The connection between the Archimedean axiom and Zen Buddhism

[Lois]

The connection between the Archimedean axiom and Zen Buddhism is subtle but very precise. It runs not through formulas, but through an attitude toward infinity, emptiness, and process.

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1. No “ultimate grain” of reality

The Archimedean axiom states:

there is no absolutely smallest segment that is ontologically different from all others.

Zen speaks almost the same language:

there is no “final substance,”
no minimal atom of being
behind which the “true reality” is hidden.

🔹 In Zen there is no “secret layer” of the world.
🔹 Satori does not reveal a new substance — it removes an illusion.

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2. Potential infinity = the Way (道)

In Archimedean geometry:

• infinity is the possibility of continuation,

• not an already existing infinite thing.

In Zen:

• the Way (道, dō) is not a goal,

• but continuous practice with no final point.

Just as a segment can be divided endlessly,
awakening never “finishes.”

There is no moment of “I have reached infinity” — that is a mistake of the mind.

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3. Emptiness ≠ infinitesimal

A crucial distinction.

In non-Archimedean mathematics:

• emptiness is replaced by the infinitesimal,

• something “smaller than any measurable quantity.”

In Zen:

• emptiness (空, śūnyatā) is not an object,

• not a scale,

• not a quantity.

In this sense, the Archimedean axiom is closer to Zen:

if something is smaller than everything measurable,
it is not a new entity, but a conceptual error.

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4. Homogeneity of the moment

The Archimedean continuum:

• every part of the line is of the same kind as any other.

Zen:

• every moment is complete,

• there is no “more real” or “less real” moment.

One breath is no closer to truth than a thousand years of meditation.

This is a rejection of depth hierarchy.

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5. Anti-metaphysics of depth

Non-Archimedean structures assume:

• hidden layers,

• inaccessible to ordinary counting.

Zen radically refuses this:

if you search for something deeper, you have already missed it.

Truth is not in the microscopic and not in the cosmic,
but in the present.

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6. Measurement as a conventional practice

Important: Zen does not reject measurement — just as Archimedes does not.

But:

• measurement is a tool,

• not a description of essence.

Archimedes:

• measurement always works,

• but does not exhaust meaning.

Zen:

• practice works,

• but does not fix truth.

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7. Koan as anti-infinity

A koan destroys the mind’s attempt:

• to find a final element,

• to grasp absolute smallness or absolute depth.

“What is the sound of one hand clapping?”
Not an infinitesimal sound,
but a halt of division.

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8. Reduced to one image

The Archimedean line and the Zen path are the same:

• no hidden levels,

• no minimal grain,

• no final point,

• only the possibility of the next step.

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9. A Zen–Archimedean formula

If something cannot be surpassed by repeating the finite,
then it is not a depth of reality,
but an attachment of the mind.