II.5 The Development of Rigor in Mathematical Analysis
Tom Archibald
1 Background
This article is about how rigor came to be introduced into mathematical analysis. This is a complicated topic, since mathematical practice has changed considerably, especially in the period between the founding of the calculus (shortly before 1700) and the early twentieth century. In a sense, the basic criteria for what constitutes a correct and logical argument have not altered, but the circumstances under which one would require such an argument, and even to some degree the purpose of the argument, have altered with time. The voluminous and successful mathematical analysis of the 1700s, associated with names such as Johann and Daniel BERNOULLI [VI.18], EULER [V1.19], and LAGRANGE [VI.22], lacked foundational clarity in ways that were criticized and remedied in subsequent periods. By around 1910 a general consensus had emerged about how to make arguments in analysis rigorous.
Mathematics consists of more than techniques for calculation, methods for describing important features of geometric objects, and models of worldly phenomena. Nowadays, almost all working mathematicians are trained in, and concerned with, the production of rigorous arguments that justify their conclusions. These conclusions are usually framed as theorems, which are statements of fact, accompanied by an argument, or proof, that the theorem is indeed true. Here is a simple example: every positive whole number that is divisible by 6 is also divisible by 2. Running through the six times table (6, 12, 18, 24,…) we see that each number is even, which makes the statement easy enough to believe. A possible justification of it would be to say that since 6 is divisible by 2, then every number divisible by 6 must also be divisible by 2.
Such a justification might or might not be thought of as a thorough proof, depending on the reader. For on hearing the justification we can raise questions: is it always true that if a, b, and c are three positive whole numbers such that c is divisible by b and b is divisible by a, then c is divisible by a? What is divisibility exactly? What is a whole number? The mathematician deals with such questions by precisely defining concepts (such as divisibility of one number by another), basing the definitions on a smallish number of undefined terms (“whole number” might be one, though it is possible to start even further back, with sets). For example, one could define a number n to be divisible by a number m if and only if there exists an integer q such that qm = n. Using this definition, we can give a more precise proof: if n is divisible by 6, then n = 6q for some q, and therefore n = 2(3q), which proves that n is divisible by 2. Thus we have used the definitions to show that the definition of divisibility by 2 holds whenever the definition of divisibility by 6 holds.
Historically, mathematical writers have been satisfied with varying levels of rigor. Results and methods have often been widely used without a full justification of the kind just outlined, particularly in bodies of mathematical thought that are new and rapidly developing. Some ancient cultures, the Egyptians for example, had methods for multiplication and division, but no justification of these methods has survived and it does not seem especially likely that formal justification existed. The methods were probably accepted simply because they worked, rather than because there was a thorough argument justifying them.
By the middle of the seventeenth century, European mathematical writers who were engaged in research were well-acquainted with the model of rigorous mathematical argument supplied by EUCLID’S [VI.2] Elements. The kind of deductive, or synthetic, argument we illustrated earlier would have been described as a proof more geometrico—in the geometrical way. While Euclid’s arguments, assumptions, and definitions are not wholly rigorous by today’s standards, the basic idea was clear: one proceeds from clear definitions and generally agreed basic ideas (such as that the whole is greater than the part) to deduce theorems (also called propositions) in a step-by-step manner, not bringing in anything extra (either on the sly or unintentionally). This classical model of geometric argument was widely used in reasoning about whole numbers (for example by FERMAT [VI.12]), in analytic geometry (DESCARTES [VI.11]), and in mechanics (Galileo).
This article is about rigor in analysis, a term which itself has had a shifting meaning. Coming from ancient origins, by around 1600 the term was used to refer to mathematics in which one worked with an unknown (something we would now write as x) to do a calculation or find a length. In other words, it was closely related to algebra, though the notion was imported into geometry by Descartes and others. However, over the course of the eighteenth century the word came to be associated with the calculus, which was the principal area of application of analytic techniques. When we talk about rigor in analysis it is the rigorous theory of the mathematics associated with differential and integral calculus that we are principally discussing. In the third quarter of the seventeenth century rival methods for the differential and integral calculus were devised by NEWTON [VI.14] and LEIBNIZ [VI.15], who thereby synthesized and extended a considerable amount of earlier work concerned with tangents and normals to curves and with the areas of regions bounded by curves. The techniques were highly successful, and were extended readily in a variety of directions, most notably in mechanics and in differential equations.
The key common feature of this research was the use of infinities: in some sense, it involved devising methods for combining infinitely many infinitely small quantities to get a finite answer. For example, suppose we divide the circumference of a circle into a (large) number of equal parts by marking off points at equal distances, then joining the points and creating triangles by joining the points to the center. Adding up the areas of the triangles approximates the circular area, and the more points we use the better the approximation. If we imagine infinitely many of these inscribed triangles, the area of each will be “infinitely small” or infinitesimal. But because the total involves adding up infinitely many of them, it may be that we get a finite positive total (rather than just 0, from adding up infinitely many zeros, or an infinite number, as we would get if we added the same finite number to itself infinitely many times). Many techniques for doing such calculations were devised, though the interpretation of what was taking place varied. Were the infinities involved “real” or merely “potential”? If something is “really” infinitesimal, is it just zero? Aristotelian writers had abhorred actual infinities, and complaints about them were common at the time.
