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[The Real and the Complex: A History of Analysis in the 19th Century (Springer Undergraduate Mathematics Series)] [By: Gray, Jeremy] [October, 2015]
This book contains a history of real and complex analysis in the nineteenth century, from the work of Lagrange and Fourier to the origins of set theory and the modern foundations of analysis. It studies the works of many contributors including Gauss, Cauchy, Riemann, and Weierstrass. This book is unique owing to the treatment of real and complex analysis as overlapping, inter-related subjects, in keeping with how they were seen at the time. It is suitable as a course in the history of mathematics for students who have studied an introductory course in analysis, and will enrich any course in undergraduate real or complex analysis.
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First published November 23, 2015
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Jeremy Gray
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July 31, 2021
In many ways, the eighteenth century can be likened to the childhood of mathematics. Like a boy presented with a set of wooden building blocks by his parents for Christmas, mathematicians took the calculus they received from the previous century and began to explore its possibilities, without being overly concerned about rigor—engaging in daring manipulations with infinitesimals and divergent series etc. Only later, in their maturity, did the impulse arise to make precise and justify informal concepts of numbers and functions, which in turn stimulated ever more brilliant discoveries. The seasoned historian of mathematics Jeremy Gray tells the story of this adventure in the present work on real analysis and complex function theory in the nineteenth century. Gray’s efforts in this direction turn out to be more successful than his Plato’s Ghost, on the rise of a modernist movement in the mathematical community during the last decade of the nineteenth century and the first half of the twentieth, because less ambitious (q.v., this recensionist’s review elsewhere on this site); his métier lies in close reconstruction of the intellectual history from a mathematician’s point of view, not in drawing out connections with wider culture, whether literary or philosophical (not to mention theological). There is not even very much reference to physics except as motivation. As the title implies, Gray’s theme is the interplay between real and complex analysis during the formative years of the nineteenth century, which we today distinguish sharply but the actors at the time did not.
His method throughout is narrative: he leads off each section with a biographical précis before delving into the equations, but by no means is Gray detained by the kinds of sociological preoccupations of much of current historiography; the emphasis stays focused strictly on the mathematics itself and what it says about methodology and internal programmatic goals. Gray’s awareness as a historian enables him to point out how the participants’ perspective differs from our own, derived with hindsight. Why should one revisit the record even today, when we have so much more adequate and powerful tools at our disposal? It depends on whether one views oneself as a student of mathematics or merely as a technician—if the latter, there would naturally be no need to pore over the history of the subject; one can very well acquire technical proficiency while knowing only the barest sketch of its past. But that route leads to the tunnel vision all too common in our day, when (so one would infer) only superannuated professors past their prime care about philosophical or epistemological issues. As a result, the overwhelming majority of papers appearing nowadays are unreadable; even if one could summon the energy to learn all the background and go through them in detail, ordinarily the authors themselves leave no clue as to why one should want to do so. In other words, they are devoid of vision. If, however, one wishes to do something profound rather than merely to crank out enough minimum publishable units to get tenure, there is no substitute to meditating on what mathematics itself is, and then a knowledgeable historian such as Gray will prove a most welcome guide.
Our view of complex analysis is too retrospective. For instance, we learn Cauchy’s integral and residue theorems without appreciating where they came from and, for us, when approaching the subject we already know that complex functions are of great interest, whereas they had to discover this. Gray recounts the long, halting process by which real-valued functions were generalized to complex-valued functions – the latter were used initially for performing definite integrals of real functions; it was not necessarily clear at first why the condition of holomorphicity matters (viz., the so-called Cauchy-Riemann equations; as the attribution suggests, their real importance did not become apparent until Riemann). Rather, the field we now call complex analysis has its roots in concrete problems whose solutions show special properties. It takes mathematical genius to discern in these a herald of a new domain of phenomena for which there may be an elegant general theory. Nobody suspected as much until elliptic functions arrived on the scene; they provide the first hint of transcendental functions possessing non-trivial properties, beyond the standard examples (say, the square root, the exponential, the logarithm and the trigonometric functions). The modern curriculum causes us to tend to view elliptic functions as an advanced topic in complex analysis appropriate for a second semester, just the reverse of the historical order! Elliptic integrals were first extensively investigated by Legendre towards the close of the eighteenth century, but not until decades later was the full potential of the field opened up by the pioneering researches of Abel and Jacobi, who had the vital idea of inverting the integral and viewing it as a complex-valued function of a complex argument (meanwhile, as is so often the case, Gauss was secretly working on them all along, too). Among the surprising phenomena elliptic functions display are (double) periodicity, poles and multivaluedness.
