Asymptotic Intentionality

Frank Chouraqui's technical term (borrowed from Leibnizian infinitesimal calculus) for the structure of intentionality shared by Nietzsche and Merleau-Ponty: a linear movement structured by two end-points it never reaches but approaches indefinitely. For Merleau-Ponty, the two horizons are the subjective and objective poles of perception; for Nietzsche, the last human and the overhuman (and, more fundamentally, subject and object). The structure explains why intentionality has a direction without having an achievable object — the end-points are "extracted by the analysis of intentionality," not pre-given in it. This is not Kantian teleology.

Key Points

  • Intentionality is linear, directional, and has end-points — but the end-points are unreachable horizons, not actual targets
  • The end-points are not pre-thematized by intentionality; they are derived by analysis. Intentionality does not aim at its end-points; they are "regulative horizons" in a non-Kantian sense
  • For Merleau-Ponty: the subjective and objective poles are asymptotic horizons of perception's "prospective activity" toward determinacy
  • For Nietzsche: "the possibilities of human existence are spread over a line that stretches asymptotically toward the overhuman on the one end and the last human on the other" (Ch. 1). More deeply, subject and object are asymptotic fictions drawn from the experience of reversibility
  • The structure precludes self-identity: if the end-points are never reached, no actual term is ever fully self-identical — only modally, along the line
  • Asymptotic intentionality is the structural form of self-differentiation as intentionality
  • It is the reason perception is essentially indeterminate and history is essentially eventful

Details

The Leibnizian borrowing

Chouraqui takes the term from Leibnizian infinitesimal calculus, where an asymptote is a line approached arbitrarily closely but never reached. This is a specific mathematical structure: the two end-points define the line (without them there would be no direction), yet the line never actually reaches them. The analogy is not metaphorical — Chouraqui treats it as a precise model for the structure of intentionality.

What distinguishes the asymptotic model from more familiar teleological models:

  • Kantian teleology: the end is pre-thematized in the faculty (e.g., reflective judgment's regulative ideas). The asymptotic model rejects this. The end-points are extracted from the line by analysis, not present in it as an implicit representation.
  • Hegelian teleology: the end is both presupposed and achieved through historical movement. The asymptotic model rejects achievement — the end is never reached.
  • Aristotelian teleology: the end is the what-for of the movement, its internal principle. The asymptotic model rejects this too — the end is not internal but derived.

The asymptotic model therefore preserves directionality without commitment to a representable, attainable, or achievable end.

Merleau-Ponty's version: determinacy as horizon

For MP, perception is a "prospective activity" (Phenomenology of Perception terminology) directed toward total determinacy. The famous example of the ship on the shore: as I approach, "I did not perceive resemblances or proximities which finally came together to form a continuous picture... I merely felt that the look of the object was on the point of altering, that something was imminent in this tension" (PP 20/30).

This imminence-without-achievement is the asymptotic structure. Perception is always in the movement toward determination but can never attain it: "The absolute positing of a single object is the death of consciousness" (PP 78/88). Full determinacy would abolish the perception that moves toward it. The horizon is both necessary (to give perception its direction) and inexistent (to keep perception alive).

Chouraqui connects this to MP's critique of polar thought throughout Ch. 4. The subjective and objective poles, the visible and invisible, the presence and absence — all are asymptotic horizons, not terms. They structure the line of intentionality by being unreachable.

Nietzsche's version: the range of human existence

For Nietzsche, the asymptotic structure shows up most strikingly in Ch. 1's treatment of the last human and the overhuman. Both are "unattainable horizons structuring the range of possible modes of existence" (Ch. 1). Neither is an actual possibility:

  • The last human would have eradicated chaos (internal opposition of drives), but "every activity is an overcoming of difficulties and resistances" — so no actual existence can be fully the last human.
  • The overhuman would have achieved full externalization of drives, but externalization takes time, and the healthy individual must still survive, which requires some internalization. So no actual existence can be fully the overhuman.

The structure is asymptotic: the line of actual human existences is spread between these unreachable horizons, which define the range without being within it.

More fundamentally, Chouraqui argues that subject and object play the same role for Nietzsche: they are "abstractions and radicalizations arising from the experience of a reality that is not bipolar but asymptotic" (Ch. 1). The actual experience of reversibility — I am simultaneously subject and object in the experience of resistance — gets idealized into the polar subject/object relation, which is then imagined as a pair of actual terms. Nietzsche's whole critique is that these terms are asymptotic fictions, not real entities.

Why asymptotic, not teleological

The distinction between asymptotic and teleological is crucial for Chouraqui. Kant's teleology (and Hegel's) assumes that intentionality represents its end — that the faculty of judgment, or Geist, has an intrinsic thematization of the end point. Chouraqui's asymptotic intentionality denies this:

"'Asymptotic' is not equivalent to Kantian teleology insofar as it does not assume that intentionality possesses an intrinsic thematization of its end point. On the contrary, it is the analysis of intentionality that figures it as teleological." (Introduction)

The upshot: intentionality is structured by end-points without representing them. The end-points are real only analytically, as the horizons we extract from the movement. They are not in the movement itself as intentional objects.

This matters because it preserves the phenomenological insight that intentionality does not require a thematic object (MP) or a representational will-to (Nietzsche) to be directional. Intentionality has a direction without having an aim.