Newton, Leibniz, and their immediate followers provided mathematical arguments to justify these methods. However, the introduction of techniques involving reasoning with infinitely small objects, limiting processes, infinite sums, and so forth meant that the founders of the calculus were exploring new ground in their arguments, and the comprehensibility of these arguments was frequently compromised by vague terminology or by the drawing of one conclusion when another might seem to follow equally well. The objects they were discussing included infinitesimals (quantities infinitely smaller than those we experience directly), ratios of vanishingly small quantities (i.e., fractions in, or approaching, the form 0/0), and finite sums of infinitely many positive terms. Taylor series representations, in particular, provoked a variety of questions. A function may be written as a series in such a way that the series, when viewed as a function, will have, at a given point x = a, the same value as the function, the same rate of change (or first derivative), and the same higher-order derivatives to arbitrary order:
f(x) = f(a) + f′(a)(x - a) + 
f″(a)(x - a)2 + · · ·
For example, sin x = x - x3/3! + x5/5! + · · ·, a fact already known to Newton though such series are now named after Newton’s disciple BROOK TAYLOR [VI.16].
One problem with early arguments was that the terms being discussed were used in different ways by different writers. Other problems arose from this lack of clarity, since it concealed a variety of issues. Perhaps the most important of these was that an argument could fail to work in one context, even though a very similar argument worked perfectly well in another. In time, this led to serious problems in extending analysis. Eventually, analysis became fully rigorous and these difficulties were solved, but the process was a long one and it was complete only by the beginning of the twentieth century.
Let us consider some examples of the kinds of difficulties that arose from the very beginning, using a result of Leibniz. Suppose we have two variables, u and v, each of which changes when another variable, x, changes. An infinitesimal change in x is denoted dx, the differential of x. The differential is an infinitesimal quantity, thought of as a geometrical magnitude, such as a length, for example. This was imagined to be combined or compared with other magnitudes in the usual ways (two lengths can be added, have a ratio, and so on). When x changes to x + dx, u and v change to u + du and v + dv, respectively. Leibniz concluded that the product uv would then change to uv + udv + vdu, so that d(uv) = udv + vdu. His argument is, roughly, that d(uv) = (u + du) (v + dv) - uv. Expanding the right-hand side using regular algebra and then simplifying gives u dv + v du + du dv. But the term du dv is a second-order infinitesimal, vanishingly small compared with the first-order differentials, and is thus treated as equal to 0. Indeed, one aspect of the problems is that there appears to be an inconsistency in the way that infinitesimals are treated. For instance, if you want to work out the derivative of y = x2, the calculation corresponding to the one just given (expanding (x + dx)2, and so on) shows that dy/dx = 2x + dx. We then treat the dx on the right-hand side as zero, but the one on the left-hand side seems as though it ought to be an infinitesimal nonzero quantity, since otherwise we could not divide by it. So is it zero or not? And if not, how do we get around the apparent inconsistency?
At a slightly more technical level, the calculus required mathematicians to deal repeatedly with the “ultimate” values of ratios of the form dy/dx when the quantities in both numerator and denominator approach or actually reach 0. This phrasing uses, once again, the differential notation of Leibniz, though the same issues arose for Newton with a slightly different notational and conceptual approach. Newton generally spoke of variables as depending on time, and he sought (for example) the values approached when “evanescent increments”—vanishingly small time intervals—are considered. One long-standing set of confusions arose precisely from this idea that variable quantities were in the process of changing, whether with time or with changes in the value of another variable. This means that we talk about values of a variable approaching a given value, but without a clear idea of what this “approach” actually is.
2 Eighteenth-Century Approaches and Critiques
Of course, had the calculus not turned out to be an enormously fruitful field of endeavor, no one would have bothered to criticize it. But the methods of Newton and Leibniz were widely adopted for the solution of problems that had interested earlier generations (notably tangent and area problems) and for the posing and solution of problems that these techniques suddenly made far more accessible. Problems of areas, maxima and minima, the formulation and solution of differential equations to describe the shape of hanging chains or the positions of points on vibrating strings, applications to celestial mechanics, the investigation of problems having to do with the properties of functions (thought of for the most part as analytic expressions involving variable quantities)—all these fields and more were developed over the course of the eighteenth century by mathematicians such as Taylor, Johann and Daniel Bernoulli, Euler, D’ALEMBERT [VI.20], Lagrange, and many others. These people employed many virtuoso arguments of suspect validity. Operations with divergent series, the use of imaginary numbers, and manipulations involving actual infinities were used effectively in the hands of the most capable of these writers. However, the methods could not always be explained to the less capable, and thus certain results were not reliably reproducible—a very odd state for mathematics from today’s standpoint. To do Euler’s calculations, one needed to be Euler. This was a situation that persisted well into the following century.
Specific controversies often highlighted issues that we now see as a result of foundational confusion. In the case of infinite series, for example, there was confusion about the domain of validity of formal expressions. Consider the series
1 - 1 + 1 - 1 + 1 - 1 + 1 - · · · .
In today’s usual elementary definition (due to CAUCHY [VI.29] around 1820) we would now consider this series to be divergent because the sequence of partial sums 1,0,1,0, . . . does not tend to a limit. But in fact there was some controversy about the actual meaning of such expressions. Euler and Nicolaus Bernoulli, for example, discussed the potential distinction between the sum and the value of an infinite sum, Bernoulli arguing that something like 1 - 2 + 6 - 24 + 120 + · · · has no sum but that this algebraic expression does constitute a value. Whatever may have been meant by this, Euler defended the notion that the sum of the series is the value of the finite expression that gives rise to the series. In his 1755 Institutiones Calculi Differentialis, he gives the example of 1 - x + x2 - x3 + · · · , which comes from 1/(1 + x), and later defended the view that this meant that 1 - 1 + 1 - 1 + · · · = . His view was not universally accepted. Similar controversies arose in considering how to extend the values of functions outside their usual domain, for example with the logarithms of negative numbers.
