The Origins of Cauchy's Rigorous Calculus (Dover Books on Mathematics) by Grabiner, Judith V., Mathematics (2011) Paperback

by Judith V. Grabiner

This text for upper-level undergraduates and graduate students examines the events that led to a 19th-century intellectual 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. 1981 edition.

Genres: Mathematics, History

Paperback

First published April 23, 1981

Reviews

[G. Branden · Rating 3 out of 5]

It's good, but real analysis simply has to be fresh in your mind for you to get the most from it.

Not accessible at all to anyone who hasn't studied calculus in any way.

Those who have will want to review delta-epsilon proofs, continuity vs. uniform continuity, convergence vs. uniform convergence, and the several tests for convergence of a series (comparison, ratio, root, and so forth).

[William Bies · Rating 4 out of 5]

Once in a great while one encounters a scholarly monograph that, contrary to what is the rule, proves to be a sheer pleasure to read. The present opusculum by the old-school historian of mathematics Judith V. Grabiner turns out to be one such, owing to her lucid style of exposition (no digressions into cultural or social matters here!). The occasion for Grabiner’s arresting display of erudition is to study, much more circumstantially than is normally the case, the revolution in mathematical rigor brought about in the nineteenth century by that formidable giant, Augustin-Louis Cauchy (1789-1857).

Indeed, the review to follow will border on a paean to Cauchy. Why ought the student of mathematics recognize in Cauchy a veritable intellectual hero? Glance for a moment at the torrent of preprints that are released daily, especially in mathematics. It would take perhaps ten lifetimes to assemble the expertise to read intelligently just the scores of mathematical papers that appear each day. The first impression one gains upon perusing these preprints is how far the frontier of knowledge is ahead of even the most advanced textbooks available, frighteningly so in view of how many years of study it takes just to get up to the level of the textbooks. The second impression, however, is how extremely specialized these papers are; almost never do they announce results of widespread applicability or interest. One wonders why one would devote more than a decade of intensive preparation just to contemplate such minor developments of the formalism.

Cauchy, that enterprising explorer of ideas in the realm of pure mathematics, breaks the mold. For, though consummately adept in the art of his métier, he is no mere technician (all that we have left nowadays). Instead, Cauchy’s lucubrations, in stark contrast to current practice in the field, are always guided by a vision of the edifice he wishes to erect, which is to say, he concerns himself with fleshing out his ideas into a satisfyingly rounded and complete theory, architectonic in conception, rather than merely with ‘getting the answer’ to some arcane technical puzzle or other.

To begin with, let us succinctly characterize the stamp of Cauchy’s outstanding personality as a scientist. He outlines his intellectual program in a series of lectures given at the Institut catholique during 1842-1843 thus:

Is it not, in the end, to conquer the truth that one interrogates algebra, espouses all the resources of analysis, and arranges in a formula in order the grasp the laws that rule the course of the stars, or the vibrations of the last particles of matter? Yes, without doubt, the search for truth must be the unique goal of all of science. The truth is an inestimable treasure, whose acquisition is not followed by any remorse and does not trouble the peace of the soul. The contemplation of these celestial charms, of this divine beauty, suffices to compensate us for the sacrifices that we make to discover it; and heavenly bliss itself is nothing but the plain and entire possession of immortal truth….Is not exactitude an essential, necessary characteristic of all true science? Is not the object of all science the quest for or the exact knowledge of the truth? Is not the exact truth that which distinguishes the true from the false philosophy, that which distinguishes history from romance? Exactitude and order are one of the characteristics of the Almighty, who disposed of all things by order, by weight and by measure. [As quoted by Amir Alexander, Duel at Dawn: Heroes, Martyrs and the Rise of Modern Mathematics (Harvard University Press, 2010), reviewed by us here]

To descend from the plane of high abstraction to concrete particulars, why does our author Grabiner choose to focus on Cauchy? Her theme is the development of the modern standard of rigor in the calculus, and she sees Cauchy’s role to have been pivotal in steering the course of analysis along the path it did, in fact, take:

Though I shall be looking constantly for the antecedents of Cauchy’s work, I shall insist throughout on the creativity and originality of his accomplishments. Like another major innovator, Copernicus, Cauchy owed much to his predecessors. But, also like Copernicus, Cauchy contributed a change in point of view so fundamental that his science was transformed when he was done. [p. 4]

Here, we run up against a seeming paradox. Those who have inspected Cauchy’s works in the original French often find themselves surprised to find there, not quite the epsilon-delta definitions we have come to expect, but somewhat loose language, including even mention of infinitesimals! Grabiner rejoins that, despite how at first glance Cauchy’s textbooks look scarcely like what the modern mathematician expects to find in them, nevertheless upon closer examination it can be substantiated that his conceptual apparatus is essentially ours. In particular, unlike what is the case with his eighteenth-century precedents, Cauchy expressly translates his statements into inequalities and uses them to prove substantial results. Thus, we can discern that he does think in terms of the predicate calculus even though his language might not indicate so on the surface.

Grabiner credits Joseph-Louis Lagrange (1736-1813) with having restored the question as to foundations to a place of primary importance and thus with having inspired Cauchy, though, as it happens, the latter winds up adopting a very different foundation [p. 18]. The following passage illuminates what Cauchy draws from Lagrange:

The study of infinite processes may be said to have come of age in 1748, when Euler published his Introductio in analysin infinitorum, a work which studied infinite series, infinite products and infinite continued fractions. The Introductio sought to give an account of infinite ‘analytic expressions’, just as theories of equations had given an account of finite ones. Among many other achievements in this book, Euler gave infinite-series developments for all the standard functions of the time, such as quotients of polynomials, exponentials, logarithms, sines and cosines. Many contemporary mathematicians believed that Euler had shown that the functions commonly studied in the calculus, even those usually defined geometrically, could be represented by infinite series – apparently with no appeal having been made to the concepts of the differential or the integral. Deriving these series was viewed as part of algebra: the algebra or analysis of the infinite. It was the idea that there was an algebra of infinite power series which led Lagrange to originate and promulgate the program of reducing the calculus to algebra – the program adopted and successfully carried out by Cauchy, Bolzano and Weierstrass. What Euler had managed to do for particular functions in the Introductio convinced Lagrange that any function could be given an infinite power-series expansion. Since Lagrange shared the view that infinite processes were part of algebra, Euler’s work probably sufficed to convince him that the calculus could be reduced to algebra. [p. 51]

Two points about Lagrange’s attempted algebraic foundation to the calculus merit mention: first,

Though Lagrange’s Taylor-series method of reducing the calculus to algebra was not the method ultimately adopted by Cauchy, it led Lagrange directly to several insights which were important to nineteenth-century analysts [p. 53]….Cauchy, unlike Lagrange, came to see that it was the algebra of inequalities, not of equalities, which could provide a basis for the calculus [p. 54];

and second,

In the last third of the eighteenth century, there was a shift in interest among algebraists from merely deriving approximation procedures to deriving precise error estimates and measures of speed of convergence [p. 59]….Even today, error bounds are considered worthy of study in their own right. Nevertheless, eighteenth-century approximation techniques are important because of their byproducts. In particular, their study produced a developed algebra of inequalities. And the study of approximations with error bounds made clear that inequalities could be used not only to compute speed of convergence, but – given Cauchy’s new definition of convergence – to prove the very fact of convergence. [pp. 68-69]….Besides believing that the calculus could be made rigorous if reduced to algebra, Cauchy appreciated how the algebra of inequalities used in approximations could provide a rigorous foundation for real analysis. [p. 75]

But Grabiner’s main task in the present work is to delineate what Cauchy himself does with the material he receives from the eighteenth-century tradition. Let it be observed that,

Unlike most of his predecessors, Cauchy did much more than simply define the derivative; he used his definition to prove theorems about derivatives, and thus created the first rigorous theory of derivatives….Cauchy knew exactly what he meant by ‘the derivative is the limit of the quotient of infinitesimal differences’, and he was really the first person in history to know this. [p. 115]

