuation itself has decisive and formally understandable consequences for the reality of structure and the logical basis of number themselves.  With the formulas of sexuation as structured by this basic and irresolvable duality, in particular, Lacan produces a kind of intra-logical inscription of the incompleteness that is radically shown as well in the outcomes of the failed twentieth-century projects of logicism, formalism, and logical positivism.  Lacan reads Gödel’s incompleteness theorems, for example, as showing the inherent possibility of articulating in arithmetic some truth that cannot be reduced to purely logical demonstration, and thereby as supporting Lacan’s own privileging of the real as the logically impossible which is nevertheless inscribed as impossible through the writing of the failed resolution of the deadlock of sexuality in the formulas of sexuation themselves (154).  As Chiesa points out, this mode of inscribing the real – a possibility of inscription that, as Lacan famously emphasizes in seminar XX, can only be understood as possible on the basis of the impasse of formalization – raising the further important question, however, of its own linguistic or meta-linguistic status.  In particular, if the truth of incompleteness is itself inscribed by psychoanalysis as the radical claim that “there is no meta-language” – or, in other terms, that there is no complete “truth of truth” that is accessible on the level of a complete and consistent discourse – then what is the status of this (apparently meta-linguistic or meta-logical) claim itself?  This problem is what, Chiesa suggests, Lacan tried to mark with the claim that the truth of incompleteness can itself only be (at best) “half-said”, or that any statement of it bears a kind of self-refuting character that inherently problematizes its own establishment.   But as Chiesa argues, it is just this “half-saying” or inherent incapacity to be completely expressed that is, for Lacan, constitutive of the possibility of language – of the logical production of “the necessity of a discourse” itself.  Connecting back to the themes of a fundamental in-difference at the root of both natural and subjective/linguistic differentiation, Chiesa argues that in this way the production of a discourse at the point of logical or meta-logical impasse can itself be seen as the re-inscription of what was “there before it” (155) and thus as producing logic itself as what “seems” to have always already been structurally/differentially “there” as a prior condition.  But at the same time, it also just as much evidences how discourse and structured language can be seen to actually be produced or realized through a wholly contingent (and thus indifferent) “realization” that itself relies decisively on the impossibility of ever demonstrating the completeness and consistency of this prior condition.

In the brief concluding chapter, Chiesa returns to the dilemma that is thus apparently posed by Lacan’s attempt, at the very limits of his ontological thought, to develop a logically grounded formal writing of the truth of incompleteness itself.  If a direct inscription of this truth is impossible—since any such inscription would risk presenting itself as the (complete) “truth about truth” and thus betraying its own content—then how is it nevertheless possible for psychoanalysis to bear witness to it on the level of its (para-)ontology (as it, indeed, must if this ontology is to take due account of the fundamental fact of sexual non-relation)?  Here, Chiesa reiterates the problematic alternative between an “absolute” truth of cosmological incompleteness, and an absolute claim and a paradoxical affirmation of a cosmological not-One that itself functions, in paradoxical fashion, as a new (ontologized) One.  On the level of cosmology, both hypotheses are equally, and deeply, problematic, as both would attempt to present an absolute “truth about the truth” of incompleteness that would essentially betray itself, and the alternative between the two is strictly undecidable.  However, even if this cosmological deadlock is irreducible, it remains possible to consider in a more practical register what we can take as our “true truth”—that is, the “true truth” that matters for our living and practice as sexed living subjects—in view of the fundamental “not-One” that psychoanalysis illuminates.  While it is indisputable that “the not-One is our true truth” (179), this still leaves the dilemma between incompleteness being an ultimate and irreducible truth or its being raised into the (contradictory) truth of a new One.  This alternative, in fact, remains undecidable, but Chiesa suggests that on the level of practical consequences, both of its terms nevertheless point to the same conclusion: that of a practical affirmation of non-unity and a correlative opening of freedom “to act as if the One/God did not exist” (74).  For even if the truth of non-Unity is itself elevated into a (contradictory) One, this One would itself anyway consist in nothing more than its own undecidability and incompleteness (74-75); on either alternative, then there is thus no practical alternative to the practical and lived affirmation of the non-One as, in any case, “our true truth.”

With this conclusion, Chiesa opens the fascinating prospect of a further development of the “para-ontology” suggested by Lacan in its practical, ethical, and political dimensions, as these are articulated in the formal and logical writing that bears witness to what is, for psychoanalysis, the fundamental truth of our linguistic subjectivity—that of the “not-two” of sexual and cosmological in-difference—itself.  Such an investigation would pursue what certainly are today, as Chiesa has convincingly argued here, the most significant “philosophical” problems and questions opened by Lacan’s thinking for us, but would also, like the current book, cross disciplinary boundaries between philosophy, psychoanalysis, biology, and logic in ways sure to be revealing and transformative for all of these fields.  With its continuation of the deep engagement already begun here with the strictly philosophical problematic of the One—the problem of unity and multiplicity whose inaugural development in the Western tradition is marked by the names of Parmenides and Plato—it would profoundly articulate the contemporary status of this problematic and its prospects for further development in the contemporary context marked by some of the main logical, scientific, and philosophical developments of the twentieth century.  But as the “para-ontology” of the fundamentality of incompleteness, it would also, essentially and necessarily, displace this problematic, as Lacan avers and Chiesa verifies, away from the philosophical project of the metaphysics of the world, irreducibly and transformatively inflecting it with the clinical ethico-practical concern which is, as Lacan repeatedly insists, nothing other than the very truth of psychoanalytic discourse itself.