A World Ruled by Number: William Stanley Jevons and the Rise of Mathematical Economics [Course Book ed.] 9781400861514

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A World Ruled by Number

A World Ruled by Number W I L L I A M S T A N L E Y J E V O N S AND T H E R I S E OF M A T H E M A T I C A L ECONOMICS

Margaret Schabas

PRINCETON PRINCETON,

UNIVERSITY NEW

JERSEY

PRESS

Copyright © 1990 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, Oxford All Rights Reserved Library of Congress Catahging-in-Publication

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Schabas, Margaret, 1954A world ruled by number : William Stanley Jevons and the rise of mathematical economics I Margaret Schabas. p. cm. Based on the author's thesis (Ph.D.)—University of Toronto, 1983. Includes bibliographical references. ISBN 0-691-08543-9 (alk. paper) 1. Jevons, William Stanley, 1835-1882. 2. Economics, Mathematical. I. Tide. HB103.J5S33 1990 330.15'7—dc20 89-70245 This book has been composed in Linotron Primer Princeton University Press books are printed on acid-free paper, and meet the guidelines for permanence and durability of the Committee on Production Guidelines for Book Longevity of the Council on Library Resources Printed in the United States of America by Princeton University Press, Princeton, New Jersey 10

9 8 7 6 5 4 3 2 1

Frontispiece: William Stanley Jevons, aged 22, from a photograph taken in Australia, 6 February 1858

To my parents, Ann and Ezra

Not without reason did Pythagoras represent the world as ruled by number. Into almost all our acts of thought number enters, and in proportion as we can define numerically we enjoy exact and useful knowledge of the Universe. —Jevons, Principles of Science

Contents

Preface

xi

CHAPTER ONE

Mathematical Pursuits

3

CHAPTER TWO

Jevons's Life

12

CHAPTER THREE

The Mathematical Theory of Economics The Theory of Value The Theory of Exchange The Theory of Labor The Theory of Rent and Capital Conclusion

31 34 39 43 47 52

CHAPTER FOUR

Logic and Scientific Method Formal Logic Induction Scientific Progress Conclusion

54 57 65 72 78

CHAPTER FIVE

Markets and Mechanics Theory and Evidence Mechanical Analogies Statics and Dynamics The System Generalized Conclusion

80 81 84 89 92 96

CHAPTER SIX

Response to the Theory Cairnes Mill The Economic Historians The Moderates Conclusion

98 100 103 110 113 117

χ · Contents CHAPTER SEVEN

Mathematical Economics Takes Hold Darwin Wicksteed Edgeworth Marshall

119 119 121 123 126

CHAPTER EIGHT

Mathematical Hegemony

135

Notes

141

Select Bibliography

111

Index

187

Preface

LIKE SO MANY first books, this one stems back to my student days. During my first semester in graduate school at Indiana University, I had the good fortune to enroll in Scott Gordon's course on the history and philosophy of the social sciences, and was soon struck by the peculiar attributes of economics. It seemed to offer a perfect blend of logic, ethics, and practical reasoning, or, in Robert Heilbroner's apt words, a "worldly philosophy." My studies in the history and philosophy of science continued at the University of Toronto, under the guidance of Samuel Hollander and Trevor Levere. Both had that uncanny sense of knowing when to impose discipline on my prose and ideas and when to let me loose to explore on my own. As I now guide others, my respect for them both as teachers and scholars has grown ever greater. Many others have subsequently read my manuscript and offered guidance. I wish particularly to thank R.D.C. Black, Scott Gordon, Warren Samuels, Robert Nye, I. B. Cohen, Daniel Hausman, Craufurd Goodwin, and Theodore Porter. Mary Morgan, Ann Schabas, S. S. Schweber, and Michael Green have provided helpful suggestions on specific chapters. Countless others have directed me to important sources or patiently allowed me to air my views. Credit is due particularly to David Hollinger, Tracy Mott, David Mapel, Timothy Alborn, Craig Fraser, Eric Gorham, Thomas Archibald, and Judith Margies. Special thanks also go to the members of the 1986 Summer Institute at the Center for Advanced Study in the Behavioral Sciences at Stanford University for subjecting my ideas to considerable scrutiny. The diagrams are courtesy of illustrator Kandis Elliot. Research fellowships from the Social Sciences and Humanities Research Council of Canada (1981-1983) and from the Andrew W. MelIon Foundation (1987-1988) helped me begin and complete this work. But for the most part, my bread and butter was earned in the classroom and, like many of my generation of aspiring scholars, I have become acquainted with the rituals of more than one American college town. Since completing my Ph.D., I have held teaching appointments at Michigan State University, the University of Michigan, the University of Colorado, Harvard University and, last but not least, the University of Wisconsin. More present and former colleagues than can be named here have offered their support and encouragement and I thank them one and all.

χϋ · Preface Portions of this book have already appeared in print, in Victorian Studies (1984), History of Political Economy (1985), and Isis (1989), and I am grateful to the editors of each journal for allowing me to reproduce the material here. Peter McNiven, archivist at the John Rylands University Library at Manchester, was most helpful in arrang­ ing my two visits to the Jevons Archives, and archivist Barbara Geller allowed me to peruse the Seton-Jevons Papers at Seton Hall Univer­ sity, New Jersey. But there is no question that the most invaluable resource for this study is the handsome seven-volume edition of the Papers and Correspondence of William Stanley Jevons, edited by Pro­ fessor R. D. C. Black of the Queen's University, Belfast. Every time I pore over one of his copious notes, I am reminded of how great my debt is to his efforts. Let me also thank the Macmillan Publishing Company for permission to quote extensively from their volumes, and Anthony Hayter, executor of Rosamond Konekamp's estate, for per­ mission to cite the unpublished Jevons material and reproduce Jevons's photograph. Mrs. Konekamp, who passed away in 1986, was Jevons's granddaughter and his last direct descendant. I was first drawn to the period of the early neoclassical economists because it offered the allure of a scientific revolution, replete with si­ multaneous discoveries, new methods, and theoretical dislocations. Over time, I became convinced that the 1870s was the period during which Western economic thought underwent its most dramatic trans­ formation. Jevons appealed to me not only as a leading instigator of the revolution, but also for his remarkably broad range of contribu­ tions to logic, the history and philosophy of science, and economics. And, while there are many short assessments of his contributions, this is the first full-length study of his life and work. His primary program of mathematizing economics raises numerous questions on the epistemological status of economics and mathemat­ ics. I have found, over the course of six or seven years, that my views on these subjects have changed considerably and that Jevons had an even richer set of insights than I had initially suspected. I hope that the reader will also come to appreciate his very distinctive grasp of the economic order while at the same time savoring his Victorian sensi­ bility. That the world is ruled by number may still be subject to dis­ pute, but not the element of simplicity and beauty in Jevons's vision of mind and matter. He was, in short, a Pythagorean without the mysticism.

A World Ruled by Number

CHAPTER ONE

Mathematical Pursuits Mathematical economics isridinghigh. It seems much longer ago than 1871 that Jevons should feel compelled to offer a variety of defences for his primitive symbols. —George Stigler

EVEN A brief perusal of a shelf of contemporary economics journals would convince someone unfamiliar with the field that economics is a subject steeped in mathematics. But this was not always the case. Although political economy emerged by the mid-eighteenth century as a distinct discipline, it was not until the last third of the nineteenth century that mathematics took hold within it, and much later, somewhere in the 1930s or 1940s, that economic theorists began to study the calculus as a matter of course. For some, mathematical renditions of economics are nothing more than attempts to dress the subject in the garb of the more revered science of physics, and thereby mask what are essentially apologetics for capitalism. There may be a grain of truth in this, in that all knowledge is imbued with the ideology of its time and its practitioners may be said to seek ways, perhaps unwittingly, to sustain the power that comes with expertise. No one would deny that economic thought of the last three or four centuries is by and large a product of the rise of Western capitalism and, insofar as much of the subject matter is drawn from extant economic phenomena such as banking, private property, and labor markets, is undoubtedly a reflection of the specific characteristics that mark this epoch in world history. Prima facie, this seems to be the case far more than in the natural sciences. But it does not follow that economists of the past one hundred years have conspired to dupe the public, or even that they have been engaged in selfdeception with regard to their scientific credentials. Mathematics might appeal for its scientific sheen, and as a barrier to entry into the profession, and yet nonetheless enrich our understanding of the economy, capitalist or noncapitalist. If mathematics offers the allure of a hard science, why did economists take so long to enlist its aid? Since Hobbes, social theorists have sought to emulate the exact sciences of astronomy and mechanics. In

4 · Chapter 1

many respects, the human sciences were the focal point of Enlightenment thought, particularly in France and Scotland. Starting in the eighteenth century, one finds various attempts to infuse the moral sciences with elements of geometry, algebra, or the theory of probability; the example of Condorcet is legendary, but there were many others. Each attempt, however, either failed to garner a following or was explicitly denounced by such prominent figures as Jean-Baptiste Say and Auguste Comte.1 Perhaps what was lacking was a judicious use of mathematics, although the case of Augustin Cournot (1838) suggests otherwise. Or perhaps few economists knew enough mathematics to make significant inroads. It so happens, however, that a number of prominent nineteenth-century economists, such as Thomas Robert Malthus and John Stuart Mill, had studied mathematics without succumbing to its charms. A knowledge of mathematics, while a necessary condition for the transformation under consideration here, was clearly not a sufficient one.2 The most important figure in shifting the tide was William Stanley Jevons (1835-1882), and much of this study will focus on his contributions to the rise of mathematical economics. Trumpeting the virtues of mathematics was easy. What set Jevons apart from numerous predecessors, and enabled him to secure a following where others before him had failed, were his analyses of logic and scientific methodology, his assimilation of new ideas in mathematics, and the concrete details of his economic theory. Other factors, which will only be touched on here and there, also helped his cause: advances in mathematical education and the growth of universities, stagnation and strife among economists of the post-Millian era, and the more widespread shift to professionalization starting in the 1880s. Jevons is usually discussed within the context of the so-called Marginal Revolution, which is said to have begun in 1871 with the publication of Jevons's Theory of Political Economy and Carl Menger's Grundsatze der Volkswirtschaftskhre. Subsequent reinforcement purportedly came in 1874 with Leon Walras's Elements d'e'conomie politique pure. Each of these professors of political economy—Jevons at Manchester, Menger at Vienna, and Walras at Lausanne—had independendy arrived at the principle of diminishing marginal utility and proclaimed this insight to be the cornerstone of a significandy new science of economics. For this reason, the Marginal Revolution has traditionally been characterized as a recasting of the theory of value. Whereas the classical theory of Smith, Ricardo, and Mill had focused upon costs as the determinant of exchange-value, the early neoclassical theory emphasized utility. Or so the story is told in countless texts on the history of economic thought.

Mathematical Pursuits · 5 Since the 1970s, however, scholars have begun to question this account. The impetus for their reinterpretation derived more, it appears, from careful historical research than from collateral debates on the structure of scientific revolutions, though these have also left their mark. Studies of various pre-1870 economists established quite conclusively that the principle of diminishing marginal utility had been discovered several times throughout the last century, and recognized as a central concept from which to derive an explanation of market prices.3 Other scholars have challenged the received view since Joseph Schumpeter that "Jevons, Menger, and Walras taught essentially the same doctrine." William Jaffe, for one, has argued that each of these economists, working in isolation from one another, built upon distinct intellectual traditions and drew different implications from the principle of marginal utility. Erich Streissler has also done much to separate Menger from Walras and Jevons, primarily because of his disdain for mathematics. The difficulties in treating Jevons, Menger, and Walras as a united front have led some historians, such as Mark Blaug, to deny categorically that a revolution happened at the time.4 If one looks only at Britain, however, a strong case can be made for a "Jevonian Revolution."5 As Terence Hutchison has argued, "in the space of a few years in the late 1860s and early 1870s the classical structure of 'theory' underwent a remarkably sudden and rapid collapse of credibility and confidence, considering how long and authoritative had been its dominance in Britain." Until Jevons, British economic thought from Locke to Mill formed one long, unbroken tradition. Even the occasional injection of foreign notions, Say's law most saliently, grew from seeds planted by Adam Smith. For the most part, nineteenth-century British economists wrote in a state of "splendid isolation," bolstered by appeals to such indigenous philosophical currents as utilitarianism. Jevons's own allegiance to British thought and exposure to foreign works is a case in point. Although he read French well and German poorly, he was schooled almost exclusively on British texts and was totally unaware of explorations similar to his own on the Continent. With time, he came to realize that the "truth lay with the French" rather than the British, and helped to foster intellectual trade across the channel. Jevons never learned of Menger's existence and, although thenwork was subsequently lumped together, more probing scholarship has tended to set them apart.6 Certainly, the Austrian School that Menger spawned has retained a distinct identity well into this century. Although Walras corresponded with both Jevons and Menger, he failed to establish a close rapport with either man and was relatively unappreciated within French-speaking circles until the second or

6 · Chapter 1

third decade of this century.7 He will be treated here as part of the Jevonian Revolution, as one who, like Alfred Marshall, helped to secure the practice of mathematics within economics. Not that he was a subordinate theorist. If anything, late twentieth-century economists hark back to Walras rather than to Jevons. But the movement depicted here, the emergence of mathematical economics in the late nineteenth century, is one in which Walras's role was that of supporter rather than leader. There is little doubt that Jevons regarded mathematics as the means to further the scientific standing of economics. A reading only of the introductory sections to his Theory of Political Economy certainly gives the impression of unfettered opportunism. Yet, as our study of his Principles of Science will demonstrate, he was acutely aware, far more than previous economists, of the many limitations of scientific knowledge. The contrast with John Stuart Mill is particularly striking. Jevons, by and large, marks the shift toward fallibilism and conventionalism that preoccupied the fin-de-siecle methodologists Ernst Mach, C. S. Peirce, and Henri Poincare\ During the nineteenth century, mathematics lost its supreme epistemological standing and came to be seen more as a convenient tool than as an essential key to unlock the mysteries of the universe. Jevons was one of the first in Britain to appreciate and disseminate these findings. In fact, he was the first to respond to Helmholtz's celebrated paper on non-Euclidean geometry and one of the first to assimilate Boole's contributions to mathematical logic. All of this, we shall see, placed him in the timely position to argue that mathematics had as legitimate a place in economics as it did in the physical sciences. One implicit aim of this book is to assess from an historical perspective the extent to which economic theory is amenable to mathematical analysis. A priori, there is no obvious reason why economists should ever have adopted mathematics. For many it is still difficult to understand how numbers link up with human behavior. But one might make a similar case for the behavior of physical objects. It is possible to imagine, albeit with more difficulty, a history of physics in which mathematics does not appear. The physics that we do possess, however, is fundamentally mathematical. Its central concepts of matter and force are only fully explicable in the language of mathematics.8 Merely recognizing that this is the cumulative result of independent actors does not render it accidental. Similarly, economics could hardly have become the densely mathematical discipline it is at present were it not for some fruitful anchoring of mathematics at the theoretical level. To suppose otherwise is to assume a widespread conspiracy on the part of thousands of seemingly intelligent men and women. No

Mathematical Pursuits · 7 doubt, the majority of economists might admit that at times they are excessive in their use of mathematics, often at the expense of empirical relevancy. At least, such indulgences have been a frequent source of humor, even from within the ranks.9 But there are now numerous instances where mathematics has yielded interesting results, and recent cases where economists have contributed to certain branches of mathematics, such as set theory and topology.10 The use of geometry or curves, which also dates back to Jevons, has become so much a part of the economist's craft as rarely to be noticed when the subject of mathematics is broached. Historians of economic thought have often assumed that nineteenth-century science was organized along the same lines as the present. But historians of science now paint a very different picture. To be sure, the nineteenth century was a time of convergence and conceptual unification, as well as one of increasing specialization and professionalization. In many respects, however, practitioners of science had more in common with the savant of the Enlightenment than with the scientist of the mid-twentieth century. Polymaths abounded: John Herschel, Hermann von Helmholtz, and Louis Pasteur to name just a few. It would not be misleading to assert that virtually every nineteenth-century contributor to a major field recognized today— physics, astronomy, chemistry, biology, geology—left his or her mark in another, seemingly disparate, field. This suggests an obligation on the part of the historian to resist imposing current categories on the past, even the fairly recent past of the nineteenth century. As Thomas Kuhn has argued, there were two quite separate traditions in the physical sciences—the experimental and the mathematical—up until the 1830s.11 And whereas the experimental tradition ranged over problems we would now readily place within chemistry (or even physiology), the mathematical tradition had long included astronomy. Similarly, there were two separate strands in the life sciences—natural history, which encompassed morphology as well as geology; and physiology, which blended seamlessly into medical practices. The two converged toward the end of the last century, but until the 1860s or thereabouts were kept quite distinct. Political economy's flirtation with various fields, whether sociology, law, or history, was thus in keeping with much of early nineteenthcentury science. In many respects, it was no more or less diffuse than any other branch of knowledge. Efforts were made to rename the field or to formulate a narrower domain of discourse, such as Richard Whately's neologism "catallactics" for a science of exchange. But by and large, there was considerable consensus over the definition and scientific status of political economy. Virtually every treatise began

8 · Chapter 1

with a declaration to the effect that political economy is the "Science which treats of the Nature, the Production, and the Distribution of Wealth." According to J. R. McCulloch, political economy "admits of as much certainty in its conclusions as any science founded on fact and experiment can possibly do."12 It is worth emphasizing the extent to which political economy, by the middle of the last century, was recognized as a legitimate, if not exemplary science, since historians have had a tendency to project current insecurities over the status of economics back into the past. One source of confusion, perhaps, are the frequent appeals by classical economists to Newtonian mechanics. But this was de rigueur of every scientific tract. Indeed, whereas political economists could readily proclaim Adam Smith as the Newton of his age, chemists were divided between Lavoisier and Davy. Another source might be the general conflation of science with technology in our century and the tendency to appraise science in terms of its engineering feats. But in the time of Smith and Ricardo, science was primarily a branch of philosophy, the search for truth rather than a technical mastery of the world. Many in the natural sciences, John Herschel and William Whewell most notably, had a high esteem for, if not active interest in, political economy.13 Charles Babbage led a successful campaign in the early days of the British Association for the Advancement of Science to secure a place—the notorious Section F—for political economy and statistics. At Cambridge, the tripos in the moral sciences was established at the same time as the one for the natural sciences (1838). We have tended to lose sight of the immense impact Smith, Malthus, and Quetelet had with their insights into the social order. But there is reason to believe that Darwin and Maxwell, among others, worked under their spell. As historians are beginning to recognize, one cannot fully assess Victorian science without considering the impact and influence of political economy.14 By early nineteenth-century standards, political economy was about as bona fide a science as chemistry. According to the received view, both emerged in their modern guise in the 1770s. But it was not until the 1830s that these subjects were widely recognized in the form of university professorships. Until that time, chemistry was treated as an adjunct of the medical faculty, while political economy, particularly on the Continent, was taught as a branch of law. As a factor in gaining greater autonomy, chemistry had the additional advantage of serving industry. Its rise in university curricula matched almost year for year the transformation of Britain into the workshop of the world.15 A steady increase in the teaching of political economy, at the uni-

Mathematical Pursuits · 9 versities and Mechanics Institutes, also mirrored the commercial growth of London and the Midlands. But it had the drawback of being less overtly practical. Despite widespread concern for poverty and labor unrest, the general lesson of laissez-faire did not suggest a distinct career path. Jevons's life, quite appropriately, instantiates all of these generalizations. He studied chemistry in preparation for a career in industry but, upon switching to political economy, found himself without obvious employment. Nowadays, the two subjects are probably comparable stepping-stones into the business world. Early nineteenth-century political economy had a substantial body of texts, a set of laws, and established methods for advancing the subject. Mill's oft-quoted phrase, that "happily, there is nothing in the laws of Value which remains for the present or any future writer to clear up," captures not complacency as much as pride in what the field had achieved up to that point. His Principles of Political Economy (1848) assimilated the many insights of a long and rich discourse on the subject of value and the distribution of wealth by such luminaries as Smith, Say, Malthus, and Ricardo. Critics existed, to be sure. But for every denunciation of political economy as the dismal science (note, however, that it was still a science), there were appeals to its contribution to Britain's economic hegemony. Nassau Senior boasted to a French visitor at the Great Exhibition of 1851, "it is a triumph of theory. We are governed by philosophers and political economists." And Walter Bagehot remarked at the centenary celebration of Smith's Wealth of Nations, "the life of almost every one in England—perhaps every one—is different and better in consequence of it."16 Clearly, few other sciences could claim such widespread benefits, or such striking confirmations. Having entrenched itself in the scientific establishment, British economic theory then languished in the 1850s and 1860s. Much of this has been attributed to the enormous success of Mill's Principles in settling any remaining controversies, as well as to the far-reaching benefits of free trade. Nonetheless, the problems of labor unrest, poverty, and a growing national debt sparked some debate. In the late 1860s and early 1870s, as Hutchison has maintained, the "four central pillars" of Mill's theory—the wages fund, labor theory of value, natural wages doctrine, and Malthusian constraints—suddenly collapsed.17 Jevons's manifesto of 1871 was thus perfectly timed and his call "to fling aside, once and for ever, the mazy and preposterous assumptions of the Ricardian School," won many of the hearts and minds of the younger generation of economists.18 Although Jevons stands at the very juncture at which classical political economy transmuted into neoclassical economics, there are no

10 · Chapter 1

books, apart from an obscure short monograph, specifically devoted to his life and work.19 Most surveys of the history of economics allot a chapter or section to his contributions, but the treatment is rarely adequate. There are two excellent accounts of Jevons's life, by John Maynard Keynes and Rosamond Konekamp, and a number of perceptive appraisals of his economics by R.D.C. Black.20 Shortly after Jevons died, his wife edited a volume of his letters and journal. But large quantities of material were omitted or missing at the time. Since the 1960s, Professor Black has tracked down and edited some seven volumes of Jevons's correspondence, lecture notes, and papers, but even this collection by no means exhausts Jevons's unpublished remains.21 Jevons was a fairly introverted man. Although he seemed to have had warm relations with his university colleagues, there are virtually no extant descriptions of his personality. His close friend of his last few years, Francis Ysidro Edgeworth, spoke of his "philosophic smile" and "serene and lofty intellect," an observation that matches the half dozen photographs of Jevons at various points in his life. A great deal more can be gleaned of Jevons's character from his journal and letters. Particularly in his formative years, he made efforts to record his inner thoughts and aspirations on paper. His letters to his siblings reveal a very sensitive, clearheaded individual, and though they are filled with deeply rooted ambitions, there is remarkably litde to entice the psychoanalyst in comparison to Mill or Marshall. Although Jevons wrote some nine books on economics, he regarded his Principles of Science (1874) as his greatest achievement.22 John K. Whitaker has maintained that Jevons's philosophy of science and work in economics, unlike those of John Stuart Mill, were "quite divorced." Ostensibly, Whitaker is right, for unlike Mill in his System of Logic (1843), Jevons in his Principles makes hardly any attempt to reveal the connection to his economics.23 But this, we shall see, does not belie the existence of a fundamentally important linkage. Thus far, attempts to identify these ties have concentrated on Jevons's applied economics and his insights into the probabilistic nature of scientific laws.24 But much remains to be said on the subject. This study will emphasize the extent to which Jevons's views on logic and scientific method shaped his program for mathematical economics and thereby enabled others to see past Mill. In the next chapter we will sketch his life and delineate the motives that led him to advocate a new system of thought. Given the task at hand, it will not be possible to consider, except in passing, his many achievements in the fields of applied economics, monetary theory, and social reform. There is simply too much material, his short life notwithstanding.25 The focus will be primarily on his two most important theoretical

Mathematical Pursuits · 11 works, The Theory of Political Economy and The Principles of Science. Chapters Three and Four examine each work in turn, as far as is appropriate for our theme, and Chapter Five develops their common ground. Chapters Six and Seven evaluate the reception and development of Jevons's program to reformulate economics. It will be argued that, as early as the mid-1880s, the majority of leading English economists welcomed the mathematization of their discipline and used mathematical tools in their writing or teaching.

CHAPTER TWO

Jevons's Life I have discovered someone whom I had not realized to be very good—namely Jevons. I am convinced that he was one of the minds of the century. —John Maynard Keynes to Lytton Strachey My whole second nature consists of one wish, or one intention, viz to be a powerful good in the world.... Not towards one or a dozen, or a hundred but towards a nation or the world. —Jevons, age twenty-two

was born in Liverpool on September 1, 1835. Both his parents were Unitarian and solidly middle-class. His paternal grandfather, William Jevons, had established an iron trade in Liverpool at the beginning of the century and by the time of Jevons's birth the firm of Jevons & Sons was prospering.1 Jevons's father Thomas was the oldest of these sons and a steadfast member of the family firm. He also wrote two short pamphlets on political reform and was one of the first to launch a seaworthy iron boat. Jevons's maternal grandfather, William Roscoe, had risen from humble beginnings to amass considerable wealth as a lawyer and banker. He then took to politics, philanthropy, and scholarship. As a Whig Member of Parliament, he championed the abolition of slavery and prison reform. Liverpool remembers him for the collection of Italian and Flemish paintings he donated to the Walker Art Gallery and for the botanical gardens he helped to found. Most of all, he is known for his many books on history and poetry. Roscoe lost his fortune in a commercial crisis of 1820 and died in 1831, before Jevons was born.2 By all accounts, Jevons's mother, Mary Anne, was a woman of exceptional intelligence and fortitude. Before her marriage at the age of thirty, she had assisted her father in his literary endeavors and published two books of her own poetry. Her marriage to Thomas Jevons brought eleven children, though only six survived infancy. William Stanley, who came ninth, was said to have most resembled her in countenance and to have absorbed from her his love for learning and music. WILLIAM STANLEY JEVONS

Jevons's Life · 13 When Jevons was very young, his family and the extended family on his father's side—grandparents, uncle, aunt, and cousins—moved to three adjoining houses on the outskirts of the city. The large common garden provided ample space to roam and play and Jevons would later recollect these years with much pleasure. We do not know at what age he first ventured from this somewhat idyllic existence and witnessed the horrors of Liverpool, but whenever the trip took place, it probably left a lasting impression. During the hungry forties, Liverpool experienced unprecedented immigration from Ireland. An estimated 300,000 arrived in 1847 alone, many of them forced to sleep in cellars or alleyways. During those years, approximately half of Liverpool's children died before the age of eleven.3 Despite Jevons's own auspicious beginnings, the story of his family is marked with much sorrow. In 1845, when Jevons was but ten years of age, his mother died following a prolonged illness. Three years later, a widespread collapse of the railway boom brought bankruptcy upon the family firm. The entire clan disbanded and took lodgings in the city. Although Jevons's father found other employment, he never recovered financially and, upon his death in 1855, left them little means of support. Jevons's oldest brother, Roscoe, having shown exceptional gifts in his adolescence, became terminally insane following the death of his mother. The next older brother, Herbert, was unable to shoulder the responsibilities of adulthood and was plagued with poor health. After a series of unsuccessful ventures, including a brief attempt at homesteading in Minnesota, he finally settled in New Zealand but died of cancer at the age of forty-two. A younger sister, Henrietta, though she lived to seventy, spent most of her adult life in an asylum. His other sister, Lucy, married happily but suffered from financial insecurities and the loss of a ten-year-old child. Only the youngest brother, Thomas, lived a conventional life as a banker in the city of New York and son-in-law of the prominent Seton family. Jevons was raised as his father had been. He was given a very fine education designed to develop gendemanly interests in science, the arts, and politics, but it was assumed throughout that he would pursue a career in business. Jevons was tutored at home until his mother died and then spent two years at the Liverpool Mechanics Institute.4 This was followed by two years at Mr. Beckwith's day school where the emphasis on classics was not to Jevons's liking. At fifteen, he was sent to the Junior School of University College, London, and then to University College proper. For two of these years, he lodged with his cousin, Henry (Harry) Enfield Roscoe, who later became one of Britain's most eminent chemists. Although only two years older, Harry

14 · Chapter 2

exerted considerable influence over Jevons and was to have a hand in many of his professional advancements.5 Jevons's studies went well and he received a number of prizes, including the gold medal in chemistry and top honors in experimental philosophy. His courses in mathematics were taught by one of the leading mathematicians of the time, Augustus De Morgan. Jevons was pleased to have placed fourth in the lower senior class.6 He also took advantage of London's rich cultural life, attending concerts, royal ceremonies, and, most memorably, the Great Exhibition of 1851. A particular treat was the opportunity to hear Michael Faraday lecture at the Royal Institution, and there to set eyes on John Herschel and Charles Lyell. He also began to study the "industrial mechanism of society," venturing into the dark alleyways of Spitalfields to observe firsthand how the poor worked and lived. Toward the end of his second year, Jevons made arrangements to secure a good starting position with a chemical firm. It is not entirely clear why he was unable to stay a third year to complete his degree. He certainly showed no lapse of interest in his studies and, in his diary that year, expressed ambitions of contributing to the theory of atomism. Then an unexpected turn of events took him to Sydney, Australia, to serve as assayer to the new mint. This came about as follows. Thomas Graham, the professor of theoretical chemistry, had nominated Harry for the position, but Harry declined in order to pursue a doctorate under Robert Bunsen at the University of Heidelberg. Harry proposed his cousin, who initially considered the offer out of the question because it would take him away from home. Jevons had already found life in London sufficiently lonely and looked forward to his move back to Liverpool. His father, however, made it quite plain that he must accept the post, not only because it meant a career with considerable security, but also because the estimated salary of five hundred pounds would be of great assistance to the family. To allay his son's apprehensions, Thomas assured him that he need stay no longer than five or ten years and would therefore not be parting from the family forever. Although Jevons later claimed that he ought to have disobeyed his father and remained in England, he also realized that the "visit to Australia, by breaking my college course & giving time to mature my powers did peculiar service."7 And he certainly never regretted the opportunity it gave him to see the world. As it turned out, Jevons might have completed his bachelor's degree and still taken the post in Australia. He spent the summer of 1853 working as an assistant to Graham and by early October was certified to be a capable assayer. While awaiting news of the completion of the

Jevons's Life · 15 Sydney mint, he spent two months in Paris where he received the diploma from the French mint. He did not sail for Australia until the following June and, on reaching Sydney, discovered that he had still arrived nine months too soon. In all, then, almost two years elapsed between his acceptance of the post and the opening of the mint. Jevons adjusted well to his new work. After a year or two, the demands for assaying at the mint slackened off and he was free to devote more time to scientific investigations. Though he did not continue with his two favorite subjects, mathematics and chemistry, a fascination for the Australian landscape led him to study the local flora, rock formations, and weather.8 For almost three and a half years he kept daily meteorological records and, for one of those years, was the only weather recorder in Sydney. During those first years in Australia, he also wrote two unpublished books, one a social survey of Australian cities and the other on music theory.9 In 1856, a dispute over the funding of the first railway for New South Wales sparked his interest in economics. Jevons began to read Adam Smith, John Stuart Mill, Thomas Robert Malthus, Richard Whately, and Harriet Martineau. His letters home reveal an immediate affinity with the discipline: "I have been much occupied lately with a new Subject viz, Political Economy, which seems mostly to suit my exact method of thought." He also praised Smith's Wealth of Nations as "an excellent though rather old book."10 By the end of 1856, he had resolved to stay in Australia for only a few more years and then return to London to study political economy and higher mathematics. That Jevons became interested in economics by reading about a railway controversy in the local papers is an oversimplification. It took his five years in Australia to come to terms with his life goals and abilities. Soon after he had arrived, he had announced to his cousin Harry: "Through the whole of science there is nothing that I should like to follow up more, & spend my whole life in than the atomic theory & theoretical chemistry, but my road seems to lie another way." Here, and in later letters, he expressed misgivings for having left his university studies. It must have been difficult for him to reconcile the fact that his cousin had been in the position to turn down the Australian post for further studies under one of Europe's leading scientists and he had not. As he related in a letter to Harry in October 1856: I myself am an awful deserter, having given up all ideas of engaging in chemistry again; indeed I doubt if I shall ever be able to call myself a scientific man, for although I am yet deep in more than one scientific subject,

16 · Chapter 2 Ifindmyself gradually spending more and more time over other more general subjects for which I believe I am equally well or even better suited.11 Although one can only guess that Jevons had political economy in mind as one of these "more general subjects," it is clear that he had decided to cast his net away from the natural sciences. Harry, who had read some of his cousin's papers on cloud formation, encouraged him to continue his meteorological research. It would seem likely too that, if Jevons had wished to conduct research in theoretical or experimental chemistry, he had the time and the laboratory at hand to do so. Instead, he spent much of his spare time at the office reading the newspapers, and during the evenings he read books or played the organ. By 1858, he announced to his sister Henrietta, "I am fond of Physical science and I believe not unsuccessful in it; but I must give it up."12 There are certain personal indications that may also account for Jevons's transition from the natural to the social sciences during his Australian sojourn. Following his father's death in 1855, Jevons experienced something of a religious crisis. Although at the time he was reluctant to express his thoughts to his more conformist sisters, he later admitted to being "still afloat and on the look out for an anchoring ground." In 1858, he confessed to his sister Henrietta: "do not misunderstand me when I say that I am in some respects an Atheist. .. . On the other hand . .. I almost Deify the love of Man."13 Like so many before and since, Jevons found in the social sciences a means to vent his "love of man." To extend and perfect the abstract or the detailed and practical knowledge of man and society is perhaps the most useful and necessary work in which anyone can now engage. There are plenty of people engaged with physical science,... but thoroughly to understand the principles of society appears to me now the most cogent business. It was at this point that Jevons acquired a sense of purpose that was to sustain him in his work for the rest of his days. To his sisters he wrote: "I have an idea, which I do not object to mention to you, that my insight into the foundations and nature of the knowledge of man is deeper than that of most men or writers. In fact, I think it is my mission to apply myself to such subjects, and it is my intention to do so."14 These internal convictions were strengthened by a growing recognition in Sydney's intellectual circles of Jevons's various scientific accomplishments. At the time, with two years of university studies to his credit, he was undoubtedly one of the educated elite of Australia.15

