This text for upper-level undergraduates and graduate students examines the events that led to a 19th-century intellectual revolution: the reinterpretation of the calculus undertaken by Augustin-Louis Cauchy and his peers. These intellectuals transformed the uses of calculus from problem-solving methods into a collection of well-defined theorems about limits, continuity, series, derivatives, and integrals. Beginning with a survey of the characteristic 19th-century view of analysis, the book proceeds to an examination of the 18th-century concept of calculus and focuses on the innovative methods of Cauchy and his contemporaries in refining existing methods into the basis of rigorous calculus. 1981 edition.
Cauchy and the Nineteenth-Century Revolution in Calculus
The Nineteenth-Century Ideal
When a nineteenth-century mathematician spoke about rigor in analysis, or in any other subject, he had several general things in mind. First, every concept of the subject had to be explicitly defined in terms of concepts whose nature was held to be already known. (This criterion would be modified by a twentieth-century mathematician to allow undefined termsâthat is, terms defined solely by the postulates they satisfy; this change is related to the late nineteenth-century tendency toward abstraction in mathematics, an important development but beyond the scope of this book.) Following Wei erstrass, in analysis this meant that every statement about equality between limits was translatable, by well-defined rules, into an algebraic statement involving inequalities. Second, theorems had to be proved, with every step in the proof justified by a previously proved theorem, by a definition, or by an explicitly stated axiom.1 This meant in particular that the derivation of a result by manipulating symbols was not a proof of the result; nor did drawing a diagram prove statements about continuous curves. Third, the definitions chosen, and the theorems proved, had to be sufficiently broad to support the entire structure of valid results belonging to the subject. The calculus was a well-developed subject, with a known body of results. To make the calculus rigorous, then, all previous valid results would have to be derived from the rigorous foundation.
Many nineteenth-century mathematicians believed themselves superior to their eighteenth-century counterparts because they would no longer accept intuition as part of a mathematical proof or allow the power of notation to substitute for the rigor of a proof.2 To be sure, even nineteenth-century mathematicians often pursued fruitful methods without the maximal possible rigor, especially in developing new subjects, and individual mathematicians differed in the importance they gave to foundations. Cauchy himself was not consistently rigorous in his research papers. Nevertheless, criteria like those listed above were constantly in Cauchyâs mind when he developed his Cours dâanalyse. When Cauchy referred in that work to the rigor of geometry as the ideal to which he aspired, he had in mind, not diagrams, but logical structure: the way the works of Euclid and Archimedes were constructed.3
Cauchy explicitly distinguished between heuristics and justification. He separated the task of discovering results by means of âthe generalness of algebraââthat is, discovering results by extrapolating from finite symbolic expressions to infinite ones, or from real to complex onesâfrom the quite different task of proving theorems. He described his own methodological ideal in these words:
As for methods, I have sought to give them all the rigor which exists in geometry, so as never to refer to reasons drawn from the generalness of algebra. Reasons of this type, though often enough admitted, especially in passing from convergent series to divergent series, and from real quantities to imaginary expressions, can be considered only ⊠as inductions, sometimes appropriate to suggest truth, but as having little accord with the much-praised exactness of the mathematical sciences.⊠Most [algebraic] formulas hold true only under certain conditions, and for certain values of the quantities they contain. By determining these conditions and these values, and by fixing precisely the sense of all the notations I use, I make all uncertainty disappear.4
These are high standards. Let us turn to Cauchyâs work and see how he met them.
But the important differewidth="" height"" width="" height"" nce between Cauchyâs definition and those of his predecessors is that when Cauchy used his definition of limit in a proof, he often translated it into the language of inequalities. Sometimes, instead of so translating it, he left the job for the reader. But Cauchy knew exactly what the relevant inequalities were, and this was a significant new achievement. For example, he interpreted the statement âthe limit, as x goes to infinity, of f (x + 1) â f(x) is some finite number kâ as follows: âDesignate by
a number as small as desired. Since the increasing values of x will make the difference f(x + 1) â f(x) converge to the limit k, we can give to h a value sufficiently large so that, x being equal to or greater than h, the dilference in question is included between k â
and k +
.â11 This is hard to improve on. (We will find a delta to go with the epsilon when we describe Cauchyâs theory of the derivative in chapter 5.)
