Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus (Judith V. Grabiner)

Introduction

• 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.

Historical Context

• 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.

Cauchy's Contributions

• 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.

Development of Rigorous Calculus

• 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.

Shift Towards Rigor

• 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.

Key Mathematical Concepts

• 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.

Legacy of Cauchy

• 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.

Conclusion

• 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.