Probably the most famous eighteenth-century critique of the language and methods of eighteenth-century analysis is due to the philosopher George Berkeley (1685–1753). Berkeley’s motto, “To be is to be perceived,” expresses his idealist stance, which was coupled with a strong view that the abstraction of individual qualities, for the purposes of philosophical discussion, is impossible. The objects of philosophy should thus be things that are perceived, and perceived in their entirety. The impossibility of perceiving infinitesimally small objects, combined with their manifestly abstracted nature, led him to attack their use in his 1734 treatise The Analyst: Or, a Discourse Addressed to an Infidel Mathematician. Referring sarcastically in 1734 to infinitesimals as the “ghosts of departed quantities,” Berkeley argued that neglecting some quantity, no matter how small, was inappropriate in mathematical argument. He quoted Newton in this regard, to the effect that “in mathematical matters, errors are to be condemned, no matter how small.” Berkeley continued, saying that “[n]othing but the obscurity of the subject” could have induced Newton to impose this kind of reasoning on his followers. Such remarks, while they apparently did not dissuade those enamored of the methods, contributed to a sentiment that aspects of the calculus required deeper explanation. Writers such as Euler, d’Alembert, Lazare Carnot, and others attempted to address foundational criticisms by clarifying what differentials were, and gave a variety of arguments to justify the operations of the calculus.
2.1 Euler
Euler contributed to the general development of analysis more than any other individual in the eighteenth century, and his approaches to justifying his arguments were enormously influential even after his death, owing to the success and wide use of his important textbooks. Euler’s reasoning is sometimes regarded as rather careless since he operated rather freely with the notation of the calculus, and many of his arguments are certainly deficient by later standards. This is particularly true of arguments involving infinite series and products. A typical example is provided by an early version of his proof that
His method is as follows. Using the known series expansion for sin x he considered the zeros of
These lie at π2, (2π)2, (3π)2, .... Applying (without argument) the factor theorem for finite algebraic equations he expressed this equation as
Now, it can be seen that the coefficient of x in the infinite sum, -
, should equal the negative of the sum of the coefficients of x in the product. Euler apparently concluded this by imagining multiplying out the infinitely many terms and selecting the 1 from all but one of them. This gives
and multiplying both sides by π2 gives the required sum.
We now think of this approach as having several problems. The product of the infinitely many terms may or may not represent a finite value, and today we would specify conditions for when it does. Also, applying a result about (finite) polynomials to (infinite) power series is a step that requires justification. Euler himself was to provide alternative arguments for this result later in his life. But the fact that he may have known counterexamples—situations in which such usages would not work—was not, for him, a decisive obstacle. This view, in which one reasoned in a generic situation that might admit a few exceptions, was common at his time, and it was only in the late nineteenth century that a concerted effort was made to state the results of analysis in ways that set out precisely the conditions under which the theorems would hold.
Euler did not dwell on the interpretation of infinite sums or infinitesimals. Sometimes he was happy to regard differentials as actually equal to zero, and to derive the meaning of a ratio of differentials from the context of the problem:
An infinitely small quantity is nothing but a vanishing quantity and therefore will be actually equal to 0. ... Hence there are not so many mysteries hidden in this concept as there are usually believed to be. These supposed mysteries have rendered the calculus of the infinitely small quite suspect to many people.
This statement, from the Institutiones Calculi Differentialis of 1755, was followed by a discussion of proportions in which one of the ratios is 0/0, and a justification of the fact that differentials may be neglected in calculations with ordinary numbers. This accurately describes a good deal of his practice—when he worked with differential equations, for example.
Controversial matters did arise, however, and debates about definitions were not unusual. The best-known example involves discussions connected with the so-called vibrating string problem, which involved Euler, d’Alembert, and Daniel Bernoulli. These were closely connected with the definition of FUNCTIONS [I.2 §2.2], and the question of which functions studied by analysis actually could be represented by series (in particular trigonometric series). The idea that a curve of arbitrary shape could serve as an initial position for a vibrating string extended the idea of function, and the work of FOURIER [VI.25] in the early nineteenth century made such functions analytically accessible. In this context, functions with broken graphs (a kind of discontinuous function) came under inspection. Later, how to deal with such functions would be a decisive issue for the foundations of analysis, as the more “natural” objects associated with algebraic operations and trigonometry gave way to the more general modern concept of function.
2.2 Responses from the Late Eighteenth Century
One significant response to Berkeley in Britain was that of Colin Maclaurin (1698–1746), whose 1742 textbook A Treatise of Fluxions attempted to clarify the foundations of the calculus and do away with the idea of infinitely small quantities. Maclaurin, a leading figure of the Scottish Enlightenment of the mid eighteenth century, was the most distinguished British mathematician of his time and an ardent proponent of Newton’s methods. His work, unlike that of many of his British contemporaries, was read with interest on the Continent, especially his elaborations of Newtonian celestial mechanics. Maclaurin attempted to base his reasoning on the notion of the limits of what he termed “assignable” finite quantities. Maclaurin’s work is famously obscure, though it did provide examples of calculating the limits of ratios. Perhaps his most important contribution to the clarification of the foundations of analysis was his influence on d’Alembert.
D’Alembert had read both Berkeley and Maclaurin and followed them in rejecting infinitesimals as real quantities. While exploring the idea of a differential as a limit, he also attempted to reconcile his idea with the idea that infinitesimals may be consistently regarded as being actually zero, perhaps in a nod to Euler’s view. The main exposition of d’Alembert’s views may be found in the Encyclopédie, in the articles on differentials (published in 1754) and on limits (1765). D’Alembert argued for the importance of geometric rather than algebraic limits. His meaning seems to have been that the quantities being investigated should not be treated merely formally, by substitution and simplification. Rather, a limit should be understood as the limit of a length (or collection of lengths), area, or other dimensioned quantity, in much the way that a circle may be seen as a limit of inscribed polygons. His aim seems primarily to have been to establish the reality of the objects described by existing algorithms, since the actual calculations he employs are carried out with differentials.
