From Shadows to Substance: Plato, Aristotle, Peirce, and the Geometry of Reality
Sungchul Ji, Ph.D.
Emeritus Professor of Theoretical Cell Biology, Rutgers University
October 2025
1 The Triadic Bridge across Millennia
Plato’s Allegory of the Cave and the modern Geometry of Reality (GOR) appear at first to belong to different worlds—one mythic, the other mathematical. Yet, as Table 1 demonstrates, both portray the same ascent: from shadow to substance, from two-dimensional illusion to three-dimensional understanding.
Plato, Aristotle, and Peirce each intuited that reality unfolds not linearly but triadically. Their visions now converge within GOR, a framework that renders ancient metaphysics visible in geometric form.
2 Plato’s Cave: The Birth of the Triad
In the cave, chained prisoners watch shadows cast by firelight and mistake them for truth. One ascends toward sunlight, discovering the real source of illumination.
Plato’s sequence—shadow → object → sunlight → the Good—is an early map of triadic transformation:
The cave’s vertical motion—darkness to light—mirrors the Z-axis emergence in GOR, elevating dyadic perception (X + Y) into triadic reality (X + Y + Z).
3 Aristotle: Matter and Form as Pre-Coordinates
Aristotle refined his teacher’s insight by grounding it in matter (hylē) and form (eidos)—two poles of potentiality and actuality.
GOR interprets these as X = Energy/Matter and Y = Information/Form.
Yet even Aristotle sensed a third principle—entelechy, the inner purpose that unites them—which today corresponds to Z = Spirit/Consciousness.
Thus, the Aristotelian dyad already hinted at the triad completed in Peirce and fulfilled geometrically in GOR.
4 Peirce: From Dyads to Irreducible Triads
Nineteenth-century logician Charles Sanders Peirce identified the fundamental grammar of reality as triadic: Sign–Object–Interpretant, or Firstness–Secondness–Thirdness.
His Irreducible Triadic Relation (ITR) [3] mirrors the ascent from shadow (Sign) to object (Object) to understanding (Interpretant).
In GOR, these map directly onto the three orthogonal axes:
Peirce provided the logical skeleton that unites Plato’s metaphysics and Aristotle’s ontology into one universal syntax of reality.
5 The Geometry of Reality: The Triadic Cube of Being
The GOR visualizes reality as a cube spanned by three axes—Energy (X), Information (Y), and Spirit (Z).
The XY plane corresponds to Plato’s shadow world: phenomena without depth.
The XYZ volume embodies true Reality: integration of substance, relation, and meaning.
The Z-axis represents the ascent—the Form of the Good—through which consciousness harmonizes energy and information.
In this geometry, awakening is not metaphorical but structural: movement from projection to participation, from image to embodiment.
6 IRVSE: The Dynamics of Becoming
The ascent through the cave parallels the universal process of IRVSE—Iterative Reproduction with Variations and Selection by Environment.
Each stage in Plato’s myth corresponds to one phase of IRVSE:
Through IRVSE, the Universe, like the prisoner, evolves from imitation to insight.
7 Codality: The Hidden Common Ancestor
Codality reveals that parallel forms share a hidden generative origin.
Homo sapiens and Pan troglodytes are codally related through their last common ancestor.
Likewise, Plato’s Cave and GOR are codally related through their shared triadic archetype—the Irreducible Triadic Relation (ITR).
This triadic code—Energy (X), Information (Y), Spirit (Z)—links biology, mind, and cosmos within one evolutionary grammar.
Together they form a continuous lineage of triadic philosophy stretching from antiquity to contemporary science.
8 The Eternal Ascent
The journey from shadow to Sun, matter to form, sign to meaning, and X to Z, describes not just human enlightenment but the universe awakening to itself.
Reality is triadic:
Matter gives structure,
Mind gives interpretation, and
Spirit gives coherence.
The cube of GOR thus completes Plato’s cave—turning the myth of ascent into a cosmological principle.