A jar holds one bacterium at 11:00. The population doubles every minute. At noon, the jar is full. How full is the jar at 11:55?
Three percent.
Five minutes from saturation, and the jar looks empty. This is the fact about exponential growth that the human mind cannot absorb. We see the curve, nod along, and keep planning as if next year will look like last year plus a little.
Albert Bartlett, a physicist who gave the same lecture 1,742 times, called this the greatest shortcoming of the species. When a lion closes distance at constant speed, you can estimate when to run. That kind of thinking kept us alive for a long time. Most of what matters now does not move at constant speed.
An exponential is a process where the output becomes the engine of more output. One factory builds two factories. Two build four. The product improves the production line. That is the whole definition.
You watch how fast the jar is filling and project forward at that rate. But the rate tomorrow is faster than the rate today, and it will be faster again the day after that. Show a trained statistician a doubling pattern and ask them to predict the next point. They undershoot. Every time. Even when warned. The flattening is reflex.
The useful unit is not the growth rate. It is the doubling time. A growth rate is a number you file and forget. A doubling time is a countdown. Ten doublings is a thousand times larger. Twenty is a million.
Every exponential you have encountered had a person between each doubling and the next. Moore’s Law doubled transistors, but people designed each new chip. The internet doubled connections, but people built the services. The doubling happened to the tool. The person still did the work.
AI is different because each generation of the system increasingly helps build the one after it. As more of the research and engineering flows through the tool itself, the person does less of the work between each doubling and the next. When the tool fully improves the toolmaker, the curve stops answering to human-scale time.
This changes what is worth working on. When the speed of change shifts by orders of magnitude, the hard problems are not harder versions of the old problems. They are different problems entirely.
If you need to get from Chicago to Portland on foot, the hard problems are provisions, route, terrain, weather, and time. You plan for weeks. You pack for every condition. If you are taking a jet, none of that matters. The hard problem is what you do when you land.
Most planning still assumes walking speed. People prepare for a pace of change that is about to become irrelevant. The problems that matter at one speed evaporate at the next. The bottleneck migrates, and the people still solving yesterday’s bottleneck do not realize it has moved.
Exponentials are polite until they are brutal. The jar at 11:55 looks empty. At 11:59 it is half full. At noon there is no room left.
The jar is at three percent. You have five minutes.