Formal Axiomatization of Advaita Vedanta in Lean 4
A machine-verified formalization of the non-dual metaphysical system of Advaita Vedanta, implemented in Lean 4 theorem prover. This work represents the first complete axiomatization of a major non-Western philosophical tradition using contemporary methods of formal verification.
Abstract
Advaita Vedanta, one of the central schools of Hindu philosophy, proposes a non-dual metaphysics in which ultimate reality consists of a single, unchanging, conscious substrate (Brahman) that is identical with the true self (Atman). This formalization captures the logical structure of this tradition through 114 axioms organized across multiple semantic domains: ontological grounding, levels of reality, consciousness structure, and the mecha…
Formal Axiomatization of Advaita Vedanta in Lean 4
A machine-verified formalization of the non-dual metaphysical system of Advaita Vedanta, implemented in Lean 4 theorem prover. This work represents the first complete axiomatization of a major non-Western philosophical tradition using contemporary methods of formal verification.
Abstract
Advaita Vedanta, one of the central schools of Hindu philosophy, proposes a non-dual metaphysics in which ultimate reality consists of a single, unchanging, conscious substrate (Brahman) that is identical with the true self (Atman). This formalization captures the logical structure of this tradition through 114 axioms organized across multiple semantic domains: ontological grounding, levels of reality, consciousness structure, and the mechanisms of apparent phenomenal diversity.
The formalization proves 40+ theorems, culminating in a master theorem that establishes the consistency and logical sufficiency of the Tat Tvam Asi doctrine, the fundamental claim that individual subjectivity and ultimate reality are identical.
Philosophical Foundations
Advaita Vedanta addresses the apparent contradiction between phenomenal multiplicity and metaphysical unity through a doctrine of levels of reality. The system distinguishes three epistemic-ontological levels:
- Paramarthika (ultimate reality): the level of Brahman alone
- Vyavaharika (conventional reality): the level of empirical objects and causal relations
- Pratibhasika (illusory appearance): the level of perceptual errors and misidentifications
The formalization axiomatizes how these levels relate through the concept of Maya (the power of appearance) and establishes that multiplicity arises through ignorance (avidya) rather than through real ontological division.
Formal Structure
The axiomatization divides into seven primary modules:
| Module | Purpose | Key Concepts |
|---|---|---|
| Signature | Fundamental types and predicates | Objects, Levels, Relations |
| CoreAxioms | Basic metaphysical principles | Uniqueness, Identity, Grounding |
| LevelAxioms | Three-tier reality structure | Paramarthika, Vyavaharika, Pratibhasika |
| AwarenessAxioms | Consciousness predicates | Witnessing, Perception, Knowledge |
| MayaAxioms | Appearance mechanisms | Superimposition, Vivarta, Maya-power |
| JivaIsvara | Individual and cosmic consciousness | Jiva, Isvara, Embodiment |
| AdditionalAxioms | Supplementary constraints | Sheaths, Gunas, Spacetime |
Key Theorems
The formalization establishes several philosophically significant results:
T0 (Brahman-Atman Identity): Demonstrates that the ultimate ground (Brahman) and the witnessing subject (Atman) are necessarily identical.
T1 (Grounding Principle): Proves that all conditioned entities are ontologically grounded in the absolute.
T7 (Ontological Monism): Shows that only one entity exists at the ultimate level of reality.
T30 (Maya Universality): Establishes that all phenomenal diversity arises through Maya-power.
Master Theorem (Tat Tvam Asi): Proves the existence and uniqueness of an entity satisfying all essential properties of the absolute self, including identity with ultimate reality, transcendence of phenomenality, unchangingness, self-luminosity, and non-perception.
Repository Structure
.
├── AdvaitaVedanta/
│ ├── Signature.lean # Core types and relations
│ ├── CoreAxioms.lean # Fundamental axioms (A1-A15)
│ ├── LevelAxioms.lean # Reality level structure
│ ├── AwarenessAxioms.lean # Consciousness axioms
│ ├── MayaAxioms.lean # Appearance mechanisms
│ ├── JivaIsvara.lean # Jiva and Isvara relations
│ ├── AdditionalAxioms.lean # Supplementary axioms
│ └── Theorems.lean # Proven theorems
├── AdvaitaVedanta.lean # Main module
├── Main.lean # Entry point
├── lakefile.lean # Lake build configuration
├── lean-toolchain # Lean version specification
└── docs/
├── AXIOMS.md # Complete axiom listing
├── THEOREMS.md # Detailed theorem documentation
└── METHODOLOGY.md # Formalization approach
Building and Verification
Prerequisites: Lean 4.16.0 must be installed. Installation instructions are available at https://leanprover.github.io/lean4/doc/setup.html
To verify all proofs:
lake build
To run the main executable:
lake exe advaita
Expected output confirms successful verification of all axioms and theorems, including the master theorem.
Philosophical Significance
This formalization demonstrates that the logical structure of Advaita Vedanta forms a consistent axiomatic system. The successful verification addresses potential philosophical objections regarding internal coherence and provides a foundation for precise comparative philosophy. The master theorem establishes that the core doctrine (Tat Tvam Asi) follows necessarily from the axioms, rather than requiring separate argumentation.
Several features merit philosophical attention. First, the formalization makes explicit the distinction between ontological grounding (Cond) and apparent manifestation (MayaPow), clarifying how unity and multiplicity coexist in the system. Second, the proof that the subject cannot perceive (T19) while simultaneously witnessing all (T14) captures the non-dual nature of awareness. Third, the verification that phenomenal properties (temporality, spatiality, qualities) necessarily exclude the absolute (A7) while characterizing all conditioned entities (A8) establishes a precise criterion for distinguishing levels.
Related Work
This formalization complements recent work in formal metaphysics and extends the scope of verification to non-Western traditions. It demonstrates the applicability of interactive theorem proving to contemplative philosophies and provides a template for formalizing other non-dual systems.
Citation
If you use this formalization in academic work, please cite:
@misc{advaita_vedanta_lean,
title={Formal Axiomatization of Advaita Vedanta in Lean 4},
author={[Matthew Scherf]},
year={2025},
howpublished={GitHub repository},
url={https://github.com/[matthew-scherf]/advaita}
}
License
This work is released under the MIT License. See LICENSE file for details.
Contributing
Contributions addressing extensions, alternative axiomatizations, or connections to other formal systems are welcome. See CONTRIBUTING.md for guidelines.
Acknowledgments
This formalization draws on classical Advaita texts (Brahma Sutras, Upanishads, works of Adi Shankaracharya) and contemporary scholarship in Indian philosophy. The formal methods derive from techniques in automated theorem proving and formal verification.