Understanding exponential growth

Or why it hardly ever happens

During the Covid-19 era numerous commentators would mindlessly parrot the definition of exponential growth but hardly any of them explained it and, of those that did, I never heard a proper explanation. In Mathematics it is always important to define your terms so let’s start with a dictionary definition.

Exponential means growing or increasing very rapidly. Collins Dictionary.

This definition was apparently written by someone who was so bad at Mathematics that they ended up in a job where they didn’t earn enough money to have use of a savings account. I say this because compound interest is the only instance of exponential growth that I can think of.

Exponential growth implies infinite resource which is a problem in a finite universe. This means that it breaks realizability, which is known to the layperson as common sense, when growth causes the variable to exceed the available resource on its way to infinity. Reductio ad absurdum. (Note that more money can always be printed.)

Everything should be made as simple as possible, but not simpler. Albert Einstein.

To get around this problem for epidemics some argue that it grows exponentially then it doesn’t. This requires some event to happen to causes the curve to change and that has to happen in an uncontrived manner. Alternatively, appealing to Occam’s razor, it doesn’t grow exponentially in the first place. This is borne out by the equations of the standard SIR model of 1927 and its subsequent variants that were used to model the spread of Covid-19.

All models are wrong, but some are useful. George Box.

Exponential growth is deterministic. i.e. It can be extrapolated. In the natural world all processes are stochastic. i.e. To a greater or lesser extent they exhibit randomness. Even though a statistician can draw a straight line on a log plot it doesn’t mean that the process is growing exponentially and any extrapolation of that curve is going to go very wrong very quickly. For example nuclear chain reactions do stop before the universe explodes.