2.2.1 Lagrange
In the course of the eighteenth century, the differential and the integral calculus gradually distinguished themselves as a set of methods distinct from their applications in mechanics and physics. At the same time, the primary focus of the methods moved away from geometry, so that in work of the second half of the eighteenth century we increasingly see calculus treated as “algebraic analysis” of “analytic functions.” The term “analytic” was used in a variety of senses. For many writers, such as Euler, it merely referred to a function (that is, a relationship between variable quantities) that is given by a single expression of the type used in analysis.
Lagrange provided a foundation for the calculus that was indebted to this algebraic viewpoint. Lagrange concentrated on power-series expansions as the basic entity of analysis, and through his work the term analytic function evolved toward its more recent meaning connected with the existence of a convergent Taylor series representation. His approach reached a full expression in his Théorie des Fonctions Analytiques of 1797. This was a version of his lectures at the École Polytechnique, a new institution for the elite training of military engineers in revolutionary France. Lagrange assumed that a function must necessarily be expressible as an infinite series of algebraic functions, basing this argument on the existence of expansions for known functions. He first sought to show that “in general” no negative or fractional powers would appear in the expansion, and from this he obtained a power-series representation. His arguments here are surprising, and somewhat ad hoc, and I use an example given by Fraser (1987). The slightly strange notation is based on that of Lagrange. Suppose that one seeks an expansion of f(x) = 
in powers of i. In general, only integer powers will be involved. Terms of the form im/n do not make sense, says Lagrange, since the expression of the function is only two-valued, while im/n has n values. Hence the series
= 
+ pi + qi2 + · · · + tik + · · ·
obtains its two values from the term , and all other powers must be integral. With fractional exponents set aside, Lagrange argued that f(x + i) = f(x) + iaP(x, i), with P finite for i = 0. Successive application of this result gave him the expansion
f(x + i) = f(x) + pi + qi2 + ri3 + · · ·,
where i was a small increment. The number p depends on x, so Lagrange defined a derived function f′(x) = p(x). The French term dérivée is the origin of the term derivative, and in Lagrange’s language f is the “primitive” of this derived function. Similar arguments can be made to relate the higher coefficients to the higher derivatives in the usual Taylor formula.
This approach, which seems oddly circular to modern eyes, relied on the eighteenth-century distinction between the “algebraic” infinite process of the series expansion on the one hand, and the use of differentials on the other. Lagrange did not see the original series expansion as based on the limit process. With the renewed emphasis on limits and modern definitions developed by Cauchy, this approach was soon to be regarded as untenable.
3 The First Half of the Nineteenth Century
3.1 Cauchy
Many writers contributed to discussions on rigor in analysis in the first decades of the nineteenth century. It was Cauchy who was to revive the limit approach to greatest effect. His aim was pedagogical, and his ideas were probably worked out in the context of preparing his introductory lectures for the École Polytechnique at the beginning of the 1820s. Although the students were the best in France in scholarly ability, many found the approach too difficult. As a result, while Cauchy himself continued to use his methods, other instructors held on to older approaches using infinitesimals, which they found more intuitively accessible for the students as well as better adapted to the solution of problems in elementary mechanics. Cauchy’s self-imposed exile from Paris in the 1830s further limited the impact of his approach, which was initially taken up only by a few of his students.
Nonetheless, Cauchy’s definitions of limit, of continuity, and of the derivative gradually came into general use in France, and were influential elsewhere as well, especially in Italy. Moreover, his methods of using these definitions in proofs, and particularly his use of mean value theorems in various forms, moved analysis from a collection of symbolic manipulations of quantities with special properties toward the science of argument about infinite processes using close estimation via the manipulation of inequalities.
In some respects, Cauchy’s greatest contribution lay in his clear definitions. For earlier writers, the sum of an infinite series was a somewhat vague notion, sometimes interpreted by a kind of convergence argument (as with the sum of a geometric series such as 
and sometimes as the value of the function from which the series was derived (as Euler, for example, often regarded it). Cauchy revised the definition to state that the sum of an infinite series was the limit of the sequence of partial sums. This provided a unified approach for series of numbers and series of functions, an important step in the move to base calculus and analysis on ideas about real numbers. This trend, eventually dominant, is often referred to as the “arithmetization of analysis.” Similarly, a continuous function is one for which “an infinitely small increase of the variable produces “an infinitely small increase of the function itself” (Cauchy 1821, pp. 34–35).
As we see from the example just given, Cauchy did not shy away from infinitely small quantities, nor did he analyze this notion further. The limit of a variable quantity is defined in a way that we would now regard as conversational, or heuristic:
When the values that are successively assigned to a given variable approach a fixed value indefinitely, in such a way that it ends up differing from it as little as one wishes, this latter value is called the limit of all the others. Thus, for example, an irrational number is the limit of the various fractions that provide values that are closer and closer to it.
Cauchy (1821, p.4)
These ideas were not completely rigorous by modern standards, but he was able to use them to provide a unified foundation for the basic processes of analysis.
This use of infinitely small quantities appears, for example, in his definition of a continuous function. To paraphrase his definition, suppose that a function f(x) is single-valued on some finite interval of the real line, and choose any value x0 inside the interval. If the value of x0 is increased to x0 + a, the function also changes by the amount f(x0 + a) - f(x0). Cauchy says that the function f is continuous for this interval if, for each value of x0 in that interval, the numerical value of the difference f(x0 + a) - f(x0) decreases indefinitely to 0 with a. In other words, Cauchy defines continuity as a property on an interval rather than at a point, in essence by saying that on that interval infinitely small changes in the argument produce infinitely small changes in the function value. Cauchy appears to have considered continuity to be a property of a function on an interval.