Jevons's life · 17 Having previously been overshadowed, first by his precocious brother Roscoe and then by his cousin Harry, he found his self-esteem growing. "My papers," he noted, "were so freely printed in the colonial papers & magazines, that I entertained no doubt it would be the same at home." And to his brother Tom, he divulged, "I begin to think I am rather more clever than I expected." But Jevons was also quite consciously motivated by an inner drive which had nothing to do with fame or fortune: "My whole second nature consists of one wish, or one intention, viz to be a powerful good in the world. . .. To be powerfully good, that is to be good, not towards one or a dozen, or a hundred but towards a nation or the world, is what now absorbs me."16 These confidential remarks to his siblings were also for a more immediate end. Jevons was placed in the difficult position of having to convince his brothers and sisters that he should give up his comfortable job in Australia and the salary which pardy supported them, in order to continue his studies in London. During his last year in Australia, the decision was made more difficult by the offer of a position in Melbourne with an annual salary of about two thousand pounds. For a man of twenty-four, with two unwed sisters and a younger brother just starting his university studies, this was a very large sum of money to forgo. Fortunately, his family accepted his plans to become a "powerful good in the world." They did not discourage him from returning to England nor, judging from the extant correspondence, did they discourage him from using some of his savings to visit other countries on the return journey, even though he gave them ample opportunity to do so. The only person who tried to dissuade Jevons from turning to the social sciences was Harry, who thought it best that he stick with the more practical work of assaying. Unfortunately, the letter in which Jevons reacted to his cousin's advice is missing. But Harry's subsequent reply was more supportive and gives some clues of Jevons's intentions. Harry wrote, "I daresay you will have interest enough left in Physical Science in spite of your taking to the Mathematics of Society." It is quite likely that Jevons, in the missing letter to Harry, had disclosed his plans to focus on mathematics so as to gain his cousin's approval. On two previous occasions Harry had written to Jevons about the fundamental importance of mathematics in the natural sciences. "A man to do anything of that sort [theoretical chemistry] must before everything be a most sound and deep mathematician." This advice must have weighed heavily on Jevons and dashed any hopes to accomplish anything in theoretical chemistry, for he had already come to terms with his limited aptitude for mathematics.17 In the spring of 1857, Jevons happened upon a copy of Dionysius

18 · Chapter 2 Lardner's Railway Economy (1850), which contained a geometrical treatment of supply and demand and notions of marginal cost and revenue. He later acknowledged it as the work to which he "was probably most indebted . . . [for] the idea of investigating Economics mathematically."18 Other remarks of his also lend weight to the view that while in Australia he had already decided to reform economics. To Henrietta he wrote in 1858: "You will perceive that Economy, scientifically speaking, is a very contracted science; it is in fact a sort of vague mathematics which calculates the causes and effects of man's industry, and shows how it may best be applied."19 And in the following year, he wrote to Harry: You know I am yet in a transition state. I told you, long since, that I intend exchanging the physical for the moral and logical sciences, in which my forte will really be found to lie.... I wish especially to become a good mathematician, without which nothing, I am convinced, can be thoroughly done. Most of my theories proceed upon a kind of mathematical basis.20 Jevons also reflected on problems of ethics and human agency. In 1856, he echoed Jeremy Bentham's views on human nature. "I regard man in reality as essentially selfish, that is as doing everything with a view to gain enjoyment or avoid pain. This self-interest is certainly the main-spring of all his actions." After reading Quetelet's Treatise on Man in 1857, he acknowledged that "each individual man must be a creature of cause & effect." From this it was only a short step to the belief that laws govern the social realm: But does it not strike you that just as in Physical Science there are general & profound principles deducible from a great number of apparent phenomena, so in treating of Man or Society there must also be general principles and laws which underlie all the present discussions & partial arguments? Is it not worth years of labour to dive into these inmost & obscurest principles (?]... To attempt to define the foundations of our knowledge of man, is surely a work worth a lifetime, and one not excelled in usefulness or interest by any other.21 In May 1857, Jevons began to write a book "on Anthropology or the general consideration of Man," only to realize within a month that a "comprehensive work on Man in Society . . . would take two lifetimes to finish." His awareness of the richness of human activity also suggested the need to focus on a more restricted domain of behavior: The causes which operate in each man, letting alone a collection of men, are so very complex that their effects supply innumerable facts for many branches of knowledge.... A perfect consideration of all these result in a

Jevons's Life · 19 determination of all effects: for instance in the case of a single person it must explain every trait of his character, every action of his life, every word he has spoken, every thought he has conceived. Of course such is the infinite complexity of causes & of effect that we cannot treat them in detail. A few of the main features of Man & Society afford plenty of occupation. Political economy appeared to him to be the queen of the social sciences, and thus the discipline to pursue. "There are a multitude of allied branches of knowlege connected with man's condition; the relation of these to political economy is analogous to the connection of mechanics, astronomy, optics, sound, heat and every other branch more or less of physical science, with pure mathematics."22 Evidently, these five years in Australia were of fundamental importance to Jevons's intellectual development. Prolonged reflection on his religious beliefs and the nature of morality fostered an interest in the human sciences. He began to study the classic texts of political economy and, almost from his first encounter with the subject, discerned its potential as a mathematical science. A growing recognition of his own abilities gave him the courage to sacrifice his comfortable existence as an assayer and risk the life of a scholar. As he mused a few years later, "it was a bold & momentous decision which brought me out of Australia. I shall not regret [it] even if my remaining days be spent in poverty."23 Jevons returned to England in 1859 by way of the Americas, so as to visit his brother Herbert. On arriving in London, he took up lodgings with his two sisters and his younger brother Tom, whom he joined as a student at University College. Jevons spent most of his first year back at the university studying mathematics and political economy, though he had to prepare eleven other subjects for the bachelor's degree. But by January of that year, his enthusiasm for his studies had waned somewhat, as he disclosed in a letter to Herbert: "I find the classes at college a litde dull—the charm is rubbed off a few things— but then one learns more and more to adore De Morgan."24 His course in political economy was particularly disappointing. Jevons claimed to have gained far more from his own readings than from the lectures. His professor, Jacob Waley, had taught political economy at University College for many years but had never published on the subject. Waley's primary interest was the law and his first degree had been in mathematics and classics. Judging from the examinations he set, his interests leaned more to applied issues such as banking and taxation than to theory. Many of the same examination questions appeared year after year, suggesting that he made litde effort to revise his lecture material. One recurrent question on the limitations of the

20 · Chapter 2

theory of rent hints at anti-Ricardian sentiments, but whether he imparted these to Jevons or not is unknown.25 In June of 1860, Jevons took the college examinations in two subjects, mental philosophy and political economy. Although he placed a joint first in philosophy, in his favorite subject of political economy he paid dearly for his unwillingness to respect Waley's views: In Political Economy I had a sad reverse, such indeed as I never had before, for in spite of having studied the subject independendy and originally and having read some dozens of the best works in it, almost neglecting other classes for the purpose, I was placed 3"1 or 4th when I felt confident of the first prize. This I can only attribute to a diffce [sic] of opinion which is perfecdy allowable having prejudiced the Profr. [sic] against my answers.26 All however was redeemed shortly thereafter. In October, he received a first for his B.A. examinations and in December won the Ricardo Scholarship of sixty pounds based on a special examination. By June 1862, following a year and a half of relendess study, Jevons was awarded the gold medal for the master of arts in logic, moral and political philosophy, the history of philosophy, and political economy. While working toward the M.A., Jevons continued to study mathematics and natural philosophy. He repeated De Morgan's course in upper senior mathematics, thereby acquiring a fair grounding in the calculus and in its applications to mechanics. During these years he published a number of articles on the natural sciences, wrote several entries for a chemistry dictionary, and presented a paper at the 1861 meetings of the Mathematics and Physics Section of the British Association.27 On two occasions during this period his former chemistry professor and assaying instructor, Thomas Graham, offered to recommend him for positions in the natural sciences, one as superintendent at the Kew Observatory and another as lecturer in natural philosophy at Anderson's Institute in Glasgow. Although honored, Jevons decided not to apply for either. Jevons also developed a passion for logic. He had read the works of Richard Whately, John Herschel, William Whewell, and John Stuart Mill while in Australia, but soon realized how little he had grasped at the time: "I conclude that I knew little or nothing about logic then & never should have done but for the new exercise of my thoughts afforded in my second college course. It seems rather late in life to be learning what logic is, yet it is better late than never."28 His first course in philosophy had been given by John Hoppus, who was reputedly a very poor lecturer; according to one student, he "mumbled so that only an acute ear could catch much of what he said, and those

Jevons's Life · 21 who caught something called it rot."29 Jevons had few kind things to say of his second philosophy professor, James Martineau, even though he had been a friend of Jevons's father. Martineau's lectures were mostly "jargon," and "when he does become comprehensible he generally goes palpably wrong." 30 Nevertheless, these courses provided Jevons with a thorough grounding in the history of philosophy. He particularly took to Plato and Leibniz. Although Martineau disparaged the British empiricists Jeremy Bentham and Alexander Bain, Jevons came away with an appreciation for their writings. And it was Martineau who introduced him to the seminal works of George Boole on mathematical logic. By 1861, Jevons had begun to develop his own system of logic, based on what was to be called the principle of the substitution of similars. His first breakthrough was expressed with considerable conciseness and ,enthusiasm: "Philosophy would be shown to consist solely in pointing out the likeness of things." Within a year, Jevons confided in his journal: "My logical speculations give me most confidence. I cannot disbelieve, yet I can hardly believe, that in the principle of sameness I have found that which will reduce the whole theory of reasoning to one consistent lucid process." These ideas would, later undergo considerable refinement, particularly in the years 18661869. But they had crystallized sufficiently by 1863 to see print in the form of a small book entitled Pure Logic. He sent copies to Mill, Boole, and De Morgan, but the book, as he put it, "made no noise." 31 Only four copies were sold in the first six months. By this time, Jevons had made even greater strides in economics, though again with no recognition. During his first year back at the university he had discovered the essential elements of what was to become his mathematical theory of economics. Entries in his diary for February 1860 read as follows: February 3rd-5th including Sat. & Sun.—was almost entirely engaged in commencing a work on Pol. Econ. to be established on a demonstrative basis, in the form of connected and distinct propositions. Value to be established on the basis of labour and the problems of rent wages interest etc., to be solved as mathematical functions. Am very sanguine of its success. 11 February A short walk in Kensington Gardens. Some work at Latin, Greek and Pol Econ. 16 February Math. & Pol. Econ.

22 · Chapter 2 18 February Geometry, Econ. etc. 19 February At home all day & working chiefly at Economy, arriving as I suppose at a true comprehension of Value regarding which I have lately very much blundered. 25 February Drill in the morning for 13th time. Platoon exercise as in square. De Morgan. Latin, & Pol. Econ. rest of day.32 That these insights came so readily to him at this point in time supports the view that he had already begun to map out a mathematical version of economics while still in Australia. But it seems unwarranted to infer that Jevons had formulated the principle of the final degree of utility by February 1860, let alone while in Australia. 33 We only know that, in his eyes, he had "blundered" in his initial decision to establish value "on the basis of labour." By June 1860, however, Jevons had most certainly discerned the main elements of his utility theory of value. To his brother Herbert he wrote: During the last session I have worked a great deal at Pol. Economy; in the last few months I have fortunately struck out what I have no doubt is the true theory of Economy so thorough-going and consistent, that I cannot now read other books on the subject without indignation. While the theory is entirely mathematical in principle, I show at the same time how the data of calculation are so complicated as to be for the present hopeless. Nevertheless I obtain from the mathematical principles all the chief laws at which Pol. Econts have previously arrived only arranged in a series of Definitions Axioms and Theorems almost as rigourous and connected as if they were so many geometrical problems. One of the most important axioms is that as the quantity of any commodity, for instance plain food, which a man has to consume increases, so the utility or benefit derived from the last portion used decreases in degree. . . . And I assume that on an average the ratio of utility is some continuous mathematical function of the quantity of commodity. This law of utility has in fact always been assumed by Pol. Econ. under the more complex form and name of the Law of Supply & Demand. But once fairly stated in its simple form it opens up the whole of the subject. Most of the conclusions are of course the old ones stated in a consistent form—but my definition of Capital and law of the Interest of Capital are as far as I have seen quite new. I have no idea of letting these things lie by till

Jevons's Life · 23 somebody else has the advantage of them—and shall therefore try to publish them next Spring.34 His indignation at other treatments of the subject, indeed, the entire passage, sounds very much like a Kuhnian conversion. A subsequent letter to Herbert, dated November 1860, expands on these discoveries: I expect every success from my theory of political economy, which seems to develop itself with that facility which is a proof of its soundness. It assumes the form of a complicated mathematical problem, from which all the common laws with due limitation flow. Independendy, however, of the mathematical form, it has led me to a new view of the action of capital, which affords a determining principle for interest, profits of trade, wages; and I now perceive how the want of knowledge of this determining principle throws the more complicated discussions of economists into confusion. The common law is that demand and supply of labour and capital determine the division between wages and profits. But I shall show that the whole capital employed can only be paid for at the same rate as the last portion added; hence it is the increase of produce or advantage, which this last addition gives, that determines the interest of the whole.35 These passages are remarkably incisive. Jevons had not only worked out the principle of the final degree of utility but also the relationship between profits and doses of capital. It is also apparent, however, that Jevons had not initially set out to apply the calculus. Although he speaks of his theory as "a complicated mathematical problem," there is no explicit use of the calculus in his account. At best, he saw the possibility of algebraic functions arranged in an axiomatic fashion. This suggests, as do his allusions to geometry, a format more akin to Newton's Principia (1687) than to the model of constrained maximization found in Lagrange's Mochanique Analytique (1788). From the above passages it is also clear that Jevons found his new principles of rent, wages, and interest to be at least as important as his theory of utility, which had "in fact always been assumed" under the laws of supply and demand. Jevons was concerned at this point not with an attempt to found economics on the utility theory of value, but rather to formulate the entire doctrine in mathematical terms. In his estimation, the most interesting result was his explication of capital and interest. Contrary to what Terence Hutchison has claimed, Jevons did not derive the idea of applying mathematics from the Benthamite notion of maximizing utility. Nor, as a logical reconstruction might suggest, was he bent on resolving the water-diamond paradox.

24 · Chapter 2 A published acknowledgment that his insights on utility solved this longstanding anomaly did not appear until the 1879 edition of his Theory. However attractive either account might seem, neither one conforms to the actual record. 16 Jevons did not immediately publish his principle of the final degree of utility. An entry in his private journal for December 1861 hints at the reason for the delay: The subjects which had pressed themselves upon me as my proper sphere of employment, viz. pol. econ.—& the Social Sci. seem opening before me by degrees in a manner exceeding my first hopes—But it is of course always true that we can have no idea of what is to be found out & not yet known. I cannot avoid also paying some attention to philosophy proper, in addition to what is required for my degree—and I begin to understand things, which were utterly beyond me, some years ago.!T In September 1862, with his master's degree completed, the time had come for him to tell the world: I have resolved however at last to let out my theory of Economy, & have accordingly written a short paper entitled "Notice of a general mathematical theory of Economy" which will I hope be read at the Brit. Assoc, meeting at the beginning of next month. Although I know pretty well the paper is worth perhaps all the others that will be read there put together, I cannot pretend to say how it will be received, whether it will be read at all, or whether it wont [sic] be considered nonsense. His premonition proved correct; the paper was "received without a word of interest or belief." He was dismayed and discouraged. The experience, he confided to his journal, "convinced me that success in my line of endeavour is even a slower achievement than I thought. This year has taken much youthfulness out of me." 38 Jevons lasted one year as an independent researcher at the British Museum. He had hoped to emulate the career of his cousin-in-law, Richard H. Hutton, a freelance writer, editor of the Economist, and an examiner in political economy at University College. Difficulties encountered in finding a publisher for some statistical diagrams, however, made it clear to him that the life of a "hack writer" was a constant struggle. 39 One is reminded of the poverty that beset Karl Marx, who was also frequenting the British Museum in these years while working on Das Kapital. In 1863, after several months of despair, Jevons decided to accept his cousin Harry's offer to serve as a tutor at Owens College, Manchester. The position, though it involved long hours at poor pay, promised to lead to better things, quite possibly, Jevons speculated,

Jevons's Life · 25 40

a professorship itself. In the summer of 1863, Jevons moved to Manchester. Shortly after Jevons settled in Manchester, his first book, A Serious Fall in the Value of Gold, published earlier that year, began to gain recognition. Jevons had sent copies to several leading economists, notably John Elliott Cairnes and Henry Fawcett. Both responded favorably.41 Notices of the work appeared in the Economist and the Times. By January 1864, Jevons recorded in his journal: "My pamphlet on the gold question has had a degree of success that must surely be allowed to be beyond my highest hopes." The praise was well-deserved. His assessment of the impact of the Australian and Californian gold discoveries on prices has since been regarded as one of the most original forays into the problem of indexing in the history of economic thought.42 With the memory of his recent failures still vivid, Jevons decided to continue with applied economics: "I am better in theory than I am in fact; but theorists have a bad odour until their soundness is established by the slowest possible process. Hence it is a good thing to begin by diagrams, tables of prices, and such things, so that you can never be charged with arguing without a reference to or knowledge of facts."43 This strategy proved successful. The Coal Question (1865), which predicted in a somewhat Malthusian fashion the depletion of Britain's coal reserves and subsequent economic decline, brought him notoriety within a year. Mill praised its impeccable logic in a Parliamentary speech of April 1866. A Royal Commission was established to investigate the matter and, in light of Jevons's pessimistic findings, Gladstone, then Chancellor of the Exchequer, proposed reducing the national debt. Inevitably, Jevons was attacked in the popular press. By May 1866, he admitted to being somewhat drained by the publicity: "I have had quite enough fame for the present, and should not be altogether sorry to retire in safety."44 His professional standing also improved significantly. In 1864 he had been elected to the Statistical Society of London, and in May 1865 he secured the Professorship in Political Economy and Logic at Queen's College, Liverpool. In the same year, he also took on the task of lecturing in political economy, logic, and philosophy at Owens College. By May 1866, with supporting letters from De Morgan, Herschel, Mill, and Herbert Spencer, Jevons was appointed Cobden Professor of Political Economy and Professor of Logic, Mental and Moral Philosophy at Owens College. Thus, in a period of three years, he had advanced from the lowly rank of a tutor, at an annual salary of one hundred pounds, to that of a name professor, with three times the income. He was to occupy the chair for ten years, until his return to

26 · Chapter 2

University College, London as Professor of Political Economy in 1876.45 Following the triumphs of the Coal Question, Jevons's thoughts turned once again to logic. According to his journal, he awakened one morning in March 1866 with an exhilarating feeling: "There was a consciousness on my mind . . . that I was the discoverer of the true logic of the future." But it was not until December of that year that he "began thinking about logic again seriously." It was at this time that he formulated his idea of the substitution of similars: "I suddenly met with what seems to me the great and universal principle of all reasoning that some things may be treated identically or that whatever we may say of one member of an identity we may say of the other. .. . It would be worth while to spend years in developing a system of logic on this basis." Over the course of the next month, his confidence in the soundness of these ideas grew ever firmer. "My notions of a new logic become clearer and clearer as I think about the subject now & then. By working gradually and steadily I know not at what I may not arrive. For six years I have thought about it & my thoughts have gradually penetrated."46 His strength of mind was gready enhanced in this very year by marriage to Harriet Ann Taylor (no relation to the Harriet Taylor whom Mill had married). She was also a Unitarian and, as the daughter of the first proprietor of the Manchester Guardian, came from a progressively minded background similar to his own. Harriet's subsequent role as editior of his posthumous papers suggests that she not only gave him comfort and emotional support but also shared in his intellectual work.47 They had three children, the oldest of whom, Herbert Stanley, followed in his father's footsteps to become an economist. Shordy after his marriage, Jevons began to write his Principles of Science, which built directly upon his insights on logic. It was to take seven years to finish, due to frequent bouts of illness and a six-month interruption, in 1871, to write his Theory of Political Economy.48 Apparendy, he rushed off the Theory for fear of losing some of the glory to Fleeming Jenkin, who in 1870 had sent Jevons a short but incisive paper on mathematical economics.49 After completing and publishing his Principles in 1874, Jevons resumed work on applied economics. His next book, Money and the Mechanism of Exchange (1875) was well-received and became a standard text in university curricula well into the early part of our century. In 1875, he also published his celebrated sunspot theory of business cycles, though the original notion dates back to his student days.50 Over the next seven years, until his premature death in 1882, Jevons was to write two more texts on logic, Primer of Logic (1876) and Studies in Deductive Logic (1880), and

Jevons's Life · 27 another on economic policy, The State in Relation to Labour (1882). Most of his remaining essays and manuscripts on economic topics were published posthumously: Methods of Social Reform (1883), Investigations in Currency and Finance (1884), Pure Logic and Other Minor Works (1890), and The Principles of Economics (1905). Jevons was pleased with the reception of his Principles of Science, but found it necessary to devote a considerable amount of energy to the promotion of his Theory.51 In 1874, he had come into contact with L6on Walras at the University of Lausanne, who had independendy arrived at similar analyses of value and distribution and who also advocated the adoption of the calculus. Although somewhat piqued at finding a rival to his claims, Jevons accepted Walras as an ally and promised to spread news of his Elements. Initially, Jevons followed through on his word by citing Walras in an address that same year to the Manchester Statistical Society.52 But, for reasons that remain obscure, Jevons later reneged on his promise on at least two similar occasions. Also peculiar was his avoidance of any personal encounter. They arranged to meet in 1877 when Walras came to London, but Jevons begged off on account of his wife's health. Jevons also went to France and Switzerland for several vacations, but never accepted WaIras's standing invitation to visit him in Lausanne. Jevons and Walras corresponded from time to time on their progress toward reforming economics. As Jevons noted in his first letter to Walras in 1874: "I confess that I have always in my own mind attached much importance to this mathematical theory of economy, believing it to be the only basis upon which an ultimate reform of the science of political economy can be founded." And in 1875 he wrote: "I have no doubt whatever about the ultimate success of our efforts, but it will take some fighting."53 Nevertheless, there was far less collaboration between the two than might easily have been managed. By the mid-1870s the name of one precursor after another was brought to Jevons's attention. He became disheartened. To his brother Tom he wrote in 1878: There are, in fact, a whole series of books, hitherto quite unknown, even on the Continent, in which the principal ideas of my theory have been foreshadowed. I am, therefore, in the unfortunate position that the greater number of people think the theory nonsense, and do not understand it, and the rest discover that it is not new. Jevons soon reasoned his way around this apparent impasse. He grew to recognize that the many independent discoveries of the principle of the final degree of utility helped to confirm his own findings. To H. S. Foxwell he wrote: "The Theory . . . has in fact been independently

28 · Chapter 2 discovered 3 or 4 times over & must be true." He also realized that it was pointless to insist on his own originality. As he wrote to Henry Sidgwick: I have for some time past been inquiring into the history of the mathematical treatment of Economics, and the truth gradually dawns upon me that the mathematical method is as old as the science of Economics itself. The details are curious & in the long course of time I may perhaps hope to publish them. But it is quite plain that no recent writers have any claim to novelty, or priority of method.54 As a result, the preface to the second edition of his Theory (1879) emphasized its continuity with the past, as did an appended bibliography of works on mathematical economics since the early eighteenth century.55 But Jevons made little effort to rework the theoretical core, despite a number of criticisms and a familiarity with related ideas in the works of A. A. Cournot, Hermann Heinrich Gossen, and Johann Heinrich von Thiinen. Other areas of Jevons's research suggest a predilection for breaking new ground rather than for revising ideas he believed to be basically sound. As Professor Black has suggested, Jevons may not have been able to move much beyond his initial flashes of brilliance.56 His later correspondence, particularly with George Darwin, John Shadwell, and Walras, suggests that he had difficulty in comprehending their theoretical extrapolations. As he reluctantly acknowledged, his command of mathematics was not up to the task. Jevons had additional reasons for moving on to other areas of research. As he noted in his Principles, "when we examine the history of scientific problems, we find that one man or one generation is usually able to make but a single step at a time."57 His step had been taken in 1860-1861, when he discovered the powers of marginalism and the essentials of his system of logic. Those were undoubtedly his most creative years, and it is fair to say that much of his remaining twenty years was spent developing his early insights and insuring their posterity. Jevons was an active citizen in scientific organizations. As T. S. Ashton has noted, he was the mainstay of the Manchester Literary and Philosophical Society and the Manchester Statistical Society during the late 1860s. In 1870, he served as president of Section F of the British Association for the Advancement of Science and, in 1872, was appointed a member of the Royal Society.58 Upon moving back to London in 1874, Jevons became an active member of the Political Economy Club and in 1879, through the efforts of Herbert Spencer, joined the Athenaeum Club, where he brushed shoulders with London's intellectual elite. Though he never relished being in the public eye, he

Jevons's Life · 29 dutifully addressed various academic and industrial associations throughout England. On several occasions he met and advised government officials on such issues as international coinage, taxation, and census taking, and engaged in various debates in newspapers and periodicals. He also continued to publish, most often for the journal Nature, on a variety of scientific topics, such as comets, Brownian motion (which he named pedesis), heredity, geology, and non-Euclidean geometry. Although he had self-consciously rebelled against classical economic theory, on political issues Jevons was only moderately progressive and shared few of Mill's leanings toward socialism.59 In his view, there was an important but limited role for government intervention on a range of social issues, such as sanitation, housing, and recreation. He specifically advocated more public education and public libraries, and various forms of labor legislation, such as prohibiting the employment of mothers of young children. On a number of occasions he displayed sympathy for the plight of the working class, but resolutely maintained that trade unions should only serve noneconomic needs. Wages and working hours were to be determined, quite simply, by the marketplace. Like so many economists before and since, what started as a grand vision to better the world ended up as guarded faith in market forces.60 Jevons taught at University College for only five years. Poor health led him to resign in 1881 and to decline other public engagements. Although his health continued to deteriorate, he lived long enough to have the satisfaction of witnessing the widespread approval of his innovations in economic theory. It is interesting to recall his thoughts after his setback at the British Association meeting in 1863: It is quite obvious to me that it is useless to go on printing works which cost great labour much money [and] are scarcely noticed by any soul. I must begin life again & by another way, ingratiating myself where & when I can—only after long years of slow progress can ones [sic] notions be brought out with any chance of being even examined by those capable of judging of them. But in these last years of his life he acquired a sizable following and befriended many of the leading economists of the time, including Cliffe Leslie, H. S. Foxwell, Henry Sidgwick, and F. Y. Edgeworth. In 1881, Jevons wrote enthusiastically to Walras: "I am glad to say I think the Math, view of Economics is making much progress in England and is fully recognized by those competent to judge."61 The fol-

30 · Chapter 2

lowing year, having disobeyed his doctor's orders not to swim, Jevons drowned off the coast of Devon.62 Whether nineteen years is indeed "slow progress" is another matter. But, in Jevons's own estimation and in the estimation of those who followed him, he did, most assuredly, leave his mark on the world.

CHAPTER THREE

The Mathematical Theory of Economics When at length a true system of Economics comes to be established, it will be seen that that able but wrong-headed man, David Ricardo, shunted the car of Economic science on to a wrong line—a line, however, on which it was further urged towards confusion by his equally able and wrongheaded admirer, John Stuart Mill. —Jevons, Theory of Political Economy

in economic theory has, more often than not, been discussed and judged in terms of contemporary theory rather than on its own terms. This may be attributed in part to the Whiggish tendencies of many commentators and in part because he was, after all, one of the first to set down the basic components of what became neoclassical economics. Like other pathbreakers, however, his exposition left many loose ends. In his analysis of Jevons's theory of capital and interest, Ian Steedman has demonstrated that the entire system is indeterminate. But he is only one of a long line of critics, from Alfred Marshall to George Stigler, to note the many oversights and inconsistencies in Jevons's theoretical work. Our assessment here will be more backward- than forward-looking. Rather than fault Jevons for lacking the prescience to devise a general equilibrium model, we will focus on those aspects of his work that steered economics in a new direction, particularly his many appeals to greater abstraction and generality. More immediately, we will attempt to understand how he applied mathematics to economics, given that he had so few examples to emulate. One question to be kept in mind is the extent to which he moved away from the classical theory of value, which had always ascribed a place for utility as a prerequisite for exchange. As Jevons notes in the second edition of the Theory, "the mathematical treatment of Economics is coeval with the science itself." His bibliography of "mathematico-economic" writings covers about a hundred entries, many of them dating from the eighteenth century. But when Jevons delivered his 1862 paper to the British Association, he only knew of two forerunners, William Whewell and Dionysius Lard-

JEVONS'S WORK

32 · Chapter 3

ner. In the first edition of his Theory he identified a few more—Francis Hutcheson, John Edward Tozer, Henry Dunning Macleod, Fleeming Jenkin, and Richard Jennings—and later claimed of them that "it is from the works there mentioned, if from any, that I derived the idea of investigating Economics mathematically."1 It is difficult to find a firm imprint of any of the British predecessors in Jevons's work. We noted in the previous chapter that he had first read Lardner's Railway Economy while in Australia and had admired its mathematical bent. But when it comes to specific details, Lardner's influence appears to have been fairly slight. Jevons had a very poor opinion of Whewell's forays into mathematical economics, and snowed no hesitation in dismissing them as "nonsense." To some extent, this is a just pronouncement. For someone of Whewell's stature in mathematics, the results are most uneven. But there are some points in common, particularly the many analogies to the science of mechanics and disdain for Ricardian economics. Jevons might also have garnered a rough formulation of the price elasticity of demand from Whewell's last paper (1850).2 Why Jevons took umbrage at Whewell's work is still something of a mystery, unless it stemmed from his more general contempt for the status quo. Jevons also had a low opinion of Macleod for harping on about the importance of negative numbers in economics. Coincidentally, Macleod had presented his first paper on the subject in 1862, at the same session of the British Association meetings as Jevons. But Jevons had already articulated the notion of negative utility and thus any acknowledgment of Macleod on this point was totally gratuitous.3 Tozer wrote two papers on mathematical economics, apparently under the inspiration of Whewell's work.4 But Whewell ignored them, possibly because of a personal dislike for Tozer but more likely because Whewell himself had little interest in developing a school of mathematical economics. There is nothing in the papers themselves to suggest any direct influence on Jevons's work. Neither Hutcheson nor Jennings used any mathematics, but they may well have shaped Jevons's theory as much or more than Whewell, Tozer, or Macleod put together. Both addressed the subject of pleasure and pain in quantitative terms. In Jevons's estimation, Hutcheson's Inquiry into the Original of our Ideas of Beauty and Virtue (1720) "thoroughly anticipates the foundations of Bentham's moral system."5 Jennings's Natural Elements of Political Economy (1855) emphasized the importance of reducing economics, first to psychology, and then to physiology. Although there is no record of when Jevons first read Jennings, there is reason to believe that he served as an important source of inspiration. Jenkin's 1870 paper, "The Graphical Representation of the Laws of

Mathematical Theory of Economics · 33 Supply and Demand," may have been most responsible for precipitating the Theory. By superimposing utility curves, he demonstrated that in the case of pure barter there was nothing to induce two traders to agree upon a ratio of exchange that corresponded to the maximum utility for both parties. Jenkin's reasoning is still sound. As Black has noted, however, Jevons skirted the problem, both in his 1862 paper and in his Theory, by treating his traders, even in pure barter, as price takers.6 Nevertheless, Jenkin's criticisms may have done some service in sharpening Jevons's understanding of the topic. As Schumpeter noted, "in very few other cases . . . is it so difficult to speak of 'roots' as it is in the case of Jevons."7 Only after 1871 did he learn of the highly impressive works of Cournot, Dupuit, Gossen, and von Thiinen. Each one had used the calculus, thus preempting what Jevons considered his most original contribution and the key to the true theory of economics. Had he known of even one of them in advance, he most likely would not have raised his revolutionary banner so vehemendy. But the fact that he did is in part what makes him and not the others loom so large in the annals of economics. A few words are in order pertaining to the scope of mathematics as understood by Jevons and his contemporaries. Most saliently, it did not include statistics. Although Jevons avidly promoted the collection and analysis of market data, the actual use of statistical methods did not, in his view, constitute mathematical reasoning. As Theodore Porter has shown, techniques such as the method of least squares were developed primarily within the context of specific disciplines, such as astronomy or heredity. Only in this century were the properties of statistical distributions grounded in the theory of probability.8 Mathematics was primarily taken to be comprised of geometry, algebra, and the calculus. In England the subject was usually taught in its "mixed" rather than "pure" form, that is, within the context of problems in the physical sciences. At Cambridge, despite some protestations in the 1810s, Euclidean geometry dominated the curriculum well into the latter half of the century. Preparation for the mathematics tripos culminated in a study of Newton's Principia, rather than the Continental texts on the calculus. This seemingly antiquated program survived for so long because it was thought to best serve the objectives of a liberal education. A study of traditional mathematics instilled an appreciation for the nature of truth itself and thus prepared a young man for the clergy or the courtroom. The very few who wished to advance mathematics or physics had to learn their skills elsewhere.9 Jevons was fortunate to have studied under Augustus de Morgan and thus been exposed to more recent branches of mathematics.

