Cause and Effect: 'Every Cause Has Its Effect' as Deterministic Functions

2026-januaro-02

BY NICOLE LAU

The sixth Hermetic Principle, from The Kybalion:

"Every Cause has its Effect; every Effect has its Cause; everything happens according to Law; Chance is but a name for Law not recognized; there are many planes of causation, but nothing escapes the Law."

This sounds like philosophical determinism. But it's actually mathematics—a precise statement about functions, deterministic mappings, and causal structure.

In this article, I'll prove that the Hermetic Principle of Cause and Effect is mathematically equivalent to the definition of a function—the foundation of deterministic systems.

Not metaphorically. Exactly.

The Mathematical Translation: Functions

Hermetic Version:
"Every Cause has its Effect; every Effect has its Cause."

Mathematical Version:

A function f: X → Y is a mapping where:

∀x ∈ X, ∃!y ∈ Y: y = f(x)

For every input x (cause), there exists exactly one output y (effect).

This is the mathematical formalization of causality: deterministic mapping from causes to effects.

What is a Function?

A function is a relation between sets where each input has exactly one output.

Key properties:

1. Well-defined: Every input has an output
2. Deterministic: Same input always produces same output
3. Unique: Each input maps to exactly one output

This is causality: given the same cause, you always get the same effect.

Examples:

• f(x) = 2x: Cause x → Effect 2x
• F = ma: Cause (mass, acceleration) → Effect (force)
• E = mc²: Cause (mass) → Effect (energy)

Determinism: Laplace's Demon

Pierre-Simon Laplace (1814) articulated perfect determinism:

"An intellect which at a certain moment would know all forces that set nature in motion, and all positions of all items of which nature is composed, if this intellect were also vast enough to submit these data to analysis, it would embrace in a single formula the movements of the greatest bodies of the universe and those of the tiniest atom; for such an intellect nothing would be uncertain and the future just like the past would be present before its eyes."

Mathematical formulation:

State at time t₀ + Laws of physics → State at any time t

s(t) = F(s(t₀), t)

Where F is a deterministic function.

This is the Hermetic principle: perfect causality, no randomness, everything determined by law.

Newton's Laws: Deterministic Causality

Newton's Second Law:

F = ma

Or more precisely:

F = dp/dt (force = rate of change of momentum)

This is a differential equation—a deterministic causal law.

Given:
• Initial position x(t₀)
• Initial velocity v(t₀)
• Force F(x, v, t)

You can calculate:
• Position x(t) at any future time
• Velocity v(t) at any future time

Cause (initial conditions + forces) → Effect (future state)

Perfectly deterministic. Perfectly causal.

Causal Chains: Function Composition

The Hermetic text mentions "many planes of causation." Mathematically, this is function composition:

If f: X → Y and g: Y → Z, then:

g ∘ f: X → Z

(g ∘ f)(x) = g(f(x))

Causal chain:

Cause₁ → Effect₁ = Cause₂ → Effect₂ = Cause₃ → Effect₃

Example:
• Rain (cause₁) → Wet ground (effect₁/cause₂) → Slippery road (effect₂/cause₃) → Car accident (effect₃)

Mathematically: f₃(f₂(f₁(rain))) = accident

Causal chains are function compositions.

"Chance is but a name for Law not recognized"

The Hermetic claim: Randomness is just hidden determinism.

This is controversial, but there are mathematical frameworks supporting it:

1. Chaos Theory

Deterministic systems can appear random:

• Lorenz attractor (weather)
• Double pendulum
• Three-body problem

These are deterministic but unpredictable:

• Governed by deterministic equations
• Sensitive to initial conditions (butterfly effect)
• Appear random due to exponential divergence

Small uncertainty in initial conditions → large uncertainty in predictions

But the underlying law is deterministic. "Chance" is our ignorance of initial conditions.

2. Hidden Variables in Quantum Mechanics

Standard quantum mechanics is probabilistic. But some interpretations propose hidden variables:

• Bohm's pilot wave theory: Deterministic, but with hidden variables
• Many-worlds interpretation: Deterministic evolution of universal wave function

If hidden variables exist, quantum "randomness" is actually deterministic law we don't yet understand.

The Hermetic claim: "Chance is but a name for Law not recognized."