More important, dâAlembertâs and de la Chapelleâs two restrictions on the variableâs approach to the limit, italicized in my citation of their definition, are too strong for mathematical usefulness. If a magnitude never surpasses its limit, then a variable cannot oscillate around the limit. How then could we use this definition to define the limit of the partial sums of the series 1 â 1/2 + 1/3 â 1/4 +
, or to evaluate the limit, as x goes to zero, of x2 sin 1/x? And if a magnitude can never equal its limit, how can the derivative of the linear function f(x) = ax + b be defined as the limit of the quotient of differences? Abandoning these restrictions was necessary to make the definition of limit sufficiently broad to support the definitions of the other basic concepts of the calculus.
Cauchyâs definition of limit, of course, has a history. But my main point here has been to exhibit the contrast between the usual understanding of the limit concept in the eighteenth century and that brought about by Cauchy. This contrast exemplifies both the nature and quality of Cauchyâs innovations. It will be seen later how Cauchy applied his concept of limit to establish a rigorous theory of convergence of series, to define continuity and prove the intermediate-value theorem for continuous functions, and to develop delta-epsilon proofs about derivatives and integralsâin short, to provide an algebraic foundation for the calculus. (For a sample of Cauchyâs work, the reader may consult the selected texts translated in the appendix.)
Cauchy and Bolzano
Cauchyâs achievements, though outstanding, were not unique. His contemporary Bernhard Bolzano made many similar discoveries. Though Bolzanoâs impact on the mathematics of his time appears to have been negligible,14 his work was nonetheless excellent. Regardless of his relative lack of influence, it will be worthwhile to be aware of some of his achievements. For if a key to understanding Cauchy is an understanding of the influence his predecessors had on him, the importance of these predecessors will be even clearer if they can be shown to have influenced Bolzano in the same wa...
Table des matiĂšres
Cover
Title Page
Copyright Page
Dedication
Contents
Preface
Abbreviations of Titles
Introduction
1Â Â Cauchy and the Nineteenth-Century Revolution in Calculus
2Â Â The Status of Foundations in Eighteenth-Century Calculus
3Â Â The Algebraic Background of Cauchyâs New Analysis
4Â Â The Origins of the Basic Concepts of Cauchyâs Analysis: Limit, Continuity, Convergence
5Â Â The Origins of Cauchyâs Theory of the Derivative
6Â Â The Origins of Cauchyâs Theory of the Definite Integral
Conclusion
Appendix: Translations from Cauchyâs Oeuvres
Notes
References
Index
Normes de citation pour The Origins of Cauchy's Rigorous Calculus
APA 6 Citation
Grabiner, J. (2012). The Origins of Cauchyâs Rigorous Calculus ([edition unavailable]). Dover Publications. Retrieved from https://www.perlego.com/book/111552/the-origins-of-cauchys-rigorous-calculus-pdf (Original work published 2012)
Chicago Citation
Grabiner, Judith. (2012) 2012. The Origins of Cauchyâs Rigorous Calculus. [Edition unavailable]. Dover Publications. https://www.perlego.com/book/111552/the-origins-of-cauchys-rigorous-calculus-pdf.
Harvard Citation
Grabiner, J. (2012) The Origins of Cauchyâs Rigorous Calculus. [edition unavailable]. Dover Publications. Available at: https://www.perlego.com/book/111552/the-origins-of-cauchys-rigorous-calculus-pdf (Accessed: 14 October 2022).
MLA 7 Citation
Grabiner, Judith. The Origins of Cauchyâs Rigorous Calculus. [edition unavailable]. Dover Publications, 2012. Web. 14 Oct. 2022.