The horizon/principle distinction (Chouraqui's Husserl-critique)

Surfaced by the 2026-04-21 motif re-ingest as a structural finding the April 11 extraction underweighted: the asymptote-motif has a phenomenological corollary in Chouraqui's reading of MP's horizon-concept, developed in Chapter 6 (pp. ~190–200, raw lines 2004–2038). The cardinal MP-passage is the one Chouraqui anchors at V&I 237/286:

"Every concept is first a horizonal generality... there is no longer a problem of the concept... when one has understood that the sensible itself is invisible, that the yellow is capable of setting itself up as a level or a horizon." (MP, V&I 237/286)

Chouraqui's reading extracts a distinction that becomes the sharpest formulation of the book's critique of Husserl: horizons vs. principles. Horizons are the asymptotic end-points of intentionality — never reached, defining the direction of the movement without being achieved. Principles are sedimented horizons — horizons that have mistakenly been taken as actual, as grounds beneath the movement rather than as the analytic poles that structure it.

The structural claim: the origin of error is taking the horizon for a principle. Husserl's transcendental idealism is exactly this mistake — Husserl's apodictic ground (pure consciousness, the transcendental ego, the constituting subject) is a horizon that Husserl treats as a principle. It is reached analytically from the movement of intentionality but does not precede that movement. Once the horizon is converted into a principle, the structure of intentionality is inverted: what was a pole becomes a foundation.

The horizon-as-intensive-not-extensive (MP's refinement at V&I 149/193) reinforces this:

"The texture [of the horizon] itself is horizontal." (MP, V&I 149/193)

Horizons are not distant meeting-points (extensive horizons) but textures of existence (intensive horizons). They do not lie far away and become reachable with more effort; they are the horizontal dimensionality within which anything can appear at all. "Lines of flight or horizons" (MP, V&I 137/177) — the lips of the visage are themselves horizons, in this sense, not edges that bound the face from elsewhere.

This is where the asymptote-motif meets the shadow-philosophy method: the "horizon" is what shadow philosophy discloses as the real structure of a thinker's thought, while "principle" is what the thinker's explicit philosophy substitutes for the horizon (see shadow-philosophy § "Husserl's Shadow Philosophy"). Husserl's explicit principle-philosophy is shadowed by Husserl's own horizon-phenomenology — which is what MP reads Husserl for.

The cross-author consequence: Nietzsche's own rejection of "principle-thinking" in favor of "perspective-thinking" is the same move in different vocabulary. Nietzsche's "no facts, only interpretations" and MP's "horizons are not principles" are structurally parallel — both dissolve the grounds by showing that what looked like a foundation was the projected endpoint of a movement. Asymptotic intentionality is the name for the shared structure.

Consequences for the ontology

Asymptotic intentionality is the structural form of self-differentiation as intentionality. Because the end-points are never reached, actual moments are always less-than-determinate (MP) or less-than-self-identical (Nietzsche). This is not a limitation but a constitutive feature: the indeterminacy is what keeps the movement alive.

For MP, this yields flesh as less-than-determinate Being, and the movement of sedimentation as the essence of Being. For Nietzsche, it yields the impossibility of the teleological cosmology (no pyramidal unification of drives is possible) and the eternity of becoming.

The Nietzsche-MP parallel as structural

Chouraqui's thesis that Nietzsche and MP share the asymptotic structure is what makes his "juxtaposition" method work. The parallel is not a thematic coincidence (they both talked about intentionality) but a structural necessity: if you take the phenomenon-of-truth seriously, you are forced into an ontology that requires asymptotic intentionality, because the phenomenon is the structural fact that experience has directionality without a thematized aim. Anyone who takes the question of truth ontologically must end up with some form of the asymptotic structure.

Connections

  • is the intentional form of self-differentiation — asymptotic intentionality is how self-differentiation shows up in intentionality
  • is the structure of perceptual-faith — perceptual faith is itself asymptotic
  • is how self-falsification moves — the movement of falsification is asymptotic, not teleological
  • grounds the impossibility of total determinacy in MP and self-identity in Nietzsche
  • contrasts with Kantian teleology — asymptotic intentionality does not pre-thematize its end-points
  • contrasts with Hegelian dialectic — no synthesis, no achievement
  • contrasts with Aristotelian teleology — end not internal but derived
  • is a case of horizonal structure (MP) — see visible-invisible, flesh-as-element
  • is articulated by ecart as the structural distance that keeps the movement open
  • explains why the last human and overhuman are not actual possibilities
  • explains why "the absolute positing of a single object is the death of consciousness" (PP 78)

Open Questions

  • How exact is the calculus analogy? Chouraqui calls it exact but does not work out the mathematics. Can all features of infinitesimal calculus be transferred to ontology, or is "asymptote" only a structural metaphor?
  • Does the asymptotic structure require infinite time? If so, how does it relate to Nietzsche's eternal recurrence (which is infinite in a different sense)?
  • Is there an asymptotic structure for values, or only for cognition? Nietzsche suggests the former; MP is ambivalent.
  • Chouraqui does not engage with Deleuze's concept of "differential," which has its own Leibnizian roots. Is asymptotic intentionality a rival or a cognate?

Sources

  • chouraqui-2014-ambiguity-and-absolute — the term is introduced in the Introduction and used throughout. Ch. 1 develops it for Nietzsche (especially "Asymptote," "End Types," and the discussion of last human/overhuman as unattainable horizons). Ch. 4 develops it for MP (the "teleology of determinacy," the ship-on-the-shore example, the prospective activity of perception). Ch. 6 shows how less-than-determinacy is the ontological consequence of the asymptotic structure