An offering of the Aadya Meditation Centre and Academy (AMCA), presented by its founder, Arun Singha Mahapatra.

Brahman as the Ultimate Invariant

Advaita Vedanta, Mathematics, and the Logic of the Infinite

Focus Key phrase: Advaita Vedanta, Mathematics, and Brahman

Introduction: Non-Duality and the Language of Mathematics

Advaita Vedanta stands as one of the most careful and rigorous inquiries into the nature of reality. It clearly states that Brahman alone is real and that there is no second reality apart from it. This claim is not poetic imagination or mystical exaggeration. Instead, the Upanishads arrive at this conclusion through direct insight and careful reasoning. Again and again, they explain that Brahman goes beyond time, space, causation, and all categories of thought. At the same time, they affirm that these very categories arise and function within Brahman itself.

Because of this, Advaita Vedanta directly challenges the ordinary habit of the human mind. The mind prefers to divide reality into parts, levels, and hierarchies. However, non-duality asks us to see reality as whole, indivisible, and without internal separation. Therefore, Advaita does not deny the appearance of diversity; rather, it questions the belief that diversity exists independently of its ground.

Non-Duality and the Search for Invariant Structures

In a different field, modern mathematics follows a similar search for what remains stable beneath complexity. It looks for basic structures that do not change, even when forms and relations vary. For this reason, ideas such as infinity, zero, limits, symmetry, and universal inclusion play a central role in mathematics. These ideas are not just technical tools. Instead, they help the intellect approach what cannot be fully understood by counting or measurement alone.

In this context, this essay brings Advaita Vedanta and mathematics into a conceptual dialogue. It does not try to reduce one discipline to the other. Rather, it allows each to shed light on the other. In particular, the analogy of mathematical set theory helps explain how many forms can arise within an all-inclusive unity without breaking non-duality. Building on this idea, the essay examines key mantras from the Isha Upanishad and the Katha Upanishad through a mathematical lens. These mantras describe Brahman as moving yet unmoving, smaller than the smallest and greater than the greatest. Although these statements appear paradoxical, they actually point to a deeper logic that goes beyond ordinary measures of space and time. Thus, the discussion invites reflection that is both clear to the intellect and open to contemplation.

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Brahman in Advaita Vedanta: The One Without a Second

In Advaita Vedanta, Brahman is understood as one, indivisible, and beyond time and space. All that is experienced—gross or subtle, visible or invisible, causal or manifest—arises from Brahman, abides in Brahman, and ultimately dissolves back into Brahman.

The Chandogya Upanishad declares with striking simplicity:

Sarvam khalvidam Brahma
“All this indeed is Brahman.”

Brahman is not a thing among things. It is not located somewhere, nor does it exist alongside others. It is the ground of existence itself—pure consciousness (Chaitanya), self-luminous, and self-established. Time and space themselves arise within it, just as waves arise within the ocean without altering the ocean’s essential nature.

The Brihadaranyaka Upanishad states:

Neha nanasti kinchana
“Here, there is no multiplicity whatsoever.”

This non-duality is not a numerical oneness but an absolute absence of a second.

This vision of fullness and completeness is explained in detail through Bhuma Vidya, where the Chandogya Upanishad reveals Brahman as the infinite, all-inclusive reality beyond limitation. Readers interested in this perspective may refer to
Essence of Chandogya Upanishad: Bhuma Vidya (Part 7A)
https://arunsingha.in/2025/02/17/essence-of-chandogya-upanishad-bhuma-vidya-part-7a/

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1. The Upanishadic Vision: Brahman Beyond All Measures

Advaita Vedanta begins not with belief, but with radical inquiry into the nature of reality. Brahman is described as that from which everything emerges, in which everything exists, and into which everything dissolves—yet Brahman itself remains unchanged.

The Isha Upanishad opens with a vision that already transcends ordinary logic:

Ishavasyam idam sarvam yat kincha jagatyam jagat
“All this—whatever moves in this moving universe—is pervaded by the Lord.”

Here, pervasion does not mean spatial filling. It points to Brahman as the ground of existence.

This idea echoes the classical Vedantic assertion that knowledge of the ultimate reality renders all other knowledge secondary. A related exploration of this theme, in dialogue with modern cosmology, can be found in
By Knowing Which Everything Is Known: Vedanta and the Big Bang
https://arunsingha.in/2025/10/10/by-knowing-which-everything-is-known-vedanta-and-the-big-bang/

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2. “It Moves and It Moves Not”: Stillness, Motion, and Quantum Analogy

One of the most profound statements in world philosophy occurs in the Isha Upanishad:

Tadejati tannaiyati
“It moves, and it moves not.”

