Polarity: 'Everything is Dual' as Symmetry Group Theory

2 janvier 2026

BY NICOLE LAU

The fourth Hermetic Principle, from The Kybalion:

"Everything is Dual; everything has poles; everything has its pair of opposites; like and unlike are the same; opposites are identical in nature, but different in degree; extremes meet; all truths are but half-truths; all paradoxes may be reconciled."

This sounds like mystical paradox. But it's actually mathematics—a precise statement about symmetry, group theory, and the structure of complementary pairs.

In this article, I'll prove that the Hermetic Principle of Polarity is mathematically equivalent to symmetry group theory, particularly the concept that every element has an inverse.

Not metaphorically. Exactly.

The Mathematical Translation: Group Theory

Hermetic Version:
"Everything is Dual; everything has its pair of opposites."

Mathematical Version:

In group theory, for every element x in a group G, there exists an inverse x⁻¹ such that:

x ∘ x⁻¹ = e

Where:
• ∘ is the group operation
• e is the identity element

This is the mathematical formalization of duality: every element has a complementary opposite.

What is a Group?

A group (G, ∘) is a set G with an operation ∘ satisfying:

1. Closure: ∀a,b ∈ G: a ∘ b ∈ G

2. Associativity: (a ∘ b) ∘ c = a ∘ (b ∘ c)

3. Identity: ∃e ∈ G: e ∘ a = a ∘ e = a

4. Inverse: ∀a ∈ G, ∃a⁻¹ ∈ G: a ∘ a⁻¹ = a⁻¹ ∘ a = e

The fourth property—inverse—is the mathematical expression of polarity. Every element has a dual.

Examples of Mathematical Duality

Example 1: Integers Under Addition

Group: (ℤ, +)

• Element: 5
• Inverse: -5
• Operation: 5 + (-5) = 0 (identity)

Every positive number has a negative dual. Polarity.

Example 2: Non-Zero Reals Under Multiplication

Group: (ℝ*, ×)

• Element: 2
• Inverse: 1/2
• Operation: 2 × (1/2) = 1 (identity)

Every number has a reciprocal dual.

Example 3: Symmetry Transformations

Group: Symmetries of a square

• Rotation 90° clockwise
• Inverse: Rotation 90° counterclockwise
• Operation: Rotate 90° CW then 90° CCW = no rotation (identity)

Every transformation has an inverse transformation. Duality.

Physical Examples of Polarity

Example 1: Particle-Antiparticle Pairs

Every particle has an antiparticle:

• Electron (e⁻) ↔ Positron (e⁺)
• Proton (p⁺) ↔ Antiproton (p⁻)
• Quark ↔ Antiquark

When they meet: annihilation → energy

e⁻ + e⁺ → 2γ (two photons)

This is literal polarity: complementary opposites that cancel to identity (pure energy).

Example 2: Charge

Electric charge comes in pairs:

• Positive (+) ↔ Negative (-)
• Total charge is conserved
• Opposite charges attract, like charges repel

Magnetic poles:

• North (N) ↔ South (S)
• Always appear in pairs
• No magnetic monopoles (isolated N or S)

Polarity is fundamental to electromagnetism.

Example 3: Matter and Antimatter

The universe contains:

• Matter (particles)
• Antimatter (antiparticles)

Why more matter than antimatter? The baryon asymmetry problem—one of physics' great mysteries.

But the principle holds: for every type of matter, antimatter exists as its dual.

Symmetry and Conservation Laws

Noether's Theorem (1915) connects symmetry to conservation:

Every continuous symmetry corresponds to a conservation law.

Examples:

• Time translation symmetry → Energy conservation
• Space translation symmetry → Momentum conservation
• Rotation symmetry → Angular momentum conservation
• Gauge symmetry → Charge conservation

Duality (symmetry) is fundamental to physics. The Hermetic principle, proven by Noether.

Wave-Particle Duality

Quantum mechanics reveals ultimate duality:

All matter exhibits both wave and particle properties.