It is striking how not until the early 1850’s was Cauchy clear on what complex analysis was, as a subject in its own right—an index of how challenging the new ideas were. Nevertheless, if one pays attention, one can gain an impression of why, for once, common opinion is right in ascribing the lion’s share of the credit for putting the calculus upon a rigorous foundation to Cauchy and Weierstrass, as opposed, say, to Dirichlet, Dedekind or Riemann (Gray himself does not remark upon this). A good illustration of the light that the historian can throw upon a subject is the following: everyone knows about Cauchy’s ‘error’ about uniform continuity, yet this is not how people at the time thought about it. Of course, they were aware that Cauchy’s formulation left something to be desired (Abel’s counterexample) but the counterexamples were seen as exceptional anyway. For someone brought up on modern analysis, Gray helpfully points out that our definition of continuity at a point, as opposed to continuity over an interval, is not a notion one might stumble across very immediately. Only much later, as examples accumulated, did the need for the condition of uniform continuity in contrast to pointwise continuity everywhere become apparent. Thus, a familiarity with the research practices of the time explains why events took the course they did.
Another area where Gray’s erudition comes in helpful is the question as to why infinitesimals, which Cauchy still used in the Cours d’analyse of 1821, fell out of favor by Weierstrass’ time. The concept of quantity in the analysis of the early nineteenth century was intuitive and fairly flexible. As such, it was perfectly compatible with infinite or infinitesimal magnitudes. Once Weierstrass invented a very precise concept of real number in the 1860’s (although his would be supplanted by Dedekind’s), one discovered that the role infinitesimals played could be just as well filled by limiting processes with finite quantities, the newly-minted real numbers. Thus, infinitesimals fell out of favor. These days, one often sees proposals from would-be reformers to restore infinitesimals in the teaching of freshman calculus, because they are supposedly simpler and more intuitive—as if a freshman could take in non-standard analysis before even knowing standard analysis! But their view rests upon a misunderstanding of the mathematics itself. True, infinitesimals work nicely for analytic functions, but, during the course of the nineteenth century, mathematicians came to appreciate that the concept of a function can be much broader than analytic and that, when one expands the range of functions one is willing to consider, many subtle and counterintuitive phenomena emerge, for which more precise definitions of the terms of art of real analysis have to be devised. Weierstrassian real analysis is conceptually very clean and lends itself to effective problem-solving technique, once one surmounts the initial barrier and becomes accustomed to its strange jargon of nested quantifiers and so on. For instance, it makes possible measure theory and Lebesgue’s integral. Hence, reinstating infinitesimals, though superficially appealing, would be a step backwards. Can it really serve the student’s long-term interest to hide from him the full complexity of behavior a real-valued or complex-valued function can display and to pretend as if all functions were analytic, or at worst smooth? If one were to try to apply the transfer principle to, say, an arbitrary measurable function or a holomorphic function with an essential singularity, it would yield non-standard objects of unimaginable complexity, and this would defeat the purpose of returning to infinitesimals in the first place.
Other topics Gray covers: 1) why Fourier series are more than just an interesting application, but central to the evolving understanding of real analysis; also the distinction between trigonometric versus Fourier series; 2) the controversial Dirichlet principle; 3) the evolving function concept. People started out assuming with Legendre that every function must be locally expandable in power series up to possibly a small number of jump discontinuities. In fact, continuity and differentiability are very different properties a function can have, as had to be learned. The reader will be pleased to find in Gray’s text explicit formulae for functions exhibiting phenomena he may have heard about, but never seen spelled out in detail. For instance, Riemann’s example of an integrable function with a dense set of discontinuities; Weierstrass’ example of a continuous nowhere-differentiable function (arrived at by considering the natural domain of an analytic function, to which one is led by analytic continuation, Weierstrass’ forte, unlike Riemann’s); and Seidel’s counterexample showing that the sum of a series of continuous functions need not be continuous. As naïveté about what the typical function looks like subsided, the need for a more precise definition of the integral became apparent. Hence, the significance of Riemann’s integral and its advantages and disadvantages (it is defined not just for continuous or sectionally continuous functions, but allows for well-behaved oscillations as long as they integrate to something reasonable). Although Riemann’s integral was better than anything that had gone before, it does behave somewhat strangely. Everyone who has studied real analysis up to the Lebesgue integral will know about this, but may not perhaps have worked out many explicit examples of counterintuitive behavior. Thus, the reader will welcome Gray’s discussion of a non-integrable limit of integrable functions and Darboux’s example of the failure of term-by-term integration applied to a series.
