Idempotence in Life: What Changes When We Repeat?

Many people misunderstand mathematics at a fundamental level. They think mathematics is simply the manipulation of formulas. In fact, mathematics reflects the structure of the world as seen through human reason. Formulas are only abstract expressions of that structure.

Idempotence is a particularly interesting mathematical concept.

A function is idempotent if it satisfies

f(f(x)) = f(x)

In plain language, performing the same operation once or many times leads to the same final result.

The concept becomes equally interesting when applied to life. Some people appear to have lived through a great deal, yet in practice they have been repeating the same experience. More time has passed, but their own state has changed very little.

There is an old joke about professional experience. Someone says that they have twenty years of experience. The reply is that they may not have twenty years of experience at all; they may have repeated one year of experience twenty times.

The joke distinguishes time from experience. More time does not automatically become more experience. If someone keeps encountering the same problems, responding with the same methods, and allowing none of those encounters to alter their understanding or ability, then twenty years may indeed amount to little more than repetition.

Other people keep encountering new problems. Each experience changes their previous state, and that changed state becomes the starting point for the next action. This is closer to mathematical iteration:

x_{n+1} = f(x_n)

Today begins from yesterday, and tomorrow begins from a today that has already changed. Time is no longer mere repetition. It becomes a continuous evolution of state.

Of course, being non-idempotent does not mean making progress. Change may be for the better or for the worse. It may even consist of oscillating among a few recurring states. What matters is not change by itself, but whether an experience genuinely enters our internal structure and becomes part of the basis for whatever happens next.

Nor is repetition necessarily without value. Piano practice involves a great deal of repetition. A pianist may play the same passage many times, but if each repetition adjusts the ear, the touch, and the understanding of the music’s structure, the outward action is being repeated while the internal state continues to change.

The distinction that matters, then, is not between repetition and challenge. It is whether anything changes through repetition.

Some people repeat actions. Others accumulate change.

Mathematics is more than calculation and the manipulation of formulas. It reflects the structure of the world from the perspective of human reason. An abstract mathematical concept can therefore help us understand parts of life that are otherwise difficult to put into words.

What ultimately matters is not how many years we have been through, but whether, after those experiences, we are still the person we were before.