• Light: Wave (interference) AND particle (photon)
• Electron: Particle (charge, mass) AND wave (diffraction)
• Everything: Complementary aspects, not contradictory

Heisenberg uncertainty principle:

Δx Δp ≥ ℏ/2

You can't know position (particle property) and momentum (wave property) simultaneously with perfect precision.

Complementarity (Bohr): Wave and particle are complementary descriptions. Both true, both necessary.

This is Hermetic Polarity: "opposites are identical in nature, but different in degree."

Yin-Yang as Mathematical Symmetry

The Taoist yin-yang symbol encodes mathematical duality:

Properties:

• Two complementary regions (black/white)
• Rotational symmetry (180° rotation maps yin → yang)
• Each contains seed of the other (dots)
• Together form complete circle (unity)

Mathematical interpretation:

• Yin and Yang are group elements
• Rotation by 180° is the group operation
• Each is the inverse of the other
• Together they form the identity (Tao)

Yin-Yang = visual representation of group theory duality.

Dialectics as Mathematical Process

Hegel's dialectic:

Thesis + Antithesis → Synthesis

This is a mathematical operation:

• Thesis: Element a
• Antithesis: Inverse a⁻¹
• Synthesis: Higher-order structure containing both

Example in physics:

• Thesis: Particle
• Antithesis: Wave
• Synthesis: Quantum object (both/neither)

Dialectics = group theory + emergence.

Creation and Annihilation Operators

In quantum field theory, particles are created and destroyed by operators:

Creation operator: â†
Annihilation operator: â

Commutation relation:

[â, â†] = ââ† - â†â = 1

These are dual operators:

• â† creates a particle
• â destroys a particle
• Together they define the quantum field

This is Hermetic Gender principle (next article), but it's also Polarity: complementary operations that are inverses.

Extremes Meet: Topology of Opposites

The Hermetic claim: "extremes meet."

Mathematical interpretation: Circular topology

On a circle, if you go far enough in one direction, you return from the opposite direction.

Examples:

• Temperature: Absolute zero (0 K) and infinite temperature both represent maximum entropy states
• Angles: 0° and 360° are the same
• Phase: 0 and 2π are identical
• Modular arithmetic: In mod 12, 0 ≡ 12

Opposites meet when the space is circular/periodic.

All Truths Are Half-Truths

The Hermetic claim: "All truths are but half-truths."

Mathematical interpretation: Complementarity and incompleteness

Gödel's Incompleteness Theorems:

In any consistent formal system:

1. There exist true statements that cannot be proven within the system
2. The system cannot prove its own consistency

No single perspective captures complete truth. You need complementary viewpoints.

Quantum complementarity:

Wave description is "half-truth."
Particle description is "half-truth."
Both together approach complete truth.

Practical Applications

1. Error Correction

Use duality for redundancy:

• Parity bits (even/odd)
• Checksums
• Dual-rail logic

2. Optimization

Dual problems in linear programming:

• Primal problem: Maximize objective
• Dual problem: Minimize constraints
• Solutions are related

3. Signal Processing

• Time domain ↔ Frequency domain (Fourier duality)
• Position ↔ Momentum (Quantum duality)

4. Balance and Homeostasis

Biological systems use opposing forces:

• Sympathetic ↔ Parasympathetic nervous system
• Insulin ↔ Glucagon (blood sugar)
• Flexor ↔ Extensor muscles

The Hermetic Insight Validated

The Hermeticists claimed:

"Everything is Dual; everything has its pair of opposites."

Modern mathematics has discovered:

• Group theory: Every element has an inverse
• Symmetry: Transformations come in pairs
• Particle physics: Every particle has an antiparticle
• Quantum mechanics: Wave-particle duality
• Noether's theorem: Symmetry ↔ Conservation

The convergence is exact:

"Everything is dual" = "∀x ∈ G, ∃x⁻¹: x ∘ x⁻¹ = e"

Same claim. Different language. Perfect convergence.