The one major topic that gets rather short shrift from Gray is the arithmetization of the continuum, or construction of the real numbers. It is described succinctly but accompanied by little philosophy: what, for instance, would Frege, Russell, Weyl, Brouwer have to say about it? For such a fundamental development in modern real analysis, an oversight such as this is something of a surprise.
Some minor quibbles: there seem to be a lot of typos. Along the same lines, although for the most part Gray’s prose is fluent and supple, solecisms crop up more often than one would expect from an academic, such as the failure to employ the subjunctive even when ordinary mathematical language demands it; a particularly egregious instance can be found on p. 25: ‘This is typical of the way complex considerations could enter mathematical arguments at the time, but only on the condition that the final expressions are [sic] purely real. He insisted that the modulus c and amplitude ϕ were [sic] real, and that c<1’. The last few chapters are loosely organized and wandering. The author intends to whet his students’ appetite to follow developments in mathematical analysis into the twentieth century, but his grab-bag of curiosities almost degenerates into the hype characteristic of more popular literature and, in any case, his recounting of highlights is too hurried to convey anything of substance.
As is usual for writings of a historian, the present work conveys no normative thesis, but is content to retell what others felt; e.g. Weierstrass’ reception of Riemann and Cauchy. To have something non-trivial to observe on this score would require another order of thought than the historian per se is capable of. One wishes, however, that Gray could have hazarded at least a few critical obiter dicta on the present situation in the world of mathematics in light of what his researches have uncovered about the past. To supply a little of what he does not, let us suggest one lesson for us today: the string theorists resemble nothing so much as the mathematicians of the eighteenth century in their employment of heuristic methods in order rapidly to hit upon many amazing propositions (one cannot call them ‘results’ as they are not necessarily known to be true). In some cases, once suspected they can be confirmed by conventional lines of proof, but theoretical physicists appear to have uncovered a deeper secret that is responsible for generating their insights, even though, just like the calculus before the late nineteenth century, for reasons that are imperfectly understood. Will these ever become theorems according to the modern standard of rigor? Perhaps, but one should not look for this to happen anytime soon; in almost fifty years, string theorists themselves have evinced little, if any, interest in justifying their sprawling patchwork of conjectures. The foreseeable scenario is for it to collapse under the sheer weight of its speculative excess, rather as did the Italian school of algebraic geometry in the early decades of the twentieth century; who knows how long we will have to wait before a solitary genius along the lines of a Grothendieck arrives to put some order into the chaotic remains that will then be left behind? Why doesn’t Gray draw from his store of wisdom to comment upon an eventuality such as this?
But within these limitations, Gray’s scholarship is very good. It is nice to have many explicit formulae written out so that one can tell what the exposition is about, also numerous portraits of the players and figures from which one can get an idea of the convergence properties of the series under discussion. Recommended to the serious student of mathematics (at the undergraduate level or higher) who wants to improve his intuition, if not to the dilettante, for whom the amount of detail may prove a discouragement.
His method throughout is narrative: he leads off each section with a biographical précis before delving into the equations, but by no means is Gray detained by the kinds of sociological preoccupations of much of current historiography; the emphasis stays focused strictly on the mathematics itself and what it says about methodology and internal programmatic goals. Gray’s awareness as a historian enables him to point out how the participants’ perspective differs from our own, derived with hindsight. Why should one revisit the record even today, when we have so much more adequate and powerful tools at our disposal? It depends on whether one views oneself as a student of mathematics or merely as a technician—if the latter, there would naturally be no need to pore over the history of the subject; one can very well acquire technical proficiency while knowing only the barest sketch of its past. But that route leads to the tunnel vision all too common in our day, when (so one would infer) only superannuated professors past their prime care about philosophical or epistemological issues. As a result, the overwhelming majority of papers appearing nowadays are unreadable; even if one could summon the energy to learn all the background and go through them in detail, ordinarily the authors themselves leave no clue as to why one should want to do so. In other words, they are devoid of vision. If, however, one wishes to do something profound rather than merely to crank out enough minimum publishable units to get tenure, there is no substitute to meditating on what mathematics itself is, and then a knowledgeable historian such as Gray will prove a most welcome guide.
