45｜The Logico-Mathematical Formalization of the Mode of Presence

20.4 The Logico-Mathematical Formalization of the Mode of Presence

Since the Mode of Presence is an entirely novel concept, it is prone to conflation with the Linguistic Mode. To articulate the distinction with crystalline precision, it is essential to design a formal logico-mathematical structure for the Mode of Presence—one that permits deduction and comparison. This serves as a necessary theoretical calibration, insulating the inquiry against the pitfalls of mere "poetization" or psychological reductionism.

We shall complete this calibration in three movements:

1. Minimal Formal Language
2. Structure Definition
3. Mapping Relations (The Correspondence between P and L)

I. Minimal Formal Language

The Mode of Presence represents a cognitive operational mode fundamentally distinct from the Linguistic Mode. We begin by defining a "Cognitive State Space":

Let $C = \{ \text{Cognitive states of the subject} \}$.

Within this space, we distinguish two primary modes:

• L-mode (Linguistic mode): Cognitive operations centered upon concepts, judgment, and objectification.
• P-mode (Presence mode): Cognitive operations centered upon openness, holistic manifestation, and the dissolution of the subject.

The minimal definition states that the Mode of Presence is a specific subset: $P \subset C$. Crucially, the algebraic rules of operation for $P$ and $L$ are non-identical ($P$ and $L$ possess different operational algebras).

II. Structure Definition

1. The Structure of the Linguistic Mode (For Comparison):

The Linguistic Mode may be formalized as:

$$L = (O, D, J)$$

Where:

• $O$ (Objectification): Rules for fragmenting the world into discrete objects.
• $D$ (Distinction): Rules for categorization and differentiation.
• $J$ (Judgment): The structure of formal logic and predication.This is the standard paradigm of human Reason: Fragment ➔ Classify ➔ Judge ➔ Logicalize. This is the domain of Wittgenstein, Kant, and the Analytic tradition.

2. The Structural Definition of the Mode of Presence

I propose the following formal mathematical structure for the Mode of Presence:

$$P = (A, M, E, S)$$

Where:

• (A) Attenuation of Subjectivity:Mathematically, the weight of the subjective structure ($Sbj$) diminishes: $w(Sbj) \to 0^+$. The subject does not vanish; it simply ceases to be the organizing center of experience.
• (M) Merger of Field:The manifestation of the world no longer satisfies the rules of objectification. $Object(x)$ is no longer valid, and the rule of Distinction ($D$) is suspended ($D(x, y)$ is not executed). The world manifests as a single, holistic Field ($F$):$$F = \langle \text{world-as-presence} \rangle$$
• (E) Emotional Intensity as Structure:Emotion ceases to be a subjective function $emotion = f(subject, object)$ and is transformed into a constituent of the field: $emotion \in F$. Emotion becomes a dimension of the manifestation itself.
• (S) Suspension of Linguistic Operations:The three fundamental operations of language—Naming ($N$), Division ($D$), and Judgment ($J$)—enter a state of suspension: $N = 0, D = 0, J = 0$. Note that the linguistic faculty remains latent but is temporarily shielded.

III. The Mapping Relations between P and L

To be integrated into a total philosophical system, we must define the relationship between $P$ and $L$. We define a "Re-expression Map" from the Mode of Presence to the Linguistic Mode:

$$f: P \to L$$

Crucially, this mapping is non-invertible; $f^{-1}$ does not exist.

The philosophical implications are profound:

• The Mode of Presence is partially expressible (it can be translated into $L$).
• The Linguistic Mode cannot reach the totality of Presence (the mapping is irreversible).This structure differs fundamentally from Derrida’s différance. While différance represents an internal incompleteness within Language, the Mode of Presence represents the Inherent Non-Invertibility between Language and that which lies beyond it.