Why Is Calculus Still Hard to Learn Even Though It Was Established Centuries Ago?


Because you’re passively receiving this knowledge. You might not be interested in finding the area of an ellipse or calculating the work done by a variable force. You care more about practical, tangible applications. You lack the motivation.

But when Newton and Leibniz created calculus, they genuinely wanted to solve the unsolved problems of their era. They had powerful motivation. In production practice, situations requiring the handling of irregular shapes or other applications of integral principles were everywhere, yet mathematics was still stuck in the “age of constants,” unable to address these basic calculus problems — it was practically a “disgrace” for the mathematics of that time. At the same time, the concept of limits had already matured. Even without Newton, Leibniz, Euler, and others, someone else would have built a methodology to tell us how to find the area under the curve of e^x, how to derive the brachistochrone equation, and how to formulate the equations of fluid mechanics to study the properties of fluids. This was an inevitable path in the development of human civilization.

Our own era is filled with great scientific questions as well. Various conjectures in pure mathematics, the exploration of dark matter and dark energy in cosmology, the attempt to unify relativity with quantum field theory; industrially, turbulence in fluids still lacks a satisfactory mathematical description, and the interpretability of deep learning remains far from resolved. And so on. Yet most people couldn’t care less about these things. They take today’s level of technology for granted — just as people before the Industrial Revolution took the pastoral idyllic era for granted. They don’t look up at the stars above; they have no longing or imagination for the future. In every age, it is only a handful of scientific giants who, driven by their own curiosity, keep breaking through the ceiling of scientific understanding of their time. Ordinary people today certainly possess far more knowledge than ordinary people in Newton’s day, but it is hard to argue that their mindset has fundamentally changed. When college students today study advanced mathematics, they only feel “it’s too hard” and “I can’t learn this”; they cannot appreciate the intense desire of the scientific pioneers to explore and conquer the unknown — desires that drove them to spend years, decades, or even a lifetime building and perfecting a single theory. The gap between ordinary people and great minds is not simply a matter of IQ. It is a gap in the level of thinking, and above all, in motivation.