34 · Chapter 3 THE THEORY OF VALUE

Jevons begins the Theory with a quantitative analysis of the feelings of pleasure and pain. Of the seven Benthamite "circumstances" associated with pleasure and pain, Jevons selects intensity and duration as the most fundamental dimensions of feeling. For clearly, "every feeling must last some time, and . .. while it lasts, it may be more or less acute and intense." The quantity of feeling, then, is just the product of its intensity and duration: "the whole quantity would be found by multiplying the number of units of intensity into the number of units of duration. Pleasure and pain, then, are quantities possessing two dimensions, just as superficies possesses the two dimensions of length and breadth."10 What Jevons had in mind as these "units" is never specified. There are certain remarks, however, to suggest that he did not regard them as totally beyond our grasp. He makes it clear from the start that economics treats only "the lowest rank of feeling," or "the ordinary wants of man." In short, the "lower calculus" of the economist could be used without granting full consideration to the "higher calculus of moral right and wrong." Nevertheless, he also insists that "every mind is thus inscrutable to every other mind, and no common denominator of feeling seems to be possible."11 Whenever Jevons speaks of comparing two feelings, it is always the case that one of the two variables, either duration or intensity, is held constant. Thus, if the intensities of two feelings are the same, but one of the feelings persists for a longer period of time, the "quantity of feeling" is greater in the one with a longer duration. And since one only compares feelings two at a time, it could be argued that, for the case just presented, the intensity acts as a unit in that particular comparison. As he notes in the introduction, "we only employ units of measurement in other things to facilitate the comparison of quantities; and if we can compare the quantities direcdy, we do not need units. Now the mind of an individual is the balance which makes its own comparisons, and is the final judge of quantities of feeling." And, since the "theory turns upon those critical points where pleasures are nearly, if not quite, equal," a well-defined notion of a unit may not be necessary.12 It may seem quite arbitrary that Jevons chose intensity and duration as the dimensions of feeling, particularly at the exclusion of certainty and propinquity which, one could argue, have just as much bearing on the notion of a feeling. He was, however, intent on constructing an algebraic function that could give, by integration, the quantity of feeling for any given time interval. He therefore not only restricted his

Mathematical Theory of Economics · 35 analysis to two dimensions, but chose those dimensions that are most amenable to being treated as continuous variables. Given that the practice of measuring time in units and of treating time as a continuous variable had the secure backing of classical physics, duration presents few if any problems. It is intensity that raises difficulties. He asserts, through an appeal to introspection, that it would be "artificial" to treat intensity discretely. Rather, "the intensity of feeling will change from moment to moment. Incessant variation characterises our states of mind." As these moments are made "infinitely short," we find that "we must treat the intensity as varying continuously."13 In other words, given his assumption that the intensity of a feeling is a function of time, the perfecdy legitimate process of treating time as a continuous variable in some way compels us ("we must") to treat intensity as continuous. Jevons makes no attempt to justify his assumption that intensity is a function of time, though it is not so implausible. One could argue that the recognition of a change in the intensity of a feeling depends upon a recognition of time. But then, philosophers have often attributed an awareness of time to some variation in the intensity of sensations. Jevons reflected on this at length and came to the conclusion that time is the more primitive notion.14 Duration may also have been chosen as the independent variable on mathematical grounds, since it offers the least dubious interval for integrating under the "feeling curve." Nonetheless, without a means to establish a unit of intensity and without a means of determining the geometric properties of the "feeling function," any estimate in terms of "intensity-seconds" remains meaningless. By including a diagram of this curve (Figure 3.1), however, Jevons gives the impression that it is knowable. The whole feeling is defined as the area under the feeling curve (pq) for any time interval (ran). Moreover, a positive or negative sign can be ascribed to a particular feeling depending on whether it is pleasurable or painful. In this way, feelings can be added to or subtracted from one another, with the goal, needless to say, of having pleasure as the net result. Only a brief attempt is made to establish that pleasure and pain are commensurable, let alone capable of being treated "as positive and negative quantities are treated in algebra." Citing Alexander Bain, Jevons draws an analogy to the process by which acids are neutralized by alkalis. In the same manner, "when the pleasure exacdy assuages the pain, we say that the two are equivalent." What he fails to consider is the possibility that there is no such thing as a neutral feeling. If this is the case, then one pleasure could never nullify the equivalent amount of pain.15 Jevons defines a commodity as "any object, substance, action, or

36 · Chapter 3

a

η

χ

duration

Feeling Curve service, which can afford pleasure or ward off pain." And "whatever can produce pleasure or prevent pain may possess utility." The last clause is significant, for whereas a commodity is objectively deter­ mined, utility can only be ascertained subjectively, by "the will or in­ clination of the person immediately concerned."16 The theory of utility thus rests direcdy upon the more fundamental theory of pleasure and pain. The classical economists had long recognized utility as a neces­ sary but not sufficient condition for an object to qualify as a commod­ ity, but had not attempted to ground this in Bentham's pleasure-pain calculus. In this respect, Jevons has brought together two hitherto separate strands of thought, and probed more deeply into the psycho­ logical roots of economic action.17 Since utility is a relationship, an abstract quality between a com­ modity and a person and not an inherent property of any one thing, it is meaningless to speak of utility as rigidly proportional to the quantity of an object. In fact, identical objects can have varying amounts of utility for different persons and even for one person, depending on the time and place. A quart of water, for example, does not have an inher­ ent amount of utility. Rather, the utility of any one quart depends upon the availability of other quarts and the number of quarts already consumed in the recent past. To a person in a desert, an initial quart or even a fifth quart may possess infinite utility, whereas to a person in London, with readily available sources close at hand, a given quart may have a utility close to zero. In the case of a flood, a quart of water clearly has negative utility, or disutility. The more meaningful rela­ tionship to analyze, then, is the variation of the degrees or additional increments of utility with respect to the varying amounts of a partic­ ular commodity already available. To establish this relationship in mathematical terms, Jevons per­ forms the same operation he used to quantify feelings. He sets up a graph with the quantity of a good on the abscissa and the intensity of

Mathematical Theory of Economics

·

37

utility on the ordinate (see Figure 3.2). Here, the intensity of utility is taken to be the net outcome (pleasure minus pain) produced from the consumption of the good in hand. Utility therefore incorporates both positive and negative feelings; it is "the sum of the pleasure created and the pain prevented."18 For each unit of good, there is a corre­ sponding amount of intensity; the product of these two equals the quantity of utility for any given unit of a good. For the first couple of units, the utility "will be infinitely great." Although, as additional amounts of the good are consumed, the total utility will increase, each additional increment will be less than the one preceding. Jevons then suggests that the units of the independent variable be taken as infinitesimally small increments, thereby establishing a con­ tinuous function. There is, moreover, empirical support for this pro­ cedure. Although "the notion of infinitely small quantities of food may seem absurd," in the case of a large number of individuals, such as a nation, it is plausible to believe that consumption can "increase or di­ minish by quantities which are, practically speaking, infinitely small compared with the whole consumption."19 Jevons then defines the "degree of utility" in mathematical terms. By adding a very small amount of the good χ (i.e., Δχ), an additional amount of utility (Au) will be generated, equal to the area of the rectangle (abb'a') with Δχ as the base. The degree of utility, then, or intensity of this small ad­ dition, is defined as the height of this rectangle, or AuIAx. The limit of this ratio is, for Jevons, du/dx, which he calls the "degree of utility," or, more specifically, "the differential coefficient of u considered as a function ofx."20 From the assumption that "all our appetites are capable of satisfac­ tion or satiety sooner or later," Jevons states a "general law" now known as the principle of diminishing marginal utility: "the degree of utility varies with the quantity of commodity, and ultimately de-

o

m

Figure 3.2

a a Utility Curve

η

χ

commodity

38 · Chapter 3

creases as that quantity increases." Thus, the total utility is not the relevant concept. Rather, the degree of utility, or, to be even more precise, the "final degree of utility," is the concept on which the "Theory of Economics will be found to turn."21 Using this idea of the final degree of utility, Jevons considers the problem of distributing a good, for example barley, among alternative uses, such as beer, bread, and cattle feed. The most satisfactory distribution of the barley is one where no further redistribution would yield additional utility. This implies that the final degrees of utility for each of the two or more uses to which the barley is put are equal. The principle is presented in algebraic terms. If barley, s, can be used for either beer, x, or bread, y, and s = x + y, then the most efficient distribution is one where du/dx = duldy. Jevons acknowledges the fact that this equation fails to hold in cases where it would be more advantageous to apply the good exclusively to one use. In such a circumstance, it could be said that "theoretically, . . . the final degree of utility in this employment always exceeds that in any other employment." The utility equation then becomes an inequality.22 In a somewhat unsystematic fashion, Jevons spells out his underlying assumptions about the behavior of economic agents. "It is the inevitable tendency of human nature to choose that course which appears to offer the greatest advantage at the moment." Since perfect knowledge, however, is never possible, what may appear to offer the greatest advantage may not in fact maximize utility. Only a "perfectly wise being" would always succeed in maximizing utility. The actions of ordinary people only approximate a state of maximum utility. Nevertheless, it is the role of economics to "point out the mode of consuming it [a good] to the greatest advantage—that is, with a maximum result of utility."23 Hence, whereas the appraisal of the final degree of utility is subjective, the optimal distribution of utility is in some sense objective. There exists, in every case, the most rational distribution, which the perfectly wise being (or economist!) could determine without fail. More can be gleaned of Jevons's views on human nature from a later essay on "Utilitarianism." He there takes issue with the empiricist notion of a mind as a tabula rasa. In his view, "human nature is one of the last things which can be called 'pliable.' Granite rocks can be more easily moulded than the poor savages that hide among them."24 He also inveighs against Mill's firm belief that the majority of persons, either by lethargy or a series of mishaps, have been led astray from higher pursuits such as poetry to setde for a life of pushpin. Jevons opts to "hold to the dry old Jeremy" on the fundamental equivalence of all forms of pleasure. As for social or altruistic feelings,

Mathematical Theory of Economics · 39 they differ from lower ones only in degree, not kind. To support this, Jevons turns to Spencer and Darwin, who had demonstrated how strong a grip nature has over our behavior.25 Morality is more like an animal instinct than a cultivated virtue. The philosopher and street porter have in effect become as disparate as the greyhound and the mastiff. THE THEORY OF EXCHANGE

In order to formulate the exchange equation, Jevons first sets out a number of definitions and assumptions. He starts by constructing a market and a trading body, both of which, he readily grants, only approximate conditions in the real world. A perfect market is defined without reference to any particular institutional setting or locality; it exists under the proviso that "all traders have perfect knowledge of the conditions of supply and demand, and the consequent ratio of exchange." Insofar as markets, such as the British corn market of the nineteenth century, publish and distribute information about the prices and quantities available, "the theoretical conception of a perfect market is more or less completely carried out in practise."26 A trading body is defined as any individual or group of individuals engaged in exchange. Such a body can range in size from a single person to an entire continent. The breadth of this definition is justified on the grounds that the principles of exchange are universally applicable and must therefore be the "same in nature, however wide or narrow may be the market considered."27 With little ado, Jevons also notes two other important assumptions, the classical postulates of self-interest and perfect competition. "Every individual must be considered as exchanging from a pure regard to his own requirements or private interests, and there must be perfecdy free competition, so that any one will exchange with any one else for the slightest apparent advantage." Given that both of these assumptions play an essential role in his theory of exchange, it is peculiar that they are embedded in a passage on the definition of the market, rather than placed in a more prominent light. Jevons may simply have regarded them as universally recognized assumptions. But it seems more plausible to attribute this sleight-of-hand to an attempt, on his part, to gloss over the similarities between his own work and that of the classical authors and to highlight his analysis of a market and a trading body, which was very original.28 Jevons's next assumption is the "Law of Indifference," which he considers to be quite new. A theoretical rather than empirical generalization, the law states that "in the same open market, at any one

40 · Chapter 3 moment, there cannot be two prices for the same kind of article."29 But this is nothing more than the classical principle of uniform mar­ ket prices presented with a new name. In fact, given a system of prices, the law follows directly from the more fundamental assump­ tions of perfect competition and perfect information, a point Jevons acknowledged only in his posthumous Principles of Economics.30 It might seem odd that Jevons pays so much more attention to this lower-level principle at the expense of such basic postulates as perfect competition and economic self-interest. But this law permits him to derive an important mathematical relationship which he thinks he needs in order to solve the exchange equations. If two identical goods command the same unit price, if a trader is completely indifferent be­ tween one portion of a stock and another, then "the hst increments in an act of exchange must be exchanged in the same ratio as the whole quantities exchanged." In mathematical terms, this is expressed as: dy/dx = ylx, where χ and y are quantities of commodities such that "more or less can be given or received in infinitely small quantities." His grasp of the economics of the situation is strictly correct. If a good can be divided into many small increments, and each portion is iden­ tical to any other, then one would evaluate them all at the same, low­ est ratio of exchange. But the equation dy/dx = ylx is full of difficul­ ties. Jevons would have done better to have expressed the principle as a ratio rather than as an equation (i.e., dy/dx : y/x).31 To analyze exchange, Jevons takes two trading bodies, each of whom is endowed with different commodities (a and b). Trader A of­ fers portion χ of a, in exchange for portion y of b from Trader B. How are χ and y determined? Provided there are no impediments to the transaction, exchange will take place up to the point where neither party would benefit from further exchange. In other words, exchange ceases when the degrees of utility of the portion given and the portion received are, for each trader, equal in their own respective minds. That is to say, if an infinitely small additional amount of the commod­ ity were to be exchanged, it would bring neither a gain nor a loss in utility. Thus the fundamental principle of exchange is stated as fol­ lows: "The ratio of exchange of any two commodities will be the recip­ rocal of the ratio of the final degrees of utility of the quantities of commodity available for consumption after the exchange is com­ 32 pleted." For Jevons, provided that his assumptions about human nature are correct, this proposition is "necessarily true." Exchange will cease at the point when the final degrees of utility are equal. Here, then, is the second equation needed to solve for the two unknowns, χ and y. The first equation derives from the identity that in equilibrium what is

Mathematical Theory of Economics · 41 bought by A is sold by B, hence the χ and y of one equation must equal the χ and y of the other. This algebraic rule of thumb—that the number of unknowns must equal the number of equations—was in his mind of great significance and was first expounded in his 1862 paper. Jenkin, Marshall, and Edgeworth also used it to much advan­ tage, and may well have come to appreciate its cutting force in Jevons's work. In algebraic terms, exchange may be represented as follows: φ1(α-χ)·άχ

= tyiydy

where, for Trader A, the final degree of utility of commodity a is de­ noted by φ](α - χ) and the final degree of utility of commodity b is denoted by ψ^. This is the point at which the increments, dx of a and dy ofb, have just been exchanged. From the law of indifference, how­ ever, we know that dy/dx = ylx. The final equation of exchange thus takes the form: φ ^ α - χ ) _y_ _ Ψιί/ ~ χ ~

φ2χ ^Φ-y)

where φ and ψ are the respective final degrees of utility. The exchange equation can be modified to apply to other, more complicated situations. For example, if there are transaction costs, Jevons shows how they can be incorporated into the equations. And given more than two traders and commodities, one simply augments the number of equations to account for every possible transaction. By a change of sign, the equation can also represent the exchange of one or more commodities with disutility (discommodities). For example, if the amount y of b has disutility to A and, in order to dispose of y, A must pay with quantity χ of a, the equation becomes: - φ(α-χ)·οίχ + ty(b-y)-dy = 0 The general form of the equation, which covers all types of exchange, whether it be a commodity for a discommodity or even the exchange of two discommodities, is written as: φ(α±χ)·(2χ + \\i(b±y)-dy = 0 Jevons also considers the imaginary case in which the substances have neither utility nor disutility, what he calls inutility. The ratio of exchange is then indeterminate: dy/dx = 0/0 . The mathematics thus reinforces the view that utility (either positive or negative) is a neces­ sary condition for exchange. As Jevons puts it, "there can be no com­ parison in our theory between things which do and those which do not possess utility." In short, the mathematics offers a foolproof means

42 · Chapter 3

for detecting limiting cases. "The failure of exchange will be indicated by the failure of the equations."33 Jevons remained unsure of his use of the calculus, and for good reason. Others drew to his attention that the mere presence of infinitesimal quantities in his equation does not constitute differentiation or integration. By 1874, Jevons conceded that his exchange equation simply equates ratios of infinitesimal quantities. He inserted the following justification in the second edition of the Theory: It is remarked that the equations in question continually involve infinitesimal quantities, and yet they are not treated as differential equations usually are, that is integrated.... I will here show that the equations employed do not differ in general character from those which are really treated in many branches of physical science. Whenever, in fact, we deal with continuously varying quantities, the ultimate equations must lie between infinitesimals. The process of integration, if I understand the matter aright, only ascertains other equations, the truth of which follows from the fundamental differential equation.34 We will return to this point when we examine his analogy to the principle of virtual velocities. Suffice it to say that he had found an appropriate anchor for the calculus in his analysis of exchange, but lacked the means to secure it firmly in place. One important consequence of the theory of exchange is that "a person distributes his income in such a way as to equalise the utility of the final increments of all commodities consumed." This implies that "in expending a person's income to the greatest advantage, the algebraic sum of the quantities of commodity received or parted with, each multiplied by its final degree of utility (after exchange), will be zero." This "equimarginal principle" lends additional support to the principle of free trade that the classical economists had preached so vigorously. Since exchange only occurs when a net increase in utility results, "so far as is consistent with the inequality of wealth in every community, all commodities are distributed by exchange so as to produce the maximum benefit." In order to attain this, trade should take place with as few restrictions as possible: "perfect freedom of exchange, therefore, tends to the maximising of utility." Jevons's new approach thus reinforced one of the central tenets of the classical theory.35 Implicit in these remarks is the possibility of aggregating from individuals to an entire nation. Yet, as Blaug has rightly noted, "since Jevons denied the possibility of making interpersonal comparisons of utility, the conclusion that 'a perfect freedom of exchange must be to the advantage of all' has no clearly assignable meaning." Jevons has

Mathematical Theory of Economics · 43 also been chastised for failing to explicate the link between marginal utility curves and individual demand curves, even though he must have stumbled upon such figures in Jenkin's work.36 As two scholars have recently shown, Jevons constructed aggregate demand curves, using data from annual corn purchases, in his lectures at Owens College.37 Moreover, as he recognized in his posthumous Principles of Economics, "the total demand is the sum of individual demands." He thus had glimpses of the procedure subsequendy established. What held him back may have been the very astute realization that the law of demand is not only subject to exceptions but, like other functions in economics, is "seldom or never really linear."38 As Samuel Bostaph and Yeung-Nan Shieh have demonstrated, Jevons realized that, until adequate data could be unearthed, individual demand curves must remain purely hypothetical. THE THEORY OF LABOR

Jevons defines labor in terms of negative utility, as "any painful exertion of mind or body undergone partly or wholly with a view to future good."39 It therefore has the same dimensions as utility, namely duration and intensity. With one fell swoop, Jevons has done away with the classical distinction between productive and unproductive labor. Any labor, whether it be bricklaying or balletdancing, insofar as it yields utility to the consumer, is productive.40 The central problem of economics thereby becomes: to "satisfy our wants with the least possible sum of labour." With considerable ingenuity, Jevons attempts to solve this maxima-minima problem geometrically (see Figure 3.3). He recognizes that, for an initial period of time, labor can be pleasurable before the pain sets in. But since labor ceases at some point, it clearly has negative utility. At some point, "the exertion has become so painful as to nearly balance all other considerations." Hence, the labor-intensity curve shown below (curve abed) lies primarily in the negative quadrant, the quadrant of pain. The curve that depicts the degree of utility of the commodity produced by that labor (curve pq) lies, of course, in the positive quadrant. The point at which labor ceases is at m (where qm = dm), the point at which, for a single person, the "pleasure gained is exactly equal to the labour endured."41 In algebraic terms, this point is formulated as follows. Given an instantaneous rate of production, dxldt, with a final degree of utility duldx for commodity x, and the degree of painfulness of labor, dlldt, labor should cease precisely when dlldt = (dx/dt)(du/dx~). If the rate of production is uniform, as in some types of mechanized labor, the

44 · Chapter 3 pleasure

V

m

χ

pain

Figure 3.3

Labor Curve

equation can be simplified to dlldx = duldx. Thus, as Jevons pro­ claims emphatically, we arrive at "the final equivalence of labour and utility"42 The distribution of labor between the production of two commodi­ ties, χ and y, is a straightforward problem to solve. The laborer would spend each successive increment of labor, dl, on the commodity which yields the higher degree of utility. In equilibrium, the incre­ ments of utility of the last portions of χ and y produced must be equal: dux dx

dx _ dug a\ CU1 dy άΙ%

In this way, "the utility produced is at its maximum." Once again, Jevons derives a solution by equating ratios of infinitesimal quantities rather than by using the calculus to determine the first-order and sec­ ond-order conditions for an extremum solution. This line of analysis leads "directly to the well-known law, . . . that value is proportional to the cost of production." Not having introduced costs and wishing to avoid the word value, Jevons suggests that a pref­ erable version of this law would be "that the ratio of exchange of com­ modities will conform in the long run to the ratio of productiveness, which is the reciprocal of the ratio of the costs of production."43 Jevons offers a "proof" of this law in algebraic terms. Adopting the symbol omega (ω) to represent the rate of production, dlldx, and the same symbols previously used to denote the respective degrees of utility, the above equation can be expressed as: φχ · (O1 = ψΐ/ · G)2

Mathematical Theory of Economics · 45 where ω! and ω 2 represent the quantities of the two different com­ modities (x and y) produced by an increment of labor. From the exchange, however, we know that: φ% _ dy ψι/ dx and from the law of indifference we know that: d]L = R dx χ the ratio of the quantities exchanged. We thus have "the all-important equation": R = 2½ (O 1

χ

Having now established that the ratio of exchange "conforms" to the ratio of the rate of productiveness, Jevons draws a conclusion that is tantamount to a labor theory of value: "thus we have proved that com­ modities will exchange in any market in the ratio of the quantities produced by the same quantity of labour."44 Prices are at last officially introduced, though with reservations. A price, Jevons maintains, is almost as ambiguous as the concept of value. But we do know that in exchange: ϊ = Ei χ p2 where px and p 2 are the per unit prices of χ and y respectively. We thus arrive at the final set of equations: R = Ei χ p2

=

Φ* = i2a ω y ι

Here then is the final equivalence between the degrees of productive­ ness and the degrees of utility. Why, then, did Jevons place such emphasis on utility? One answer stems from his belief that labor differs "infinitely in quality and effi­ ciency." Although the duration of labor may be readily measured, the intensity, as in the case of positive utility, can only be known subjec­ tively. Contrary to Ricardo, Jevons contends "it to be impossible to compare a priori the productive powers of a navvy, a carpenter, an iron-puddler, a schoolmaster, and a barrister."45 Jevons, however, faced the same problem of interpersonal compari­ sons of utility in the commodity market. Although it may be impossi­ ble a priori to make such comparisons in terms of the quantity of util-

46 · Chapter 3

ity, his entire case rests on the fact that the ratios of exchange are manifestations of these subjective appraisals. The same approach must surely hold for the price and quantity of labor. The very existence of a wage scale for separate trades overrides any claim to the contrary. Just as prices for goods correspond to the final degrees of utility, so too wages correspond to the final productiveness of labor inputs. Even more to the point, Jevons encouraged the scientific investigation of the intensity and productiveness of labor. He refers the reader to studies undertaken by Charles Babbage, S. Haughton, and himself on the mechanical measure of work in terms of pounds lifted per second.48 Implicit in these researches is the possibility of comparing the labor of different persons and trades. Clearly, Jevons's enthusiasm for mechanical and quantitative methods got the better of him. The comparison, it appears, of different kinds of labor is as or even more feasible than the comparison of individual feelings. As is well known, Jevons dispensed with the Malthusian mechanism for varying the supply of labor, at least in the short run.47 A population is given exogenously and human reproduction is no longer said to be at the mercy of a subsistence wage. The supply of labor is taken to be a positive function of the real wage, though in some professions, at a sufficiently high salary a person might actually substitute leisure for additional income.48 Jevons was thus one of the first to articulate the notion of a backward-bending supply curve of labor. Jevons is adamant that labor can never be the cause or origin of value. Given the existence of such goods as rare coins and paintings, of which the value is "wholly independent of the quantity of labour originally necessary to produce them," the general rule must be that utility, and not labor, is the source of all value. How, then, can labor in some sense measure value? In those cases in which the quantity of the goods produced can be varied, supply must play a role in the determination of value. Thus labor, though "never the cause of value, . . . is in a large proportion of cases the determining circumstance."49 In the second edition, Jevons added the following "catena": Cost of production determines supply; Supply determines final degree of utility; Final degree of utility determines value.

From his work in logic, Jevons had a firm grasp of the principle of transitivity. In this case, it follows that the cost of production also determines value. How does he square this with his insistence that "value depends solely on the final degree of utility"?50 It seems that he has conflated value in the long run and the short run without ex-

Mathematical Theory of Economics • 47 planation. To give him the benefit of the doubt, he may have inserted this aphorism primarily for pedagogical purposes rather than as an accurate summary of his position. From another perspective, however, Jevons appears to have gone full circle. Insofar as labor can be treated as negative utility, value does indeed depend entirely upon utility. In the preface to the second edition of the Theory, Jevons admits that "the whole theory might probably have been put in a more general form by treating labour as negative utility, and bringing it under the ordinary equations of ex­ change," but time and energy prevented him from carrying out this revision.51 Hence, he envisioned a more complete model, in which the relative contributions of positive and negative utility in the determi­ nation of exchange value would be given equal weight. Capital is also, as we shall see, linked to the concept of utility. The fact that he did not formulate this revision by giving equal emphasis to costs strongly suggests that he wished to highlight positive utility as the more novel element of his theory, as the element which appeared to overturn the Ricardo-Mill theory of value. This lends some credibility to Alfred Mar­ shall's accusation that Jevons twisted "his own doctrines so as to make them appear more inconsistent with Mill's and Ricardo's than they really were."52 THE THEORY OF RENT AND CAPITAL

On the subjects of rent and capital, Jevons claims to be in "fundamen­ tal agreement" with the classical authors, including Ricardo. Because the theory of rent already embodies the notion of a margin and, more­ over, has "a distincdy mathematical character, . .. it seems to give a clue to the correct mode of treating the whole science."53 For this rea­ son, the concept of rent "needs little or no alteration to adapt it to expression in mathematical symbols."54 Increments of labor (ΔΖ) yield increments of produce (Ax) that, once decreased infinitesimally, give ratios of the productiveness of labor in terms of the differential calcu­ lus (dx/dl). The principle of diminishing returns implies that a final dx/dl will be reached which also gives the rate at which labor is rec­ ompensed. Rent will be the difference between the total product (x) and the payment to labor (I • dx/dl). Jevons illustrates this geometri­ cally (Figure 3.4), where the area under the curve ape represents the produce, mm' the final increment of labor, orpm the payment to labor, and rap the rent. Jevons tends to view rent intensively, as a return to additional in­ vestments of labor or capital in a piece of land. In contrast to his pre­ decessors, rent is "determined by the excess of produce in the most

48 · Chapter 3 land y a b

r

ο Figure 3.4

mm' η χ Rent Diagram

profitable employment," rather than as the return to the landlord in the next to least arable strip of land. Moreover, he insists that it be included whether the land be used for agriculture or industry. What was the exception for Mill now becomes the rule. For the sake of even greater generality, he claims that rent exists whether one works on one's own land or leases it from someone else. "This expression [for rent] represents the advantage he derives from the possession of land in affording him more profit than other methods of employing his la­ bour. It is therefore the rent which he would ask before yielding it up to another person, or equally the rent which he would be able and willing to pay if hiring it from another." The opportunity cost implies that rent is no more eliminable than wages or interest. Indeed, "the parallelism between the theories of rent and wages is seen to be per­ fect in theory, however different it may appear to be in the details of application. Precisely the same view may be applied, mutatis mutan­ dis, to the rent yielded by fixed capital, and to the interest of free cap­ 55 ital." In his theory of capital, Jevons retains two central propositions: the rate of profit tends to equal the rate of interest, and the rate tends to diminish over time. The interest rate is determined by the final degree of capital. Given a fixed amount of labor, and a function Ft whereby additional increments of capital produce with time additional incre­ ments of a commodity, the interest of capital is simply "the rate of increase of the produce divided by the whole produce." In algebraic terms, this is represented as F't/Ft, where the numerator is the first derivative, with respect to time, of Ft. 5 6 Recall that this insight was first discerned in 1860, and deemed to be "quite new." Both capital and labor are essential to production, yet neither is able to reap its full value to society, for this would be "indefinitely great." Given normal market conditions, the monetary returns to labor and

Mathematical Theory of Economics · 49 capital are driven down to equal the gain from the final degree. "The interest of capital has no relation to the absolute returns to labour, but only to the increased return which the last increment of capital allows." It follows, then, that "the wages of a working man are ultimately coincident with what he produces after the deduction of rent, taxes, and the interest of capital." Workers are said to forfeit a portion of their produce to capitalists in return for important services that they could take on themselves: "Every labourer ultimately receives the due value of his produce after paying a proper fraction to the capitalist for the remuneration of abstinence and risk."57 Jevons recognized that "to obtain a correct notion of capital is the greatest difficulty in Political] E[conomy]."58 Such skepticism seems to have seeped into his own analysis, for he oscillates back and forth between some bold stabs and a reliance on more conventional notions. Jevons explicitly repudiated the wage fund theorem, whereby the level of wages is determined by the fixed fund of variable capital (commonly seen as the annual harvest of corn) divided by the number of workers. But he allowed it to hold in a temporary manner for industries that have yet to acquire adequate cost data and thus be in a position to calculate the relevant final degrees of productivity. Moreover, as Black has noted, there are other vestiges of the wage fund theory in Jevons's account. His key innovation was the extension of the time period beyond the conventional year, but he still retained the idea of a fixed stock.59 Jevons also takes issue with the idea of capital as embodied labor. Future valuations, due to commercial fluctuations, may have litde to do with prior costs. For Jevons, "industry is essentially prospective, not retrospective." People undertake economic activities with an eye to future gain. Potential or anticipated utility drives the system. As a result, past labor counts for nought in economic calculations. "Labour once spent has no influence on the future value of any article: it is gone and lost for ever. In commerce bygones are for ever bygones."60 For this reason, he impugns Mill's fourth proposition on capital, that "demand for commodities is not demand for labour."61 For Jevons, production is always the production of utility. The capitalist or merchant is merely a catalyst, anticipating what the consumer will purchase: But the whole purpose and end of the business is to meet the demand of this or some equivalent consumer. The capitalist is merely a kind of assistant in the process, anticipating for his own profit the wants of the consumer. .. . Mill wholly fails to bear in mind, ... that capital is nothing but advances. . . All is done for the consumer and must be paid for by him.62

50 · Chapter 3 Jevons sides with Nassau Senior and others in attributing the origin of capital to abstinence. "If there is any enjoyment which we could procure by present labour, but we abstain from enjoyment and use the labour for a distant but greater end, here is proper capitalisation." Capital is in effect a certain kind of relationship between objects and persons. A loaf of bread can sustain "the hardworking navvy, the idle beggar, the well-to-do annuitant." But only in the first case is it capital, for only there is it intended for the production of additional wealth. From this it follows that "there is really nothing which marks off certain commodities as being by nature capital as compared with other commodities which are not capital. The very same bag of flour may have to change its character according to the mental changes of its owner."63 Jevons has thus moved a considerable distance from the traditional notion of capital as a set of tools and store of grain. In fact, he is disinclined to view machinery, tools, and modes of transport as capital at all. "I would not say that a railway is fixed capital, but that capital is fixed in the railway. The capital is not the railway, but the food of those who made the railway." Nor does he regard the laborer as creating capital. "Labour creates nothing." The true source of capital is the passage of time, coupled with the intentions of the owner. Education is another form of capitalization, the cost of which will be equal to the income forgone while engaged in one's studies. Attending law school (his example) is in fact no different than the classic case of the deer hunter taking time from his hunting to improve his bow and arrow.64 As with rent, Jevons considers the problem from the perspective of a single individual. "The whole of the phenomenon of capital might go on in the case of a single family's industry where there was nobody else to hand it over to."65 The division of labor and frequency of exchange simply introduce "irrelevant complications." Indeed, the existence of separate economic classes is like a wrench thrown into a welloiled machine: "the fact that workmen are not their own capitalists introduces complexity into the problem."66 Jevons has hereby shed the class distinctions that figured so significantly in the classical theory. His subjective theory of costs places every individual on equal footing. It is their mental deliberations, both of factors of production and the final degree of utility, that drive the entire system. Jevons's propensity for greater generality is also apparent in his criticisms of the classicist's exclusion of certain goods as capital. His emphasis on utility and satisfaction enables him to see that housing contributes to the sustenance of workers as much as food or clothing. "Capital is sunk in farming that we may have bread, in cotton mills