Causal Graphs and Bayesian Networks

Modern causal inference uses directed acyclic graphs (DAGs):

• Nodes = variables
• Directed edges = causal relationships
• A → B means A causes B

Bayesian networks quantify probabilistic causality:

P(B|A) = probability of B given A

Even when causality is probabilistic (not deterministic), the structure is causal: causes precede effects, effects depend on causes.

This is the Hermetic principle in probabilistic form.

Correlation vs. Causation

The Hermetic principle distinguishes true causation from mere correlation.

Correlation: A and B happen together
Causation: A causes B

Mathematical test for causation (Pearl's do-calculus):

P(B | do(A)) vs. P(B | A)

• P(B | A): Probability of B given we observe A (correlation)
• P(B | do(A)): Probability of B given we intervene to set A (causation)

If P(B | do(A)) ≠ P(B | A), then A causes B.

This formalizes "every cause has its effect" vs. "things just correlate."

Limits of Determinism

1. Quantum Mechanics

Standard interpretation: Quantum mechanics is fundamentally probabilistic.

Wave function collapse is random:

|ψ⟩ = α|0⟩ + β|1⟩ → |0⟩ with probability |α|²
→ |1⟩ with probability |β|²

No hidden cause determines which outcome occurs.

This challenges strict determinism. But:

• The wave function evolves deterministically (Schrödinger equation)
• Only measurement is probabilistic
• Some interpretations (Bohm, Many-Worlds) restore determinism

2. Gödel's Incompleteness

In formal systems, some truths are unprovable. Does this mean some effects have no cause (within the system)?

Not quite. It means the system can't prove all causal relationships, not that they don't exist.

3. Free Will

If everything is deterministic, is free will an illusion?

Philosophical debate. But mathematically:

• Compatibilism: Free will is compatible with determinism
• Libertarian free will: Requires indeterminism
• Hard determinism: Free will is illusion

The Hermetic principle leans toward determinism, but doesn't resolve the free will question.

Practical Applications

1. Prediction

If causality is deterministic, we can predict:

• Weather (chaotic but deterministic)
• Planetary motion (Newtonian mechanics)
• Chemical reactions (stoichiometry)
• Economic trends (causal models)

2. Control

If we know causal relationships, we can intervene:

• Medicine: Treat cause, not just symptoms
• Engineering: Design systems with desired effects
• Policy: Implement causes that produce desired effects

3. Causal Inference

Discover causal relationships from data:

• Randomized controlled trials
• Natural experiments
• Instrumental variables
• Regression discontinuity

4. Debugging

Find root causes of problems:

• Software debugging (trace cause of bug)
• Medical diagnosis (find cause of symptoms)
• Failure analysis (determine cause of failure)

Objections and Responses

Objection 1: "Quantum mechanics is random, not deterministic."

Response: Standard interpretation, yes. But:
• Wave function evolution is deterministic
• Some interpretations restore full determinism
• Even probabilistic causality has causal structure

Objection 2: "Chaos theory shows determinism doesn't mean predictability."

Response: Correct! Determinism ≠ predictability. The Hermetic principle claims causality (deterministic laws), not perfect predictability.

Objection 3: "This denies free will."

Response: Not necessarily. Compatibilism argues free will is compatible with determinism. The Hermetic principle is about causal structure, not about consciousness or agency.

The Hermetic Insight Validated

The Hermeticists claimed:

"Every Cause has its Effect; every Effect has its Cause; everything happens according to Law."

Modern mathematics has discovered:

• Functions: y = f(x) (deterministic mapping)
• Differential equations: Deterministic evolution
• Causal graphs: Structural causality
• Bayesian networks: Probabilistic causality
• Pearl's do-calculus: Formal causal inference

The convergence is exact:

"Every cause has its effect" = "For every x, there exists exactly one y = f(x)"

Same claim. Different language. Perfect convergence.

Conclusion

The sixth Hermetic Principle—Cause and Effect—is not mysticism.

It's function theory: the mathematics of deterministic mappings.

Validated by:

• Function definition: ∀x ∈ X, ∃!y ∈ Y: y = f(x)
• Newton's laws: F = ma
• Laplace's determinism
• Causal graphs and Bayesian networks
• Pearl's causal inference

Everything happens according to law because:
• Physical laws are deterministic functions
• Causal structure is mathematical
• Even apparent randomness may be hidden determinism

The Hermeticists discovered function theory and deterministic causality 2,000 years before modern mathematics.

Hermetic Mathematics, validated.