Why were early mathematicians critical of infinitesimals?

Early mathematicians were critical of infinitesimals primarily because the concept lacked a rigorous logical foundation. When Isaac Newton and Gottfried Wilhelm Leibniz developed calculus, they relied on heuristic methods that treated infinitesimals as quantities smaller than any positive real number yet not equal to zero.123

This "nebulous" idea faced significant scrutiny, most notably from the philosopher George Berkeley. In his work The Analyst, Berkeley famously mocked these values as "the ghosts of departed quantities," arguing that they were neither finite, nor zero, nor truly infinitesimal. He challenged mathematicians to justify their reliance on such inconceivable concepts, which lacked a coherent rational explanation at the time.2

This unease eventually drove the mathematical community to develop the formal concept of a limit, which allowed for the rigorous definition of derivatives and integrals without requiring the existence of infinitesimals. It was not until the 20th century that Abraham Robinson demonstrated that infinitesimals could be made logically consistent through nonstandard analysis.132