Mathematical Theory of Economics · 51 that we may be clothed, and why not in houses that we may be lodged? If land yields an annual revenue of corn and wool, milk, beef, and other necessaries, houses yield a revenue of shelter and comfort."67 Furthermore, the ownership of such property is irrelevant to its status as variable capital. "Whenever one person provides the articles and another uses them and pays rent, there is capital. Surely, then, if the same person uses and owns them, the nature of the things is not fundamentally different." In fact, most people do not own the things they consume. They live on credit: There is hardly ... a man of fashion in London who walks in his own clothes; and the tailorsfindin the practice of giving credit a very profitable investment for capital. Except among the poorer classes, and often among them, food is seldom paid for until after it is consumed. Interest must be paid one way or another upon the capital thus absorbed. Whether or not these articles in the consumers' hands are capital, at any rate they have capital invested in them—that is, labour has been spent upon them of which the whole benefit is not enjoyed at once.68 Capital, in short, consists of "all useful objects which, in supplying a labourer's ordinary wants and desires, enable him to undertake works of which the result will be deferred for a greater or less space of time."69 This conception of economic activity, as a myriad of simultaneous but unequal acts of abstention, constitutes a distinct shift away from the classicist's adherence to a rigid calendar. From his earliest writings, Jevons conceived of capital as commensurate to utility. The "amount of employment of capital is the amount of utility multiplied by the number of units of time during which its enjoyment is deferred." In his discussion of the dimensions of capital, he compares abstinence to labor, noting that both can be thought of in terms of utility and thus formalized in the same manner. Labor and capital are the means for the production of utility, while consumption "is simply the destroying of utility." Marshall was to underscore this new perspective: "just as man can produce only utilities, so he can consume nothing more."70 Jevons has taken a sizable step away from the classical approach to capital and labor, which were thought to be governed by physical or natural processes and thus to transcend the contingencies of human institutions. Jevons identifies the three requisites of production as natural agents, labor, and capital, but since he grounds capital and labor in his calculus of pleasure and pain, natural agents must fall into a very different category. Somewhat paradoxically, he has swung back to the Physiocratic distinction between land as a gift from nature, and

52 · Chapter 3 labor and capital as artificial components of the economy. But Jevons did not thereby privilege physical nature. Quite the contrary, natural resources are taken as an exogenous contraint, and human wants and ingenuity as the source of wealth. If there are laws to discover in the economic realm, they stem from human psychology. In this respect, the contrast with Enlightenment philosophers of wealth could not be greater. CONCLUSION

It is not difficult to find examples, such as the central exchange equation, to lend weight to A. A. Young's accusation that Jevons gave a "mathematical garb to results reached by non-mathematical reasoning." Marshall made a similar criticism. "In general, indeed, he [Jevons] makes but little use of mathematical methods of reasoning."71 Jevons himself was aware that his mathematics was awkward and unsatisfactory: "mathematical readers may perhaps think that I have explained some elementary notions . . . with unnecessary prolixity." But he excused himself on the grounds that he was an economist writing for other economists, not for professional mathematicians: "In short, I do not write for mathematicians, nor as a mathematician, but as an economist wishing to convince other economists that their science can only be satisfactorily treated on an explicidy mathematical basis."72 In sum, the body of the text of the Theory of Political Economy does not fulfill the expectations set out in the prefaces. There are neither formal proofs nor a systematic recognition of all of the various underlying assumptions. The presentation of the material is often irregular and thus less general in scope than it might otherwise have been. Nor does Jevons sound convincing when he posits certain variables, for example time and intensity, and then sets out an algebraic function. Understandably, his attempts to quantify utility by integration led nowhere. And rather than apply the differential calculus to the functions he posits, he simply equates ratios of infinitesimals. As an example of applied mathematics, the Theory is sketchy and incomplete. If one were so inclined, one could also level similar criticisms at Newton's Principia, yet no one would deny the brilliance of that achievement. What is remarkable about Jevons's work was his ability to shed the classical analytical apparatus and to incorporate ever greater abstraction and generality. And for these steps, the heuristic value of mathematics was uppermost. The calculus guided him toward the marginal analysis of prices and costs as well as the equimarginal principle. His use of geometry, particularly in the analysis of la-

Mathematical Theory of Economics · 53 bor, his recognition of the dimensions of the variables of his theory, and the discussion of limiting cases and exceptions to the equations all stemmed from a mathematical approach. Much credit is also due his attempt to summarize barter exchange in the form of a single equation. And, by casting the key economic variables, including labor, in terms of utility, he envisioned an economy of unprecedented plasticity. Jevons was the one who released economics from its material constraints and shifted the focus to the deliberations of the mind. The economist would henceforth look to psychology rather than to physics for his fundamental principles.

C H A P T E R FOUR

Logic and Scientific Method The mathematician is only strong and true as long as he is logical, and if number rules the world, it is logic which rules number. —Jevons There are few books, so superficial in argument yet suggesting so much truth, as Jevons' Principles of Science. —John Maynard Keynes

philosophical writings fall squarely within the tradition of British empiricism, the tradition of Bacon, Locke, Berkeley, and Hume. But the century that separates Hume from Jevons spawned several new schools of thought. Without question, the most influential of these was utilitarianism, ably developed by Jeremy Bentham, William Paley, and the two Mills. Mid-nineteenth century Britain also witnessed a major revival in logic. With the publication of works by John Herschel, William Whewell, and John Stuart Mill in the period 18301843, new ground was broken in the philosophy of inductive logic, or what is now known as the philosophy of science. And in 1847, two books, Augustus De Morgan's Formal Logic and George Boole's Mathematical Analysis of Logic, ushered in the modern era of symbolic logic.1 Jevons was primarily engaged in weaving together these various strands of British philosophy. His Principles of Science (1874; 1877) was the first work to synthesize the informal inductive logic of Mill with the formal deductive logic of Boole and De Morgan, as well as to attempt to reconcile some of the disputes between Mill and Whewell on scientific methodology. For this reason, it has generally been regarded as a commentary on Victorian philosophy rather than as an original work in its own right and, to a large extent, this is a just appraisal. But there are also some ideas of considerable originality, as we shall see. Most impressive is Jevons's grasp of the limitations of science and his marshalling of evidence to support his claims. Of all his immediate predecessors, Jevons the philosopher was most directly indebted to John Herschel. He had first read Herschel's Preliminary Discourse during his Australian sojourn and, even after an

JEVONS'S

Logic and Scientific Method · 55 exposure to Kant and Hegel during his university studies, remained a loyal adherent. Because of a comparable faith in the physical sciences, neither Herschel nor Jevons felt compelled to defend their adherence to empiricism. Whereas Mill and Whewell analyzed in depth the mechanisms by which ideas are acquired, and made full-fledged attempts to anticipate objections to their respective positions, both Herschel and Jevons brushed over various epistemological problems that would have threatened their empiricist stand. Not only did Jevons assume outright that "all knowledge proceeds originally from experience," but problems of perception were promptly dismissed on the grounds that scientific research necessarily presupposed the objectivity of perception. Jevons, like Herschel, was first and foremost the practical scientist with litde patience for philosophical problems that might impede the task at hand.2 Herschel, Whewell, Mill, and Jevons were, each in their own way, intent on defending scientific practice as a legitimate means for the acquisition of knowledge. The main area of contention, though not always explicitly recognized as such, centered around Hume's socalled problem of induction. Hume recognized that the mere constant conjunction of two events does not in itself demonstrate the existence of a causal law. Indeed, a consistent empiricist cannot rationally justify any belief in causation (although empiricism may provide an explanation for why we, as creatures of habit, hold this belief). Such skepticism did not sit well with English philosophers, accustomed as they were to the triumphs of Newtonian mechanics. Jevons, however, was more aware of the limitations of scientific knowledge. Perhaps more than any of his contemporaries, he moved toward the tenets of fallibilism. Much of the preliminary groundwork for the philosophy of science, specifically the nature of induction and deduction in the acquisition of knowledge, had already been threshed out by Whewell and Mill. One of the major criticisms Whewell levelled at Mill pertained to the proper method of inquiry and justification in the study of scientific methodology.3 Whewell insisted that any claim about scientific method must be supported by historical evidence. It was essential, in his view, that what is prescribed by the philosopher be at the same time a description of actual practice. He went even further by insisting that the philosopher must induce the properties of scientific method from the historical data rather than simply use examples which happen to suit a particular claim. Although Whewell made a valiant effort to comply with these directives, his two major works, History of the Inductive Sciences (1837) and Philosophy of the Indue-

56 · Chapter 4

tive Sciences, Founded upon Their History (1840), readily attest to the open-endedness of such an undertaking. Mill recognized that providing an example or even series of examples to illustrate a point did not constitute justification. Although he by no means opposed the use of historical data to support a claim, he tended to argue at the level of abstractions. It was more important, in Mill's view, to abide by the rules of inductive logic, rules that were to be arrived at on strictly logical grounds, than to provide descriptive summaries of the history of science. Thus, whereas Whewell criticized Mill for failing to demonstrate that his rules of reasoning were practiced by scientists, Mill chastised Whewell for neglecting to derive a general theory of proof. To a large extent this conflict between Whewell and Mill, while stemming from a more fundamental clash of epistemologies, was due more immediately to the fact that each was taking a one-sided approach to the study of the philosophy of science. Whewell was primarily concerned with the "logic of discovery," a topic that tends even to this day to be descriptive and historical in approach. Mill, on the other hand, put forth a "logic of justification" for scientific laws; he attempted to show on purely logical grounds how beliefs are warranted. That Mill's exposition was not brimming with examples from the sciences was no doubt due as much to a dearth of supporting evidence as to his poor grounding in the history of science.4 His rules of induction, the celebrated four methods of experimental inquiry, were (and still are) primarily prescriptive. Jevons attempted to strike neutral ground in this dispute by analyzing inductive inferences first on logical, then on empirical grounds. Although he ran into some difficulties in bridging the two approaches, he succeeded in handling them in a way that gave a semblance of greater compatibility. Whewell's call for an initial survey of the history of science was not heeded, since even an exhaustive study of the past would, in Jevons's view, not yield a general understanding of scientific method. It was more important to start by studying the most fundamental branch of knowledge, logic, from which first mathematics and then the rest of science were derived. "In attempting to give an explanation . . . of Scientific Method, I have first to show that the sciences of number and quantity repose upon and spring from the simpler and more general science of Logic."5 And when it came time to examine the intricacies of empirical science, Jevons drew on numerous examples from the history of science to support his insights, whether on measurement, experimentation, or the extension of theory. Mill's conception of logic as the science of proof or evidence was considered to be too broad. Logic was for Mill in some sense an em-

Logic and Scientific Method · 57 pirical science that judged not only the form of an argument but the content as well. Jevons adopted what has become the more orthodox view. Logic, as a purely formal and abstract science, is independent of concrete existence. Indeed, he maintained that "nothing is more certain than logical truth," precisely because it makes no reference to the existence of concrete objects. For Jevons, there was but one scientific method for the natural and social sciences alike. "The application of Scientific Method cannot be restricted to the sphere of lifeless objects. We must sooner or later have strict sciences of those mental and social phenomena, which, if comparison be possible, are of more interest to us than purely material phenomena."6 He makes no attempt, however, either in the Principles or in a separate work, to justify his position. Given his sustained interest in reforming economic method and his general hostility toward Mill, this is a curious oversight. Perhaps Jevons found Mill's celebrated argument for methodological monism sufftciendy convincing to be accepted without qualification. As he grants in the introduction to the Theory of Political Economy, "Mill is substantially correct in considering our science [of economics] to be a case of what he calls the Physical or Concrete Deductive Method."7 Jevons sets himself apart from Mill by making exclusive use of the physical sciences to illustrate his methodological claims. "It is the proper course of reasoning to proceed from the known to th^ unknown. . . . The physical sciences may therefore be properly made the practise-ground of the reasoning powers, because they furnish us with a great body of precise and successful investigations."8 Unlike Mill, Jevons did not regard the political economy of Smith and Ricardo as an exemplary branch of knowledge. As far as he was concerned, there was no mature social science. Nor, given the general appeal of the Principles, would it have been appropriate to set out his new theory of mathematical economics and hope to convince his readers that it confirmed his methodological claims. In addition to the Principles, Jevons's philosophical writings include a number of essays and elementary texts on formal logic, a series of five essays on the philosophy of John Stuart Mill, including the tenets of utilitarianism, and various short articles on miscellaneous topics. We will be primarily concerned with the contents of the Principles, with reference, as appropriate, to his other works. FORMAL LOGIC

Formal logic is defined as "an objective science, like mathematics or mechanics," and not to be mistaken for the logic that both William

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Hamilton and Mill deemed a branch of psychology. Although Jevons grants that psychology might shed light on the foundations of logic, that it might in particular contribute to an understanding of the basic powers of the mind, logic could be studied independently of such considerations. He refers to other sciences, such as optics, where one might ignore foundational problems, the nature of light itself, yet still form sound principles at the descriptive level. He thereby excuses himself without further ado from dealing with the more intractable problems of psychology.9 Citing Alexander Bain as his authority, Jevons attributes three fundamental powers to the mind, the power of discrimination, of detecting identity, and of retention. From these powers arise, in some unspecified manner, the three fundamental laws of thought, the law of identity (whatever is, is), the law of contradiction (a thing cannot both be and not be), and the law of duality (a thing must either be or not be). These laws are of critical significance. Insofar as they enter in one form or another into virtually every act of reasoning, they are three of the most certain and pervasive laws yet known. But do the laws of thought belong to the mind or to material nature? Science, he declares, is clearly in the mind and not in the things: it is an artificial system that classifies natural phenomena according to somewhat arbitrary rules.10 With regard to logic, however, Jevons is more inclined toward realism. Logic, because it provides an essential framework for all reasoning, is higher up on the epistemic ladder than science. That is to say, "before the mind can perceive or reason at all, it must have the conditions of thought impressed upon it." It is impossible, therefore, to prove the basic laws "by any process of reasoning or observation." They are the necessary presuppositions for any such proof. One is therefore led into a vicious circle, as Jevons is willing to admit; "this is one of the most difficult questions of psychology and metaphysics which can be raised."11 Jevons would not accept intuition as a source of knowledge. The fundamental laws of thought must have been adopted in an empirical manner. They must, therefore, reside in nature; in fact, they must pervade nature throughout. It is not a matter of finding specific confirming instances. Once the mind has grasped these basic canons, there is no further need, as Boole had already recognized, for additional sources of verification. It just so happens that nature and the mind are in such a union that the laws of thought hold true.12 The laws of thought, however, are not necessary laws. It is possible to violate them. "They are the laws which the mind ought to obey rather than what it always does obey."13 Whewell had defined a necessary law as one for which the contradiction would be inconceivable.

Logic and Scientific Method · 59 He granted that one must first acquire a clear and distinct understanding of a necessary law. A neophyte might well reason erroneously. But once a person had grasped the law of the excluded middle in its fullest clarity, it would be impossible to conceive of its contradiction. By contrast, Jevons insists that anyone, even the most astute scientist, could commit fallacies. In this respect, then, his laws did not carry the same powers of necessity as Whewell's. Nevertheless, insofar as they capture reality in its most essential form, it might have been more consistent to have conceded Whewell's position. According to Jevons, all inference, whether it be inductive or deductive, is based on the principle of the substitution of similars, the principle which states that "so far as there exists sameness, identity, or likeness, what is true of one thing will be true of the other."14 Jevons claims to have arrived at the principle through an analogy to mathematics, more specifically, to the process of substituting a term for its equivalent in an algebraic equation. It was, for him, patently obvious that all inference in mathematics is based on the recognition of quantitative equality. One need only look at the cornerstone of geometry, Euclid's first axiom, that things equal to the same thing are equal to each other. Conversely, "two inequalities give no ground of inference whatever."15 The force of analogy between logic and mathematics was particularly strong for Jevons. It led him, on this issue, to rebel against the views of De Morgan, who had refused to use the equation sign because the analogies were not sufficiendy compelling. As early as 1863, Jevons sought "to show that the prop[ositio]n is really an equation, analogous in most of its properties to an equation in quantity."16 In his view, "the true clue to the analogy of mathematics and logic has not hitherto been seized."17 He expanded on this in the Principles: To denote the relation of sameness or identity I unhesitatingly adopt the sign =, so long used by mathematicians to denote equality. The meaning of the sign has however been gradually extended beyond that of equality of quantities; . . . The force of analogy has been so great that writers in most other branches of science have employed the sign.... I hold that the common use of this sign = in so many different meanings is really founded upon a generalization of the widest character and of the greatest importance—one indeed which it is a principal purpose of this work to explain.18 This in turn led him to devise a new interpretation of deduction. He retained the standard view of deduction as a truth-preserving form of inference: "from one or more propositions called premises to draw such other propositions as will necessarily be true when the premises are true." But instead of a multitude of rules to handle the scholastic

60 · Chapter 4 moods of the syllogism, deduction is carried out by one uniform and unambiguous operation: "for any term occurring in any proposition substitute the term which is asserted in any premise to be identical with it." In short, one substitutes in precisely the same way as in algebra. With this principle in hand, Jevons set out to reconstruct a much more general logic: A true reform in logic must consist, not in explaining the syllogism in one way or another, but in doing away with all the narrow restrictions of the Aristotelian system, and in showing that there exists an infinite variety of logical arguments immediately deduciblefromthe principle of substitution of which the ancient syllogism forms but a small and not even the most important part.19 Great attention is paid to the invention of a "clear, brief, and appropriate system of symbols." By culling the more straightforward operations from the logics of Boole and De Morgan, Jevons succeeded in devising a system of symbolic logic that outlived several generations of university teaching.20 Following De Morgan, he adopted the use of capital (type) letters to represent terms (either nouns or adjectives) and italicized lowercase (type) letters to represent the negation of terms. For the proposition "hydrogen is the least dense element," he writes A = B, where A represents hydrogen and B the least dense element. And to convey the fact that the subject is only a subset of the predicate rather than identical to it (as in the proposition "copper is a metal"), he writes A = AB. No additional sign for the conjunction of terms is employed. Only two more symbols are needed to complete the system, "0" (zero) to represent nothing or that which is inconceivable, and " ·|· "to represent disjunction. Although this set of symbols cannot convey as much information, the end result is a simpler, more elegant language than that put forth by either Boole or De Morgan.21 With these symbols, Jevons transcribes the fundamental laws of thought, as well as the basic operations—commutation, distribution, and transitivity22—as follows: Law of Identity: Law of Contradiction: Law of Duality: Law of Commutativeness: Distribution: Transitivity:

A=A Aa = 0 A = AB ·|· Ab AB = BA; IfA = B, B = A; A ·|· B = B ·|· A A(B |- C) = AB ·|· AC A = B and B = C; therefore A = C

There are two additional laws that do not mimic algebraic operations. These are the Law of Unity (A ·|· A = A) and the Law of Sim-

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plicity (AA = A). Jevons briefly refers to another symbol, ~, to represent complete nonidentity between terms, but shows that it can be represented just as well by an equation sign and the negation of the second term. Thus, A-B becomes A = Ab. Occam's razor favors the affirmative form, since it does away with the tilde. But then the system cannot represent double negation, something Jevons neglects to discuss at any length.24 By introducing a number of constraints in his system, Boole's logic permitted all four of the simple algebraic operations: addition, subtraction, multiplication, and division. Jevons not only did away with subtraction and division, but, in light of his laws of unity and simplicity, also restricted the interpretation of addition and multiplication.25 As we have seen, he upheld certain parallels with algebra, the operations of commutation, association, and transitivity, as well as the use of the equation sign. But in general he was less prone than Boole to import the elements of algebra into his logic. To represent what Jevons calls "direct deduction," take the premises "iron is a metal" or A = AB, and "a metal is an element" or B = BC. From this we conclude, "iron is an element," A = ABC, by substituting in the first premise the equivalent of B, namely BC. This very simple case of deduction, known in the scholastic tradition as the mood Barbara, makes the whole process appear trivial. A more complicated example, in the mood Camenes, uses the disjunctive and the law of duality and thus demonstrates the powers of his system. Take the premises (1) "all monarchs are human beings" or A = AB and (2) "no human beings are infallible" or B = Bc (all humans are fallible). From the law of duality, we can expand C such that C = aC ·|· AC. Substituting in this last expression the equivalent of A, namely, AB, and the equivalent of B, namely Bc, we have C = aC ·|· ABcC. But from the law of contradiction, we know that Cc = 0. Thus, we can conclude that C = aC or all infallible beings are nonmonarchs.26 Standard fallacies are readily detectable by this procedure since illegitimate substitutions are prevented as long as one keeps to the rules. Take, for example, the premises "hydrogen is an element" or A = AB, and "all metals are elements" or C = CB. From this we can draw no conclusion about the metallic nature of hydrogen. It is not possible to factor out the B in either premise and thus no substitution can be made. As long as the predicate is quantified correcdy (i.e., one writes A = AB rather than A = B), fallacies cannot be committed. Insofar as one can make a legitimate substitution, one reasons deductively. Moreover, because commutation holds, it does not matter on which side of the equation one makes the substitution or in what order a compound term is formalized. The simplicity and elegance of

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Jevons's system, as well as its pedagogical value, should be readily apparent. Indirect deduction, which Jevons claims to be of comparable impor­ tance to direct deduction, also involves the substitution of similars. For a given term A, expansion by the law of duality yields A = ABC ·|· ABc ·|· AbC ·|· Abe. With premises such as A = AB and B = BC he can arrive at the conclusion A = ABC by substituting the equivalent of A and the equivalent of B in the expanded disjunctive form of A. Thus we have A = ABC ·|· ABCc ·|· ABbC ·|· ABbCc. By the law of contradiction we eliminate every term except ABC, thereby deriving the conclusion. One of the most appealing features of this method is that the pro­ cess can be extended to any degree of complexity. For η terms, one obtains 2" members of the expanded disjunctive form, which Jevons entitles the "Logical Alphabet." Substitution can be carried out me­ chanically, as can the elimination of contradictory terms. "Logical de­ duction, even in the most complicated questions, becomes a matter of mere routine, and the amount of labour required is the only impedi­ ment." 27 This gave him the idea of constructing several logic machines, which resembled small keyboard instruments and are now honored as early computers.28 Jevons's logical piano could handle up to four terms and their negations (one per key) and provide all of the possible com­ binations. Its operating principle was one of the most basic modes of thought, the law of duality (A = AB ·|· Ab). Given one to four terms, the machine, using this law, would expand, eliminate those with con­ tradictions, and churn out the conclusion. In this way, "the actual pro­ cess of logical deduction is thus reduced to a purely mechanical form, and we arrive at a machine embodying the Laws of Thought."29 Jevons's system of logic has certain limitations. For one, he does not attempt to incorporate conditional propositions or recognize that by the rule now known as implication, one can capture if-then state­ ments with the disjunctive (pDq = —pvq, or, in Jevons's terminology, α ·|· B). He is aware that most hypothetical propositions can be ex­ pressed in categorical form. For example, "if iron contains phospho­ rus, it is britde" could be expressed as "iron, containing phosphorus, is brittle" or AB = ABC. But Jevons did not take the matter any fur­ ther. As we have seen already, he cannot represent the rule of double negation; by the law of simplicity, aa is equal to a, not A. Other, more complicated types of logic, such as the logic of relations, were also beyond his reach. By excluding a symbol to represent the "universe of discourse" (such as Boole's "1"), Jevons also limited the range of op­ erations. Nevertheless, as a system it is complete and rigorous, its

Logic and Scientific Method · 63 greatest virtue being, as Jevons himself recognized, that there are "no uninterpretable or anomalous results. .. . Every process is of self-evident nature and force, and governed by laws as simple and primary as those of Euclid's axioms."30 Jevons took Boole to task on several accounts. Most importantly, he criticized Boole for claiming that his system of logic represented the "laws of thought." Quite correcdy, Jevons recognized that Boole had created not a system of logic but a new branch of algebra, and in that sense did not represent "the natural system of human thought." Simply on account of its many restrictive interpretations, Boole's system could not possibly represent the "laws of thought" in their entirety. Jevons also contested Boole's acceptance of division in his system, since it led to such indeterminate entities as 0/0 and 1/0. Yet another area of dispute was over the interpretation of the disjunctive. Following De Morgan, Jevons argued that the disjunctive should be taken in the inclusive sense, as either "A or B or both A and B," rather than in the exclusive sense expounded by Boole. Because Jevons, rather than Boole, popularized symbolic logic, we to this day favor the inclusive reading.31 In his Pure Logic (1864), Jevons sought to demonstrate, in opposition to Boole, that: "All and more than all the ordinary processes of logic may be combined in a system founded on comparison of quality only, without reference to logical quantity." By this he meant that the terms of logic could be interpreted solely with respect to their intent, to the properties or qualities connoted, without reference to their extent, to the number or quantity of individuals denoted by the predicate: "The primary and only present meaning of a name or term is a certain set of qualities, attributes, properties, or circumstances, of a thing unknown or pardy known."32 The intent and extent of a term are inversely related. The greater the number of qualities connoted, the fewer the number of individual objects denoted in the universe. Jevons favors the intensive or qualitative view because it is more enduring. For example, he points out that the extensive meanings of the terms "planet" and "element" had both changed during the nineteenth century, but that the intensive meanings, the properties connoted by these terms, had remained the same. In the Principles, Jevons is less insistent that the terms of logic be taken without reference to their extent, an indication that he may have conceded the fact that the distinction was not as solid a one as he had hoped. Logicians have since shown that a system of logic such as the one put forth by Jevons is in fact unaltered by an acknowledgment of the extension of terms.33 Nevertheless, for Jevons, the idea of a pure logic, of a "logic of quality apart from quantity," was of great

64 · Chapter 4 significance. He wished above all to put Boole on his head, to demonstrate that logic was more fundamental than mathematics rather than the reverse. "In my opinion," he wrote, "logic is the superior science, the general basis of mathematics as well as of all other sciences."34 It was essential, therefore, to purge from logic any reference to quantity, the defining characteristic of mathematics. The first step in the reduction of mathematics to logic was to establish a criterion that could demarcate the two branches of knowledge. This, we have just seen, was the recognition of logic as the study of "quality apart from quantity." The second step, and in this respect Jevons's intuitions were sound, involved an analysis of the concept of a number. "Number," he wrote, "is but another name for diversity. Exact identity is unity and with difference arises plurality."35 From the law of unity, we see that exactly identical terms do not generate a number: A ·|· A ·|· A = A. It is only through qualitative differences, say between three similar but slightly different objects, that plurality can be conceived. The number three could be formed by A' + A" + A'", although Jevons preferred to use his logical alphabet. Expansion of A gives us AB, aB, Ab from which the notion of three allegedly emerges. No justification for this conceptual leap, alas, is provided. Jevons says nothing more than that the concept of a number involves some form of "logical discrimination." Mathematics, in some mysterious way, is thereby derived from the basic laws of logic via the principle of number. And since, according to Jevons, "nothing is more certain than logical truth," the further mathematics strays from logic, the more subject it is to uncertainty. However appealing his logicist thesis, his failure to provide an analysis of the conceptual foundations of numbers, let alone explicate how numbers give rise to mathematics, did little to win disciples.36 The idea that mathematics and logic share a common conceptual base was derived from his conviction of the unity of knowledge. In the same chapter in which he argues that "number is but logical discrimination, and algebra a highly developed logic," he claims: I believe that all the sciences meet somewhere. No part of knowledge can stand wholly disconnectedfromother parts of the universe of thought; it is incredible, above all, that the two branches of abstract science, interlacing and co-operating in every discourse, should rest upon totally distinct foundations. I assume that a connection exists, and care only to inquire, What is its nature? Does the science of quantity rest upon that of quality; or vice versa?37 In contrast to Boole, Jevons opted for the former interpretation. To summarize thus far, we have seen that Jevons devised a system

Logic and Scientific Method · 65 of formal logic that, by expressing the proposition in the form of an equation, exemplified his principle of the substitution of similars. It was a system, moreover, that blended and simplified the notation and innovations of Boole and De Morgan. Most importandy, his logic established a criterion for the demarcation of logic from mathematics, which, we will see, had direct consequences for his mathematical treatment of economics. And while quantity had long been viewed as a defining property of mathematics, until it was linked to logic and broader epistemological conceptions, such claims were relatively innocuous. Jevons was one of the first to envision the reduction of mathematics to a pure logic and, for this reason, his treatment of mathematics as the science of quantity was of the utmost significance. INDUCTION

Traditionally, deduction had been viewed as the only truth-preserving form of inference. With induction, one aspired to preserve truth, but always with some liability. Mill broke with this tradition by arguing that all inference is in fact inductive. We only reason from one particular to another. In the case of the syllogism, what appears to be the deduction of a particular from a generalization is in fact merely the retrieval of the same particular that had previously entered into the formulation of the general statement. Mill readily acknowledged the need to assume the uniformity of nature, though he also believed that this assumption is amply supported by the many laws of science. Skepticism could be surmounted once it became plain that one does not validate induction by the standards of deduction.38 As an empiricist, Jevons agreed with Mill that induction must be the initial reasoning process. And in his early work, Pure Logic, he maintained that Mill's "argument from similars to similars is the identical process which I have called the substitution of similars." But he changed his mind in the Principles, and issued an outright condemnation of Mill's thesis: "I shall feel bound to state, in a separate publication, my very deliberate opinion that many of Mill's innovations in logical science, and especially his doctrine of reasoning from particulars to particulars, are entirely groundless and false."39 Jevons here implies that all inference involves discerning universals. It was impossible, in his view, to move direcdy from one particular to another without passing through an inductive generalization. What, then, does Jevons mean by the statement that induction is the inverse of deduction? Once again, he draws an analogy to mathematics. Given two numbers, it is a simple operation to find their product. When given a large number, however, and particularly one that

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is the product of two large prime numbers, it is a long and tedious task to find its factors. Similarly, in the calculus, while differentiation is straightforward and unambiguous, most integration involves a process of trial and error, of educated guesses. It is for this reason that one must first understand the direct, deductive-like methods of differentiation and multiplication, before turning to the indirect, inductivelike methods of integration and division. Induction, moreover, like integration, is considerably more difficult than its counterpart. There are no firm rules to follow and often no solution or algorithm to be found. Jevons also makes a distinction between perfect and imperfect induction, the former being the case where every instance on which the generalization is based is known; for example, the propositions of the third book of Euclid pertain only to circles. These inferences are important primarily as a means of simplifying and storing knowledge. Imperfect induction, on the other hand, involves those cases in which it would be impossible to investigate all of the particulars covered by the generalization. It is in this case that we infer the course of the future based on observations of the past, where we appear "to reap where we have never sown." Nevertheless, induction is no more ampliative than deduction: Of imperfect induction itself, I venture to assert that it never makes any real addition to our knowledge . . . As in other cases of inference, it merely unfolds the information contained in past observations; it merely renders explicit what was implicit in previous experience. It transmutes, but certainly does not create knowledge. The only source of knowledge, the only means by which we may expand upon our present knowledge, is through experience; "it is quite impossible to make any real additions to the contents of our knowledge except through new impressions upon the senses."40 Induction "merely unfolds the information contained in past observations." To illustrate this claim, Jevons turns to his logical alphabet. Since it provides us with an exhaustive list of every possible combination that follows deductively from a given set of premises, logical induction consists quite simply of finding those premises from which a given alphabet can be formed. In order to educate the reader in this method, Jevons provides a series of examples and problem sets of increasing difficulty.41 He shows that it is possible, given a series of various combinations of a finite number of terms (preferably a small number of terms), to infer the premises by induction. Throughout his book Jevons emphasizes the infinite variety of experience: "Nature is a spectacle continually exhibited to our senses in

Logic and Scientific Method · 67 which phenomena are mingled in combinations of endless variety and novelty." He discusses at length the vast number of combinations or permutations that can be extracted from a small class of similar elements, for example the eight notes of an octave. It is important, he claims, to have a good understanding of the range of combinations, for only then can one realize "how hopeless it would be to attempt to treat nature in detail." Science must restrict itself to the study of only a fraction of the possible combinations found in nature. But, by using the logical alphabet as a guide, the organization of these combinations is, in principle, straightforward. Jevons, in an oft-quoted phrase, draws an analogy to the process of drawing different colored balls from a box to make his point. "Nature is to us like an infinite ballotbox, the contents of which are being continually drawn, ball after ball, and exhibited to us. Science is but the careful observation of the succession in which balls of various character present themselves."42 This suggests a very passive view of nature; nature presents itself to us indiscriminately. Although Jevons recognizes limits to perception, he believes that they can be identified and handled accordingly. Ultimately, there are no grounds for concern simply because mind and nature are united by the three fundamental laws of thought. Induction thus becomes, like the integral calculus, a process of trial and error. "Induction is the decyphering of the hidden meaning of natural phenomena. Given events which happen in certain definite combinations, we are required to point out the laws which govern these combinations." Induction, Jevons somewhat naively believed, is simply a matter of translating the phenomena of nature into the ABC's of the logical alphabet. Jevons agrees with Mill that everything in nature is subject to law. The future is completely determined by the past: "In nature the happening of a physical event has been predetermined from the first fashioning of the universe. Probability belongs wholly to the mind." Anything that appears to be random or not subject to a known law, is simply a sign of our ignorance: "where chance seems to work it is our own deficient faculties which prevent us from recognizing the operation of Law and of Design." Knowledge is possible, then, precisely because the world is orderly. As Jevons puts it: "Happily the Universe in which we dwell is not the result of chance." Jevons suggests that one could conceive of a completely chaotic universe, "in which no two things should be associated more often, in the long run, than any other two things." Such a situation, however, would never enable human knowledge to arise. "In such a world knowledge would be no more than the memory of past coincidences, and the reasoning powers, if they existed at all, would give no clue to the nature of the pres-

68 · Chapter 4 ent, and no presage of the future."43 Hence we are led to the belief in a completely law-governed universe. But Jevons skirts around the problem of causality.44 Although habit usually leads us to assume that some things are consistently associated with others, we look in vain, Jevons claims, for causes defined as "necessary or invariable antecedents of events." To know the cause of an event implies that we know for certain that the event will invariably follow. But such certainty, simply because of the finite nature of our own experiences, is not attainable. Suitably armed, Jevons launches an attack on Mill's celebrated four methods of induction. "Not only does Mill treat the Laws of Causation as almost coextensive with science, but he so uses the expression as to imply that when once we pass within the circle of causation we deal with certainties." Science consists rather in noting the combinations in which the phenomena occur and in determining the "probable conditions, or the antecedents" from which an event probably follows. If science is to use the term cause, it must recognize that any metaphysical implications must be excluded: "there is no particular difference between knowledge of causes and our general knowledge of the succession of combinations . . . found in experimental inquiry." Jevons may be convinced that nature is wholly deterministic, but righdy renounces any claim to justify his conviction: We are in the position of spectators who witness the productions of a complicated machine, but are not allowed to examine its intimate structure. We learn what does happen and what does appear, but if we ask for the reason, the answer would involve an infinite depth of mystery. The simplest bit of matter, or the most trivial incident, such as the stroke of two billiard balls, offers infinitely more to learn than ever the human intellect can fathom.45 Induction must be accepted and used simply because it works. Although Jevons believes that the future is dictated by the past, imperfect inductions can never be granted the certainty attributed to deductive and perfect inductive inferences. "All predictions, all inferences which reach beyond their data, are purely hypothetical, and proceed on the assumption that new events will conform to the conditions detected in our observations of past events." As observers we are limited both spatially and temporally from the assurance not only of projecting from past to future, but also of knowing that at any given instant we have grasped the phenomena in their entirety: There is thus a double uncertainty; even supposing the Universe as a whole to proceed unchanged, we do not really know the Universe as a whole . . .