This definition emphasizes the importance of jumps in the value of the function for the understanding of its properties, something that Cauchy had encountered early in his career when discussing THE FUNDAMENTAL THEOREM OF CALCULUS [I.3 §5.5]. In his 1814 memoir on definite integrals, Cauchy stated:
If the function 
(z) increases or decreases in a continuous manner between z = b′ and z = b″, the value of the integral 
′ (z) dz] will ordinarily be represented by (b″) - (b′). But if… the function passes suddenly from one value to another sensibly different… the ordinary value of the integral must be diminished.
Oeuvres (volume 1, pp. 402–3)
In his lectures, Cauchy assumed continuity when defining the definite integral. He considered first of all a division of the interval of integration into a finite number of subintervals on which the function is either increasing or decreasing. (This is not possible for all functions, but this appeared not to concern Cauchy.) He then defined the definite integral as the limit of the sum S = (x1 - x0)f(x0) + (x2 - x1)f(x1) + · · · + (X - xn-1)f(xn-1) as the number n becomes very large. Cauchy gives a detailed argument for the existence of this limit, using his theorem of the mean and the fact of continuity.
Versions of the main subjects of Cauchy’s lectures were published in 1821 and 1823. Every student at the École Polytechnique would have been aware of them subsequently, and many would have used them explicitly. They were joined in 1841 by a version of the course elaborated by Cauchy’s associate, the Abbé Moigno. They were referred to frequently in France and the definitions employed by Cauchy became standard there. We also know that the lectures were studied by others, notably by ABEL [VI.33] and DIRICHLET [VI.36], who spent time in Paris in the 1820s, and by RIEMANN [VI.49].
Cauchy’s movement away from the formal approach of Lagrange rejected the “vagueness of algebra.” Although he was clearly guided by intuition (both geometric and otherwise), he was well aware that intuition could be misleading, and produced examples to show the value of adhering to precise definitions. One famous example, the function that takes the value e-1/x2 when x ≠ 0 and zero when x = 0, is differentiable infinitely many times, yet it does not yield a Taylor series that converges to the function at the origin. Despite this example, which he mentioned in his lectures, Cauchy was not a specialist in counterexamples, and in fact the trend toward producing counterexamples for the purpose of clarifying definitions was a later development.
Abel famously drew attention to an error in Cauchy’s work: his statement that a convergent series of continuous functions has a continuous sum. For this to be true, the series must be uniformly convergent, and in 1826 Abel gave as a counterexample the series
which is discontinuous at odd multiples of π. Cauchy was led to make this distinction only much later, after the phenomenon had been identified by several writers. Historians have written extensively about this apparent error; one influential account, due to Bottazzini, proposes that for various reasons Cauchy would not have found Abel’s example telling, even if he had known of it at the time (this account appears in Bottazzini (1990, p. LXXXV)).
Before leaving the time of Cauchy, we should note the related independent activity of BOLZANO [VI.28]. Bolzano, a Bohemian priest and professor whose ideas were not widely disseminated at the time, investigated the foundations of the calculus extensively. In 1817, for example, he gave what he termed a “purely analytic proof of the theorem that between any two values that possess opposite signs, at least one real root of the equation exists”: the intermediate value theorem. Bolzano also studied infinite sets: what is now called the Bolzano-Weierstrass theorem states that for every bounded infinite set there is at least one point having the property that any disk about that point contains infinitely many points of the set. Such “limit points” were studied independently by WEIERSTRASS [VI.44]. By the 1870s, Bolzano’s work became more broadly known.
3.2 Riemann, the Integral, and Counterexamples
Riemann is indelibly associated with the foundations of analysis because of the Riemann integral, which is part of every calculus course. Despite this, he was not always driven by issues involving rigor. Indeed he remains a standard example of the fruitfulness of nonrigorous intuitive invention. There are many points in Riemann’s work at which issues about rigor arise naturally, and the wide interest in his innovations did much to direct the attention of researchers to making these insights precise.
Riemann’s definition of the definite integral was presented in his 1854 Habilitationschrift—the “second thesis,” which qualified him to lecture at a university for fees. He generalized Cauchy’s notion to functions that are not necessarily continuous. He did this as part of an investigation of FOURIER SERIES [III.27] expansions. The extensive theory of such series was devised by Fourier in 1807 but not published until the 1820s. A Fourier series represents a function in the form
on a finite interval.
The immediate inspiration for Riemann’s work was DIRICHLET [VI.36], who had corrected and developed earlier faulty work by Cauchy on the question of when and whether the Fourier series expansion of a function converges to the function from which it is derived. In 1829 Dirichlet had succeeded in proving such convergence for a function with period 2π that is integrable on an interval of that length, does not possess infinitely many maxima and minima there, and at jump discontinuities takes on the average value between the two limiting values on each side. As Riemann noted, following his professor Dirichlet, “this subject stands in the closest connection to the principles of infinitesimal calculus, and can therefore serve to bring these to greater clarity and definiteness” (Riemann 1854, p. 238). Riemann sought to extend Dirichlet’s investigations to further cases, and was thus led to investigate in detail each of the conditions given by Dirichlet. Accordingly, he generalized the definition of a definite integral as follows:
We take between a and b an increasing sequence of values x1,x2, . . . ,xn-1, and for brevity designate x1 - a by δ1,x2 - x1 - x1 by δ2, . . . , b - xn-1 by δn and by 
a positive proper fraction. Then the value of the sum
S = δ1f(a + 1δ1) + δ2f(x1 + 2δ2) + δ3f(x2 + 3δ3) + · · · + δnf(xn-1 + nδn)
depends on the choice of the intervals δ and the quantities . If it has the property that it approaches infinitely closely a fixed limit A no matter how the δ and are chosen, as δ becomes infinitely small, then we call this value 
f(x) dx.
In connection with this definition of the integral, and in part to show its power, Riemann provided an example of a function that is discontinuous in any interval, yet can be integrated. The integral thus has points of nondifferentiability on each interval. Riemann’s definition rendered problematic the inverse relationship between differentiation and integration, and his example brought this problem out clearly. The role of such “pathological” counterexamples in pushing the development of rigor, already apparent in Cauchy’s work, intensified greatly around this time.