Our view of complex analysis is too retrospective. For instance, we learn Cauchy’s integral and residue theorems without appreciating where they came from and, for us, when approaching the subject we already know that complex functions are of great interest, whereas they had to discover this. Gray recounts the long, halting process by which real-valued functions were generalized to complex-valued functions – the latter were used initially for performing definite integrals of real functions; it was not necessarily clear at first why the condition of holomorphicity matters (viz., the so-called Cauchy-Riemann equations; as the attribution suggests, their real importance did not become apparent until Riemann). Rather, the field we now call complex analysis has its roots in concrete problems whose solutions show special properties. It takes mathematical genius to discern in these a herald of a new domain of phenomena for which there may be an elegant general theory. Nobody suspected as much until elliptic functions arrived on the scene; they provide the first hint of transcendental functions possessing non-trivial properties, beyond the standard examples (say, the square root, the exponential, the logarithm and the trigonometric functions). The modern curriculum causes us to tend to view elliptic functions as an advanced topic in complex analysis appropriate for a second semester, just the reverse of the historical order! Elliptic integrals were first extensively investigated by Legendre towards the close of the eighteenth century, but not until decades later was the full potential of the field opened up by the pioneering researches of Abel and Jacobi, who had the vital idea of inverting the integral and viewing it as a complex-valued function of a complex argument (meanwhile, as is so often the case, Gauss was secretly working on them all along, too). Among the surprising phenomena elliptic functions display are (double) periodicity, poles and multivaluedness.
It is striking how not until the early 1850’s was Cauchy clear on what complex analysis was, as a subject in its own right—an index of how challenging the new ideas were. Nevertheless, if one pays attention, one can gain an impression of why, for once, common opinion is right in ascribing the lion’s share of the credit for putting the calculus upon a rigorous foundation to Cauchy and Weierstrass, as opposed, say, to Dirichlet, Dedekind or Riemann (Gray himself does not remark upon this). A good illustration of the light that the historian can throw upon a subject is the following: everyone knows about Cauchy’s ‘error’ about uniform continuity, yet this is not how people at the time thought about it. Of course, they were aware that Cauchy’s formulation left something to be desired (Abel’s counterexample) but the counterexamples were seen as exceptional anyway. For someone brought up on modern analysis, Gray helpfully points out that our definition of continuity at a point, as opposed to continuity over an interval, is not a notion one might stumble across very immediately. Only much later, as examples accumulated, did the need for the condition of uniform continuity in contrast to pointwise continuity everywhere become apparent. Thus, a familiarity with the research practices of the time explains why events took the course they did.
Another area where Gray’s erudition comes in helpful is the question as to why infinitesimals, which Cauchy still used in the Cours d’analyse of 1821, fell out of favor by Weierstrass’ time. The concept of quantity in the analysis of the early nineteenth century was intuitive and fairly flexible. As such, it was perfectly compatible with infinite or infinitesimal magnitudes. Once Weierstrass invented a very precise concept of real number in the 1860’s (although his would be supplanted by Dedekind’s), one discovered that the role infinitesimals played could be just as well filled by limiting processes with finite quantities, the newly-minted real numbers. Thus, infinitesimals fell out of favor. These days, one often sees proposals from would-be reformers to restore infinitesimals in the teaching of freshman calculus, because they are supposedly simpler and more intuitive—as if a freshman could take in non-standard analysis before even knowing standard analysis! But their view rests upon a misunderstanding of the mathematics itself. True, infinitesimals work nicely for analytic functions, but, during the course of the nineteenth century, mathematicians came to appreciate that the concept of a function can be much broader than analytic and that, when one expands the range of functions one is willing to consider, many subtle and counterintuitive phenomena emerge, for which more precise definitions of the terms of art of real analysis have to be devised. Weierstrassian real analysis is conceptually very clean and lends itself to effective problem-solving technique, once one surmounts the initial barrier and becomes accustomed to its strange jargon of nested quantifiers and so on. For instance, it makes possible measure theory and Lebesgue’s integral. Hence, reinstating infinitesimals, though superficially appealing, would be a step backwards. Can it really serve the student’s long-term interest to hide from him the full complexity of behavior a real-valued or complex-valued function can display and to pretend as if all functions were analytic, or at worst smooth? If one were to try to apply the transfer principle to, say, an arbitrary measurable function or a holomorphic function with an essential singularity, it would yield non-standard objects of unimaginable complexity, and this would defeat the purpose of returning to infinitesimals in the first place.