Logic and Scientific Method · 69 We cannot be sure, then, that our observations have not escaped some fact, which will cause the future to be apparendy different from the past.46 All that we can do, at any given point in time, is select a series of balls from the "infinite ballot-box of nature." Although there is no dif­ ficulty in identifying the characteristics of a given ball and in applying inductive logic to the data received, it would be impossible both to attain a complete knowledge of the proportion of the different kinds of balls in the box and to know that the contents of the box have re­ mained unchanged. Jevons is thus claiming unequivocally that we can never know for certain that the future will resemble the past. To remedy this undesirable state of affairs, Jevons turns to the the­ ory of probability, a theory he had first learned to appreciate under the tutelage of De Morgan. Probability theory is envisioned as the means of salvation for the inductive sciences, and is praised accordingly: "It is impossible to expound the methods of induction in a sound manner, without resting them upon the theory of probability .. . This theory appears to me the noblest creation of intellect. . . To eulogise the the­ ory ought to be as needless as to eulogise reason itself."47 Jevons interprets probability as the measure of the comparative amounts of knowledge and ignorance of a given proposition. It mea­ sures what he calls "rational expectation"; probability "teaches us to regulate our actions with regard to future events in a way which will, in the long run, lead to the least disappointment."48 Whereas previous logicians had attempted to justify induction as a valid means of infer­ ence, Jevons held that induction could, at best, yield only partial knowledge.49 One can never know for certain that χ will follow y, de­ spite a metaphysical conviction that nature is deterministic. One can, however, assign a measure of rational expectation to the belief that χ will follow y and utilize this information to guide future decisions. Given his empiricist allegiances and high regard for probability the­ ory, one might expect Jevons to have favored the frequency approach to the estimation of probability values. Following Laplace and De Mor­ gan, however, he adopts the classical or a priori interpretation. The empirical approach, he argues, is prone to error. For example, a die or coin, if it is heavier on one side or another, may give a distorted mea­ sure that could not be applied to other cases. A rational or a priori calculation, however, gives us the general and most perfect distribu­ tion of outcomes to which the actual cases approximate in the long run. 5 0 As Jevons realized, this measure is of no predictive value in the individual instance. We cannot predict, on the throw of a penny, whether the outcome will be heads or tails. We do know, however, that either outcome, given a fair coin, is equally likely, and this in itself

70 · Chapter 4 constitutes valuable information. We have partial, if not perfect knowledge. Furthermore, since the laws of probability are a priori and, in Jevons's view, stem from the fundamental principles of reasoning, they cannot be revised on empirical grounds: "The Laws of Probability rest upon the fundamental principles of reasoning, and cannot be really negatived by any possible experience." Once again, "Probability belongs wholly to the mind." Whereas in nature everything is subject to a fixed law, probability simply measures "our ignorance of the causes at action." The rational interpretation is therefore the better one since we know in advance that in the long run, the effects produced by a given law will be perfectly distributed.51 Jevons held some peculiar views on the subject. For example, he maintained that in the case of complete ignorance one may assign the probability 1/2 to the alternative outcomes: "if A and C are wholly unknown things, we have no reason to believe that A is C rather than A is not C. The antecedent probability is then 1/2." He tries to justify this position by arguing that it would be misleading to favor a belief in the falsity of either proposition in advance. It is one thing, however, to assign values to a situation, such as the toss of a coin, in which the alternative outcomes are known, and quite another to attempt to measure the truth or falsehood of a proposition about a state of affairs of which nothing is known, not even the mutual exclusivity of the events. Jevons was duly criticized for endorsing this so-called Principle of Indifference.52 Jevons may have retained this outmoded view in part because of a stubborn loyalty to Laplace and De Morgan. But there was also an independent source of appeal. Jevons was very drawn to the idea that numbers could be applied to virtually everything, but particularly to beliefs and actions. He was direcdy indebted, on this matter, to Condorcet: The theory of probabilities, I repeat, is the noblest, as it will in course of time prove, perhaps the most fruitful branch of mathematical science. It is the very guide of life, and hardly can we take a step or make a decision of any kind without correctly or incorrectly making an estimation of probabilities.53 It would therefore be fitting that some number, such as the impartial 1/2, be assigned to two alternative outcomes even in the case of the complete absence of prior knowledge. Moreover, one of the key operations in his system of logic, the law of duality, insures that all possible alternatives could be formulated. Jevons goes on to show that one can assign probability values to every

Logic and Scientific Method · 71

member of the logical alphabet: if x,y,z are the probabilities of A,B,C respectively, then (1 -x),(l-y),(l-z) are the probabilities of a,b,c. It is simply a matter of compounding these values to find the probability value of any given member in the alphabet. Thus, for example, Ak; has the probability x ( l - j / ) ( l - z ) . Beyond this, however, Jevons has nothing to add. The transition from the logical alphabet to the real world in terms of probability assignments is not specified. All that he has claimed at this point is that "the whole cogency of inductive reasoning rests upon probabilities."54 Jevons then outlines his philosophy of inductive inference. Betterknown today as the hypothetico-deductive method, this method was proposed as a replacement for the more widely accepted, purportedly Baconian procedure of induction by enumeration. Scientific reasoning, Jevons argues, does not consist of the blind accumulation of facts but rather of the theoretical anticipation of natural laws. "I hold that in all cases of inductive inference we must invent hypotheses, until we fall upon some hypothesis which yields deductive results in accordance with experience."55 Jevons was by no means the first to recognize the significance of this "inductive method." Whewell is better-known as an early advocate of the hypothetico-deductive method and even Mill, despite his partiality toward induction by enumeration, endorsed the method under certain circumstances. But Jevons made a number of important modifications to the method through the application of the theory of probability. The "inductive-probabilistic method" proceeds as follows: (1) We frame a hypothesis; (2) We deduce the probability of various series of possible consequences; (3) We compare the consequences with the particular facts, and observe the probability that such facts would happen under the hypothesis. One may then ascertain the probability of the hypothesis itself by the "inverse method," which Jevons defines as follows: If an event can be produced by any one of a certain number of different causes, all equally probable a priori, the probabilities of the existence of these causes as inferred from the event, are proportional to the probabilities of the event as derived from these causes.56

Given an event E with only three conceivable causes, A, B, or C, the probability that A is the cause is: pa I pa + pb + pc. Jevons shows that these rules can be applied to his ballot-box metaphor, provided one knows in advance the number of balls in the box and wishes simply to know, from a series of drawings with replacement, the distribution of white and black balls. One can arrive at a specific measure of the

72 · Chapter 4 probability that a particular color would be drawn next. Probability theory can thus be used to estimate the likelihood of any one future event.57 Laplace's general rule for assigning a probability to the next occur­ rence of a particular event is p(E) = m + Hm + η + 2, where m is the number of times the event has been known to occur and η the num­ ber of times it has been known to fail. This method was severely crit­ icized by Boole and Venn for its ad hoc features. Others have since shown that it leads to absurd calculations.58 For Jevons, however, the method, particularly because it carried the authority of Laplace, was a most welcome addition to his system. Moreover, the fact that one could work backwards from the probability of the consequences to the probability of the hypothesis, lent even more weight to the thesis that induction is the inverse of deduction. His approach to induction brings out an essential aspect of the sci­ entific enterprise—the need to decide between competing hypotheses. Mill had adopted, in effect, an eliminative strategy whereby the one surviving hypothesis of the test was deemed true. As Strong has noted, Jevons's major advance over Mill was the realization that, in principle, an infinite number of possible hypotheses could be proposed for any given set of observations: "Having no a priori knowledge of the conditions of the phenomena in question, there is no limit to the variety of hypotheses which might be suggested."59 Since Jevons's inductive logic is essentially prescriptive, it is not surprising that there are few historical examples in his treatment of the subject. But in his study of the methods of scientific measure­ ment, one finds a wealth of examples and insights on the nature of scientific reasoning. Alas, there are few explicit links to the inverse method of probability, though the reader is reminded, from time to time, that all scientific knowledge is inherently probabilistic. More in­ teresting are his discussions of the growth of scientific knowledge and the more informal rules of thumb that guide scientific inquiry. SCIENTIFIC PROGRESS

There are no absolute rules for the investigation of nature, but there is reason to believe that Baconian induction is relatively ineffective. A study of the history of science suggests the need to anticipate nature and reason deductively: "Whether we look . . . to Newton and Des­ cartes, Leibnitz and Huygens,.. . we find that discovery was achieved by the opposite method to that advocated by Bacon." Unlike Whewell, however, Jevons did not intend to imply that sci­ ence begins with innate ideas. Although he grants that "mental con-

Logic and Scientific Method · 73 ditions . . . enter into all acts of observation," science is an empirical endeavor through and through: The investigator begins with facts and ends with them. He uses facts to suggest probable hypotheses; deducing other facts which would happen if a particular hypothesis is true, he proceeds to test the truth of his notion by fresh observations. If any result prove different from what he expects, it leads him to modify or to abandon his hypothesis; but every new fact may give some new suggestion as to the laws in action. Even if the result in any case agrees with his anticipations, he does not regard it as finally confirmatory of his theory, but proceeds to test the truth of the theory by new deductions and new trials.60 Nature can unmistakably respond to and direct our attempts to discover its laws. Scientific observations are analogous to the black and white balls drawn from an infinite ballot-box.61 For this reason, Jevons insists time and again that "agreement with fact is the sole and sufficient test of a true hypothesis." Although he also notes that a hypothesis must both agree with the facts and conform to other scientific laws, it is safer to abide by our experimental results than to adapt to other scientific laws, if only because "not one of the inductive truths which men have established, or think they have established, is really safe from exception or reversal." If one tries to adapt a hypothesis to the extant edifice of knowledge, one might go even more astray.62 Again, looking to the history of science, one finds that science is rarely a simple process of conjectures and refutations. Jevons articulates what is now known as the Duhem-Quine problem: Seldom, indeed, shall we have a theory free from difficulties and apparent inconsistency with facts. Though one real inconsistency would overturn the most plausible theory, yet there is usually some probability that the fact may be misinterpreted, or that some supposed law of nature, on which we are relying, may not be true. But one may be able to privilege one part of a theory over another if it tends to lead in unforeseen ways to new or different sets of observations. A good hypothesis will help "to anticipate deductively series of facts which are not already connected and accounted for by any equally probable hypothesis." And while there is no a priori way to decide when to endorse a given theory, if it survives a prolonged period of testing one may be able to put aside certain doubts. "It is only when difficulty after difficulty has been successfully explained away, and decisive experimenta crucis have, time after time, resulted in fa-

74 · Chapter 4 vour of our theory, that we can venture to assert the falsity of all objections."63 If a part of a theory encounters a severe counterexample or discordance with another branch of science, Jevons offers the following advice. The scientist might check the experimental data more thoroughly, or reexamine the validity of the other laws invoked in the explanation or the numerical constants used in the calculations, or search for some regular unknown cause that may have led to the unexpected refutation. Otherwise, the falsity of the theory must be accepted. Jevons notes that scientists have not always proceeded as rationally as one might think. As he wisely remarks, "men require no incitement to generalise; they are too apt to draw hasty and ill-considered inferences. . . . Hasty and superficial generalisations have always been the bane of science." But some scientists, such as FranQois Arago, have displayed the admirable quality of suspending judgment when faced with insufficient data. It is such impartiality, such caution, that in Jevons's view will ultimately lead to the most fruitful results.64 Like the majority of his contemporaries, Jevons never seriously doubted that scientific knowledge, at least since the seventeenth century, had progressed. He was, after all, a great admirer of the leading Victorian proselytizer of progress, Herbert Spencer. But he insisted that science had not made nearly as much progress as was commonly thought. No science, not even Euclidean geometry or Keplerian astronomy, was perfect or complete. The principle of inertia, the method of the composition of forces, even the measurement of time, were subject to doubt. "In science and philosophy nothing must be held sacred." Nor should we expect science to answer all queries. We have mistakenly come to believe "that chemistry can resolve the mysteries of existence," when in fact almost nothing is known about the nature of matter or force.65 Jevons also emphasizes the accidental path by which knowledge is acquired. It would appear, given his image of drawing different colored balls from the infinite ballot-box of nature, that the odds are strongly against the discovery of even a single law. Moreover, from any set of observations the scientist could formulate an almost endless number of hypotheses. The initial breakthroughs, Jevons suggests, must be serendipitous: "Chance then must give us the starting point; but one accidental observation well used may lead us to make thousands of observations in an intentional and organised manner, and thus a science may be gradually worked out from the smallest opening."66 Scientists can develop various strategies, however. For example,

Logic and Scientific Method · 75 they can pay attention to the more vulnerable parts of a given theory: "it is among such anomalies that we must look for the clues to new realms of facts worthy of discovery. They are like the floating waifs which led Columbus to suspect the existence of the new world." But one must also be prepared to jump ship: "perfect readiness to reject a theory inconsistent with fact is a primary requisite of the philosophic mind." Yet other examples, such as Faraday's work on magnetism and light, illustrate the importance of "the pertinacity with which a favourite theory may be pursued, so long as the results of experiment do not clearly negative the notions entertained." Thus, while one should always remain receptive to new ideas, it is still important to cling to some guiding hypotheses.67 One of the most important heuristic devices is the discernment of analogies, either within one branch of science or between different sciences. Although analogical reasoning was widely appreciated by nineteenth-century scientists, Jevons was one of the first to treat it as part of a system of logic. For him, it was but another version of his principle of the substitution of similars. "All science . . . arises from the discovery of identity, and analogy is but one name by which we denote the deeper-lying cases of resemblance." Scientists have and should continue to recognize the importance of drawing analogies as a manifestation of this most fundamental rule of reasoning. Whether one turns to Descartes, Faraday, or Boole, it is patendy obvious that "the discovery of an unsuspected analogy between two branches of knowledge has been the starting-point for a rapid course of discovery."68 One of the best indicators of scientific progress, for Jevons, is the increasing refinement of the quantitative character of science. "As physical science advances, it becomes more and more accurately quantitative. Questions of simple logical fact after a time resolve themselves into questions of degree, time, distance, or weight." Considerable evidence from the history of physics is presented in support of this claim. Indeed, the evidence is so convincing that, by analogy, the same developmental path must await all other sciences: "there can be little doubt, indeed, that every science as it progresses will become gradually more and more quantitative." And, as soon as the quantitative character of a science emerges, it becomes a mathematical science, simply by virtue of the fact that, for Jevons, quantity and mathematics are one and the same. Eventually, scientific theories may reach such a degree of sophistication that the scientist and mathematician are indistinguishable from one another. "As science progresses, its power of foresight rapidly increases, until the mathematician in his library acquires the power of anticipating nature, and

76 · Chapter 4 predicting what will happen in circumstances which the eye of man has never examined."69 Jevons is careful to point out, however, that the degree of certainty of a given theory is never increased by the use of mathematics. Although mathematics may clarify or add rigor to rudimentary scientific ideas, it can never add to the initial truth of the theoretical claims. Not only is truth solely an empirical matter, a matter of finding statements which correspond to the real world, but mathematics itself, as it strays further from strictly logical considerations, becomes increasingly more dubious. As he emphasizes, "many processes of mathematical reasoning are of the most doubtful validity." In any event, the degree of certainty of any empirical claim is not enhanced by a mathematical rendition: Many persons may be mislead by the expression exact science, and may think that the knowledge acquired by scientific methods admits of our reaching absolutely true laws, exact to the last degree. There is even a prevailing impression that when once mathematical formulae have been successfully applied to a branch of science, this portion of knowledge assumes a new nature, and admits of reasoning of a higher character than those sciences which are still unmathematical.70 Nothing, Jevons insists, could be further from the truth. His system of logic served to diminish the certainty traditionally ascribed to mathematics. This view is made plain in his assessment of Euclidean geometry. The only aspect of it that generates certainty is the use of deduction. The content of the set of axioms, definitions, and postulates is strictly a matter of convention. His appreciation of this was the product of two recent developments, Mill's advocacy of geometry as an empirical science and Helmholtz's celebrated paper on non-Euclidean geometry. Whereas Jevons chastised the first, he praised the second. It was preposterous to verify the axioms and postulates of geometry in one's mind's eye. The sole attribute of a good geometry was consistency. Moreover, as he realized from Helmholtz, the actual world may be better understood in non-Euclidean terms. "If, in the course of time," Jevons wrote, "the curvature of our space should be detected, it will not falsify our geometry, but merely necessitate the extension of our books upon the subject."71 Ideally, a science develops in such a way that the "measuring apparatus and mathematical theory should advance pari passu." There is no stipulation, however, that the measurements be exact or complete. Throughout the Principles, Jevons promotes the idea that an empirical science can never become perfectly exact. Even the so-

Logic and Scientific Method · 77 called exact sciences, such as astronomy, necessarily involve approximations. In attempting to prove Kepler's first law, it is only possible "to show that the motion of an unperturbed planet approaches very nearly the form of an ellipse." Even if the planets are assumed to be homogeneous spheres, the calculation of the simultaneous attraction between three or more bodies must remain an approximation. The physical sciences, in that they involve assumptions of perfect homogeneity, rigidity, elasticity, and so forth, only approximate the conditions of the real world. "In truth men never can solve problems fulfilling the complex circumstances of nature. All laws and explanations are in a certain sense hypothetical, and apply exacdy to nothing which we can know to exist . . . When we probe the matter to the bottom physical astronomy is as hypothetical as Euclid's elements."72 Mathematics, then, neither makes a science more certain nor more exact. Both certainty and exactitude are functions of the degree of accordance between fact and theory. Mathematics, however, can assist the discernment of the degree of exactitude: "the approximate character of physical science will be rendered more plain if we consider it from a mathematical point of view." This is explained by a procedure he calls quantitative induction. In essence, the scientist strives to join together various experimental facts or simple observations into a mathematical function: It is by an inverse process of reasoning [i.e., induction] that we can alone discover the mathematical laws to which varying quantities conform. By well-conducted experiments we gain a series of values of a variable, and a corresponding series of values of a variant, and we now want to know what mathematical function the variant is as regards the variable. Although it is preferable that the variables be precisely measurable, this is not essential to the task at hand: "even when we have no means of accurately measuring the variable quantities we may yet be convinced of their connection, if one always varies perceptibly at the same time as the other."73 According to Jevons, all functions are of the form: y = A + Bx + Cx2 + Dx3 + . .. and the degree of approximation is measured, quite simply, by the number of terms. Most scientific laws, Jevons recognizes, are only at the first stage of approximation, y = A + Bx. But as the techniques of measurement become more sophisticated, and as more factors become evident, the function becomes increasingly more complex, adding terms of higher and higher orders. Theoretically, every function has an infinite number of terms. But in practice, one can seldom go beyond two or three terms. Such celebrated laws as the inverse-square law of gravitation

78 · Chapter 4

may, as both Euler and Clairaut had noted, be more complex. One may construe it as follows: F = . . . + a/D + b/D2 + c/D3 + .. . , and maintain that the coefficients a and c have yet to be detected. Perhaps, as the measuring apparatus of science improves, they will be found to exist and thus account for the known discrepancies in the theory, such as the perturbations of the planets. Jevons shies away from a full commitment to this claim. He recognizes that any disturbance emanating from a given point in space will vary inversely with the square of the radius on geometric grounds alone. But the alternative, more complicated function cannot be ruled out. "The calculations of physical astronomy, however accurate, could never show that the other terms of the above expression were absolutely devoid of value. It could only be shown that they had such slight value as never to become apparent."74 Given that there are an indefinite number of such terms, a scientist can never obtain a precise grasp of some part of the world. Exactitude must always remain a matter of degree. Mathematical functions, therefore, are viewed as the end product of quantitative induction. Not only do such functions help to reveal the approximate nature of all scientific laws, but because they connect phenomena without reference to causes, they help to reinforce the fact that scientific knowledge must necessarily fail to discover the vera causae of nature. CONCLUSION

Jevons's treatise on the content and methods of science stands out alone in his generation. Just as Keynes deemed Jevons's Theory the "first modern book in economics," so one might say of the Principles that it imparts, for the first time, the mode of argumentation we find in contemporary texts on the philosophy of science. It was not until the end of the century, with the writings of Ernst Mach, Karl Pearson, C. S. Pierce, and Pierre Duhem, that anything appeared to rival it. More impressive, perhaps, is its performance among economists. Jevons's clear grasp of the fundamental principles of the various natural sciences, ranging from astronomy, crystallography, optics, mechanics, and meteorology to chemistry, geology, botany, and evolutionary biology, and his ability to discern the deeper problems and experimental limitations of science in general, make him unique among economists. It would be virtually inconceivable, in this age of specialization, to find such wide-ranging expertise in the natural and social sciences. More concretely, we have seen that Jevons devised a relatively novel method for dealing with the thorny problem of induction. On essentially Humean grounds, he discarded the attempts by Herschel and

Logic and Scientific Method · 79 Mill to develop methods for the detection of causal laws. And although he endorsed Whewell's insight into the workings of induction—the hypothetico-deductive method—he would not entertain the possibility that such a method could produce genuine knowledge. Whewell had placed great stock in this method precisely because the mind was endowed, a priori, with the appropriate guiding concepts. As a strict empiricist, Jevons found this tainted with idealism. To establish an inductive inference as certain, one must have an infinite knowledge of nature. At best, then, we can assign specific probabilities to particular hypotheses and tentatively accept the one which is most probable. Although we can never justify induction in the full sense of the word, and although we can never hope to know more than an infinitesimal amount of the goings-on in nature, we must be content with the inductive approach as the method which allows us to take hold, however feebly, of our surroundings.

C H A P T E R FIVE

Markets and Mechanics [Jevons's] application of Mathematics to Economics was no accident, but stood in close relation to the general cast of his mind and scheme of his constructive thought. It was the same impulse that impelled him to contrive his logic machine, to attempt to ground the principles of science on the doctrine of chances, to look for the source of commercial crises in the supposed cycle of meteorological phenomena of which the spots on the sun were an indication, and to apply the principles of the differential calculus to the theory of value. —Philip Henry Wicksteed

W E ARE now in a position to examine the extent to which Jevons's program for mathematical economics was shaped and justified by his philosophical convictions. Like many before him, he firmly believed that a unified method held for all of the sciences and that nature itself was ultimately uniform and law-governed. Hence, the economist was free to explore analogies to the natural sciences and to read them as meaningful clues to more fundamental harmonies. Where Jevons differed from his predecessors was in discerning more detailed similarities at the conceptual level that went beyond simple appeals to abstraction and the hypothetico-deductive method. Of comparable importance was his grasp of recent shifts in the epistemological status of logic and mathematics. Together, these enabled him to see a little more deeply into the fundamental principles of political economy. Perhaps the most simplistic of Jevons's arguments was his appeal to the quantitative complexion of economics. Since logic, by his definition, could only analyze qualitative relationships, as soon as it was apparent that economics dealt with quantitative relations, mathematics must be adopted: The greater or less accuracy attainable in a mathematical science is a matter of accident, and does not affect the fundamental character of the science. There can be but two classes of sciences—those which are simply logical, and those which, besides being logical, are also mathematical. If there be any science which determines merely whether a thing be or be

Markets and Mechanics · 81 not—whether an event will happen, or will not happen—it must be a purely logical science; but if the thing may be greater or less, or the event may happen sooner or later, nearer or farther, then quantitative notions enter and the science must be mathematical in nature, by whatever name we call it.1 This may seem to be a somewhat feeble line of reasoning to us, but to Jevons and his contemporaries, it was perfectly cogent. The science of economics "must be mathematical in nature," simply by virtue of its quantitative subject matter. Jevons also made it quite clear that this would in no way increase the degree of certainty of the theoretical claims. There was no danger, then, of ascribing to mathematical economics greater validity than was already to be found in its logical form. Mathematics could assist in the many "slippery and complicated processes of reasoning," and possibly yield new insights, but truth, as we have seen, could only be acquired by increasing the correspondence between theory and reality. THEORY AND EVIDENCE

Jevons's Principles of Science shows an extensive comprehension of the many limitations of scientific knowledge. As he recognized, "the more we have explained, the more there is to explain." Unfortunately, social theorists have had a tendency to overestimate their achievements: In the writings of some recent philosophers, especially of Auguste Comte, and in some degree John Stuart Mill, there is an erroneous tendency to represent our knowledge as assuming an approximately complete character. At least these and many other writers fail to impress upon their readers a truth which cannot be too constantly borne in mind, namely, that the utmost successes which our scientific method can accomplish will not enable us to comprehend more than an infinitesimal fraction of what there doubdess is to comprehend.2 Jevons draws attention to several episodes in the history of the physical sciences, such as the triumph of the wave theory of light, to demonstrate that even the best of minds could hold erroneous views. Newton's belief in a corpuscular theory of light, for example, "retarded for nearly a century the progress of inquiry."3 Similarly, in chemistry, Lavoisier had mistakenly defined oxygen as the defining ingredient of acids, a confusion that persisted for some two decades until the work of Humphry Davy. Surely, if the beliefs of such great men of science as Newton and

82 · Chapter 5 Lavoisier could be set aside, it was wrong for political economists to uphold the views of Ricardo and Mill solely on the grounds of authority. That, unfortunately, was the general state of affairs of the discipline when Jevons wrote the Theory, or so he believed.4 As a result of Mill's preeminence, "the science [of political economy] has become far too much a stagnant one, in which opinions rather than experience and reason are appealed to." On the opening page of the first edition of the Theory, Jevons takes Mill to task for having once claimed that the laws of value had reached their final form. In Jevons's view, such a claim would be foolhardy for any scientist to maintain. As far as he is concerned, "in the republic of the sciences sedition and even anarchy are [more] beneficial in the long run" to the pursuit of truth. Economists would get nowhere, then, unless they recognized that their science was still very underdeveloped. The mere fact that economists appealed constandy to the authority of the classical writers suggested the need for a Galileo. In several places, Jevons sounds very similar to that great polemicist: Show us the undoubted infallible criterion of absolute truth, and we will hold it as a sacred inviolable thing. But in the absence of that infallible criterion, we have all an equal right to grope about in our search of it, and no body and no school nor clique must be allowed to set up a standard of orthodoxy which shall bar the freedom of scientific inquiry.... I protest against deference for any man, whether John Stuart Mill, or Adam Smith, or Aristode, being allowed to check inquiry.5 The history of science demonstrates very convincingly that scientists have often developed theories before consulting the facts: "physicists are, of all men, most bold in developing their mathematical theories in advance of their data." Indeed, such an approach may be the only means to advance scientific knowledge. "Had physicists waited until their data were perfectly precise before they brought in the aid of mathematics, we should have still been in the age of science which terminated at the time of Galileo."6 It so happened that as the physical sciences increased their mathematical profile, they also tended to become more exact. But, as we saw in the Principles, this does not imply that a mathematical science need be exact. A widespread confusion over this distinction is also noted in the Theory: Many persons entertain a prejudice against mathematical language, arising out of a confusion between the ideas of a mathematical science and an exact science. They think that we must not pretend to calculate unless we

Markets and Mechanics · 83 have the precise data which will enable us to obtain a precise answer to our calculations; but in reality, there is no such thing as an exact science, except in a comparative sense.7 The application of mathematics, we have seen, helps to expose the degree of exactness, particularly through the formulation of algebraic functions. But this formulation does not presuppose the accumulation of precise data. Economists ought to take heart particularly from the example of the theory of electricity. At the time of Adam Smith, "electricity was a vague phenomenon, which was known, indeed, to be capable of becoming greater or less, not measured nor calculated: it is within the last forty or fifty years that a mathematical theory of electricity, founded on exact data has been established."8 From his readings of Faraday, Maxwell, Thomson, and Tait, Jevons recognized the many interesting lags between theory and confirmation, between mathematical formulation and experimentation, that transpired in the development of the theory of electricity and magnetism. In economics, he argued, theorizing must also precede observation. A scientist must anticipate reality, preferably with sound conceptual tools. As a result, the Theory contains only occasional references to economic data: "Before we attempt any investigation of facts, we must have correct theoretical notions; .. . I make hardly any attempt to employ statistics in this work." Jevons, however, is convinced that the data are there to be found: "But where," the reader will perhaps ask, "are your numerical data for estimating pleasures and pains in Political Economy?" I answer, that my numerical data are more abundant and precise than those possessed by any other science, but that we have not yet known how to employ them. The very abundance of our data is perplexing. There is not a clerk nor bookkeeper in the country who is not engaged in recording numerical facts for the economist. The private-account books, the great ledgers of merchants and bankers . . . are all full of the kind of numerical data required to render Economics an exact mathematical science. ... It is pardy the very extent and complexity of the information which deters usfromits proper use. But it is chiefly a want of method and completeness in the vast mass of information which prevents our employing it in the scientific investigation of the natural laws of Economics.9 Jevons thus drew a sharp division between the formulation of the mathematical theory of economics and its verification. Although he

84 · Chapter 5 has indicated the means by which his theory could be confirmed, he is concerned only with the first task: It is no part of my purpose in this work to attempt to trace out... the results of the theory given in the preceding chapters. When the view of the nature of Value, and the general method of treating the subject by the application of thefluxionalcalculus, has received some recognition and acceptance, it will be time to think of results.10 Notwithstanding this one-sided approach to the science, Jevons insists that economics uses the same inductive method as the physical sciences, that "it combines observation, deduction, and induction in the most complete and perfect way." But, since he cannot complete all of the steps of this method, he must agree with Mill that "Political Economy tends to be more deductive than many of the physical sciences, in which closely approximate verification is often possible."11 Only when statistics has advanced will economics incorporate the requisite amount of induction. MECHANICAL ANALOGIES

Jevons reinforces the classical axiomatic approach, again by analogy to mechanics. "A few of the simplest principles or axioms concerning the nature of the human mind must be taken as its [i.e., economics] first starting-point, just as the vast theories of mechanical science are founded upon a few simple laws of motion."12 According to Jevons, these axioms are usually "known to us immediately by intuition, or, at any rate, they are furnished to us ready made by other mental and physical sciences." Examples are that "every person will choose the greater apparent good; that human wants are more or less quickly satiated; that prolonged labour becomes more and more painful."13 From these, one can deduce the laws of economics, and then attempt to verify them in the data of everyday commerce. Jevons is aware of the difficulties involved in carrying out this set of procedures. But, as he notes in the Principles, it is often very difficult to test directly the basic postulates of Newtonian mechanics. Since there is no body in nature that is not subject to some force, it is impossible to confirm directly the principle of inertia. And even if such a situation could be found, there is no absolute standard of time that could determine a given motion to be truly uniform: "inasmuch as the measure of motion involves time, and the measure of time involves motion, there must be ultimately an assumption." In fact, all of the basic dimensions of physics—space, time, and mass—are defined and measured on the basis of certain questionable assumptions. "When

