In his book How Not to Be Wrong, Jordan Ellenberg identifies a cognitive bias that he calls “lineocentrism”: the reflexive belief that every trend in the universe is a straight line.

We see a slope, we calculate the rate of change, and we assume that $y = mx + b$ will govern the system until the end of time.

Ellenberg’s favorite example of this “linear regression to the point of madness” comes from a 2008 study in the journal Obesity. Researchers plotted the growth of the overweight population in the U.S. from the 1970s through the early 2000s.

By drawing a perfectly straight line through the data and projecting it forward, they concluded that by the year 2048, 100% of Americans would be overweight.^[1]

If you extend that same line just a few more years, the math suggests that by 2060, more than 100% of the population will be overweight.

It is a mathematically sound extrapolation of a linear trend, but it is a biological and logical impossibility. Systems have ceilings, and reality is rarely a straight line.

The Math Behind the Trap

The reason our brains prefer lines is rooted in Local Linearity. Calculus teaches us that if you zoom in far enough on any smooth curve $f(x)$, it begins to look like a straight line. This is the definition of the derivative $f'(x)$, the slope of the tangent line at a specific, infinitesimal point.

$$f(x + \Delta x) \approx f(x) + f'(x) \Delta x$$

This approximation is a powerful tool for short-term engineering, but it becomes a trap when we mistake the local slope for a global law. In the obesity study, the researchers saw a slice of a curve that looked linear and assumed it would never bend.

In reality, most natural systems eventually hit a “carrying capacity” ($L$), shifting the growth from linear to a Logistic Curve:

$$P(t) = \frac{L}{1 + e^{-k(t-t_0)}}$$

The Laffer Trap

Ellenberg connects this to one of the most famous non-linear concepts in economics: the Laffer Curve. In the early 1980s, the “linear” argument for Reaganomics was simple: if you lower taxes, people have more incentive to work, which generates more revenue.

This works perfectly if you are at a 90% tax rate. But if you are already at a 15% rate, lowering taxes further will eventually lead to a total collapse in revenue. The relationship between “tax rate” and “revenue” isn’t a line; it’s a curve that peaks and then falls.

As Ellenberg notes, “Which way you should go depends on where you already are.”^[2] Doing more of a “good” thing eventually leads to diminishing, and then negative, returns.

The Newton Fallacy

Even the pioneers of calculus haven’t been immune to the allure of the straight line. Sir Isaac Newton famously lost a fortune in the South Sea Bubble of 1720. He entered the market early, saw the linear-looking “moon mission” of the stock’s price, and assumed the momentum was a constant of the universe.

After cashing out early for a profit, he watched the “line” keep going up, panicked, and jumped back in at the absolute peak, just before the crash. He later remarked, “I can calculate the movement of stars, but not the madness of men.”^[3] Newton could calculate the orbit of a planet because gravity is a consistent force, but he failed to realize that market trends are curves driven by finite resources and human psychology, both of which have breaking points.

The Lesson of the Curve

For the modern engineer or the student of innovation, recognizing the “straight line” fallacy is a requirement for high-fidelity thinking. We see this in the hardware world with Moore’s Law. For decades, we could assume a linear-logarithmic increase in transistor density, but we are now hitting the physical “bend” in the curve where heat dissipation and quantum tunneling prevent further linear progress.^[4]

In our own lives and projects, we must learn to distinguish between momentum and trajectory. Doubling your output for three months is a local trend; assuming that output will double every quarter for five years is a failure to account for system saturation.

The most important question we can ask is not “what is the slope?” but “where is the ceiling?” If we don’t account for the moment the curve starts to bend, we aren’t performing an analysis, we’re just drawing lines on a map that doesn’t exist.

References

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1. Wang, Y., et al. (2008). Will All Americans Become Overweight or Obese? Obesity. The study that serves as the centerpiece for Ellenberg’s critique of linear extrapolation. ↩︎

2. Ellenberg, J. (2014). How Not to Be Wrong: The Power of Mathematical Thinking. Penguin Press. See Chapter 3: “Everyone is Obese.” ↩︎

3. Odlyzko, A. (2001). Newton’s Financial Misadventures in the South Sea Bubble. Notes and Records of the Royal Society. ↩︎

4. Waldrop, M. M. (2016). The Chip that Finally Explains Why Moore’s Law is Ending. Nature. ↩︎