The Quaternion Person

Abstract

The term imaginary is often used dismissively in philosophy to mean illusory, subjective, or unreal. This paper rejects that conflation. Drawing on the mathematical distinction between the imaginary and the fictive, it argues that several indispensable dimensions of human existence are imaginary in the formal sense—non-somatic, non-measurable, yet objectively real because they perform irrefutable explanatory work.

After establishing that the intellectual and relational dimensions of man already function as such imaginary axes, the paper introduces the quaternion as a disciplined analogy: a formal structure in which one real axis and multiple non-commuting imaginary axes are required to model rotation and orientation in physical reality. The quaternion’s mathematical properties—orthogonality, non-commutativity, and irreducibility—are shown to map coherently onto the structure of human action and identity.

The paper concludes that once the intellectual and relational axes are admitted as real-but-immaterial dimensions, parity of reasoning forces the admission of a third imaginary axis: volition. This volitional axis is not representational, predictive, or causal in the ordinary sense, but is irreducible and necessary to account for authorship, obligation, and binding across time. Denying it while accepting the others is shown to be logically incoherent.

1. “Imaginary” Is Not “Illusory”

The word imaginary has suffered a rhetorical collapse.

In ordinary discourse:

• Imaginary means unreal, fictional, or merely psychological.
• To call something imaginary is to dismiss it.

In mathematics and formal science, however:

• Imaginary means orthogonal to direct measurement.
• It denotes a dimension that cannot be isolated as a physical magnitude, yet is indispensable for correct description and prediction.

The square root of −1 (i) is not an illusion.
It is not optional.
It is required.

Complex numbers became indispensable not because they were metaphysically asserted, but because models without them failed. Electrical engineering, wave mechanics, and control theory all rely on imaginary components to represent phase, potential, and stored energy—features that are not directly measured but are undeniably real in their effects.

This paper adopts that formal meaning.

Imaginary will mean:
non-somatic, non-directly-measurable, yet objectively necessary for explanatory completeness.

2. The First Irrefutable Imaginary Axis: Intellect

The intellectual axis of man already satisfies this definition.

Logic, mathematics, language, and symbolic reasoning are:

• not physical objects,
• not measurable forces,
• not reducible to neural activity without remainder,

yet they are objectively binding.

One cannot deny:

• that contradictions are invalid,
• that proofs constrain conclusions,
• that symbols refer beyond themselves,
• that arguments can succeed or fail independently of preference.

Mathematics represents the physical world without being the physical world.
A map represents territory without being territory.
Argumentation ethics can demonstrate the existence of individual agency without measuring it.

The intellectual axis is therefore:

• immaterial,
• imaginary in the formal sense,
• and already universally accepted.

Any attempt to deny imaginary axes must begin by denying mathematics and logic themselves—which is self-defeating.

3. The Second Irrefutable Imaginary Axis: Relation

Equally undeniable is the relational axis.

Human beings exist:

• in networks,
• under recognition,
• within trust, reputation, obligation, and expectation.

Property, contract, promise, betrayal, and authority are not individual phenomena.
They are intersubjective realities.

The relational axis is:

• not somatic,
• not purely cognitive,
• not directly measurable,

yet it is constitutive of identity and action.

A person is not merely a body-plus-brain.
A person is also:

• a bearer of reputation,
• a counterparty,
• a member,
• a debtor,
• a witness.

This axis is as real as the intellectual one—and just as immaterial.

4. Why a Quaternion Is the Right Analogy

At this point, we already have:

1. A real axis: the somatic (physical embodiment and causation)
2. An imaginary axis: the intellectual (symbolic representation)
3. An imaginary axis: the relational (intersubjective recognition)

The question is how these axes interact.

Here the quaternion is not metaphorical flourish, but a disciplined formal analogy.

A quaternion consists of:

• one real component,

• three imaginary components (i, j, k),
  with crucial properties:

• Orthogonality: each axis captures something irreducible to the others.

• Non-commutativity: the order of operations matters.

• Completeness: rotations cannot be modeled correctly without all components.

This matters because quaternions were not invented for philosophy.
They were adopted because simpler representations failed.

5. Non-Commutativity and Human Action

Quaternion multiplication is non-commutative:

ij ≠ ji

This property maps cleanly onto human structure.

• Cognition does not generate commitment.
• Social pressure does not equal authorship.
• Volition can override cognition and relation.
• Somatic force without the others is violence.

Order matters.

A promise understood intellectually but not willed is empty.
A commitment willed but not recognized relationally is inert.
A social role without volition is coercive.

These axes interact asymmetrically.
Reduction fails.
Sequence matters.

This is not poetic resemblance.
It is structural correspondence.

6. The Missing Axis: Volition

Now the critical move.

If we already accept:

• an immaterial intellectual axis,
• an immaterial relational axis,

then denying a volitional axis is no longer conservative.
It is selective.

The volitional axis explains what neither intellect nor relation can:

• why commitments bind across time,
• why promises survive preference changes,
• why guilt is experienced as obligation rather than error,
• why repentance reverses trajectory instead of updating policy,
• why authorship persists beyond cognition or social pressure.

Volition is not:

• metacognition,
• representation,
• prediction,
• optimization.

It is binding.

In quaternion terms, it is the imaginary axis that carries directional continuity—the analogue of momentum or phase in human action.

7. Volition Is Imaginary, Not Illusory

Volition fits the same formal category as intellect and relation:

• immaterial,
• not directly measurable,
• inferred from effects,
• indispensable for coherent explanation.

Denying it requires one to:

• re-describe promise as threat,
• re-describe guilt as conditioning,
• re-describe obligation as prediction,
• re-describe law as force.

That is not parsimony.
It is explanatory collapse.

8. What the Quaternion Model Does—and Does Not—Claim

This model does not claim:

• that humans are literally mathematical objects,
• that ontology reduces to algebra,
• that measurement can capture volition.

It claims only this:

Just as physical rotation cannot be modeled without multiple imaginary axes, human action cannot be modeled without multiple immaterial dimensions.

The quaternion does not mystify reality.
It disciplines description.

9. Conclusion

Calling something imaginary is not a refutation.
It is often an admission that one has not learned how to formalize it.

The intellectual and relational axes of man are already accepted as imaginary-but-real.
Once that is conceded, the volitional axis follows by parity of reasoning.

Denying it does not preserve rigor.
It amputates the model.

The quaternion analogy does not inflate metaphysics.
It reveals structure.

And once that structure is seen, the volitional axis is no longer speculative.
It is the missing coordinate.

Final Compression

• Imaginary does not mean illusory.
• Intellect is imaginary and real.
• Relation is imaginary and real.
• Quaternion structure explains their interaction.
• Volition is the remaining irreducible imaginary axis.
• Denying it is incoherent once the others are admitted.

This is not theology.
It is model completeness.