Markets and Mechanics · 85 we get down to primary notions, like those of space and time, the mind must meet a point of mystery beyond which it cannot penetrate."14 It may well be the case that the basic axioms of economics are as sound epistemologically as those of classical physics. In this respect, economics as a deductive science based on a few assumptions, is as "sure and demonstrative as that of kinematics or statics."15 Jevons often refers to his theory of economics as "the mechanics of utility and self-interest." Although the term mechanics was widely used during the nineteenth century to describe systems which had little or no connection to classical particle mechanics, in this particular case the analogy has substance. Economics, like physics, is a composite discipline with a system of mechanics at the core: [Economics] is no more one science than statics, dynamics, the theory of heat, optics, magneto-electricity, telegraphy, navigation, and photographic chemistry are one science. But as all the physical sciences have their basis more or less obviously in the general principles of mechanics, so all branches and divisions of economic science must be pervaded by certain general principles. It is to the investigation of such principles—to the tracing out of the mechanics of self-interest and utility, that this essay has been devoted. The establishment of such a theory is a necessary preliminary to any definite drafting of the superstructure of the aggregate science.16 Jevons's vision of a mechanistic economics resonated well with his goal to mathematize. The mechanical philosophy of the seventeenth century that engendered classical mechanics had also motivated the belief that the book of nature was written in the language of mathematics. While the leading adherents, Galileo, Descartes, and Huygens, differed widely on the nature of the specific properties and distribution of matter, they unilaterally sought to explain the given of experience by reduction to a quantitative order. The immense variety and range of phenomena encountered in the world were the fleeting effects of a more permanent world of matter in motion.17 The primary properties of matter, since they are unobservable, must be ascertained by an appeal to reason. It was thought that, if the fundamental particles or atoms were stripped of their sensible qualities, only quantitative properties would remain. This view attained its most definitive form in Book One of Newton's Principia, in which the problems of mechanics were treated in terms of a geometry of pointmasses. The mechanical philosophy, if only because it attempts to explain the visible by the invisible, has traditionally found comfort in rationalism, and thus in mathematics. Jevons belongs to this tradition, not only for his privileging of mathematics, but also for his predilection for reductionism. Recall that his

86 · Chapter 5

first intellectual aspirations lay in the theory of atomism, an interest he sustained in later years.18 His celebrated theory of business cycles is another example. Even in psychology, he envisioned the eventual reduction of mental states to physical processes: Life seems to be nothing but a special form of energy which is manifested in heat and electricity and mechanical force. The time may come, it almost seems, when the tender mechanism of the brain will be traced out, and every thought reduced to the expenditure of a determinate weight of nitrogen and phosphorus. No apparent limit exists to the success of scientific method in weighing and measuring, and reducing beneath the sway of law, the phenomena both of matter and mind.19 To Jevons, then, a primary aim of science was the reduction of the observable to a more fundamental material world. In his theory of economics, Jevons took the phenomena of exchange, the prices and quantities observed in the market place, as the starting point of all economic inquiry. For centuries, moral philosophers had grappled with the problem of economic value, and had attempted either directly or indirectly to explain the system of relative prices in terms of some more fundamental or absolute form of value. Jevons's new approach consists of taking prices as phenomena, pure and simple, as the only positive form of value. As he maintains in the Principles, "among the most unquestionable rules of scientific method is that first law that whatever phenomenon is, is. We must ignore no existence whatever, .. . if a phenomenon does exist, it demands some kind of explanation."20 It is not that prices, due to market disturbances, fluctuate about some natural or absolute value that reflects costs. Rather, every price that appears in the market place, no matter how short-lived, is to be taken as the effect of more fundamental motive forces. Every existing price "demands some kind of explanation." As early as 1862, Jevons maintained that "a true theory of economy can only be attained by going back to the great springs of human action—the feelings of pleasure and pain." As time went on, he believed all the more strongly that economic actions stem from an internal imbalance between pleasure and pain: "we cannot make a choice, or manifest the will, without indicating thereby an excess of pleasure in some direction." Prices, as the ratios of exchange for any two goods, are the manifestation of internal balancing acts of the mind. Economic phenomena are reducible to mental states; "every event in the outward world is represented in the mind by a corresponding motive." Economics has become, in effect, a branch of psychology; "the theory

Markets and Mechanics · 87 presumes to investigate the condition of a mind, and bases upon this investigation the whole of Economics." 21 Although this scheme proposes the reduction of observable data to something as inscrutable and subjective as the human mind, an analogous procedure, Jevons claims, can be found in classical mechanics: A unit of pleasure or pain is difficult even to conceive; but it is the amount of these feelings which is continually prompting us to buying and selling, borrowing and lending, labouring and resting, producing and consuming; and it is from the quantitative effects of the feelings that we must estimate their comparative amounts. We can no more know nor measure gravity in its own nature than we can measure a feeling; but, just as we measure gravity by its effects in the motion of a pendulum, so we may estimate the equality or inequality of feelings by the decisions of the human mind. The will is our pendulum, and its oscillations are minutely registered in the price lists of the markets.22 Given the precedent established in physics, it is perfectly legitimate to infer the operation of these forces by the phenomena they beget. We find the argument repeated and extended in an 1872 letter to John Elliott Cannes: The fundamental objection which you make to my theory of exchange is that I define value by utility and then propose to use prices to measure the variation of utility. This seems a vicious circle—but I do not think you will find it to be so really, and the method seems to me exactly analogous to that employed in other theoretical subjects such as that of light, heat, electricity, &c. The properties of the basis of light undulations are entirely hypothetical. Scientific men assume those properties as they like, calculate what would happen on such conditions and then by comparison with facts ascertain whether they are correct. They have no means of measuring the properties of the ether except by arguing back from observation. So there is no means of measuring pleasure & pain directly, but as those feelings govern sales and purchases, the prices of the market are those facts from which one may argue back to the intensity of the pleasures concerned. Even gravity is known only in its effects.23 To carry out this reduction of prices to the feelings of pleasure and pain, it is first necessary to deduce, from a few psychological postulates, the theory of exchange. Jevons's exchange equation asserts that prices are in direct proportion to the final degrees of utility. In the concrete, however, in the actual case of the reduction of a specific set of prices to a specific utility function, the economist must identify what Jevons calls "the numerical determination of the laws of utility":

88 · Chapter 5 We cannot really tell the effect of any change in trade or manufacture until we can with some approach to truth express the laws of the variation of utility numerically. To do this we need accurate statistics of the quantities of commodities purchased by the whole population at various prices. The price of a commodity is the only test we have of the utility of the commodity to the purchaser; and if we could tell exacdy how much people reduce their consumption of each important article when the pricerises,we could determine, at least approximately, the variation of the final degree of utility—the all-important element in Economics.24 Given the various problems involved in the measurement of utility, particularly the problem of treating commodities as independent of one another, this program remained on the drawing board. From his own attempts to collect data, he recognized that "the subject is too complex to allow of our expecting any simple precise law like that of gravity."25 But he upheld the belief that the laws of utility are in principle numerically determinant, and, moreover, that these laws could be inferred from market prices. Jevons has thereby redefined the methods and aims of economic analysis. The problem of economics is no longer to account for the origin of value, or to explain the relationship between market prices and some preordained natural value, but rather to determine the laws of utility which give rise to relative price changes. Economic analysis starts with a system of prices and a supply of goods as the given of experience. Exchange, the central activity of the economy, is assumed to be carried out by individual utility maximizers in the face of these market phenomena. Ideally, then, an explanation in pure economic theory would consist of the reduction of the phenomena of the economy, the movement of the prices and quantities in the market, to individual mental states, to the mechanics of utility and self-interest: "the main branches of the science of political economy resolve themselves into the theory of the equilibrium of utility."26 Jevons is often praised for recognizing the importance of identifying the dimensions of the economic variables. This set of insights appear to have arisen while working on the Principles of Science, since they were added to the second edition of the Theory. The justification for this line of analysis is again through an appeal to mechanics: "Now, if such a theory of dimensions is requisite in dealing with the precise ideas of physical magnitudes, it seems to be still more desirable as regards the quantities with which we are concerned in Economics." As his two initial dimensions, Jevons selects mass and time to represent the supply of commodities. Prices or ratios of exchange, as epiphenomena, have no dimension. They are simply numbers. Whereas

Markets and Mechanics · 89 mass and time are taken in the same sense as they are used in physics, there is no parallel dimension for the most basic economic quantity, the feelings of pleasure and pain: "the ultimate quantities which we treat in Economics are Pleasures and Pains, and our most difficult task will be to express their dimensions correcdy."27 Jevons postulates the dimension U, to represent the intensity of the feeling produced instantaneously by an infinitesimal quantity of a commodity. But the dimension of a feeling must also incorporate duration UT, and thus, some continuous supply of a commodity MIT. The quantity of utility is the product of these MXJTIT or MU.28 Mass is ascribed to commodities, though Jevons must stretch the notion somewhat to incorporate services. As for time, precisely because it is so ubiquitous, it is usually factored out of the key economic relationships, as in the analysis of the dimension of utility just illustrated. These appeals to mass and time are yet another example of the parallels between economics and classical mechanics. STATICS AND DYNAMICS

As we have seen, Jevons constructs his analysis of exchange with two traders, who enter a market, each in possession of a different commodity, and face a prevailing ratio of exchange. Since the commodities are assumed to be infinitely divisible, the exchange is said to be carried out at a continuously changing ratio of exchange: Theoretically speaking, it would not usually be possible to buy two portions of the same commodity successively at the same ratio of exchange, because, no sooner would the first portion have been bought than the conditions of utility would be altered . . . In theory this effect of exchange upon the ratio of exchange must be conceived to exist in some degree, however small may be the purchases made.29 If transactions are made in sufficient magnitudes, the infinitesimal fluctuations in the price become visible. For Jevons, this is most evident in the case of a single buyer who, by purchasing a substantial proportion of the total supply of a given commodity available in the market, raises the price against himself. Similarly, large numbers of transactions, each on a relatively small scale, would generate a new observed price. But prices tend to move by small increments. "Though the price of the same commodity must be uniform at any one moment, it may vary from moment to moment, and must be conceived as in a state of continual change."30 Jevons has only suggested in a rather vague manner the mechanism by which individual decisions could give rise to a macroscopic

90 · Chapter 5 variation in a particular price. The entire problem, he argues, is dynamic in nature and must therefore be put aside until the much simpler problem, the analysis of an equilibrium price, has first been solved: The real condition of industry is one of perpetual motion and change. Commodities are being continually manufactured and exchanged and consumed. If we wished to have a complete solution of the problem in all its natural complexity, we would have to treat it as a problem of motion—a problem of dynamics. But it would surely be absurd to attempt the more difficult question when the more easy one is yet so imperfecdy within our power. It is only as a purely statical problem that I can venture to treat the action of exchange.31 Exchange is thus treated not as a continuous process over time, but as a lump-sum transfer. "Holders of commodities will be regarded not as continuously passing on these commodities in streams of trade, but as possessing certain fixed amounts which they exchange until they come to equilibrium."32 In this way, the actual act of exchange does not alter the ratio of exchange. Every price that appears is an equilibrium price which reflects not only the equilibration of the final degrees of utility for every pair of traders, but the collective agreement or acceptance, by all of the traders, of the prevailing price. Jevons has failed to give an adequate causal explanation for the motion of prices. His analysis demonstrates at best that individual estimations of the degrees of utility correspond to a single market price. Once again, Jevons justifies his method of analysis, his inability to treat exchange in anything but a state of equilibrium, by an analogy to mechanics: It is much more easy to determine the point at which a pendulum will come to rest than to calculate the velocity at which it will move when displaced from that point of rest. Just so, it is a far more easy task to lay down the conditions under which trade is completed and interchange ceases, than to attempt to ascertain at what rate trade will go on when equilibrium is not attained. And he grants, as he did in settling for a static analysis, that these assumptions do not match conditions in the real world; "in practise, no market ever long fulfils the theoretical conditions of equilibrium." But, as he argued at length in the Principles, every branch of physics involves abstracting from the complexities of the actual world. Economists are therefore equally entitled to make simplifying assumptions, to formulate a theory which, for the time being at least, fails to map

Markets and Mechanics · 91 onto the world in every detail. Economists, Jevons urges, "can never be free from difficulties unless they distinguish between a theory and the application of a theory."33 His approach to exchange in terms of static equilibrium also pro­ vides a means for demonstrating the role of the calculus. In his view, "there is no statical problem which is not subject to the principle of virtual velocities." This principle, which analyzes a given system, such as a lever, in terms of infinitesimal displacements from a position of equilibrium, had been regarded, most notably by Lagrange in his cel­ ebrated Mochanique Analytique, as one of the most fundamental prin­ ciples in mechanics. Jevons, it seems, believed that this was still the case in the 1870s: "The mode in which mechanics is usually treated in elementary work tends to disguise the real foundation of the sci­ ence which is to be found in the so-called theory of virtual veloci­ ties."34 The source of this belief can be traced directly back to a course he took in natural philosophy, in which the principle, but not the more advanced systems of Lagrange and Hamilton, was studied as part of the section on classical mechanics. 35 The analogy to economics is readily apparent. Exchange is just like the balancing of a lever: Equilibrium of exchange of goods resembles in mathematical conditions the equilibrium of weights upon a lever of the first order. In the latter case one weight multiplied by its arm must exactly equal the other weight mul­ tiplied by its arm. So, in an act of exchange, the commodity given multiplied by its degree of utility must equal the quantity of commodity received mul­ tiplied by its degree of utility. The theory of economy proves to be, in fact, 36 the mechanics of utility and self-interest. The analogy was heightened by Jevons's demonstration that the ex­ change equation took precisely the same form as the analysis of a lever in terms of the principle of virtual velocities. "I illustrate the mathematical character of the equations of exchange by drawing an exact analogy between them and the equations applying to the equi­ 37 librium of the lever." If P and W are forces applied to the respective ends of lever AB, centered on fulcrum C, then the resulting infinitesimal displace­ ments, if the lever remains in equilibrium, are the virtual velocities AA' and BB'. Since these displacements, however, are 'exacdy propor­ tional' to the arms of the lever (AC and BC), the equilibrium condition can be represented as in Figure 5.1 or, algebraically, as: P χ AA' = W x BB'

92 · Chapter 5 The form of this equation, then, is identical to the form of the ex­ change equation: W/P = AA'/BB' = ACIBC

φ χ % = dyldx = ylx

The parallels between the principle of virtual velocities in physical statics and his own solution to the problem of exchange in economic statics, assured Jevons that he had discovered, in the concept of an infinitesimal amount of utility, the path by which to establish conclu­ sively the need for the calculus in economics. In Jevons's mind, the concept of the final degree of utility was inseparably bound to the mathematical treatment of economics. As he wrote to Herbert S. Foxwell in 1875, "what I contend is that my notion of utility is the correct one & the only sound way of laying the foundations for a math, [sic] theory."38 This helps to explain why Jevons did not regard Cournot as a fullfledged precursor. Although Cournot had used the calculus with greater facility, he failed to reduce economics to the more fundamen­ tal theory of utility and thus to establish a mechanical scheme that demonstrated the indispensability of mathematics in economics. As he noted in the Theory, "Cournot does not recede to any theory of utility, but commences with the phenomenal laws of supply and de­ mand." 39 Thus, as the anchor point for his mathematics, the utility theory acquired a privileged position. In the final analysis, both were part of a more general program to found a mathematical system of mechanics in economics, a "calculus of pleasure and pain," which would serve as the basis for all future work in the science. THE SYSTEM GENERALIZED

Another striking parallel between Jevons's theory of exchange and mechanics is the unqualified use of aggregation. In classical mechan-

dy AA·

Figure 5.1

Lever of Exchange

Markets and Mechanics · 93 ics, as Jevons notes, it is legitimate to isolate a given system, such as the attraction between two bodies, to analyze it as an independent system, and then to construct much more complicated systems from pairs of bodies. This analytical manipulation openly conflicts with the theory of gravitation, that "every particle of matter in the universe attracts every other particle." But as a simplifying assumption, it gready advanced our understanding of the planetary system. Had this assumption not been made, Jevons claims, "the calculations of physical astronomy would have been infinitely more complex than they actually are, and the progress of knowledge would have been much slower."40 Jevons adopts the same tactic in his theory of exchange. He cannot actually treat, simultaneously, the mutual exchange between more than two bodies, though these bodies could be as large as entire nations. In the case of three or more traders and/or commodities, he must aggregate from independent pairs of exchange: We have hitherto considered the Theory of Exchange as applying only to two trading bodies possessing and dealing in two commodities. Exactly the same principles hold true, however numerous and complicated may be the conditions . . . The exchanges in the most complicated case may thus always be decomposed into simple exchanges, and every exchange will give rise to two equations sufficient to determine the quantities involved.41 But, in reality, the three-body problem exists as much in economic analysis as it does in astronomy: If we are to apply scientific method to morals, we must have a calculus of moral effects, a kind of physical astronomy investigating the mutual perturbations of individuals. But as astronomers have not yet fuDy solved the problem of three gravitating bodies, when shall we have a solution of the problem of three moral bodies?42 Jevons adopts another simplifying device which has a direct counterpart in mechanics. Just as physicists construct a many-bodied system from separate two-bodied systems, so economists can construct a market out of the interactions of pairs of traders. For Jevons, this also implied that large trading bodies could be decomposed into individual minds. "The principles of exchange are the same in nature, however wide or narrow may be the market considered. Every trading body is either an individual or an aggregate of individuals, and the law, in the case of the aggregate, must depend upon the fulfilment of law in the individuals."43 Jevons refers the reader of the Theory (1879) to the direct mechanical analogue of the average man as discussed in his Principles. It is

94 · Chapter 5 the center of gravity of a body. The weight of a body, like the price in a marketplace, is an aggregate. "The weight of a body is the sum of the weights of infinitely small particles, each acting at a different place, so that a mechanical problem resolves itself, stricdy speaking, into an infinite number of problems."44 It was Archimedes, Jevons notes, who discovered, with the aid of geometry, the analytical device of the center of gravity. "The beautiful idea that one point may be discovered in a gravitating body such that the weight of all the particles may be regarded as concentrated in that point, and yet the behaviour of the whole body will be exactly represented by the behaviour of this heavy point."45 As a purely fictitious point, the center of gravity corresponds to no one single particle. Similarly, in economics, "the economic laws representing the conduct of large aggregates of individuals will never represent exactly the conduct of any one individual." This approach not only reinforces the role of statistics in the process of verification, but suggests a means by which the economist could indirectly estimate the laws of the human mind. "Accordingly, questions which appear, and perhaps are, quite indeterminate as regards individuals, may be capable of exact investigation and solution in regard to great masses and wide averages."46 Since the laws of motion hold universally, the same must be true in economics: The theory of science consists of those general laws which are so simple in nature, and so deeply grounded in the constitution of man and the outer world, that they remain the same throughout all those ages which are within our consideration. But though the laws are the same they may receive widely different applications in the concrete. The primary laws of motion are the same, whether they be applied to solids, liquids, or gases, though the phenomena obeying those laws are apparently so different. Just as there is a general science of mechanics, so we must have a general science or theory of economy.47 Jevons's own efforts to formulate an economic mechanics of universal scope are evident throughout the Theory. The fundamental postulates of his theory hold true, he claims, for all human beings and possibly for much more. "The first principles of political economy are so widely true and applicable, that they may be considered universally true as regards human nature . . . [Indeed] I should not despair of tracing the action of the postulates of political economy among some of the more intelligent classes of animals."48 Since the laws of economics are purportedly independent of historical conditions, economic history is relegated to the task of verifying the laws arrived at by the pure theory of economics. "Historical polit-

Markets and Mechanics · 95 ical economy, so far from displacing the theory of economy, will only exhibit and verify the long-continued action of its laws in most widely different states of society." Similarly, Jevons insisted that the laws of economics were not contingent on specific political institutions. As a result, he proposed that the adjective political be dropped from the name of the discipline.49 In the Principles, Jevons devotes a chapter to the heuristic value of exceptional phenomena: We are not to assume . . . that there is any limit to the wonders which nature can present to us.. .. Now every strange phenomenon may be a secret spring which, ifrightlytouched, will open the door to new chambers in the palace of nature. To refuse to believe in the occurrence of anything strange would be to neglect the most precious chances of discovery.50 In the classical theory of political economy, Jevons points to several exceptions to the rule that, under his own theory, fall into place. The classical economists had often acknowledged the problem of explaining the value of certain types of commodities, such as rare paintings, under the cost-of-production theory of value. With the utility theory of value, Jevons maintains, these types of goods are no longer anomalous. "The mere fact that there are many things, such as rare ancient books, coins, antiquities, etc., which have high values, and which are absolutely incapable of production now, disperses the notion that value depends on labour."51 The classical distinctions between productive and unproductive consumption and labor, we have seen, were also collapsed. For Jevons, "all reasoning and all operations of scientific method really involve classification." Progress consists in expanding the pigeonholes. "It comes to pass that almost every classification which is proposed in the early stages of a science will be found to break down as the deeper similarities of the objects come to be detected." The reason, quite simply, is mental economy, an insight later associated with the work of Ernst Mach. In Jevons's words, "the facilitation and abbreviation of mental labour is at the bottom of all mental progress."52 Another aspect of Jevons's search for a more general theory is his overall avoidance of references to specific economic and social institutions. Not that he was disinterested in these subjects. His many books and papers on applied economics attest to a thorough knowledge of economic institutions. In his Theory, however, no attempt is made to specify a medium of exchange or to discuss the function of government. Markets are depicted in the most abstract fashion. A system of technology is posited from the start. Similarly, population growth is treated, contrary to tradition, as an exogenous variable:

96 · Chapter 5 The doctrine of population . . . forms no part of the direct problem of Economics . . . The problem of Economics may, it seems to me, be stated thus: —Given, a certain population, with various needs and powers of production, in possession of certain lands and other sources of material: required, the mode of employing their labour which will maximise the utility of the produce.53

The central problem confronting the economist is here couched in ahistorical terms. It has become a mathematical problem of maximizing utility under given constraints, and is thus ripe for the differential calculus. CONCLUSION

Jevons justified his reconstruction of economics along mathematical lines in a number of ways. From his revision of Boole's logic, in which he maintained that quantity was the sole criterion that demarcated mathematics from logic, he insisted time and again that economics, simply because it dealt with quantitative variables, must be mathematical in character. There would be various advantages to the explicit use of mathematics, in addition to the most obvious ones of clarity and precision. The use of equations, wherever possible, helped to identify the pervasive powers of his principle of the substitution of similars. Although the use of mathematics per se would not increase the degree of certainty or exactitude of a scientific theory, it would aid in the recognition of the degree of approximation between theory and fact. This would be particularly true when the time came for economists to identify the specific parameters of the key mathematical functions. As part of his system of logic, Jevons believed that analogical reasoning was not only a legitimate form of reasoning but also possibly the most fruitful means for discovery. This insight gave him license to develop extensive analogies between economics and the physical sciences. His impressive knowledge of the historical development and explanatory structure of the physical sciences enabled him to recognize the fundamental role of the theory of mechanics. Economics, as the most mature branch of the social sciences, demanded a similar system. In this way, Jevons reinforced the classical emphasis on the construction and elaboration of a relatively abstract theory deduced from a limited set of postulates. But he went much further. Like its counterpart in physics, the new system of economic mechanics would be reductionist, universal in scope, and, above all, mathematical. In his provocative chapter on the Jevonian Revolution, Maurice

Markets and Mechanics · 97 Dobb remarked of the emergence of mathematical economics that "the cutting knives of economic discussion became sharper—whether they were used to cut so deeply is another matter."54 Jevons, we have seen, most certainly intended his mathematics to cut more deeply, but his tools met many obstacles along the way. It is also fair to say that he did not have a particularly firm grip on his knives. Schumpeter made this point with his characteristic brevity: Jevons's "performance was not up to his vision."55 In sum, Jevons's program for the mathematization of economics took hold more because of his persuasive appeals to the scope and methods of science in general than from his specific achievements in the mathematical analysis of economic theory.

CHAPTER SIX

Response to the Theory Of course, Jevons reads fresh and stimulating even where he utters a platitude; Mill never reads fresh and stimulating even where he speaks valuable wisdom. —Joseph A. Schumpeter

was disappointed with the initial response to his Theory of Po­ litical Economy. To Walras he wrote in 1874 that his theory of ex­ change had been "either neglected or criticized" in England.1 But it is fair to say that Jevons overreacted, probably because the reviews appeared while he was hard at work on the Principles of Science. By May 1872, he was so strained and exhausted by this project that his doctor insisted he take a three-month holiday.2 We know from his "sad reverse" on the university examination in political economy that Jev­ ons had, in his own words, set out to "avenge" the world with his new theory of economics. But he had few or no illusions about the broad appeal of the Theory when it finally took shape. When he sent the manuscript to Macmillan, he described it as "a small work on Political Economy" which would be "far from being a popular book," though he hoped to sell at least one edition. Jevons was thus somewhat aware that the Theory was not a definitive work, that it left itself open to criticism, and that it would by no means become as popular as Mill's Principles.3 In fact, Jevons's Theory received wide coverage in the popular and academic press. This is made plain if one compares his case to that of Walras, who did not receive a single respectable review of his ΕΙέments, despite his own preview of the work in a notice for the Journal des Economistes.4 Jevons's book was given many favorable notices in the daily press, most notably in the Manchester Guardian (albeit there were familial ties), as well as several favorable though brief notices in such periodicals as the Athenaeum, the British Quarterly Review, and the Westminster Review.5 Simon Newcomb, one of the most promi­ nent American scientists of the time, gave him a very laudatory report 6 in the North American Review. Moreover, it would have been difficult to find two more qualified reviewers than John Elliott Cairnes and Al­ fred Marshall, publishing in the Fortnightly Review and the Academy

JEVONS

Response to the Theory · 99 7

respectively. Yet another lengthy though anonymous appraisal appeared in the Saturday Review that contained, Jevons admitted, a number of constructive criticisms.8 In view of the complete silence that followed Jevons's first statement, his 1862 "Brief Account of a General Mathematical Theory," and given the difficult and novel character of the material covered by the book, it is somewhat surprising that Jevons was so despondent. To be sure, not one of the scholarly reviews was entirely glowing. Considering, however, that Jevons had openly attacked the status quo, and given Mill's eminence in the field, this was fully predictable. For the young Alfred Marshall, it was aspersions cast on Ricardo rather than on Mill that induced his "youthful loyalty" to "boil over" as he read the Theory.9 It should also be borne in mind that Jevons was relatively isolated at Owens College, and had for a number of years regarded himself as a lone adversary. His combative frame of mind is apparent in the following remark to his brother, written shortly before submitting his 1862 paper to the British Association: "I am very curious, indeed, to know what effect my theory will have both upon my friends and the world in general. I shall watch it like an artilleryman watches the flight of a shell or shot, to see whether its effects equal his intentions."10 Much of the Theory is cast in the same provocative mode. The tide began to turn as early as 1875. Jevons's confidence was boosted first by his appointment to the Royal Society, then to the professorship at University College, London, and by the largely favorable reception of his Principles of Science. By March 1878, Jevons not only undertook preparations for a second, revised edition of the Theory, but also, optimistically, made arrangements for future editions. Thus, in a little over six years, Jevons had succeeded in making a place for himself in the area of theoretical economics. As William Ashley later noted in his 1907 address to the British Association for the Advancement of Science: "among the diverse lines of thought which converged upon the old orthodoxy for its destruction in 1870-80, that represented by Jevons has for the time had the widest influence."11 In order to substantiate this reading of the record, we will examine in some detail the individual responses toward Jevons, his Theory, and his program for mathematical economics as a whole. Although it is not possible to treat all of the British economists contemporary with Jevons, we will treat the more important ones.12 They have been placed in four groups, the first comprising the classical theorists John Elliott Cairnes and John Stuart Mill. Next we examine the response of the economic historians, most notably Thomas Edward Cliffe Leslie and John Kells Ingram. The third group, which I have designated the

100 · Chapter 6 moderates because of their concern for settling methodological disputes, includes Walter Bagehot, Henry Sidgwick, John Neville Keynes, and Herbert Somerton Foxwell. And, in the next chapter, we will assess the extent to which mathematical economics took hold in the English-speaking world, focusing in particular on George Darwin, Philip Henry Wicksteed, Francis Ysidro Edgeworth, and Alfred Marshall. CAIRNES

John Elliott Cairnes was the first to promote Jevons's 3 863 study on the value of gold, with notices in the Economist and the Times, and Jevons responded gratefully.13 This initiated a correspondence between them which lasted until Cairnes's death in 1875, although their exchange did not end quite as amicably as it had begun. In Jevons's eyes, Cairnes's 1872 review of his Theory was a total repudiation.14 Since Cairnes was by that time second only to Mill among British economists, the review could singlehandedly do significant damage.15 To his ally, George Darwin, Jevons confessed "much regret that Cairnes should have raised such absurd objections to the theory proceeding entirely from misapprehension. His remarks may temporarily prejudice the theory."16 Cairnes's misapprehension was primarily due to his inability to understand the mathematical passages of the Theory, a shortcoming to which he readily confessed, though not in the review itself.17 He chose to attack the Theory on different fronts. First, he took Jevons to task for having suggested that "anarchy" would be an improvement over the supposed "tyranny" of Mill and Ricardo. Although Cairnes maintained that only a few economists of the time could be said to be subservient to Mill, as the best-known disciple of Mill his case was not wholly convincing. Jevons, in his reply to Cairnes, reminded him that the university examinations were by and large based on Mill's work. For Jevons, this was virtually equivalent to the "noxious influence of authority" once associated with Aristode.18 The second major criticism by Cairnes was more substantial. The utility theory of value, he argued, was circular: "exchange-value depends upon utility, and utility is measured and can only be known by exchange-value." Although there is some truth to Cairnes's claim, Jevons, we have seen, went to great lengths to demonstrate that the procedure of measuring utility by its effects on prices is direcdy analogous to some of the most established canons of classical physics. In his letter to Cairnes, Jevons repeated his case but with no apparent success.19 Cairnes, though he professed to have "taken every pains to

Response to the Theory · 101 understand it [the final degree of utility]," upheld his accusation of circularity in his next work, Some Leading Principles of Political Economy.20 Fortunately for Jevons, other economists recognized that Cairnes had misconstrued the utility theory and offered assistance. George Darwin in particular did Jevons a great service in his review of Cairnes's 1874 book by demonstrating that the classical cost-of-production theory of value could be deduced from the more fundamental utility theory of value. This did much to boost Jevons's spirits. To Leon Walras, he wrote in 1875: Much more assent to the theory is now apparent. The last number of the Fortnightly Review (February) contains an article on the subject . .. by George Darwin. It is a defence of our theory against Cairnes and will have considerable effect, owing to the reputation of the name of Darwin, George Darwin being the son of the celebrated Charles Darwin.21 Jevons also exposed Cairnes's inability to understand the distinction between the total and the final degree of utility in his November 1874 address, "The Progress of the Mathematical Theory of Political Economy." Although the paper was originally intended as a promotion of Walras's parallel development of mathematical economics, it quickly turned to the subject of value in an open rebuttal to Cairnes. On this particular charge, Jevons was characteristically blunt: So long as Mr. Cairnes fails to distinguish between the total useful effect of a commodity and its remaining degree of utility, and so long as he interprets my employment of the term utility in a way which is totally opposed to my own use of it, there is no common ground of discussion between us. Jevons was therefore fully confident that Cairnes's critique, because it was based on a misconception, was of no lasting significance. To his Dutch admirer, Johan d'Aulnis de Bourouill, Jevons wrote in 1874: "Cairnes, as you truly say, has failed to seize the idea of the theory, and his objections are consequently of no weight, though he is usually a most able economist."22 Perhaps the most striking feature of Cairnes's 1872 review is the noticeable absence of a developed critique on the use of mathematics.23 We learn from an October 1871 letter to Mill that this was in fact Cairnes's primary objection to the work: "I own I have no faith in the development of economic doctrines by mathematics."24 Cairnes closed his review with the allowance "that some of the doctrines of political economy may be exhibited mathematically," but in principle he regarded "the practice with profound distrust."25 He promised to develop his objections in a subsequent work, but not until 1875, with the sec-

