paper:doi-10-2139-ssrn-6691298A Mathematical Elucidation of Buddhist Enlightenment
Frameworks (4)
- Differential GeometryThird lens; used to elucidate the self-as-coordinate-system and enlightenment as intrinsic geometry.
- Information GeometrySecond lens; used to elucidate attachment and suffering via KL divergence.
- Mādhyamaka SchoolThe primary Buddhist philosophical position (Nāgārjuna's) adopted as the paper's doctrinal frame.
- Phenomenal Category (P)Category-theoretic model introduced by this paper where objects are phenomena and morphisms are conditioning relations.
Claims (16)
- This structural resonance does not mean mathematics proves Buddhism; it shows that human intelligence approaching the same structural truth from different directions can produce overlapping contours.
Closing epistemic disclaimer distinguishing structural resonance from proof.
- Enlightenment is the transition to a state in which identification with the ego-centric coordinate system φp0 is dissolved, enabling direct description of the intrinsic geometry of (M,g).
Proposition 4.4, differential-geometric elucidation of enlightenment.
- The Zen koan 'What is your original face before your parents were born?' may be read as asking for the intrinsic geometry of the manifold that exists prior to any coordinate system.
Interpretive geometric reading of a classical Zen koan.
- Proposition 4.2 establishes that no fixed self exists, yet the self that 'understands' this proposition appears to remain — a paradox Nāgārjuna addresses as śūnyatā-śūnyatā.
Acknowledged self-referential paradox left unresolved by the formal elucidation.
- Under the Dependent Origination Axiom, no object in the phenomenal category possesses svabhāva (its representable functor is never isomorphic to a constant functor).
Proposition 4.1, category-theoretic elucidation of śūnyatā.
- Enlightenment is the structural transformation of cognition arising when the reference frames organising experience are transparently recognised as reference frames.
The paper's proposed one-sentence unified description of enlightenment.
- Rather than falling silent at the limits of formal language, the paper chose to articulate what can be articulated and make explicit the boundary of what cannot.
Explicit closing methodological stance regarding the outer boundary of formal language.
- Every correspondence between mathematical and Buddhist concepts is stated explicitly, maintaining the distinction between 'is similar to' and 'is identical to'.
Proposition 2, guarding against conflating analogy with identity.
- Mathematics is not a guarantor of truth but a language for describing structure; the mathematical concepts function as lenses, not guarantors of correctness.
Proposition 1, the paper's core methodological stance.
- The conflation of description with reality — misidentifying hᴬ with Δs, Q with P, φp0 with M — is the common structure of avidyā across all three lenses.
Integration claim identifying a shared structure of ignorance across the three formal lenses.
Hypotheses (3)
- The static structural description could be extended to dynamic models of practice, e.g. formalising dissolution of the reference frame as a topological phase transition.
Future Direction 1, proposing dynamic/temporal extension of the static formal elucidation.
- Refining the connection between the free-energy principle and the information-geometric elucidation may open a circuit for empirical testing.
Future Direction 2, proposing an empirical bridge to cognitive science.
- The 'dissolution of the reference frame' framework may be applied to comparative studies with Heidegger, Wittgenstein, and Nishida Kitarō.
Future Direction 3, proposing comparative-philosophy extensions of the unifying concept.
Questions (4)
- What is your original face before your parents were born?
Classical Zen koan given a geometric reading as a question about intrinsic manifold structure prior to coordinates.
- How does transparent recognition arise?
Open question noting the elucidation describes the state of realization but not the dynamic process leading there.
- Is realization gradual or sudden?
Open question regarding whether sudden enlightenment (Zen tradition) can be mathematically described as a phase transition.
- Whose realization is it?
Open question probing the paradox that a self persists to 'understand' the non-existence of a fixed self.
Original abstract (expand)
This paper offers a structural elucidation of Buddhist enlightenment (bodhi) through three formal mathematical frameworks — category theory, information geometry, and differential geometry — as precision lenses illuminating different facets of its structure. The paper shows that Buddhist notions of śūnyatā (emptiness), pratītyasamutpāda (dependent origination), anātman (non-self), and nirvāṇa (cessation) correspond to well-defined mathematical structures, with all three frameworks converging on the dissolution of the reference frame. The paper is explicit throughout about the scope and limitations of formal description, including the ineliminable gap between structural elucidation and first-person realization.
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