Judith V. Grabiner discusses the contributions of Cauchy to the establishment of rigorous calculus.
Examines the shift from intuitive to rigorous mathematical foundations in calculus.
Calculus developed by Newton and Leibniz in the late 17th century.
Initial use of verbal definitions; rigorous delta-epsilon proofs took over 150 years to develop.
Cauchy introduced these proofs, but his verbal definitions differed from modern formulations.
Established the limit based on inequalities:
Limits defined as values that functions approach without necessarily reaching.
Introduced delta-epsilon definitions for derivatives.
Emphasis on formal definition and proved theorems around limits, convergence, continuity, derivatives, and integrals.
Examination of calculus practices from the 17th and 18th centuries:
Powerful techniques existed but lacked rigorous foundations.
Importance of the algebra of inequalities arose to create approximate solutions.
Mathematicians like Euler, d'Alembert, Lagrange contributed but did not formalize foundations.
Philosophical and practical reasons for the new interest in rigor:
Berkeley's critiques highlighted the shortcomings of non-rigorous calculus.
Limitations of 18th-century methods led to a recognized need for rigor.
Change in mathematicians' roles towards teaching necessitated clear foundations in mathematics.
Cauchy's approach to limits, specifically alternating series, marked a shift to precise definitions.
Developed the theory of convergence using limits to establish summation of series rigorously.
Defined continuity mathematically, offering systematic proofs for theorems like the Intermediate Value Theorem.
Cauchy’s work laid a foundation for future mathematicians like Riemann and Weierstrass.
Transitioned calculus from intuitive practices to a structured system of rigorous proofs.
The symbol epsilon became synonymous with precision in calculus, stemming from Cauchy’s methods.
Cauchy's definition of calculus shifted the paradigm towards a more rigorous analytical framework.
Recognizes the impact of historical methods while emphasizing the evolution of calculus into its modern form.