Beyond the Numbers: Why Math Is More Than Just Computation

Understanding the Hidden Pitfalls of Linear Thinking and Statistical Misinterpretation

*This is Part 1 in a series on How Not To Be Wrong by Jordan Ellenberg*

The word "Math" alone is enough to make most people uneasy, primarily because of their school memories—complicated integrals, Pythagoras’ theorem, and cosine functions that seemed irrelevant to real life. Yet, math is all around us, in nearly everything we do. Understanding math goes beyond just solving equations; it is a way of thinking and problem-solving—an extension of our common sense. It raises the fundamental question: what assumptions are we making, and are they justified?

Common sense in math includes fundamental properties like a + b = b + a or a × b = b × a. Explaining why these are true can be difficult, but we all intuitively understand them—even children. We accept them because they simply make sense. This is what math is: the study of truths that emerge a certain way because they logically must.

"Without the rigorous structure that math provides, common sense can lead you astray."

Consider the debate on raising taxes. To determine whether increasing taxes is a good idea, we need to establish the relationship between tax percentage and government revenue. A simplistic and common assumption is that more taxes equal more revenue. While this holds true to a point, it eventually fails. If taxes were 100%, no one would work—why labor if the government takes every cent? Revenue would drop to zero. Clearly, the optimal tax percentage lies somewhere in the middle—high enough to generate revenue but not so high as to discourage work.

This example highlights a flaw in linear thinking, where we assume that increasing one factor will indefinitely increase another. In mathematical terms, this represents a linear relationship—a straight line—but real-world relationships are often more complex. Many situations have an optimal point, beyond which more of something is actually detrimental.

We tend to default to linear thinking because it simplifies problems, forming the foundation of calculus. One of Newton’s laws of motion states that an object in motion remains in motion unless acted upon by an external force. This principle has been adapted into advanced mathematics, where we assume linearity at microscopic levels and then extrapolate to infinity.

However, linear thinking can mislead us. Linear regression, for example, involves drawing a best-fit line through data points and using it to predict future values. This method is widely used, but as the saying goes, when you have a hammer, everything looks like a nail. Jordan Ellenberg illustrates this issue with a study predicting that by 2048, 100% of Americans would be obese. The researchers extended a trend line based on past data, projecting forward until it reached 100%. But this assumption—that trends continue linearly—is flawed. Worse yet, the study also predicted that Black men wouldn’t reach 100% obesity until 2055. How could that be possible if the entire population was already obese by 2048? The contradiction exposes the danger of blindly trusting linear regression.

This issue reflects a broader problem in math education today. Schools emphasize finding precise answers, even when those answers don’t make sense. If data produces a mathematically sound answer, we often accept it without questioning whether it aligns with reality. But precision without logic is meaningless.

Another example of misleading statistical reasoning involves the law of large numbers. This law states that as a sample size increases, the observed data will approach the true average. Consider coin flipping: if ten people each flip ten coins, the results will vary significantly—some may even flip nine heads. We wouldn't conclude that they are exceptional coin flippers; the sample size is simply too small. With more flips, the proportion of heads approaches 50%. But statistics often mislead us when applied carelessly.

Ellenberg gives an example involving brain cancer rates across U.S. states. North Dakota ranked at the top, while South Dakota ranked at the bottom. The data might suggest that moving from one state to the other reduces cancer risk, but that conclusion is absurd. The real explanation? These states have small populations, making their cancer statistics highly variable. A few cases drastically skew the data. Smaller sample sizes produce greater fluctuations, making any conclusion drawn from them unreliable.

Such statistical misunderstandings also plague gambling. Imagine playing roulette and seeing six consecutive black spins. Many players believe this makes red more likely. But the probability of red or black remains 50/50, regardless of past spins. The law of large numbers ensures that results even out over time, but the roulette wheel has no memory—it does not “know” that the last spins were black.

This is the beauty of math: it illuminates the world of numbers, but we must engage critically, or we risk deceiving ourselves. Traditional school math often presents neat word problems where numbers and formulas are readily available. However, reality is never this convenient. To derive meaningful answers, we need more than numbers—we need logical and critical thinking. Only then should we reach for the calculator.

"It's only after you've started to formulate these questions that you take out the calculator. But at that point the real mental work is already finished. Dividing one number by another is mere computation; figuring out what you should divide by what is mathematics."