Other topics Gray covers: 1) why Fourier series are more than just an interesting application, but central to the evolving understanding of real analysis; also the distinction between trigonometric versus Fourier series; 2) the controversial Dirichlet principle; 3) the evolving function concept. People started out assuming with Legendre that every function must be locally expandable in power series up to possibly a small number of jump discontinuities. In fact, continuity and differentiability are very different properties a function can have, as had to be learned. The reader will be pleased to find in Gray’s text explicit formulae for functions exhibiting phenomena he may have heard about, but never seen spelled out in detail. For instance, Riemann’s example of an integrable function with a dense set of discontinuities; Weierstrass’ example of a continuous nowhere-differentiable function (arrived at by considering the natural domain of an analytic function, to which one is led by analytic continuation, Weierstrass’ forte, unlike Riemann’s); and Seidel’s counterexample showing that the sum of a series of continuous functions need not be continuous. As naïveté about what the typical function looks like subsided, the need for a more precise definition of the integral became apparent. Hence, the significance of Riemann’s integral and its advantages and disadvantages (it is defined not just for continuous or sectionally continuous functions, but allows for well-behaved oscillations as long as they integrate to something reasonable). Although Riemann’s integral was better than anything that had gone before, it does behave somewhat strangely. Everyone who has studied real analysis up to the Lebesgue integral will know about this, but may not perhaps have worked out many explicit examples of counterintuitive behavior. Thus, the reader will welcome Gray’s discussion of a non-integrable limit of integrable functions and Darboux’s example of the failure of term-by-term integration applied to a series.
The one major topic that gets rather short shrift from Gray is the arithmetization of the continuum, or construction of the real numbers. It is described succinctly but accompanied by little philosophy: what, for instance, would Frege, Russell, Weyl, Brouwer have to say about it? For such a fundamental development in modern real analysis, an oversight such as this is something of a surprise.
Some minor quibbles: there seem to be a lot of typos. Along the same lines, although for the most part Gray’s prose is fluent and supple, solecisms crop up more often than one would expect from an academic, such as the failure to employ the subjunctive even when ordinary mathematical language demands it; a particularly egregious instance can be found on p. 25: ‘This is typical of the way complex considerations could enter mathematical arguments at the time, but only on the condition that the final expressions are [sic] purely real. He insisted that the modulus c and amplitude ϕ were [sic] real, and that c<1’. The last few chapters are loosely organized and wandering. The author intends to whet his students’ appetite to follow developments in mathematical analysis into the twentieth century, but his grab-bag of curiosities almost degenerates into the hype characteristic of more popular literature and, in any case, his recounting of highlights is too hurried to convey anything of substance.
As is usual for writings of a historian, the present work conveys no normative thesis, but is content to retell what others felt; e.g. Weierstrass’ reception of Riemann and Cauchy. To have something non-trivial to observe on this score would require another order of thought than the historian per se is capable of. One wishes, however, that Gray could have hazarded at least a few critical obiter dicta on the present situation in the world of mathematics in light of what his researches have uncovered about the past. To supply a little of what he does not, let us suggest one lesson for us today: the string theorists resemble nothing so much as the mathematicians of the eighteenth century in their employment of heuristic methods in order rapidly to hit upon many amazing propositions (one cannot call them ‘results’ as they are not necessarily known to be true). In some cases, once suspected they can be confirmed by conventional lines of proof, but theoretical physicists appear to have uncovered a deeper secret that is responsible for generating their insights, even though, just like the calculus before the late nineteenth century, for reasons that are imperfectly understood. Will these ever become theorems according to the modern standard of rigor? Perhaps, but one should not look for this to happen anytime soon; in almost fifty years, string theorists themselves have evinced little, if any, interest in justifying their sprawling patchwork of conjectures. The foreseeable scenario is for it to collapse under the sheer weight of its speculative excess, rather as did the Italian school of algebraic geometry in the early decades of the twentieth century; who knows how long we will have to wait before a solitary genius along the lines of a Grothendieck arrives to put some order into the chaotic remains that will then be left behind? Why doesn’t Gray draw from his store of wisdom to comment upon an eventuality such as this?
But within these limitations, Gray’s scholarship is very good. It is nice to have many explicit formulae written out so that one can tell what the exposition is about, also numerous portraits of the players and figures from which one can get an idea of the convergence properties of the series under discussion. Recommended to the serious student of mathematics (at the undergraduate level or higher) who wants to improve his intuition, if not to the dilettante, for whom the amount of detail may prove a discouragement.
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