Nevertheless, it must be concluded that Cauchy owed a good part – though not all – of the mechanics of his applications of the derivative to geometry to Lagrange’s Fonctions analytiques. There is a major difference, however, between their works on applications of the derivative. The difference is not one of technique; it is one of context and, above all, justification. To be sure, Cauchy was not single-mindedly consistent about giving all his arguments a precise form; sometimes he used intuitive descriptions of limits or infinitesimals. But it is clear that he knows how to justify all his applications and could have done so explicitly had he wished. [p. 138]

Another issue arises that this reviewer had not appreciated before: not only does Cauchy impart the modern view of what a derivative is, he is also responsible for re-conceiving the role of the integral as well:

The integral was viewed quite differently in the eighteenth century than it is today. It has no independent definition of its own; instead, integration was defined as the inverse of differentiation. Accordingly, the indefinite integral was considered to be more fundamental than the definite integral [p. 140]….I believe that mathematical fruitfulness was the decisive factor in Cauchy’s desire for a new definition as well as in his particular choice. I think the main reason he chose to define the integral as the limit of sums was his need to make sure that the object he was defining existed….By treating the definite integral as the limit of sums, Cauchy was able to prove the existence of the integral of a continuous function, consider integrals of functions that were not derivatives of known functions, and explain the behavior of integrals along a path. Thus the integral as sum was the answer to many of the perplexing questions raised by the work of Fourier, Gauss, Legendre, Poisson and – in 1814 – Cauchy. [p. 145]

The influence on Cauchy of Euler, Lacroix and Poisson cannot be denied. Nevertheless, by far the hardest tasks, both technical and conceptual, were accomplished by Cauchy himself. Having returned to the Leibnizian view of the integral as a sum, Cauchy needed to make it more precise. It was not enough to say that the definite integral is the limit of sums. He first has to specify the sums precisely and then had to prove that the limit existed [p. 145]…. Euler and Cauchy used the same kinds of approximating sums and realized that finer subdivisions produce better approximations, but their view of the integral differed greatly. Euler believed that the integral had an existence independent of the approximation procedure. Further, he could not prove the accuracy of the approximation for a function that is not piecewise monotonic. In this is seen the characteristic attitude of the eighteenth-century approximator: the thing exists, and our job is to approximate it in the most expeditious way. Not until the work of Cauchy was the general question of the existence of the integral even raised, much less rigorously answered. [p. 149]

Grabiner then sketches Cauchy’s argument in formulaic detail. Let us remark that one will recognize Cauchy’s procedure as the kind of thing that would form a standard exercise in a graduate-level textbook in real analysis today. She tells us precisely what Cauchy gets from Lacroix and how he went beyond it [p. 152]. The same for Poisson [p. 155]. Lastly, how Bernhard Riemann’s (1826-1866) integral extends Cauchy’s [pp. 162-163].

Enough for a synopsis; one really must consult the original to arrive at a just appreciation of the author’s patient labors in reconstructing the steps along the way of this crucial juncture in intellectual history. To close, let us turn to the meaning of it all for us, would-be mathematicians in the twenty-first century. In the above, this reviewer praises Cauchy for the aesthetic merit of his intellectual complexion. It is well to let our author Grabiner weigh in on her view of what he thereby accomplishes:

What is our estimate of Cauchy’s achievement? Cauchy’s work established a new way of looking at the concepts of the calculus. As a result, the subject was transformed from a collection of powerful methods and useful results into a mathematical discipline based on clear definitions and rigorous proofs. [p. 164]

Here we can discern how the purity of Cauchy’s intention bears fruit that goes well beyond what he himself may probably have envisaged as the animating purpose behind his work. As Grabiner puts it:

The implications of this achievement go beyond the calculus. In a very important sense, it may be said that Cauchy brought ancient and modern mathematics together. He cast his rigorous calculus in the deductive mold characteristic of ancient geometry. And unlike his predecessors, he did this successfully; that is, he not only gave his work a Euclidean form but presented definitions that generally are adequate to support the desired results, proofs that are basically valid and methods that were fruitful sources for later mathematical work. Cauchy, then, brought together three elements: the major results of analysis, most of which he could now prove; some fruitful concepts and techniques from algebra (particularly algebraic approximations) and analysis; and the rigor and proof structure of Greek geometry. For a long time Greek geometry had been considered the model for all of mathematics. If the origins of modern mathematics are traced to the Renaissance, then the rigor and structure characteristic of Greek geometry first effectively became part of modern mathematics only with Cauchy’s work. Of course the late-nineteenth-century idea that mathematics is the science of abstract logical systems in general is absent from Cauchy’s work. But Cauchy’s rigorization of the calculus was an indispensable first step in that direction. [p. 164]

