Linear thinking breaks down the moment reality curves

Ellenberg opens with a caution against what he calls the 'linear model' of the world: assuming that if a little of something is good, more must always be better, in a straight, proportional line. He illustrates this with examples like tax policy debates, where people extrapolate that if cutting taxes a bit boosts revenue, cutting them further must boost revenue even more, ignoring that the relationship is actually a curve that rises and then falls. The same fallacy shows up in exercise, medication dosing, and studying for exams, where the benefit of an additional unit of effort eventually shrinks or reverses. He argues that most real-world relationships are nonlinear, and that treating them as straight lines is one of the most common and consequential reasoning errors people make. Recognizing curvature, rather than assuming constant proportional returns, is a basic mathematical habit that prevents policy and personal decisions from going badly wrong at the extremes.

Takeaway: before assuming more is always better, ask where the curve might bend back on itself.