The Origins of Cauchy's Rigorous Calculus

Judith V. Grabiner

• Editorial: Dover Publications
• Año: 2012
• ISBN:
  9780486143743
• Categoría: Mathematics
• Categoría: History & Philosophy of Mathematics

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.

1	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 terms—that 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 Cauchy’s mind when he developed his Cours d’analyse. 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 “the generalness of algebra”—that is, discovering results by extrapolating from finite symbolic expressions to infinite ones, or from real to complex ones—from 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 Cauchy’s work and see how he met them.

A First Look at Cauchy’s Calculus

It is a commonplace among mathematicians that Cauchy gave the first rigorous definitions of limit, convergence, continuity, and derivative, and that he used these definitions to give the first essentially rigorous treatment of the calculus and the first systematic treatment of convergence tests for infinite series. Many people have heard also that Cauchy’s rigorous proofs introduced delta-epsilon methods into analysis.⁵ When the mathematician opens Cauchy’s major works, he expects these beliefs to be confirmed. But on first looking into the Cours d’analyse (1821), he may be somewhat shocked to find no deltas or epsilons anywhere near the definition of limit; moreover, the words in the definition sound more like appeals to intuition than to the algebra of inequalities. Then, when he opens the Calcul infinitésimal (1823) to find the definition of a derivative, he may be even more surprised to find that Cauchy defined the derivative as the ratio of the quotient of differences when the differences are infinitesimal. (He may be surprised also at the word infinitésimal in the title.) Returning to the Corns d’analyse, the mathematician is apt to be disappointed with the treatment of convergence of series, for although Cauchy used the Cauchy criterion, he did not even try to prove that it is a sufficient condition for convergence. The mathematician may well conclude that Cauchy’s rigor has been highly overrated.⁶

But the discrepancy between what the mathematician expects and what Cauchy actually did is more apparent than real. In fact Cauchy’s definitions and procedures are rigorous not only in the sense of “better than what came before” but in terms of nearly all that the mathematician expects. One major difficulty the modern reader finds in appreciating Cauchy comes from his old-fashioned terminology, the use of which—as will be seen—was deliberate. Another source of difficulty is the fact that the two books of 1821–1823 were originally lectures given to students who planned to apply the calculus.⁷ Finally, the modern reader is likely to be unfamiliar with the more discursive style of mathematical exposition used in the early nineteeth century. Once the Cours d’analyse or the Calcul infinitésimal has been examined more closely, it will be seen that Cauchy’s achievement is as impressive as expected.⁸

To illustrate what we have just said, let us look at the central concept in Cauchy’s analysis, that of limit, on which his definitions of continuity, convergence, derivative, and integral all rest: “When the successively attributed values of one variable approach indefinitely a fixed value, finishing by differing from that fixed value by as little as desired, that fixed value is called the limit of all the others.”⁹ This definition seems at first to resemble the imprecise eighteenth-century definitions of limit more than it does the modern delta-epsilon definition. For instance, a classic eighteenth-century formulation is that given by d’Alembert and de la Chapelle in the Encyclopédie: One magnitude is said to be the limit of another magnitude, when the second can approach nearer to the first than a given magnitude, as small as that [given] magnitude may be supposed; nevertheless, without the magnitude which is approaching ever being able to surpass the magnitude which it approaches, so that the difference between a quantity and its limit is absolutely inassignable.… Properly speaking, the limit never coincides, or never becomes equal, to the quantity of which it is a limit, but the latter can always approach closer and closer, and can differ from it by as little as desired.¹⁰ [Italics mine]

But the important differewidth="" height"" width="" height"" nce between Cauchy’s 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 “the limit, as x goes to infinity, of f (x + 1) − f(x) is some finite number k” as follows: “Designate by

a number as small as desired. Since the increasing values of x will make the difference f(x + 1) − 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 −

and k +

.”¹¹ This is hard to improve on. (We will find a delta to go with the epsilon when we describe Cauchy’s theory of the derivative in chapter 5.)

Moreover, Cauchy’s defining inequalities about limits were not ends in themselves; their purpose was to support a logical structure of results about the concepts of the calculus. For instance, he used the inequality we have just mentioned in a proof of the theorem that if lim_x→∞ f(x + 1) − f(x) = k, then lim_x→∞ f(x)/x = k also,¹² and then used an analogue of this theorem as the basis for a proof of the root test for convergence of series.¹³ By contrast, definitions like the one in the Encyclopédie were not translated into inequalities and, more important, were almost never used to prove anything of substance.

In addition, the Encyclopédie definition has certain conceptual limitations which Cauchy’s definition eliminated. For instance, the eighteenth-century term magnitude is less precise than Cauchy’s term variable; rather than have one magnitude approach another, Cauchy clearly distinguished between the variable and the fixed value which is the limit of the variable. Moreover, the sense of the word approach is unclear in the Encyclopédie definition. Though it could have been understood in terms of inequalities, d’Alembert and de la Chapelle, unlike Cauchy, did not explicitly make the translation. Probably instead they were appealing to the idea of motion, as Newton had done in explaining his calculus.

More important, d’Alembert’s and de la Chapelle’s two restrictions on the variable’s 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 − 1/2 + 1/3 − 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.

Cauchy’s 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 Cauchy’s 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 integrals—in short, to provide an algebraic foundation for the calculus. (For a sample of Cauchy’s work, the reader may consult the selected texts translated in the appendix.)

Cauchy and Bolzano

Cauchy’s achievements, though outstanding, were not unique. His contemporary Bernhard Bolzano made many similar discoveries. Though Bolzano’s 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...

Índice

• Cover
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Abbreviations of Titles
• Introduction
• 1  Cauchy and the Nineteenth-Century Revolution in Calculus
• 2  The Status of Foundations in Eighteenth-Century Calculus
• 3  The Algebraic Background of Cauchy’s New Analysis
• 4  The Origins of the Basic Concepts of Cauchy’s Analysis: Limit, Continuity, Convergence
• 5  The Origins of Cauchy’s Theory of the Derivative
• 6  The Origins of Cauchy’s Theory of the Definite Integral
• Conclusion
• Appendix: Translations from Cauchy’s Oeuvres
• Notes
• References
• Index