Riemann’s definition was published only in 1867, following his death; an expository version due to Gaston Darboux appeared in French in 1873. The popularization and extension of Riemann’s approach went hand in hand with the increasing appreciation of the importance of rigor associated with the Weierstrass school, discussed below. Riemann’s approach focused attention on sets of points of discontinuities, and thus were seminal for CANTOR’S [VI.54] investigations into point sets in the 1870s and afterwards.
The use of the Dirichlet principle serves as a further example of the way in which Riemann’s work drew attention to problems in the foundations of analysis. In connection with his research into complex analysis, Riemann was led to investigate solutions to the so-called Dirichlet problem: given a function g, defined on the boundary of a closed region in the plane, does there exist a function f that satisfies the LAPLACE PARTIAL DIFFERENTIAL EQUATION [I.3 §5.4] in the interior and takes the same values as g on the boundary? Riemann asserted that the answer was yes. To demonstrate this, he reduced the question to proving the existence of a function that minimizes a certain integral over the region, and argued on physical grounds that such a minimizing function must always exist. Even before Riemann’s death his assertion was questioned by WEIERSTRASS [VI.44], who published a counterexample in 1870. This led to attempts to reformulate Riemann’s results and prove them by other means, and ultimately to a rehabilitation of the Dirichlet principle through the provision of precise and broad hypotheses for its validity, which were expressed by HILBERT [VI.63] in 1900.
4 Weierstrass and His School
Weierstrass had a passion for mathematics as a student at Bonn and Münster, but his student career was very uneven. He spent the years from 1840 to 1856 as a high school teacher, undertaking research independently but at first publishing obscurely. Papers from 1854 onward in Journal für die reine und angewandte Mathematik (otherwise known as Crelle’s Journal) attracted wide attention to his talent, and he obtained a professorship in Berlin in 1856. Weierstrass began to lecture regularly on mathematical analysis, and his approach to the subject developed into a series of four courses of lectures given cyclically between the early 1860s and 1890. The lectures evolved over time and were attended by a large number of important mathematical researchers. They also indirectly influenced many others through the circulation of unpublished notes. This circle included R. Lipschitz, P. du Bois-Reymond, H. A. Schwarz, O. Hölder, Cantor, L. Koenigsberger, G. Mittag-Leffler, KOVALEVSKAYA [VI.59], and L. Fuchs, to name only some of the most important. Through their use of Weierstrassian approaches in their own research, and their espousal of his ideas in their own lectures, these approaches became widely used well before the eventual publication of a version of his lectures late in his life. The account that follows is based largely on the 1878 version of the lectures. His approach was also influential outside Germany: parts of it were absorbed in France in the lectures of HERMITE [VI.47] and JORDAN [VI.52], for example.
Weierstrass’s approach builds on that of Cauchy (though the detailed relationship between the two bodies of work has never been fully examined). The two overarching themes of Weierstrass’s approach are, on the one hand, the banning of the idea of motion, or changing values of a variable, from limit processes, and, on the other, the representation of functions, notably of a complex variable. The two are intimately linked. Essential to the motion-free definition of a limit is Weierstrass’s nascent investigation of what we would now call the topology of the real line or complex plane, with the idea of a limit point, and a clear distinction between local and global behavior. The central objects of study for Weierstrass are functions (of one or more real or complex variable quantities), but it should be borne in mind that set theory is not involved, so that functions are not to be thought of as sets of ordered pairs.
The lectures begin with a now-familiar subject: the development of rational, negative, and real numbers from the integers. For example, negative numbers are defined operationally by making the integers closed under the operation of subtraction. He attempted a unified approach to the definition of rational and irrational numbers which involved unit fractions and decimal expansions and now seems somewhat murky. Weierstrass’s definition of the real numbers appears unsatisfactory to modern eyes, but the general path of arithmetization of analysis was established by this approach. In parallel to the development of number systems, he also developed different classes of functions, building them up from rational functions by using power-series representations. Thus, in Weierstrass’s approach, a polynomial (called an integer rational function) is generalized to a “function of integer character,” which means a function with a convergent power-series expansion everywhere. The Weierstrass factorization theorem asserts that any such function may be written as a (possibly infinite) product of certain “prime” functions and exponential functions with polynomial exponents of a certain type.
The limit definition given by Weierstrass has thoroughly modern features:
That a variable quantity x becomes infinitely small simultaneously with another quantity y means: “After the assumption of an arbitrarily small quantity ∈ a bound δ for x may be found, such that for every value of x for which |x| < δ, the corresponding value of |y| will be less than ∈.”
Weierstrass (1988, p. 57)
Weierstrass immediately used this definition to give a proof of continuity for rational functions of several variables, using an argument that could appear in a textbook today. The former notions of variables tending to given values were replaced by quantified statements about linked inequalities. The framing of hypotheses in terms of inequalities became a guiding motif in the work of Weierstrass’s school: here we mention in passing the Lipschitz and Hölder conditions in the existence theory for differential equations. The clarity that this language gave to problems involving the interchange of limits, for example, meant that previously intractable problems could now be handled in a routine way by those inculcated in the Weierstrass approach.
The fact that general functions were built from rational functions using series expansions gave the latter a key role in Weierstrass’s work, and as early as 1841 he had identified the importance of uniform convergence. The distinction between uniform and pointwise convergence was made very clearly in his lectures. A series converges, as it does for Cauchy, if its sequence of partial sums converges, though now the convergence is phrased in the following terms: the series Σfn(x) converges to s0 at x = x0 if, given an arbitrary positive ∈, there is an integer N such that |s0 - (f1(x0) + f2(x0) + . . . + fn(x0))| < ∈ for every n > N. The convergence is uniform on a domain of the variable if the same N will work for that ∈ value for all x in the domain. Uniform convergence guarantees continuity of the sum, since these are series of rational, hence continuous, functions. From this point of view, then, uniform convergence is important well beyond the context of trigonometric series (important though those may be). Indeed, it is a central tool of the entire theory of functions.