102 · Chapter 6 ond edition of his Character and Logical Method of Political Economy, did Cairnes manage to alter his line of attack from content to method. By this delay, however, he undermined the effectiveness of his overall assault on Jevons. The relatively tangential criticisms of 1872 and 1874 not only provided Jevons and Darwin with the opportunity to take him to task for misconstruing the utility theory of value but may have helped to win sympathy for Jevons's position, most notably from Alfred Marshall.26 Indeed, even Cairnes's 1875 book recognized a growing support for the mathematical treatment of economics. The preface to the Logical Method is devoted almost exclusively to the issue of "the employment of Mathematics in the development of economic doctrine," as promoted by Jevons. Although Cairnes grants that it may be possible to present economics in mathematical form, no new knowledge, he insists, could be unearthed by such methods. The reason, he claims, stems from the fact that the foundations of economic theory are borrowed from psychology. Mathematics could only secure a place in political economy if "it can be shown either that mental feelings admit of being expressed in precise quantitative forms, or, on the other hand, that economic phenomena do not depend upon mental feelings."27 Since Cairnes had already ruled out the second alternative, he had in effect played right into the hands of his opponent. By rejecting mathematics on the grounds that economics could not be a quantitative science, Cairnes had unwittingly reinforced Jevons's argument that mathematical and quantitative reasoning are equivalent. Moreover, Jevons had argued at length in the Theory that pleasure and pain differ in duration and intensity and thus vary quantitatively. It was not, however, essential to treat feelings in a precise manner, but only to grant individuals the capacity of making simple comparisons between two feelings of similar magnitude. Cairnes's objections were thus of litde consequence. In Cairnes's view "the character of an economic law [is] analogous in all respects to those laws of physical nature which are obtained by a similar process of deductive reasoning, with the important exception that it does not admit of quantitative statement."28 But to Jevons, who believed that economics unquestionably dealt with quantitative relationships of supply and demand, the obvious next step from Cairnes's statement was not the rejection but the adoption of mathematics.29 In other words, by granting that economics must be treated with the same methods as are found in mechanics and astronomy, Cairnes had laid down the very same grounds that Jevons required for his own new economic analysis. In this respect, Jevons realized, Cairnes had helped his cause. To George Darwin Jevons wrote, shordy after receiving a copy of the sec-

Response to the Theory · 103 ond edition: "I have really not read his book properly, and look forward to the pleasure of studying it with care. I expect to find it confirmatory on the whole of the mathematical theory."30 Four years later, Jevons suggested in a letter to Foxwell that any disputes with Cairnes would be laid to rest in a larger work he planned to write. It would demonstrate that "many different lines of argument including that of Cairnes converge to something quite opposite to Ricardian doctrines."31 Apart from their differences on the applicability of mathematics, the methodological position advanced by Cairnes in his second edition was remarkably similar to Jevons's own views. Cairnes not only argued at length that "political economy uses the same method and has the same aims as the physical sciences," but also placed similar emphasis on the deductive nature of economics, the empirical accuracy of the basic premises, and the limitations of verification in science as a whole.32 In an earlier essay on Comte, Cairnes had also argued that political economy followed the same developmental path as the physical sciences, a view Jevons was to marshal in support of his own general program.33 The similarities are sufnciendy conspicuous that one suspects Jevons may have been directly influenced by the first edition, which he had read in 1863. Even Cairnes acknowledged on one occasion that, apart from their "unfortunate disagreement as regards the feasibility of mathematical treatment," he and Jevons held essentially the same views on the "logical method" of economics.34 R.D.C. Black has shown from his study of the Jevons-Cairnes correspondence that, despite their differences, a mutual respect persisted between the two.35 Cairnes had consistently expressed an admiration for Jevons's work in logic and applied economics and had conceded on at least one occasion that the Theory was a remarkable and important work.36 For his part, Jevons was fully aware that his analysis of value complemented rather than contradicted the classical theory and thus magnanimously tried not to allow Cairnes's poor judgment of his theory to interfere with his overall regard for the older man. This interpretation is also reinforced by the fact that Jevons fully recognized how much of Cairnes's position on method could be used to support his own campaign for a mathematical economics. In sum, the criticisms of Cairnes, thanks largely to the rebuttals of Darwin and Jevons himself, had virtually faded to naught by the time of Cairnes's death in 1875. MILL

It is a common supposition that John Stuart Mill reacted unfavorably toward Jevons's treatment of economic theory and yet, when the evidence is closely examined, Mill's actual opposition to the Theory re-

104 · Chapter 6 mains a matter for conjecture. Neil de Marchi has attempted to account for Mill's failure, prior to Jevons, to develop along similar lines. Given Mill's background, his interests in psychology, his grounding in the Benthamite philosophy, and his study, albeit brief, of the differential calculus, it is somewhat puzzling that Mill did not arrive at or grasp the significance of the concept of diminishing marginal utility.37 De Marchi has put forth a number of plausible answers to this puzzle, the most obvious one being that Mill's Principles of Political Economy was written primarily to establish Ricardian economics as the classical doctrine rather than to forge ahead to new concepts.38 But this in itself would not have prevented Mill from recognizing the significance of marginal analysis in the work of others. More to the point, de Marchi has argued, "Mill inherited from Ricardo a bias against giving consumption a place equal to that held by production and distribution in the scheme of economic science."39 In his celebrated essay "On the Definition of Political Economy and on the Method of Investigation Proper to It," Mill categorically denied the existence of any laws of consumption.40 He also believed that "no precise and universal laws of human feeling or action are to be found." Hence the use of algebra, which Mill held to be of universal scope, would misconstrue the relativistic nature of the behavioral axioms of political economy. For these and other reasons, de Marchi has argued, Mill "reacted negatively" toward the innovations in Jevons's Theory.41 I know of no evidence which suggests that Mill, who died in 1873, ever read the Theory. Insofar as the work explicitly attacks his own, it seems highly probable that he would have responded adversely. Mill's only recorded criticism of Jevons's book was conveyed privately, in a letter to Cairnes. I have not seen Mr. Jevons's book, but as far as I can judge from such notices of it as have reached me, I do not expect that I shall think favourably of it. Jevons is a man of some ability, but he seems to me to have a mania for encumbering questions with useless complications and with a notation implying the existence of greater precision in the data than the questions admit of. His speculation on Logic, like those of Boole and De Morgan, and some of those of Hamilton, are infested in an extraordinary degree with this vice. It is one preeminently at variance with the wants of the time, which demands that scientific deductions should be made as simple and easily intelligible as they can be made without ceasing to be scientific. I look forward with much interest to seeing your notice of the book, which I am sorry not to see-in the December [1871] Fortnightly.42 There is almost no doubt that Mill, given his active support for the Fortnightly Review and close contact with Cairnes during this period,

Response to the Theory · 105 read Cairnes's review of the Theory when it appeared the following month (January 1872).43 And since there is no further mention of Jevons or the review in the extant correspondence, it seems reasonable to assume that Mill was satisfied with Cairnes's critique and, in all likelihood did not bother to read the work at all. But, as far as we know, Mill never openly attacked Jevons.44 Quite the contrary, his two public appraisals of Jevons, a testimonial for Jevons's application to serve as professor at Owens College and an 1866 Parliamentary address promoting The Coal Question, suggest that MiD had considerable esteem for the younger man.45 Although, in this one letter to Cairnes, Mill does not explicitly refer to the use of mathematics, we know from the letter to which it is a reply that Cairnes had informed him of the primary thrust of Jevons's book. It seems safe to assume, then, that Mill used the word "notation" to refer to algebraic symbols. He had reacted in a similar manner and used the same word to refer to algebra when confronted with Jevons's first work on logic.46 And it was on the basis of this early work that Mill derived his opinion that Jevons possessed a "mania" for using "a notation implying the existence of greater precision in the data than the questions admit of." For Mill, the variables of a general scientific equation must be capable of precise measurement. Why, however, did MiD react so vehemently to the use of symbols in propositional logic? The reason, it appears, lies in his belief that numbers, unlike the names of concrete objects, were taken to be of universal scope. Algebraic or symbolic statements, as distinct from mere prose, referred to aD possible entities in the universe: Propositions, therefore, concerning numbers, have the remarkable peculiarity that they are propositions concerning all things whatever; all objects, all existences of every kind, known to our experience. All things possess quantity; consist of parts which can be numbered; and in that character possess all the properties which are called properties of numbers; . . . algebraic truths are truths of all things whatever.47 AD other empirical sciences feD short in this regard precisely because it was not possible to ascertain aD of the relevant facts. In the application of algebra to the physical sciences, for example the study of mechanical forces, numbers were used as "the means of ascertaining truths not numerical," but were not directly studied in and of themselves.48 In this respect, the empirical science of mathematics, insofar as it was related to number, could attain greater certainty than the physical sciences.49 Nevertheless, aD the empirical sciences, by virtue of the Use of deduction, could partake in some degree of certainty. Some overlap between mathematics and other sciences was

106 · Chapter 6

thus always possible. Indeed, for Mill, it was imperative that scientists study mathematics in order to develop a skill in pure ratiocination. "The deductive investigation of Nature . .. is common to all science, moral and metaphysical included;. . . I may venture to say that no one ever knew what deduction is, as a means of investigating the laws of nature, who had not learnt it from mathematics."50 A brief look at Mill's views on the epistemological status of astronomy, which for him best exemplified the "concrete deductive or physical method," will serve to clarify his twofold conception of certainty as stemming from deductive rigor and empirical completeness. Initially, Mill identifies astronomy as "an exact science, because its phenomena have been brought under laws comprehending the whole of the causes by which the phenomena are influenced." But owing to planetary perturbations, "even such as Kepler's laws, are mere approximations. .. . Thus even in astronomy, perfect exactness in the mere empirical laws is not to be looked for; much less, then, in more complex subjects of inquiry."51 As a result, Mill was not sanguine about the possibility of mathematics securing a greater hold over the less developed branches of science. "In the achievements which still remain to be effected in the way of scientific generalization, it is not probable that the direct employment of mathematics will be to any great extent available: the nature of the phenomena precludes such an employment for a long time to come—perhaps for ever."52 Mill devoted very little attention in his study of methodology to the prospect of mathematizing the nonphysical sciences. In his A System of Logic, there is a brief section on the subject of the "Function of mathematical truths in the other sciences" (HI, xxiv, 9), but he does litde more than endorse the position of Auguste Comte, who maintained that mathematics, although of critical importance in training the mental faculties, had no place in the "higher" sciences of society. According to Comte, mathematics would first have to prove its worth in the lower-ranking science of biology. "As for any application of number and of a mathematical law in sociological problems, if such a method is inadmissable in biology, it must be yet more decisively so here."53 Citing Comte as his authority, Mill also ruled out the possibility of a mathematical treatment of the life and moral sciences on account of the complexity of the phenomena. "Such [mathematical] principles are manifestly inapplicable, where the causes on which any class of phenomena depend are so imperfectly accessible to our observation, that we cannot ascertain, by a proper induction, their numerical laws; . . . as in physiology, and still more, if possible, in the social science." Judging from the brevity of this section, and from the many references to Comte, it appears that on the issue of the function of

Response to the Theory · 107 mathematics in the other sciences, Mill had, for the most part, simply accepted Comte's position.54 Certain passages in this section of the Logic, however, suggest that Mill had begun, perhaps unwittingly, to break away from his mentor. For Mill also claims that mathematics, though it best serves "those deductive sciences which afford precise numerical data," is also the more feasible the more abstract the problem at hand. "Mathematical solutions of physical questions become progressively more difficult and imperfect, in proportion as the questions divest themselves of their abstract and hypothetical character, and approach nearer to the degree of complication actually existing in nature."55 To Mill's credit, he had come part way with the realization that, in fact, hypothetical physical questions are the ones most amenable to mathematization, an insight he reiterated in a later work. "For where the inaccessibility or unmanageableness of the phenomena precludes the necessary experiments, mathematical deduction often supplies their place, by making us acquainted with points of resemblance which could not have been reached by direct observation."56 Mill's stipulation that the science "afford precise numerical data," or that the "class of phenomena" be fairly "accessible" is thus, as Jevons was to argue at length, not relevant. It just so happens that those sciences that are most exact are also those that were first mathematized, but exactness of the data, Jevons emphasized, had nothing to do with the application of mathematics to a particular hypothesis. In other words, it was the general or abstract form of a problem, as Mill also recognized, that lent itself to mathematical treatment. In the opening chapter of his Theory, Jevons attempted to set the matter straight. "Many persons," he wrote, "entertain a prejudice against mathematical language, arising out of a confusion between the ideas of a mathematical science and an exact science."57 There can be litde doubt that Jevons had Mill in mind, as he did in much of his work, as the source of many unfounded prejudices. In fact, Mill's more developed study of the scope and methods of political economy, as distinct from the other social sciences (a distinction Comte was not willing to draw), conforms very closely to his conception of an ideal mathematical science. To Mill, political economy was not only a deductive science, but also, like classical physics, essentially abstract and hypothetical in character.58 Its methods were more akin to those of mechanics than of chemistry. Indeed, it was possible to start with a study of economic laws under the most ideal and simplistic conditions, such as perfect competition, and then introduce disturbing causes, in much the same way in which the physicist would gradually complicate the fundamental laws of mechanics with

108 · Chapter 6 various frictions and asymmetries. Mill also grants that all scientific knowledge, insofar as it involves the pursuit of causal laws, is ultimately quantitative and thus mathematical in principle. "All causes operate according to mathematical laws. The effect is always dependent on, or is a function of, the quantity of the agent; . . . We cannot, therefore, reason respecting causation, without introducing considerations of quantity and extention at every step."59 Mill would have to concede, given this claim, that political economy, as the study of the causes that regulate the wealth of nations, was at the very least potentially a mathematical science. Additional remarks by Mill suggest that he had some inkling of the direction economics might take. He was fully convinced that "there exist universal laws of the Formation of Character," though, since they "cannot be ascertained by observation and experiment," our knowledge of them would never approach the degree of certainty found in the laws of astronomy. Nevertheless, the principles of the science of ethology, precisely because they were to be determined by deduction, insured that the study of social phenomena could become a full-fledged science. Mill also proposed that utility be treated quantitatively: "for what is the principle of utility? . . . The truths of arithmetic are applicable to the valuation of happiness, as of all other measurable quantities."60 Furthermore, Mill maintained that the methods of economics, like the methods of algebra, were global in scope. "Though many of its conclusions are only locally true, its method of investigation is applicable universally; and as whoever has solved a certain number of algebraic equations, can without difficulty solve all others of the same kind, so whoever knows the political economy of England,. . . knows that of all nations, actual or possible."61 From this statement it is clear that Mill regarded the methods of economics as standing quite apart from the empirical content of the subject. Despite the fact that Mill had set up virtually the same premises that Jevons was to employ, he did not reach the conclusion that mathematical economics ought to be developed. For Mill believed that, by giving too much attention to the purely ratiocinative stage of the scientific method, mathematics would deter theorists from formulating sound fundamental principles. "The one really grave charge which rests on the mathematical spirit, in respect of the influence it exercises on pursuits other than mathematics . . . [is that] it leads men to place their ideal of Science in deriving all knowledge from a small number of axiomatic premises, accepted as self-evident, and taken for immediate intuitions of reason." This "abuse of deduction" was directly traced to Descartes who, for Mill, instantiated "the completest type which history presents of the purely mathematical type of

Response to the Theory · 109 62

mind." French political philosophers ever since had continued to perpetrate the crime. "The French deduce politics like mathematics from a set of axioms and definitions, forgetting that in mathematics there is no danger of partial views: a proposition is either true or it is not, and if it is true, we may safely apply it to every case which the proposition comprehends in its terms: but in politics and the social sciences, this is so far from being the case."63 Mill, after all, had devoted much of his life to promulgating an empiricist epistemology and could therefore hardly have endorsed the more rationalist views of French social philosophers. Mill's fears, however, were in Jevons's eyes unfounded. Though he subscribed to much the same extreme empiricism as Mill, Jevons, we have seen, insisted that a mathematical treatment of economics would by no means increase the degree of certainty. If anything, it would clarify the very approximate nature of all scientific laws. For all of the above reasons, it is at least plausible to suppose that, had Mill read Jevons with care, or read him at all, he might have realized that the more abstract parts of economic theory could be analyzed mathematically. Mill had broken away from Comte on other, much more fundamental issues and had also demonstrated the capacity, even in his later years, to correct himself on other inconsistent beliefs.64 Moreover, Jevons and Mill openly shared, as Jevons acknowledged in his Theory, the belief that economists used the same methods as physicists. The possibility remains that Mill might have been convinced by Jevons's development of this position, particularly by his clarification of the distinction between a mathematical and an exact science. And it seems reasonable to grant, at the very least, that a careful reader of Mill, such as the young Alfred Marshall, would have realized that Mill's basic position could be construed in such a way as to entail a mathematical development of economic theory. We have seen that on many issues regarding method, the differences between Jevons, Cairnes, and Mill were not as great as numerous remarks in the secondary literature suggest. In order to raise substantial objections to Jevons's Theory, Mill and Cairnes would have had to undermine or alter their own methodological tenets. It is for this reason, I suggest, that they failed to suppress the movement inaugurated by Jevons, however much their initial instincts were repelled by such a turn of events. Moreover, Caimes's general confusion over the concept of the final degree of utility, and his delay in challenging the use of mathematics, coupled with Mill's complete silence on the subject, significantly lessened the obstacles Jevons faced. Mill and Cairnes simply did not have much time in which to mount an attack on Jevons.

110 · Chapter 6

Mill's disciples, most notably Henry Fawcett at Cambridge, showed little interest in challenging Jevons.65 And by a strange turn of events, another faction came to Jevons's aid. While exonerating Mill, the proponents of the historical method tackled Cairnes rather than Jevons as the absolutist. In the various debates that ensued over the validity of the inductivist approach, Jevons emerged more or less unscathed. THE ECONOMIC HISTORIANS

The two main proponents of the British historical school, Thomas Edward Cliffe Leslie and John Kells Ingram, both hailed from Ireland. Although a similar movement developed somewhat earlier in Germany, the initial factors responsible for the British school of the 1870s were, by and large, indigenous. Some inspiration came from Richard Jones's unsuccessful attempts to transform economics into a historical discipline. More generally, the writings of Auguste Comte and Sir Henry Maine left their imprint. But the more proximate cause was the plight of Ireland and the disillusionment both Leslie and Ingram felt over the blind application of orthodox economic theory to the problem of Home Rule.66 The main thrust of their attack was directed against the abstract, deductive method of Ricardo. They charged the Ricardians with failing to establish the postulates of political economy by careful inductive procedures. In their view, many of the postulates, such as selfinterested economic man, were unwarranted on empirical grounds. As a result, it was misleading to attribute a universal scope to the theory of political economy. Although Leslie and Ingram did not rule out the possibility, at some future time, of universal economic laws, and although they did not deny the importance, in principle, of deductive reasoning, it was essential in their view that painstaking historical research first be carried out before any general laws could be ascertained. Leslie and Ingram regarded Adam Smith as the true founder of the historical method and John Stuart Mill as the most prominent, though not the most consistent, nineteenth-century practitioner. Mill was excused from failing to arrive at a "systematic application of the historic method" on account of his Benthamite upbringing.67 The two main opponents to their cause, insofar as they upheld the "Ricardian" belief in political economy as a neutral, universal science, were Cairnes and Robert Lowe. Jevons, though he held much stronger views about the universal applicability of economic theory, was notably absent from the main line of attack.68 In fact, both Leslie and Ingram spoke of Jevons with considerable respect, primarily because he also advocated

Response to the Theory · 111 the development of the historical method and made contributions to the field.69 Although his campaign for mathematical economics would, in their view, lead economics even further astray down the deductive path, they tended to group Jevons with Smith and Mill, as economists who had worked on both sides of the fence and thus helped their cause. Although Leslie regarded Mill as his mentor in political economy and was particularly appreciative of Mill's support on the Irish question, he nonetheless took Mill to task on several issues—the wagefund theory, the purported universality of the laws of production, and Mill's categorical denial of the existence of laws of consumption. According to Leslie, the theory of economics must begin with a correct theory of consumption. Both he and Ingram explicitly recognized Jevons for having restored consumption to its primary place in economics.70 Jevons, likewise, cited Leslie in an 1874 address and in the preface to the second edition of the Theory, as the one most responsible for having overthrown Mill's wage-fund theory.71 Their mutual respect for one another was no doubt in part a product of their kindred dislike of Ricardo and their common recognition of Mill's shortcomings.72 In his 1879 review of the second edition of the Theory, for the Academy, Leslie defended Jevons's treatment of value in direct response to the criticisms of Cairnes. Leslie not only agreed with Jevons that "value depends mainly on limitation of supply," but also endorsed Jevons's principle of the final degree of utility. Although he also acknowledged the value of deductive reasoning, he drew the line at mathematics: Were the application of mathematical processes and symbols to all economic reasoning, however, possible, it does not follow that it would be expedient . . . The Times might be printed in shorthand, and much ink and paper thereby saved, but would it conduce to the enlightenment of the public to make that economy? We regret that so much of Mr. Jevons's own reasoning is put into a mathematical form, because it is one unintelligible or unattractive to many students of considerable intellectual power and attainments.73 Despite these criticisms, however, Leslie had much praise for the Theory and encouraged others to study it with care. Ingram also showed much respect for Jevons. In his History of Political Economy (1888), he wrote: "the reputation of Jevons as an acute and vigorous thinker, inspired with noble popular sympathies, is sufficiendy established." But he took issue with Jevons for believing that it would be possible "to preserve the a priori mode of proceeding

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alongside of, and concurrently with, the historical." According to Ingram, it was only a matter of time before the historical method emerged as the sole method of economics. There could certainly be no future for mathematical economics, not only because it was deductive, but also because utility was too mysterious a concept for precise measurement. In Ingram's opinion, "we have no means of estimating quantitatively the mental impression of final, or any other, utility." For this and other reasons, "the application of mathematics in the higher sense to economics must necessarily fail."74 Like Leslie, it is apparent that Ingram did not bother to attend to the more subtle arguments made by Jevons. But they did not publicize their differences as widely as Cairnes had done. Indeed, for Ingram, this was the only occasion (to the best of my knowledge) on which he issued such criticisms in print. Considering Leslie's and Ingram's express antagonism toward the use of mathematics, their favorable regard for Jevons is somewhat surprising. Perhaps they believed it would be possible to separate the grain from the chaff. That is to say, while Jevons would undoubtedly aid their attack on Ricardo and had already made several contributions to inductive economics, the mathematical side of his work would shortly be forgotten, if only because, in their view, there was only a small readership.75 At Oxford, a group of lesser-known writers—Thorold Rogers, L. L. Price, H. L. Smith, and William Cunningham—clustered around the feet of Arnold Toynbee, the leading author on the Industrial Revolution. Toynbee's last progeny before his death in 1883, Edwin Cannan, William Hewins, and William Ashley, were the ones most responsible for establishing economic history as a distinct discipline by the turn of the century. Interestingly, the economics students at Oxford, their socialist sympathies notwithstanding, openly applauded Jevons's denunciations of Mill and Ricardo. As Cannan reminisced of his Oxford cohorts in the early 1880s, "Jevons' vigorous attacks upon Mill would have been attacks on the economics of the day, which would have caused horror on the one side and acute delight on the other. Those of us who were young then would have hailed with applause the pages which begin with the caustic observation [by Jevons] 'Mill expounds the theory of capital in four fundamental propositions, all false'."76 They saw Jevons as more palatable, both methodologically and politically, than Mill and certainly Cairnes. Although this group of writers succeeded in securing economic history within the university curriculum, they did not hinder the study of pure economic theory, let alone the use of mathematics. Nor did they counteract the shift toward the treatment of economics as a science of

Response to the Theory · 113 universal scope. Economists have long since relegated that issue to the anthropologists.77 Jevons, it seems, by encouraging both historical and mathematical methods, begat the more viable program. THE MODERATES

The views of the moderates, Bagehot, Sidgwick, Foxwell, and Keynes, were of particular importance not only because they advocated a position of compromise that tolerated both the old and the new schools of thought, but also because, possibly as a result of their impartiality, they were quite influential in the economic community of the time. Bagehot served as editor of the Economist from 1859 until his death in 1877 and was known to have considerable influence in the financial district of London. He thus represents a major link between the professional economists and the interested layman at the time. The other three moderates were all at Cambridge (though Foxwell also taught at London) and, under Marshall's direction, helped to establish it as the most prominent center in Britain for the study of economics.78 Each of these men expressed a considerable respect for Jevons's work. Bagehot wrote in his 1866 testimonial for Jevons's application to teach at Owens College: "I do not think I know any recent books which show (as far as I can judge) an equal knowledge of the abstract theory of Political Economy, and an equally accurate acquaintance with the statistics of present facts."79 Sidgwick, in the preface to his major book The Principles of Political Economy (1883), openly acknowledged his debt to Jevons. "After J. S. Mill's book, from which I first learned political economy, . . . I believe that I owe the most to Jevons' Theory of Political Economy, the leading ideas of which have been continually in my thoughts."80 After serving as joint examiners at Cambridge in 1874, Foxwell and Jevons became close friends, and took delight in accompanying one another on serious book-buying sprees. Foxwell had only glowing reports of Jevons, and did much to spread his doctrines, both in Britain and abroad.81 Neville Keynes was equally enthusiastic. In his estimation, Jevons's Theory "may be regarded as one of the most suggestive and valuable contributions to the science that have ever been made."82 The moderates have been identified by their desire to promote goodwill among their fellow economists. For Foxwell, this almost took on missionary proportions. To Jevons he wrote, "I feel that the science suffers whenever there are unnecessary antagonisms . . . and if I could do anything to reconcile them, I should feel it was perhaps worth while living to do so."83 AU four maintained that both the traditional deductive and the more recently developed historical methods

114 · Chapter 6 were fully complementary. As Bagehot put it, "rightly conceived the Historical method is no rival to the abstract method rightly conceived."84 Sidgwick attempted to demonstrate that both Smith and Mill had successfully combined the two methods into an integrated system. Foxwell, who was less favorably disposed toward the classical authors, argued, as Bagehot had done, on theoretical grounds: They [the historical school] are not in any necessary antagonism to economists of the mathematical type. If they only knew it, the mathematical economists are their most effective supporters. There is no greater safeguard against the misapplication of theory than the precise expression of it. No writers have been so truly historical in their use of theory as those who, like Cournot and Jevons, have known its exact limits.85 Off the record, he was even more forthright. To Walras he wrote in 1886, "there is no doubt Mill is dead in this country. The mathematical method for pure theory, the historical for practical questions—this is what we are all coming to agree in."86 Keynes's major treatise, The Scope and Method of Political Economy (1890), has generally been regarded as the most definitive survey of British economics at the end of the last century.87 His views may therefore be accorded considerable weight. Keynes insists in this book that, although the majority endorsed the deductive or abstract method, in his view, "no one method will be advocated to the entire exclusion of other methods."88 The choice of method, he argues, is largely a function of the problem at hand. For some topics, the historical method may be the most appropriate, and for others, the mathematical method. Nevertheless, the historical method, he argued, could never supplant the deductive method. Historical case studies were inextricably dependent upon the conceptual apparatus developed by the more abstract approach. According to Keynes, who cited Jevons as the authority on this very issue, "it is impossible to frame any general theories of value, interest, wages, rent, &c. by purely a posteriori methods of reasoning. Recourse must be had to a method, in which deduction from elementary principles of human nature occupies a position of central, though not exclusive importance."89 Sidgwick also shared this belief, noting that the self-interest axiom must underpin the analysis of specific price data.90 Although Bagehot and Sidgwick did not bend over backward to advance the cause of mathematical economics, they did little to block its development. Sidgwick once intimated that he "had occasion to dissent from many of Mr. Jevons's particular opinions," but the closest

Response to the Theory · 115 he came to disapproving of mathematical economics was a very agnostic statement in an essay on "Economic Method": I have spoken once or twice of the importance of making our thought precise. I do not mean that we should necessarily aim at quantitative exactness in our statements of economic laws. I quite agree with the writers (such as Cairnes) who have warned us against the futility of such attempts. But the more inevitable it is that our conclusions should be merely rough and approximate, the more important it becomes that we should be thoroughly aware when and how far they are wanting in exactness; and in order that we may be aware of this, we should make our conceptions as precise as possible, even when we cannot make our statements so.91 When pressed on the matter, however, he agreed to be allied with the Jevonian cause. 92 Bagehot was somewhat more equivocal. One statement by him, taken on its own, suggests that mathematics might be leading economics astray: At the very moment that our Political Economy is objected to in some quarters as too abstract, in others an attempt is made to substitute for it one which is more abstract still. Mr. Stanley Jevons, and M. Walras, . . . have worked out a "mathematical" theory of Political Economy;—and any one who thinks what is ordinarily taught in England objectionable, because it is too little concrete in its method, and looks too unlike life and business, had better try the new doctrine. Yet he insists, like Cairnes, that economics is primarily an abstract science: Political Economy in its complete form, and as we now have it, is an abstract science, just as statics or dynamics are deductive sciences. And, in consequence, it deals with an unreal and imaginary subject. Just as statics and dynamics—the science of theoretical mechanics—deal with perfectly rigid bodies, . . . with a world destitute of friction; with physical materials in short which no one ever expects to find in reality—so Political Economy deals with an immaterial subject, which in the existing world cannot be found either.93 Elsewhere, he states categorically that economics uses the same methods as physics: "I do not claim for the conclusions of English Political Economy the same certainty as for the 'laws of motion.' But I say that the method by which they have been obtained is the same." 94 Thus it appears that both Bagehot and Sidgwick, though they did not explicitly promote Jevons's cause, certainly did UtUe or nothing to op-

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pose it. In this respect, they fall very much into the same category as Mill. Foxwell and Keynes, on the other hand, openly encouraged the development of mathematical economics. In his survey "The Economic Movement in England," Foxwell attributes the most recent progress in economics to the use of mathematics: But the great advance recently made by theory is not so much due to removal of contradiction, . . . as to the use of the powerful and essentially scientific instrument of mathematical analysis, practically introduced into English economics by Jevons, in his original and masterly Theory. To the influence of this book, and of the teaching of Professor Marshall, . . . English economics owes a stimulus the full effects of which can scarcely be measured, but which it would be difficult to overrate.95 Foxwell goes on to praise Jevons for having shown "how hopelessly the old theory fell below the ordinary scientific standard." Although Foxwell never contributed to the body of mathematical economics— his ability at mathematics was not up to the task—he attempted to teach parts of the subject, using the works of Cournot, Gossen, Jevons, and Marshall.96 As he noted in his correspondence with Walras, he felt more comfortable with "Marshall's method of graphic presentation," but nonetheless "entirely agree[d] with the general spirit or method of treatment of Jevons in his discussion of real economic questions."97 Keynes's book contains the most extensive case for the development of mathematical economics.98 In direct tribute to Jevons, Keynes argues that insofar as economic theory is quantitative, it is also mathematical; "The fact that political economy is essentially concerned with quantitative relations, and therefore involves mathematical notions, needs to be insisted upon." Keynes also points to the misconceptions of Cairnes and Mill over the issue of a mathematical and an exact science. He notes that both Cournot and Edgeworth had shown that "exact numerical premisses .. . are not always essential to the employment of mathematical methods."99 Keynes then argues that there are many advantages attached to a mathematical treatment of economic theory. Mathematics helps to clarify assumptions, isolate errors in reasoning, lead to a higher standard of precision, minimize circumlocution, expose the limits of economics in general, distinguish between variables and constants in economic theory, and delineate the element of continuity in the economic phenomena themselves. Although Jevons had, in one place or another, also recognized these virtues, Keynes's lucid presentation may well have made an important difference in winning support.

Response to the Theory · 117 Keynes also points to the need to campaign for the new approach, "because to some economists the very idea of a mathematical treatment of economic problems is not only repugnant, but seems even absurd." Though not unquestionably essential to the development of the discipline, mathematics had certainly, in his estimation, been a considerable boon to the subject: Mathematics may not up to the present time have proved an absolutely indispensable instrument of economic investigation and exposition; but it would be difficult to exaggerate the gain that has resulted from the application of mathematical ideas to the central problems of economic theory.100 Apart from specific contributions to mathematical economics itself— something Keynes did only in his capacity as a father of Maynard— Jevons could not have hoped for a more persuasive ratification of his views. Keynes stood apart from the other moderates on one issue, namely the scope of political economy. Whereas Bagehot had maintained that "English Political Economy is an analysis only of industrial societies, which are to a certain extent developed,"101 Keynes argued that the pure theory of economics was, in many respects, of universal scope.102 Such principles as the variation of utility were, he argued, completely independent of geographical or historical conditions. Considering the general influence of Keynes's 1890 treatise as a study of the scope and method of economics, and his position of influence at Cambridge, it seems reasonable to assert that, on this issue as well, Keynes did much to enshrine Jevons's platform within the orthodoxy. Overall, his treatise put the capstone on the methodological debates of the previous three decades. Published in the same year as Marshall's Principles, it probably played an equal part in bringing peace and prosperity to fin de siecle economics. CONCLUSION

In contrast to the state of general stagnation that characterized midcentury economics, the years following Jevons's 1862 paper on mathematical economics were fairly turbulent ones. There was probably more dissension than consensus, at least until the early 1880s. Peace was restored either by compromise or sheer attrition, although the followers of Jevons emerged as the most influential group. Even the historicists and diehard Millians conceded points of overlap. The utility theory of value, for the historians at least, was important because it brought the consumer into center stage in the economic arena, and thereby helped to discredit the Ricardo-Mill doctrine. And the late

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classicists shared with Jevons a firm conviction that the science of economics must emulate the physical sciences. Jevons's appeal to mathematics could be seen as a natural extension of this tradition and Mill's indifference or reluctance to oppose this publicly may inadvertendy have led others to favor Jevons. Cairnes, in his last publication, made a half-hearted attempt to block Jevons, but he acted too late. Schumpeter once wrote that "nothing, in fact, proves so convincingly that, forerunners notwithstanding, it [Jevons and Marshall's work] actually was something new, as does the resistance it met with."103 But Jevons, however novel his ideas, met far less resistance than has commonly been described. A single denunciation by Mill might have sufficed to bury the Theory for many years. An organized assault by the historians on Jevons, rather than on Cairnes or Fawcett, might also have slackened its pace. As it was, Jevons witnessed the triumph of his campaign by the early 1880s. This may have been a more prolonged revolution than the ones associated with Smith or Keynes, but it may also have been more far-reaching. It remains for us in the next chapter to assess the extent to which his ideas took hold and grew within the English academic community.

