Why did George Berkeley criticize the early foundations of calculus?
This suggests that the same constant (π) appears in the formula for the circumference, 2πr, and in the formula for the area, πr2. As the number of pieces increases (from left to right), the “rectangle” converges on a πr by r rectangle with area πr2—the same area as that of the circle. This method of approximating a (complex) region by dividing it into simpler regions dates from antiquity and reappears in the calculus.
George Berkeley criticized the early foundations of calculus because he believed the methods developed by Sir Isaac Newton and Gottfried Wilhelm Leibniz lacked logical justification. In his 1734 work, The Analyst; or, A Discourse Addressed to an Infidel Mathematician, Berkeley challenged the use of "fluxions" and "infinitesimals," which were treated as quantities that were neither finite nor zero.123
Berkeley famously questioned the nature of these values, famously referring to them as the "ghosts of departed quantities". He argued that mathematicians were inconsistent for demanding rigorous proof in religious matters while accepting "inconceivable" and contradictory concepts in their own science.13
This criticism highlighted the lack of a coherent rational explanation for the calculus of the time, where utility had outweighed mathematical rigor. It was not until the 19th century that mathematicians like Augustin-Louis Cauchy and Karl Weierstrass established a rigorous foundation for calculus using the concept of limits, effectively addressing Berkeley's concerns.12