At the surface, this is paradoxical. At depth, it is precision.

Mathematical & Physical Insight

In mathematics and physics, we encounter entities that:

• Remain invariant

• While generating dynamic phenomena

For example:

• The origin in coordinate geometry does not move, yet all motion is defined relative to it.

• In quantum theory, the wave function evolves, while certain probability relations remain conserved.

Brahman is similar—but deeper. It is not an object in motion. It is the condition that allows motion to appear.

This resonates strongly with Erwin Schrodinger, who observed that consciousness cannot be divided, though experiences appear many.

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3. Lesser Than the Least, Greater Than the Greatest (Katha Upanishad)

The Katha Upanishad gives a statement that directly invites mathematical contemplation:

“Smaller than the smallest, greater than the greatest, the Self is hidden in the heart of all beings.”

Mathematical Interpretation: Limits and Infinity

In mathematics:

• Zero is not “nothing”; it is a limit point

• Infinity is not a number; it is unboundedness

Advaita Vedanta understands Brahman as the sole reality beyond time and space.

• Subtler than the smallest → beyond atomic or quantum divisibility

• Vaster than the greatest → beyond cosmological magnitude

This places Brahman outside the scale itself.

Just as:

• Zero and infinity cannot be reached by finite steps

• Brahman cannot be reached by sensory or intellectual accumulation

It must be recognized, not constructed.

The Katha Upanishad develops this paradox with great clarity by describing the Atman as subtler than the subtle and vaster than the vast. A focused discussion on this theme is available in
Nature of Atman in the Katha Upanishad
https://arunsingha.in/2021/11/22/nature-of-atman-in-katha-upanishad/

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4. Set Theory Revisited: The Universal Set and Non-Duality

In set theory:

• Elements belong to sets

• Sets belong to higher sets

• All hierarchies culminate in a universal set

But:

• The universal set cannot be a subset

• There is no “outside” to it

Likewise:

• All forms, laws, dimensions, and universes arise in Brahman

• Brahman is not one entity among others

• There is no second reality in which Brahman exists

The Chandogya Upanishad affirms:

“That which is subtle—That is the Self. That Thou Art.”

Critical Clarification

Unlike a mathematical universal set:

• Brahman is self-luminous

• It is the knower of all sets, not a member of any

This distinction preserves Advaita from reductionism.

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5. Invariance: The Common Thread of Mathematics and Vedanta

Modern mathematics seeks invariants:

• Quantities unchanged by transformation

• Structures preserved under symmetry

Physics seeks invariant laws across frames of reference, as emphasized by Albert Einstein.

Advaita Vedanta identifies the ultimate invariant:

• Consciousness remains unchanged in waking, dream, and deep sleep

• Objects change, experiences change, but the witness does not

Therefore, Advaita describes Brahman not as a concept, but as the very condition of knowing.

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6. The Limit of Formal Systems and the Leap Beyond

Mathematics itself acknowledges its limits. Kurt Godel demonstrated that no formal system can fully explain itself.

The Kena Upanishad had already declared this centuries earlier:

“That which cannot be known by thought, but by which thought is known—That alone is Brahman.”

Thus:

• Mathematics purifies the intellect

• Advaita Vedanta completes the journey by transcending intellect

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7. Why This Matters Today

This synthesis is not academic ornamentation. It addresses a deep modern hunger:

• Science seeks unity

• Philosophy seeks meaning

• Spirituality seeks realization

Advaita Vedanta offers a framework where multiplicity does not destroy unity, and mathematics offers a disciplined language to approach that insight without superstition.

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8. Conclusion: From Concept to Realization

Mathematics can take us to the edge of infinity.
Advaita Vedanta invites us to step beyond the edge.

Brahman is:

• Not large or small

• Not moving or still

• Not inside or outside

It is That by which size, motion, and location are known.

Brahma satyam, jagat mithya, jivo Brahmaiva naparah.

Contemporary reflections in science also question purely material explanations of consciousness. One such perspective is offered by Federico Faggin, who argues for consciousness as foundational. For further reading, see
CIP Framework and Consciousness by Federico Faggin
https://arunsingha.in/2025/08/07/cip-framework-federico-faggin/