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.

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