The Crisis of Continuity
Why the limit concept exists: the paradoxes it resolves and the intellectual journey that brought calculus from intuition to rigor.
The Necessity of the Limit
The study of calculus represents the most significant intellectual leap in the history of mathematics, bridging the chasm between the static world of algebra and geometry and the dynamic, changing reality of the physical universe.
At the heart of this discipline lies the concept of the limit—a sophisticated mathematical tool designed to rigorously handle the concept of "infinite closeness."
Zeno's Paradoxes (5th Century BCE)
The Greek philosopher Zeno of Elea proposed a series of paradoxes designed to prove that motion is an illusion.
The Dichotomy Paradox
"To travel any distance, one must first travel half that distance, then half the remaining distance, and so on ad infinitum."
Since this requires completing an infinite number of tasks in a finite amount of time, Zeno concluded that motion is logically impossible.
For two millennia, mathematics lacked the vocabulary to refute him effectively.
The Invention of Calculus (17th Century)
The crisis deepened with the independent invention of calculus by Isaac Newton and Gottfried Wilhelm Leibniz.
Newton treated changing quantities as "flowing" and their rates of change as "fluxions."
Leibniz used infinitesimally small increments and .
Both relied on the concept of "infinitesimals"—quantities that were infinitely small, yet non-zero. While their methods produced correct results for planetary motion and tangent lines, the logical foundation was shaky.
Berkeley's Critique (1734)
The Analyst
Bishop George Berkeley published a blistering critique of this new mathematics, famously mocking infinitesimals as the "ghosts of departed quantities."
Berkeley argued that mathematicians were guilty of flawed logic by treating these quantities as non-zero during calculation but as zero at the conclusion.
The Resolution (19th Century)
The work of Augustin-Louis Cauchy and Karl Weierstrass replaced the vague notion of "infinitesimal" with the precise arithmetic of the epsilon-delta definition.
This transition from the intuitive to the rigorous is not just a historical footnote; it is the pedagogical arc that students must traverse.
This unit is designed to guide you from the intuitive "approaching" of values to the rigorous "arbitrary closeness" that underpins modern analysis.