LESSON 3: HOW THINGS CHANGE — THE SIMPLE MATH OF SOCIAL MOTION

In Lesson 2, we built our Map of Everything — the 25 vital signs of a society, called the State Vector X(t).

Now we ask: How does X(t) become X(t+1)?
How do the vital signs change from this year to next year?

This is where we introduce our first real equations. Don’t worry — we’ll use pictures, words, and simple rules first.

1. THE IDEA OF A “SOCIAL LAW”

In physics, Newton’s 2nd Law is:

Force = Mass × Acceleration

It’s a simple rule that predicts how objects move.

In CMH, we want social laws — simple rules that predict how societies move.

Example Social Law (Demographic):

Next Year’s Population = This Year’s Population + (Births – Deaths)

That’s obvious, right? But it’s already a mathematical model. Let’s write it as an equation:

P(t+1) = P(t) + B(t) – D(t)

Where:

  • P(t) = population this year
  • B(t) = births this year
  • D(t) = deaths this year
  • P(t+1) = population next year

2. ADDING REALISM: THE “CARRYING CAPACITY” IDEA

But populations don’t grow forever. A country can only support so many people (because of food, space, jobs). This limit is called the Carrying Capacity (K).

Imagine a glass being filled with water:

  • When the glass is empty, water pours in fast.
  • When the glass is nearly full, water pours in slowly.
  • When the glass is full, water stops pouring.

The same happens with populations. This is called logistic growth, and it’s one of the most important patterns in nature (and societies).

Here’s the simple equation:

Next Year’s Pop = This Year’s Pop + (Growth Rate × This Year’s Pop × (1 – This Year’s Pop / K))

In symbols:

P(t+1) = P(t) + r × P(t) × (1 – P(t)/K)

Let’s see it in action:

Suppose:

  • K = 100 million people (the country’s max supportable population)
  • r = 0.02 (2% growth rate per year)
  • P(2024) = 60 million

Calculation:

  1. 1 – P(t)/K = 1 – 60/100 = 0.4
  2. r × P(t) × 0.4 = 0.02 × 60 × 0.4 = 0.48 million
  3. P(2025) = 60 + 0.48 = 60.48 million

See? The growth is slowed by the (1 – P(t)/K) term. When P(t) is small, growth is fast. When P(t) is near K, growth nearly stops.

3. CONNECTING TWO VITAL SIGNS: THE “UNREST ENGINE”

Now let’s connect two of our vital signs from Lesson 2 in a simple feedback loop.

Vital Signs:

  • U(t) = Unrest Index (a number from 0 to 10, made from protests, strikes, riots)
  • Y(t) = Youth Unemployment Rate (percent)

Proposed Social Law:

If youth unemployment is high this year, unrest will rise next year.
If unrest is high this year, the government might create jobs, lowering youth unemployment the year after.

We can write this as two linked equations:

1. U(t+1) = U(t) + a × Y(t)   # More unemployment → more unrest next year
2. Y(t+1) = Y(t) – b × U(t)   # More unrest → less unemployment (gov reacts)

Where a and b are small numbers (like 0.1 or 0.2) that say how strong the effect is.

Example:
Let a = 0.2, b = 0.1, U(2024) = 3, Y(2024) = 25%.

  • Step 1: U(2025) = 3 + 0.2 × 25 = 3 + 5 = 8
    (Unrest jumps because unemployment is high)
  • Step 2: Y(2025) = 25 – 0.1 × 3 = 25 – 0.3 = 24.7%
    (Unrest slightly pushes government to create jobs)

Next year, we’d plug U=8 and Y=24.7 back in and repeat. This creates a wave pattern — unrest rises and falls in cycles.

4. VISUALIZING CHANGE: THE “PHASE PORTRAIT”

Instead of staring at numbers, we can draw how U and Y change together.

Youth Unemployment (Y)
    ↑
    |     • Start here
    |    / \
    |   /   \
    |  /     \
    | /       \
    |/         \
    +----------------→ Unrest (U)

This picture is called a phase portrait. Each dot is (U, Y) in a given year. The line shows how they move together over time.

In stable societies, the dots circle in a small loop.
In unstable ones, the spiral spins outward toward crisis.

5. THE “CRISIS” THRESHOLD

Not all feedback loops are gentle. Some have tipping points.

Imagine our Unrest Index U has a crisis threshold at U = 7.

If U(t) > 7 for two years in a row → Probability of revolution = 40%

That’s a conditional rule — a “social law” with a consequence.

This is how CMH moves from description to prediction:

  1. We measure U(t) and Y(t) today.
  2. We use our social laws to compute U(t+1), U(t+2), etc.
  3. We check if U will cross the threshold 7.
  4. We output: “40% chance of crisis in 3 years.”

YOUR CMH HOMEWORK:

  1. Build a Mini-Model: Pick two vital signs from Lesson 2. Invent a simple social law connecting them (like we did with Unrest and Unemployment). Write it in words, then try to write it as a tiny equation.
  2. Predict Your Own Country: Using your gut feeling, estimate today’s Unrest Index (U) for your country on a scale of 0–10. What do you think it will be in one year? Why?
  3. Think About Tipping Points: What’s one threshold in society that, once crossed, changes everything? (Example: “Once inflation passes 50%, people stop using the national currency.”)

When you’re ready, we’ll move to Lesson 4: Probability — Why We Say “Maybe” Instead of “Yes/No.”

This is where CMH becomes truly useful — and truly honest about what it can’t know.