Weierstrass’s role as a critic of rigor in the work of others, notably Riemann, has already been noted. More than any other leading figure, he generated counterexamples to illustrate difficulties with received notions and to distinguish between different kinds of analytical behavior. One of his best-known examples was of an everywhere-continuous but nowhere-differentiable function, namely f(x) = Σbn cos(anx), which is uniformly convergent for b < 1 but fails to be differentiable at any x if ab > 1 + 
π. Similarly he constructed functions for which the Dirichlet principle fails, examples of sets constituting “natural boundaries,” that is, obstacles to continuing series expansions into larger domains, and so forth. The careful distinctions he encouraged, and the very procedure of seeking pathological rather than typical examples, threw the spotlight on the precision of hypotheses in analysis to an unprecedented degree. From the 1880s, with the maturity of this program, analysis no longer dealt with generic cases and looked instead for absolutely precise statements in a way that has for the most part endured to the present. This was also to become a pattern and an imperative in other areas of mathematics, though sometimes the passage from reasoning from generic examples to fully expressed hypotheses and definitions took decades. (Algebraic geometry provides a famous example, one in which reasoning with generic cases lasted until the 1920s.) In this sense the form of rigorous argument and exposition espoused by Weierstrass and his school was to become a pattern for mathematics generally.
4.1 The Aftermath of Weierstrass and Riemann
Analysis became the model subdiscipline for rigor for a variety of reasons. Of course, analysis was important for the sheer volume and range of application of its results. Not everyone agreed with the precise way in which Weierstrass approached foundational questions (through series, rational functions, and so on). Indeed, Riemann’s more geometric approach had also attracted followers, if not exactly a school, and the insights his approach afforded were enthusiastically embraced. However, any subsequent discussion had to take place at a level of rigor comparable to that which Weierstrass had attained. While approaches to the foundations of analysis were to vary, the idea that limits should be rigorously handled in much the way that Weierstrass did was not to alter. Among the remaining central issues for rigor was the definition of the number systems.
For the real numbers, probably the most successful definition (in terms of its later use) was provided by DEDEKIND [VI.50]. Dedekind, like Weierstrass, took the integers as fundamental, and extended them to the rationals, noting that the algebraic properties satisfied by the latter are those satisfied by what we now call a FIELD [I.3 §2.2]. (This idea is also Dedekind’s.) He then showed that the rational numbers satisfy a trichotomy law. That is, each rational number x divides the entire collection into three parts: x itself, rational numbers greater than x, and rational numbers less than x. He also showed that the rationals greater and less than a given number extend to infinity, and that any rational corresponds to a distinct point on the number line. However, he also observed that along that line there are infinitely many points that do not correspond to any rational. Using the idea that to every point on the line there should correspond a number, he constructed the remainder of the continuum (that is, the real line) by the use of cuts. These are ordered pairs (A1, A2) of nonempty sets of rational numbers such that every element of the first set is less than every element of the second, and such that taken together they contain all the rationals. Such cuts may obviously be produced by an element x, in which case x is either the greatest element of A1 or the least element of A2. But sometimes A1 does not have a greatest element, or A2 a least element, and in that case we can use the cut to define a new number, which is necessarily irrational. The set of all such cuts may be shown to correspond to the points of the number line, so that nothing is left out. A critical reader might feel that this is begging the question, since the idea of the number line constituting a continuum in some way might seem to be a hidden premise.
Dedekind’s construction stimulated a good deal of discussion, especially in Germany, about the best way to found the real numbers. Participants included Cantor, E. Heine, and the logician FREGE [VI.56]. Heine and Cantor, for example, considered real numbers as equivalence classes of Cauchy sequences of rationals, together with a machinery that permitted them to define the basic arithmetical operations. A very similar approach was proposed by the French mathematician Charles Méray. Frege, by contrast, in his 1884 Die Grundlagen der Arithmetik, sought to found the integers on logic. While his attempts to construct the reals along these lines did not bear fruit, he had an important role in his insistence that the various constructions should not merely be mathematically functional but should also be demonstrably free from internal contradiction.
Despite much activity on the foundations of the real numbers, infinite sets, and other basic notions for analysis, consensus remained elusive. For example, the influential Berlin mathematician LEOPOLD KRONECKER [VI.48] denied the existence of the reals, and held that all true mathematics was to be based on finite sets. Like Weierstrass, with whom he worked and whom he influenced, he emphasized the strong analogies between the integers and the polynomials, and sought to use this algebraic foundation to build all of mathematics. Hence for Kronecker the entire main path of research in analysis was anathema, and he opposed it ardently. These views were influential, both directly and indirectly, on a number of later writers, including BROUWER [VI.75], the intuitionist school around him, and the algebraist and number theorist Kurt Hensel.
All efforts to found analysis were based in one way or another on an underlying notion (not always made explicit) of quantity. The foundational framework of analysis, however, was to shift over the period from 1880 to 1910 toward the theory of sets. This had its origin in the work of Cantor, a student of Weierstrass who began studying discontinuities of Fourier series in the early 1870s. Cantor became concerned about how to distinguish between different types of infinite sets. His proofs that the rational numbers and the algebraic numbers are COUNTABLE [III.11] while the reals are not led him to a hierarchy of infinite sets of different cardinality. The importance of this discovery for analysis was at first not widely recognized, though in the 1880s Mittag-Leffler and Hurwitz both made significant applications of notions about derived sets (the set of limit points of a given set) and dense or nowhere-dense sets.