Therefore, the following reflection on Cauchy’s role in all this seems meet:

Cauchy’s originality in the foundations of the calculus lies in part in the use he made of what others had done. Yet, as the work of many historians of scientific ideas reminds us, there can be as much innovation in transforming old methods as in developing new ones. The fact that many of the basic techniques of Cauchy’s calculus existed in the eighteenth century should increase, not decrease, our wonder at his achievement. Cauchy was able to see – where nobody else had been able to see – how these ideas could be used to build a new rigorous calculus. We do not insist that an architect make every brick he uses with his own hands; instead, we marvel that the beauty of his creations can come from such commonplace materials. Augustin-Louis Cauchy neither began nor completed the rigorization of analysis. But more than any other mathematician, he was responsible for the first great revolution in mathematical rigor since the time of the ancient Greeks. [pp. 165-166]

Passages such as these exhibit Grabiner’s acumen as a scholar. If she were not grounded in a minute knowledge of the contemporary milieu, her admiration of Cauchy’s achievement would remain at the vague level that probably most practicing mathematicians conceive of it today. For they, too, can perceive the benefits to their own practice stemming from the revolution in rigor they attribute to Cauchy, as its principal instigator, and to Karl Weierstrass (1815-1897), as its executor. Yet Grabiner’s trained historian’s eye enables her to characterize with greater precision just what Cauchy himself teaches us, and why.

So, 4 ½ stars for an exemplary contribution to the history of mathematics. What does Grabiner’s retrospective matter to us today, in prospective? Learn from Cauchy’s example. If one is to rise above the hack work that, otherwise, inevitably prevails and make a lasting contribution, one has to set out with the mind of the artist, as an artist of ideas, so to speak, and to devote oneself to formal criteria and to the development of adequate concepts, such as does Cauchy, who is the first to propose our modern conception of limit and of the derivative. In the last analysis, it is the attitude of childlike wonder retained neotenically into adulthood that makes possible an intellectual revolution of such stature. Those atheists who mistakenly pride themselves on being grown up and thus spurn Jesus’ summons to become a child of God (Matthew 18:3) – one could have in mind here, say, G.H. Hardy – are fitted, perhaps, to implement technical developments in analysis that may indeed be stunningly impressive, taken for what they are, but will ever fall short of the genial originality of a religious man like Cauchy.

[Fernando Pestana da Costa · Rating 5 out of 5]

Augustin-Louis Cauchy was one of the giants of nineteen century's Mathematical Analysis. His importance in shaping the field and definitely steering the subject into the rigorous mathematical discipline we know today, can be gauged by the number of times his name appears connected with mathematical objects and results of present day currency (Cauchy sequence, Cauchy criterion for series, Cauchy root test, Cauchy-Hadamard theorem, the Cauchy-Riemann equations, the Cauchy integral formulas,...) this not to speak of the very notion of limit and continuity, whose rigorous definition is very much Cauchy's work, or the first rigorous definition of integral (now disused, but nevertheless of historical interest.) However great Cauchy was, he did not work alone or ab initio. He was one, admitedly a very important one, of a plethora of great mathematicians that helped build one of the most impressive of humanity's intellectual achievements: the rigorous foundations of Mathematical Analysis. The story, of course, does not end with Cauchy, but this excellent and enticing book actually centers its action on the work previous to Cauchy's as well as on Cauchy's own achievements: in it, the importance of Euler, D'Alembert, Ampère, Poisson, Lagrange (of course), and the unjustly somewhat forgotten Bernand Bolzano, is properly addressed, in addition to a very stimulating account of Cauchy's own work.