How Our Brain Mirrors the Taylor Series: From Perception to Understanding

How do we comprehend this complex world? From vague early perception to increasingly clear cognition, the process is a continual updating and refinement.

As children we label TV characters as "good" or "bad," but with age we recognize gray areas and develop a more comprehensive, accurate view.

This shift in understanding reminds us of the Taylor series , which expresses a complex function as an infinite power‑series expansion. Its core idea is to approximate a function by repeatedly differentiating it at a point and constructing a polynomial from those derivatives.

The Taylor formula can be written as:

Here, the function to be expanded is denoted by f(x), the expansion point by a, and the first, second, third, … derivatives at a are f'(a), f''(a), f'''(a), … respectively.

Through Taylor expansion we can approximate a complex function with a polynomial, gradually approaching the original function.

In the cognitive process, our brain works like a Taylor expansion of the world. Sensory input provides the initial data, which we process and analyze to form an understanding, analogous to the successive steps of a Taylor series.

The first step is initial perception . When we encounter something new, our brain makes a rough estimate—e.g., seeing a dog and labeling it as "dog."

The next step is deepening details . We examine the animal’s color, size, behavior, akin to calculating higher‑order derivatives, enriching our mental model.

The final step is pattern recognition . By continuously observing and analyzing, our brain assembles an overall pattern, similar to substituting derivative values into the Taylor formula to build a polynomial that predicts behavior.

The discoverer of the Taylor formula was the English mathematician Brook Taylor (1685‑1731).

Brook Taylor, 1685‑1731

In 1715 he published a paper on Taylor series, introducing this powerful tool that now serves mathematics, physics, engineering, and many other fields.

This also brings to mind Thales, the ancient Greek philosopher and mathematician (c. 624‑546 BC), considered a founder of Western philosophy and science.

Thales is often ranked among the greatest scientific minds for being one of the first to explain natural phenomena without resorting to myth or religion, emphasizing observation and rational thought.

Could our initial understanding be wrong? It is likely; we all make similar mistakes, but through observation, rational analysis, or scientific experimentation we can gradually correct our views and approach the truth.

Thales' way of thinking later became the foundation of the scientific method.