Cantor gradually came to the view that set theory could function as a foundational tool for all of mathematics. As early as 1882 he wrote that the science of sets encompassed arithmetic, function theory, and geometry, combining them into a “higher unity” based on the idea of cardinality. However, this proposal was vaguely articulated and at first attracted no adherents. Nonetheless, sets began to find their way into the language of analysis, most notably through ideas of MEASURE [III.55] and measurability of a set. Indeed, one important route to the absorption of analysis by set theory was the path that sought to determine what kind of function could “measure” a set in an abstract sense. The work of LEBESGUE [VI.72] and BOREL [VI.70] around 1900 on integration and measurability tied set theory to the calculus in a very concrete and intimate way.
A further key step in the establishment of the foundations of analysis in the early twentieth century was a new emphasis on mathematical theories as axiomatic structures. This received enormous impetus from the work of Hilbert, who, beginning in the 1890s, had sought to provide a renewed axiomatization of geometry. PEANO [VI.62] in Italy headed a school with similar aims. Hilbert redefined the reals on these axiomatic grounds, and his many students and associates turned to axiomatics with enthusiasm for the clarity the approach could provide. Rather than proving the existence of specific entities such as the reals, the mathematician posits a system satisfying the fundamental properties they possess. A real number (or whatever object) is then defined by the set of axioms provided. As Epple has pointed out, such definitions were considered to be ontologically neutral in that they did not provide methods for telling real numbers from other objects, or even state whether they existed at all (Epple 2003, p. 316). Hilbert’s student Ernst Zermelo began work on axiomatizing set theory along these lines, publishing his axioms in 1908 (see [IV.22 §3]). Problems with set theory had emerged in the form of paradoxes, the most famous due to RUSSELL [VI.71]: if S is the set of all sets that do not contain themselves, then it is not possible for S to be in S, nor can it not be in S. Zermelo’s axiomatics sought to avoid this difficulty, in part by avoiding the definition of set. By 1910, WEYL [VI.80] was to refer to mathematics as the science of “∈,” or set membership, rather than the science of quantity. Nonetheless, Zermelo’s axioms as a foundational strategy were contested. For one thing, a consistency proof for the axioms was lacking. Such “meaning-free” axiomatization was also contested on the grounds that it removed intuition from the picture.
Against the complex and rapidly developing background of mathematics in the early twentieth century, these debates took on many dimensions that have implications well beyond the question of what constitutes rigorous argument in analysis. For the practicing analyst, however, as well as for the teacher of basic infinitesimal calculus, these discussions are marginal to everyday mathematical life and education, and are treated as such. Set theory is pervasive in the language used to describe the basic objects. Real-valued functions of one real variable are defined as sets of ordered pairs of real numbers, for example; a set-theoretic definition of an ordered pair was given by WIENER [VI.85] in 1914, and the set-theoretic definition of functions may be dated from that time. However, research in analysis has been largely distinct from, and generally avoids, the foundational issues that may remain in connection with this vocabulary. This is not at all to say that contemporary mathematicians treat analysis in a purely formal way. The intuitive content associated with numbers and functions is very much a part of the way of thinking of most mathematicians. The axioms for the reals and for set theory form a framework to be referred to when necessary. But the essential objects of basic analysis, namely derivatives, integrals, series, and their existence or convergence behaviors, are dealt with along the lines of the early twentieth century, so that the ontological debates about the infinitesimal and infinite are no longer very lively.
A coda to this story is provided by the researches of ROBINSON [VI.95] (1918–74) into “nonstandard” analysis, published in 1961. Robinson was an expert in model theory: the study of the relationship between systems of logical axioms and the structures that may satisfy them. His differentials were obtained by adjoining to the regular real numbers a set of “differentials,” which satisfied the axioms of an ordered field (in which there is ordinary arithmetic like that of the real numbers) but in addition had elements that were smaller than 1/n for every positive integer n. In the eyes of some, this creation eliminated many of the unpleasant features of the usual way of dealing with the reals, and realized the ultimate goal of Leibniz to have a theory of infinitesimals which was part of the same structure as that of the reals. Despite stimulating a flurry of activity, and considerable acclaim from some quarters, Robinson’s approach has never been widely accepted as a working foundation for analysis.
Further Reading
Bottazzini, U. 1990. Geometrical rigour and “modern analysis”: an introduction to Cauchy’s Cours d’Analyse. In Cauchy (1821). Bologna: Editrice CLUB.
Cauchy, A.-L. 1821. Cours d’Analyse de l’École Royale Polytechnique: Première Partie—Analyse Algébrique. Paris: L’Imprimerie Royale. (Reprinted, 1990, by Editrice CLUB, Bologna.)
Epple, M. 2003. The end of the science of quantity: foundations of analysis, 1860–1910. In A History of Analysis, edited by H. N. Jahnke, pp. 291–323. Providence, RI: American Mathematical Society.
Fraser, C. 1987. Joseph Louis Lagrange’s algebraic vision of the calculus. Historia Mathematica 14:38–53.
Jahnke, H. N., ed. 2003. A History of Analysis. Providence, RI: American Mathematical Society/London Mathematical Society.
Riemann, G. F. B. 1854. Ueber die Darstellbarkeit einer Function durch eine trigonometrische Reihe. Königlichen Gesellschaft der Wissenschaften zu Göttingen 13:87–131. Republished in Riemann’s collected works (1990): Gesammelte Mathematische Werke und Wissenschaftliche Nachlass und Nachträge, edited by R. Narasimhan, 3rd edn., pp. 259–97. Berlin: Springer.
Weierstrass, K. 1988. Einleitung in die Theorie der Analytischen Functionen: Vorlesung Berlin 1878, edited by P. Ullrich. Braunschweig: Vieweg/DMV.