Mathematics of Progress: Why People Don't See the Future

Your brain is broken for exponential math. Here's why that makes you pessimistic about technology—and poor at predicting the future.

Here's a question: If I fold a piece of paper in half 42 times, how thick is it?

Go ahead. Guess.

A foot? A mile?

The answer is 440,000 kilometers. That's the distance from Earth to the Moon.

You didn't guess that. Almost nobody does. And your failure to guess correctly isn't random—it's systematic. Your brain is neurologically incapable of intuitively understanding exponential growth.

This isn't a moral failing. It's a cognitive architecture problem. And it explains why most people are completely wrong about the future.

The Exponential Growth Bias

Psychologists have a name for this: exponential-growth bias. It's the tendency to perceive exponential processes as linear ones—to see accelerating curves as straight lines.

Here's how deep this goes: In controlled studies, 96% of participants underestimate compound growth. Even when given financial calculators. Even when explicitly told the growth rate.

This isn't about intelligence or education. The bias doesn't diminish with age or training. Simple graphs don't fix it. Your brain just can't do it without deliberate, effortful calculation.

The Financial Consequences

Before we get to technology, let's talk about money—because the data here is brutal.

Research shows that people who misunderstand compound growth:

• Carry more short-term debt
• Own fewer stocks
• Save less
• Accumulate lower net worth

How much does fixing this bias matter? One study found that correcting exponential-growth bias is associated with a 62% increase in lifetime savings—controlling for income, education, and other factors.

Read that again. The same person, with the same income, saves 62% more over their lifetime if they understand compound growth correctly.

This is what cognitive bias costs you. And it's the exact same bias that makes you underestimate technology.

The Technology Curves

Now let's look at what's actually happening in the world.

Solar Energy: The 99% Drop Nobody Expected

In 1976, a solar panel cost $106 per watt. Today it costs about $0.27 per watt.

That's a 99.7% decline. Solar went from an expensive curiosity to the cheapest source of electricity in most of the world.

Here's the predictable part: this followed Wright's Law—every doubling of cumulative solar production yields a ~20% cost reduction. The curve has held for 50 years.

And yet, throughout that entire period, energy experts consistently predicted solar would remain too expensive to matter. They looked at where solar was, projected linearly, and concluded it would never compete with fossil fuels.

They were exponentially wrong.

AI: The 1000x Nobody Sees Coming

In 2021, running a GPT-3 model cost about $60 per million tokens.

By 2024, equivalent performance costs $0.06 per million tokens.

That's a 1000x price drop in three years.

But wait—maybe that's just pricing games? No. The performance per dollar is improving just as fast. Models that match GPT-3.5's capabilities saw a 280x cost reduction in just 18 months (November 2022 to October 2024).

This is exponential progress. And most people—including most AI "experts" quoted in newspapers—are predicting the future by drawing straight lines from today.

Sensors: The Invisible Revolution

Here's one most people don't even know about: sensor costs.

• Average IoT sensor: $1.30 (2004) → $0.44 (2018) — 66% drop
• LIDAR: 90% cost decline since 2015
• Accelerometers: Now essentially free at scale (pennies per unit)

This is why your phone has dozens of sensors that would have cost thousands of dollars twenty years ago. This is why autonomous vehicles went from science fiction to production cars. This is why the Internet of Things is actually happening.

Sensors got exponentially cheaper. Most people didn't notice because the change was "gradual"—which is how exponential curves feel until they're not.

Why This Makes People Pessimistic

Here's the psychological trap: if you're cognitively programmed to underestimate exponential improvement, you systematically underestimate the future.

When someone says "solar will never be cheap enough to replace fossil fuels," they're not lying. They're projecting linearly from current data.

When someone says "AI is plateauing," they're looking at the current rate of improvement and failing to account for acceleration.

When someone says "we'll never solve [hard problem]," they're imagining incremental progress from today's tools—not the tools we'll have after another decade of exponential improvement.

Linear thinking makes optimism look naive. If you draw a straight line from today, the future looks incremental—maybe 10-20% better. That's not exciting. That's not transformative.

But the actual curve isn't linear. The actual curve is exponential. And exponential curves are insane.

The Compounding Effect of Multiple Exponentials

Here's where it gets really interesting: these curves don't exist in isolation.

AI is getting exponentially cheaper. The sensors that feed data to AI are getting exponentially cheaper. The compute that runs AI is getting exponentially cheaper. The energy that powers compute is getting exponentially cheaper (solar).

When multiple exponential curves combine, the results compound. The improvement in one domain enables acceleration in others.

This is why the future arrives faster than anyone predicts. It's not just one technology improving—it's a network of reinforcing exponentials, each accelerating the others.

What This Means For You

If exponential-growth bias is a bug in human cognition, here's the patch:

1. Assume the trend continues. Every major technology trend of the past 50 years—computing, solar, batteries, sensors, genomics—has followed predictable cost curves. When in doubt, bet on continuation.

2. Think in doublings, not percentages. "20% annual improvement" sounds mild. But 20% compounds to 6x in 10 years and 38x in 20 years. Train yourself to calculate doublings.

3. Distrust linear projections. When an expert says "X will never be economical," ask: what's the learning curve? How fast is it improving? Experts are just as susceptible to exponential-growth bias as everyone else.

4. Position for the curve. The biggest opportunities are in technologies currently "too expensive" but on strong exponential trajectories. That's where solar investors made fortunes. That's where AI early-adopters are building competitive moats.

The Future Is Better Than You Think

Here's the optimistic conclusion that your brain doesn't want you to believe:

The future is going to be wildly, dramatically, almost incomprehensibly better than most people expect.

Not because of magic. Not because of wishful thinking. Because of math.

The curves are clear. The trajectories are measurable. The compounding is relentless.

The only question is whether you'll see it coming—or whether you'll be one of the 96% still drawing straight lines while the exponential curves reshape the world around you.