CHAPTER SEVEN

Mathematical Economics Takes Hold The mathematical method really began with Jevons in 1871. ... In England, Prof. Edgeworth, noted for his enthusiasm on mathematical economics, has recently been elected to the chair of pol. econ. at Oxford, while Prof. Marshall is carrying forward the same movement at Cambridge. ... From all apparent evidence the mathematical method has come to stay. —Irving Fisher, 1891

DARWIN

During the 1870s, Jevons regarded George Darwin as one of his strongest British supporters. In addition to his distinction as the son of the famous naturalist, Darwin had placed Second Wrangler at Cambridge, thus putting him in the company of such notables as James Clerk Maxwell, W. K. Clifford, and Lord Kelvin. Although he published only one article in economics, his command of the subject was sufficient to undertake an essay review of Cairnes' Leading Principles for the Fortnightly Review. Entitled "The Theory of Exchange Value," the review read as a promotion for Jevons's new program. Jevons was clearly heartened by the gesture and thanked Darwin "for so boldly taking up the cause of the Theory . . . [and for] counteracting the illconsidered criticisms of Cairnes."1 Darwin made no attempt in the essay to disguise his biases. Given that Cairnes would command a much larger audience than Jevons, not only because of his greater reputation at the time but also because of the mathematical material within the Theory, it was important, Darwin argued, to clarify and remedy misconceptions in Cairnes's work, particularly his attack on the theory of utility. Hence, approximately half of the ten-page review is devoted to a discussion of the utility function, the concept of the final degree of utility, and the analysis of exchange. Darwin's essay contains a fairly lucid defense of the utility theory of value. Perhaps the most persuasive part of his argument is his demonstration that the classical cost-of-production theory could be deduced from the more fundamental theory of Jevons:

120 · Chapter 7 The question then arises, which (if either) of the two theories is correct? I would answer, both are true; but the utility-theory goes more to the root of the matter. In fact, the cost of production theory takes as its original datum a result which is obtainable from the utility-theory . .. I therefore hold that the utility-theory is more fundamental than the other, but that both are true. Jevons had made the same argument, but not as succinctly. Darwin helped emphasize the extent to which Jevons's formulation of exchange value constituted an advance in economic analysis, while at the same time noting that it in no way undermined the classical views espoused by Cairnes. But, as he also remarked, the utility theory had the virtue of being more closely allied to the central problem of economics, namely "to make a given amount of sacrifice (whether of labour or abstinence) go as far as possible towards the satisfaction of human wants."2 By couching economic activity in such general terms, Darwin had helped to display possible links to utilitarianism. Upon reading a preliminary draft of Darwin's essay, Jevons cautioned him against using any mathematical symbols which might "frighten readers away." Darwin obliged, though he also remarked in the essay itself that it would be "impossible to do justice to Mr. Jevons's ideas without the mathematical notation."3 His previous paper, an attack on Mill entided "Commodities versus Labour," also avoided mathematics.4 Yet, as Klaus Heinrich Hennings has shown from a study of the Darwin-Jevons correspondence, Darwin had a much better command than Jevons of the mathematical treatment of production and distribution. His analysis of the interest rate, in terms of a profit-maximizing rate of return, shows in particular the mathematical gifts of a Second Wrangler. Jevons failed to understand Darwin's insights on the subject and, judging from an 1874 letter, was somewhat relieved when Darwin agreed that Jevons's version was still valid under certain assumptions.5 One reason that Darwin did not publish his mathematical theory of the interest rate, which we see worked out in his letters to Jevons, may be the dirth of interested readers. In response to a request from Walras in 1876 to translate some of his papers into English, Darwin could not have been more discouraging: "I would counsel you not to attempt to have it translated. Most English political economists are not mathematicians and most of the mathematicians are not political economists, and therefore I fear you would appeal to a very small public."6 If Jevons was perplexed by Darwin's calculations, then there was litde hope, it seemed, that others would find them illuminating. Darwin's two papers may well have helped bring attention to Jevons, and he

Mathematical Economics Takes Hold · 121 may have persisted off the record in attacking Mill and Cairnes, but his written efforts at economics were short-lived. Having studied first mathematics and then law, Darwin finally settled on a career in the natural sciences and, in 1883, was appointed Plummian Professor of Astronomy and Experimental Philosophy at Cambridge.7 WICKSTEED

Of all the economists active in the latter part of the last century, Philip Henry Wicksteed is the one most often characterized as a disciple of Jevons, though it is not clear that they ever met.8 Originally trained as a Unitarian minister, Wicksteed only turned to the study of economics in the early 1880s and first read Jevons's Theory in 1882, the year Jevons died. Although Wicksteed had previously studied mathematics under De Morgan while a student at University College in the early 1860s, he was sufficiently inspired by Jevons to hire a private tutor in order to refresh and extend his facility with the calculus. In 1887, he began lecturing in the University Extension program and before long found himself in much demand. By 1897, he resigned his post at the Litde Portland Street Chapel in London and supported his family solely by writing and lecturing.9 A socialist by inclination, Wicksteed had originally set out to apply the principles of marginalism to Marx's theory of surplus value, only to find it yielded contradictions. The result, his first publication in economics, was "The Marxian Theory of Value" for the socialist journal To-Day (1884). George Bernard Shaw responded with a piece entided "The Jevonian Criticism of Marx." This exchange probably helped to bring much notoriety to Jevons's work. And Shaw could not have been more explicit, or more witty, about the fact that the calculus was to blame. The October number of To-Day is memorable for containing an attack by a Socialist on the theory of value held by the late Karl Marx. A Roman Catholic impugning the infallibility of the Pope could have created no greater scandal... A certain weight will be removed from my mind when the attack is repulsed, and the formerly pellucid stream of the Ricardian labour value theory has deposited the mud which the late Stanley Jevons stirred up in quantities which, though expressed by differentials, were anything but infinitely small. Mr. Philip Henry Wicksteed, the assailant of Marx, has adopted the Jevonian theory. Shaw went on to confess not only his ignorance but also his deep distrust of mathematical manipulations (stemming from his schooldays). This confession was fully exploited by Wicksteed in his "Rejoinder" to

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Shaw, in which he argued at length for the superior powers of a mathematical treatment of economic theory and the good intentions of those involved in the project.10 Shaw eventually conceded and shifted over to Wicksteed's point of view. Indeed, from 1884 to 1888, Shaw and Wicksteed met with Edgeworth, Foxwell, Sidney Webb, and Marshall on a regular basis to study and discuss the Jevonian theory. Although this controversy took place before the publication of the English translation of Das Kapital (Volume One, 1887), it may have convinced those who were hostile to socialism to turn to the new theory of economics, with all its scientific overtones.11 Shaw, in his reply to Wicksteed, helped to delineate the two schools of thought, by placing Marx with the classical school. "As against Cournot, Jevons, WaIras, Professor Marshall, and F. Y. Edgeworth, Marx is undoubtedly on the side of the standard English school of Adam Smith, Ricardo, Mill, and Cairnes."12 Thus, as early as 1884, it was clear to Shaw at least that a new school of mathematical economics had emerged. In his subsequent work, Wicksteed often acknowledged the significance of the changes brought about by Jevons. For example, in his 1896 article on Jevons for Palgrave's Dictionary of Political Economy, he noted that "he seems hardly to have realised the full consequences of his method. But none the less his Theory has succeeded in its aim. When all its implications have been worked out the science will be reconstituted."13 Wicksteed, appropriately, devoted considerable attention to the clarification and refinement of Jevons's insights, particularly his analysis of exchange and the dimensions of the economic variables. Also noteworthy are his many extensions of the diagrammatic method, or economic geometry. Although his major work, The Common Sense of Political Economy, did not appear until 1910, from 1884 on he wrote a number of economic papers, including the lengthy essay "On Certain Passages in Jevons's Theory of Political Economy" for the Quarterly Journal of Economics (1889) and a favorable review of Jevons's Principles of Economics (1905).14 According to Wicksteed, Jevons "was right in declaring that certain fundamental relations and conceptions in the theory of political economy are essentially mathematical." Utility, needless to say, was the concept most amenable to mathematical treatment: "the conceptions of 'degree of utility' and 'final degree of utility' lie at the heart of the mathematical method of political economy, and their complete history would almost coincide with the history of mathematical economics." But Jevons may have "overestimated the extent to which mathematics can penetrate the body of Economic Science."15 Wicksteed was thus by no means attempting to use mathematics across the board. Neither

Mathematical Economics Takes Hold · 123 a zealot nor an eccentric, he simply plowed ahead, using the Jevonian tools when appropriate. Wicksteed also confronted many of the methodological issues. He not only argued that the mathematical method complemented the historical method, but also that pure theory could be treated in completely general terms. Such problems as the distribution of resources, once mathematized, were of universal scope. Wicksteed also acknowledged the relationship between economics and psychology, suggesting in his essay "Political Economy and Psychology" that the former might be subsumed within the latter via the theory of consumption. In general, he observed, "the direction taken by economic study in recent years tends to a more express and generous recognition of the close connection between psychology and political economy."16 We saw that, for Jevons, the acceptance of economic theory as a universal, essentially psychological science, was an integral part of his general economic mechanics. Wicksteed thus embraced and reinforced many aspects of the Jevonian program. EDGEWORTH

Francis Ysidro Edgeworth has been described as one of the most enigmatic and imaginative economists who ever lived.17 Schooled in classics and law, he delighted in weaving literary allusions or Greek aphorisms into his economics. It is still unclear when or how he learned advanced mathematics, but he had such an agile mind that one is tempted to believe he cultivated these skills on his own.18 Edgeworth exerted a major influence within the community of economists, as Drummond Professor of Political Economy at Oxford from 1891 until his retirement in 1922, and as the editor of the Economic Journal from its inception in 1891 until his death in 1926.19 His general image as an avowed mathematical economist was thus of some significance. Although overshadowed by Marshall, Edgeworth may have made greater contributions to "the analytical apparatus of economics," at least if one adopts Schumpeter's measuring rod of success.20 But despite his gift for languages, Edgeworth wrote in a manner intelligible only to the most dedicated of readers. While neighbors in London from 1876 until 1882, Edgeworth and Jevons developed a close friendship, reputedly taking frequent walks together on Hampstead Heath. As a result, there is litde record of their exchange. From Edgeworth's letter of condolence to Harriet Jevons we learn that he would "always remember with gratitude the kind encouragement and a peculiar sympathy which he [Jevons] extended to one whose studies were in the same direction however immeasurably

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behind his." Jevons had also recommended Edgeworth, who was at that point employed as a barrister, for two academic posts. But despite his praise—Jevons remarked that Edgeworth's work "manifests a distinct and unquestionable genius"—he failed to secure the desired positions.21 Edgeworth's only book in economics, Mathematical Psychics (1881), was undoubtedly written under the guidance of Jevons, but it did not turn out to be as distinct an offspring of the Theory as, say, Wicksteed's Alphabet of Economic Science. Subtided "An Essay on the Application of Mathematics to the Moral Sciences," it would be difficult to find, as one scholar has put it, another "apology in the whole of economic literature which compares with Edgeworth's passionate plea for the application of mathematics."22 Edgeworth touched on many related issues as well, including the psychological foundations of economics and policy implications of the Jevonian theory of exchange for advancing utilitarianism. He was even more carried away with mechanical analogies than Jevons. Economic man was treated, quite literally, as a "pleasure-machine." Drawing inspiration from Gustav Theodor Fechner's insights on psychology, Edgeworth developed many parallels between the mathematical treatment of energy and "hedonics."23 Like Mill's proposal for a science of ethology, mathematical psychics never found a following. But the book remains a classic in the history of economics because it made two truly outstanding contributions to economic analysis: the indifference curve and the contract curve.24 Edgeworth, though himself committed to the treatment of utility in cardinal terms, had ironically hit upon the path which later enabled economists to bypass the difficult issue of ascertaining a unit of utility and thus to treat exchange in terms of preference functions that call only for an ordinal ranking.25 Despite Edgeworth's various pleas for the mathematical treatment of economics, Jevons was not entirely pleased with the work. His review of the book for the journal Mind described it as "one of the most difficult to read which we ever came across, certainly the most difficult of those purporting to treat of economic science."26 These criticisms of Edgeworth's obscure style were fully justified. But though Jevons mentioned in passing that the book contained "new and important suggestions," he failed to acknowledge Edgeworth's often significant modifications of the utility theory of value and exchange. More surprisingly, he did not refer to Edgeworth's general arguments for "unnumerical mathematics," although these same arguments had played a central role in his own campaign. Indeed, the general tenor of Mathematical Psychics was very much in keeping with Jevons's

Mathematical Economics Takes Hold · 125 own speculations on the reduction of economics to psychology and the significance of mathematics. Although in some instances Edgeworth bordered on the absurd, there is no obvious explanation for Jevons's lack of enthusiasm for what was by and large a major reinforcement of his own campaign. Marshall, oddly enough, gave it a much more favorable review, despite his many imperatives about reaching the laity.27 Edgeworth's subsequent contributions to economics took the form of numerous articles and reviews that he later compiled into three volumes. Many of them treat problems in statistics, but there are also interesting analyses of problems pertaining to currency, international trade, and taxation. Throughout, one finds a liberal use of mathematics. Not only did he develop economic geometry, but he was the first to introduce the technique of Lagrangian multipliers, the calculus of variations, and determinants into economics.28 The sheer concentration of analytical insights found in Edgeworth's writings attests to the momentum imparted by Jevons's Theory. In his MathematicaUsychics, Edgeworth unequivocally supports a belief in the necessity of mathematics, noting that "human affairs have now reached a state of regular complexity necessitating the aid of mathematical analysis."29 But in his later writings, perhaps under the influence of Marshall, he became more reserved on the matter. In his 1889 presidential address to the British Association (Section F), entitled, quite appropriately, "On the Application of Mathematics to Political Economy," he remarked that there "seems to be a natural affinity between the phenomena of supply and demand and some of the fundamental conceptions of mathematics," but that nonetheless, "it may be doubted whether the direct use of mathematical formulae extends into the region of concrete phenomena much below the height of abstraction to which Jevons has confined himself." Edgeworth also expressed the importance of reaching a broader public. Mathematics may best "be treated as a sort of scaffolding, to be removed when the edifice of science is complete."30 His inaugural lecture at Oxford, "The Objects and Methods of Political Economy" (1891), made many of the same points. He not only emphasized the many close parallels between classical physics and economics, but also urged students to study mathematics; "the differential calculus, the master-key of the physical sciences, unlocks the treasure-chamber of the pure theory of economics." But, he goes on, mathematics is not absolutely essential for the advancement of economics; it is simply a very valuable tool: "I do not deny that the refinements of pure theory may be reached by the use of ordinary language, sufficient circumlocution being employed; the treasure-cham-

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ber has a key of its own, but it is a cumbrous one."31 In this respect, as we are about to see, Edgeworth was a Marshallian. MARSHALL

Other prominent British economists at the time, notably Arthur Bowley, Leonard Courtney, and H. Cunynghame, championed the use of mathematics.32 Joseph Shield Nicholson was more ambivalent but gave his approval initially.33 In America, Simon Newcomb and John Bates Clark expressed support for the new approach while Irving Fisher, a student of the physicist J. Willard Gibbs at Yale University, took up Jevons's cause directly with his Mathematical Investigations in the Theory of Value and Prices (1892) and A Brief Introduction to the Infinitesimal Calculus, Designed Especially to Aid in Reading Mathematical Economics and Statistics (1897).34 All of this lends considerable weight to the view that mathematical economics took hold and was widely accepted, in the English-speaking world at least, by the end of the last century. But an explicit recognition of this transformation has thus far escaped entrenchment in the secondary literature. Historians have generally been led to diminish the significance of Jevons's campaign to mathematize the subject by the main exception to its success: the contribution of Alfred Marshall. Doyen of economists, Marshall purportedly turned back to the classical theorists and resisted the urge to mathematize the subject.35 And yet, when one delves into Marshall's relationship to Jevons, considering his teaching and early writings, it appears that Marshall's role in the rise of mathematical economics was of greater significance than has commonly been supposed. These misinterpretations occur mainly because Marshall's own retrospective account cannot be taken as an accurate representation of his earlier beliefs, a problem compounded by his failure to publish many of his ideas until his later years. A study of Marshall's "memorials" has led various scholars to charge him with inconsistency, a poor memory, a fear of being found wrong, an "inadequate acknowledgement of priority," and a tendency "of being less than frank." He is also reputed to have had a "talent for generating antagonism among his contemporaries," such that their appraisals of him must also be taken with a grain of salt.36 An assessment of Marshall's part in the story is thus no easy matter. My task here will be primarily to clarify and evaluate this muddled record and to bring to light the direct lineage between Jevons and Marshall on the issue of mathematical economics. In treating the relationship between Jevons and Marshall, one must

Mathematical Economics Takes Hold · 127 bear in mind that Marshall, though only seven years younger than Jevons, did not publish his first book (The Economics of Industry, coauthored with his wife, nee Mary Paley), and an elementary textbook at that, until 1879, just three years before Jevons drowned. Although Jevons had copies of two short pamphlets by Marshall (distributed privately in 1879) that suggested the greatness of things to come, he could not have gathered from Marshall's written work that late Victorian economics would be seen as the "Marshallian Age."37 Jevons recognized Marshall as a promising teacher of the subject, and had considered appointing him as his substitute at University College in 1875; he was also quite impressed by the 1879 essays.38 During the years in which they corresponded (1875-1879), Jevons evidently regarded Marshall more as a possible recruit to his campaign than as a rival. Like Jevons, Marshall turned to economics having already studied mathematics and the natural sciences and was similarly attracted to the ethical dimensions of the subject. It is all the more surprising, then, that they did not collaborate. Following his success as Second Wrangler in 1865, Marshall was appointed a fellow of St. John's College, Cambridge, and in 1868, lecturer of political economy, a position he held until 1877. Jevons may have known of Marshall's appointment in 1868, but they did not correspond until January 1875 and did not meet until the end of that year.39 In his capacity as external examiner for the moral sciences tripos at Cambridge in 1874 and 1875, Jevons had been very impressed by the performance of Marshall's students (including Mary Paley). He duly noted their facility with graphical techniques in his first letter to Marshall, with the express hope that the latter might publish these findings. Jevons also sent Marshall a copy of his "Progress of the Mathematical Theory of Political Economy" (1874), most likely with the idea of gathering support. But Marshall, rather than enlisting, merely conceded to Jevons that "the substantive difference between us is less than I once supposed." From that point on, their few exchanges were cordial but reserved. In 1877, Jevons wrote a favorable letter of recommendation for Marshall when the latter applied for a position at Bristol, and in 1879, by way of thanking Jevons for mentioning two of his unpublished papers in the preface to the second edition of the Theory, Marshall acknowledged Jevons as "the chief author . . . of abstract quantitative reasoning."40 Neither economist, however, wrote to each other as openly or as frequently as to their mutual friends, Foxwell and Edgeworth. In fact, the content of the Jevons-Marshall correspondence and its paucity suggest that Marshall wished to remain as independent of Jevons as possible (the same is true of Marshall's

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correspondence with Leon Walras). As Maynard Keynes later remarked, Marshall was "extraordinarily reluctant to admit that he owed anything to Jevons."41 Marshall's first publication was a less-than-glowing review of Jevons's Theory of Political Economy for The Academy (April 1872). We know from the extant manuscripts that when Marshall reviewed the book he was hard at work on a mathematical treatment of economic theory. One might expect him to have welcomed a fellow practitioner of mathematical economics, as Jevons did on many occasions. But, though Marshall grants that Jevons's book is strewn with "suggestive remarks and careful analyses," he emphasizes the various inaccuracies and in places absurdities of Jevons's mathematical manipulations. In a retrospective note of the late 1890s, Marshall attributed his begrudging review to Jevons's less-than-adroit mathematics. Whereas Cournot and Johann Heinrich von Thunen "handled their mathematics gracefully," Jevons seemed "like David in Saul's armour." Moreover, the substance of Jevons's Theory failed to match its revolutionary claims. The difference between Jevons and David Ricardo, he maintains, is primarily one of form: "we continually meet old friends in new dresses." Jevons's repudiation of Ricardo and Mill was even more unforgivable: "He [Jevons] seemed perversely to twist his own doctrines so as to make them appear more inconsistent with Mill's and Ricardo's than they really were. But the genius which enabled Ricardo—it was not so with Mill—to tread his way safely through the most slippery paths of mathematical reasoning, . . . had made him one of my heroes; and my youthful loyalty to him boiled over when I read Jevons' Theory."42 It has often been suggested that Marshall's unsympathetic treatment of Jevons was due in part to the disappointment of having been anticipated. But while Marshall was not totally indifferent to issues of priority, it seems implausible, given his appreciation for Cournot and von Thunen, that Jevons would have irritated him so excessively on this account.43 Rather, Marshall initially took umbrage at Jevons's pretensions to complete novelty. He firmly believed that "the older economists had not recognized the mathematical conceptions that were latent in their own [work]," and that Ricardo was "feeling his way towards the distinction between marginal and total utility." Jevons, in Marshall's estimation, had done much "mischief by implying that the older economists were more at fault than they really were."44 Following the publication of Jevons's journal and selected correspondence in 1889, Marshall may have come to realize that Jevons had indeed broken new ground in the early 1860s without the least awareness of his Continental precursors. Marshall later expressed

Mathematical Economics Takes Hold · 129 some regret for his harsh treatment of Jevons: "I have since learnt to estimate him better . .. and I reverence him now as among the very greatest of economists."45 But it is his earlier, less favorable reactions that are most frequently recollected. While there can be no question that Jevons had developed the ideas presented in his 1871 work independently of Marshall, what of Marshall's claim to complete independence from Jevons? Marshall maintained that he had only needed Adam Smith, Ricardo, Mill, Cournot, and von Thunen in order to develop the central points of his main work, Principles of Economics (1890), and that all of this had taken place before 1872, when he reviewed Jevons's Theory. It has never been doubted that it would have been possible for a mind such as Marshall's to have used only those sources. After all, Jevons, though he failed to draw out the full implications of the new approach, had even fewer works to guide him. But scholars have suggested that Marshall's own account—written some thirty or more years after the fact and with very few supporting documents—is not wholly convincing.46 We know that Marshall read Jevons carefully in order to write his review, that he used Jevons's Theory for his lectures to women at Cambridge in 1873-1875 and perhaps on other occasions, and that he wrote the bulk of the mathematical appendix to the Principles between 1870 and 1874. Moreover, in one of his autobiographical fragments he states that his "apprenticeship to economic studies [occurred] between 1867 and 1875."47 It seems quite likely, then, that Marshall had learned something from Jevons. In a letter to Jevons (1879), Foxwell wrote that "Marshall always spoke in the highest terms of your book from its first appearance, which might prove either that he was prepared to agree with it, or that he had learnt a good deal from it, or both."48 Nevertheless, the onus is still upon those who question Marshall's assertions to point to portions of his work that reveal the direct influence of Jevons's work. In this regard, Marshall's silence during the period in question works very much in his favor. According to John K. Whitaker, one of the few documents we have to go by is Marshall's review of Jevons, which he claims "is consistent with the view that Marshall became prone in later life to exaggerate his precocity as a theorist." Whitaker has claimed, from his study of the Marshall manuscripts written between 1872 and 1879, that "even without the evidence of the explicit reference to Jevons, the deference to his ideas and terminology is sufficiently marked to indicate his influence. Marshall's stance is much more explicitly 'hedonic' here than in the remainder of his early work, a fact which helps confirm the suspicion

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that Jevons's utilitarian calculus had more influence on Marshall's development than he was willing to admit."49 Another historian, Richard S. Howey, has maintained that a comparison of Marshall's 1872 review of Jevons with his 1881 review of Edgeworth, which was both favorable and cognizant of Edgeworth's debt to Jevons, strongly suggests that Marshall had grown accustomed to the powers of marginal utility analysis in the years after 1871 rather than before. All of this reinforces Whitaker's position, namely, that it must have taken at least a decade for Marshall to refine and clarify what were, in 1871, "little more than half-formed thoughts."50 But neither historian has made a totally convincing case, one which would allow us to discredit completely Marshall's own claims to the contrary. Indeed, in a statement written around 1900, Marshall is adamant that he was not indebted to Jevons for the basic principles of marginal analysis: "on many aspects of economics I have learnt more from Jevons than from anyone else. But my obligations which I had to acknowledge in the Preface to my Principles were to Cournot and von Thunen and not to Jevons."51 Presumably, Marshall had learned much from Jevons's empirical studies, since he assigned them as reading for the new tripos in economics at Cambridge (1903). But can one detect further links, particularly in the form of mathematical techniques? According to Whitaker, "there is from the first an awkwardness and hesitancy about Marshall's efforts at mathematical economics that argues against him ever having breathed wholly freely on the pinnacles of abstraction. Both Jevons and Edgeworth seem to have dwelt more comfortably in the realm of abstract logic, despite their inferiority to Marshall in mathematical training." An overview of Marshall's work before the Principles suggests that he only gradually shed his predilections for geometry in favor of algebraic representations. In early correspondence with Edgeworth (1880), Marshall waxed enthusiastic about what "seems to be a very close agreement between us as to the promise of mathematics in the sciences that relate to man's actions," but also noted that "I intend never to use analysis when I can use geometry."52 Ten years later, his mathematical appendix to the Principles amply demonstrated the value of a judicious application of linear algebra and the differential and integral calculus. Quite possibly he came to appreciate the powers of algebraic formalizations in part by studying Jevons's work over the course of many years. It may never be possible, given the extant evidence, to resolve this controversy over the degree of Marshall's indebtedness to Jevons. But to a large extent, it does not matter. For if, as Howey and Whitaker maintain, Marshall was influenced by Jevons in the 1870s and 1880s

Mathematical Economics Takes Hold · 131 more than he admitted or even realized, then he can be viewed, together with Edgeworth, Wicksteed, Foxwell, Sidgwick, and Neville Keynes, as reinforcing the shift begun by Jevons. But even if we take Marshall completely at his word and accept his claim to have worked out his theoretical insights quite independently, then, by the same token, we may conclude that the primary catalyst for Marshall's discoveries was mathematics itself. He made this point on several occasions over a number of years, speaking, for example, of himself as a young man, "accustomed to think in Mathematics more readily than in English, and bewildered on his sudden entry into the strange land of economics." He wrote John Bates Clark in 1900: "My acquaintance with economics commenced with reading Mill, while I was still earning my living by teaching Mathematics at Cambridge; and translating his doctrines into differential equations as far as they would go; and, as a rule, rejecting those which would not go."53 This method led him to discard Mill's wage-fund theory and thereby begin to see that the determination of wages and profits is analogous to rents. The heuristic value of mathematics, he recognized, was due in part to the nature of the phenomena themselves: Our observations of nature, in the moral as in the physical world, relate not so much to aggregate quantities, as to increments of quantities, and that in particular the demand for a thing is a continuous function, of which the "marginal" increment is, in stable equilibrium, balanced against the corresponding increment of its cost of production. It is not easy to get a clear full view of continuity in this aspect without the aid either of mathematical symbols or of diagrams.54 In this regard, Jevons could not have wished for a better flag-bearer. Jevons and Marshall were also very much at one with regard to the epistemological standing of mathematical economics. Indeed, one of the most unambiguous compliments in Marshall's review of Jevons's Theory was directed at the author's "singularly good" arguments for applying mathematics to economics. Both agreed that mathematics per se would not impart greater certainty or exactitude to economic doctrines. It primarily served to clarify one's assumptions, to ensure that a determinate solution existed, but only empirical methods, particularly the use of statistics, could ascertain the degree of correspondence between theory and reality. This was precisely Marshall's point of view and, whether he liked it or not, he was beholden to Jevons (and to a lesser extent, Edgeworth) for having worked much of this out. Marshall, however, is more commonly known for his hostility toward mathematical economics. But this, I submit, is an artifact of his

132 · Chapter 7 peculiar regard for Jevons, of his delays in publishing, and of the wider publicity given to later correspondence. True, his Principles of Economics has virtually no algebraic formulas in the body of the text, due to his wish to appeal to the same general audience as had read Mill's Principles of Political Economy (1848). Nevertheless, his many footnotes display to the careful reader the value of a geometrical treatment, and he refers throughout to the relevant sections in the mathematical appendix. Marshall's peculiar claim that his diagrams were not mathematics, even though he knew full well that they had algebraic equivalents, allowed them to grace the same pages as his text. But although he insists, in the preface to the Principles, that "the argument in the text is never dependent on them [the diagrams]," he is hard-pressed to explicate a number of key points, such as derived demand or consumer's surplus, without them.55 Also puzzling is Marshall's assumption that his geometry was readily accessible to the layperson. We learn from Maynard Keynes that Marshall "always used [diagrams] freely in his lectures," but that this tended to have "choked off—more or less deliberately—the less serious students."56 Surely he must have realized, with attrition averaging close to fifty percent year after year, that his curves were just as inscrutable to the average Cambridge student as any algebraic formulas. Although Marshall elsewhere remarked that the mathematics should be kept "in the background," this does not square with either his written work or university lectures. As many have noted, the mathematics required for Marshall's analysis lie just beneath the surface. Even Marshall admitted to his readers that "there are many problems of pure theory, which no one who has once learnt to use diagrams will willingly handle in any other way."57 Moreover, one must explain why, if Marshall was so antagonistic toward mathematics, he took the trouble to include the appendixes, and spend the good part of several months insuring that they were printed correcdy? The answer is readily apparent in his first letter to Frederick Macmillan (1887). "Mathematics," he wrote, "cannot now be avoided in some branches of economics." And in the first edition of the Principles, he made a point of praising mathematics for having "compeUed a more careful analysis of all the leading conceptions of economics, and especially of demand."58 In subsequent editions he omitted this remark, though by the eighth edition (1920), he recognized a different virtue, the furthering of consensus within the profession: The new analysis is endeavouring gradually and tentatively to bring over into economics, as far as the widely different nature of the material will

Mathematical Economics Takes Hold · 133 allow, those methods of the science of small increments (commonly called the differential calculus) to which man owes direcdy or indirecdy the greater part of the control that he has obtained in recent times over physical nature.... There is a remarkable harmony and agreement on essentials among those who are working constructively by the new method.59 On many occasions, Marshall encouraged his students to study mathematics. In his words, "a training in mathematics is helpful by giving command over a marvellously terse and exact language for expressing clearly some general relations and some short processes of economic reasoning; which can indeed by expressed in ordinary language, but not with equal sharpness of outline."60 In an inaugural address to the Cambridge Economic Club (founded in 1897), Marshall acknowledged the powers of mathematics and told his audience to carry on: "this work of the new methods is far from finished: much remains for your generation to do." He also instructed his students to carry out translations into mathematics, much as he had done in his own early years, and attached considerable importance to his diagrammatic exercises in the classroom, as Maynard Keynes has emphasized. In 1902 he noted that all of his most valued students were "mathematical casuals," that is, candidates for the mathematical tripos at Cambridge.61 Certainly his most prominent pupils, Maynard Keynes, Arthur C. Pigou, and Arthur Bowley, were mathematically astute. All the same, one must take into account Marshall's frequent caveats on the misuse or hazards of mathematics, particularly in his later correspondence. According to Maynard Keynes, Marshall "arrived very early at the point of view that the barebones of economic theory are not worth much in themselves."62 There is reason to believe that he delayed publishing his major treatise (apart, of course, from long bouts of poor health) because it took a substantial amount of research to find the "flesh" for his "theoretical skeleton." It was only after having spent several years in search of the empirical foundations of his theoretical model that Marshall began to realize that economics is seldom as straightforward and simple as a mathematical treatment might lead one to believe. He then, in an oft-cited letter of 1906, expressed certain fears about the potential abuses of applied mathematics. To Arthur Bowley, he wrote that it would be better to "burn the mathematics" than to allow economic theory to stray away from real-life examples.63 Considerable weight has been placed on these remarks by subsequent economists and historians, and thus led many to discount his role in the rise of mathematical economics. John Maloney, in his oth-

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erwise excellent study of Marshall and the Cambridge school, draws this conclusion: "From Marshall's comments on 200-odd pupils, recorded between 1895 and 1902, it is clear that he regarded the mathematicians so highly not because he had any great opinion of mathematical economics as such—his correspondence with Bowley makes it clear that he did not—but because he saw their analytical sophistication as symptomatic of a generalised intelligence." But rarely is it mentioned that Marshall prefaced his remarks to Bowley with the following qualification: "I have very indistinct memories of what I used to think on the subject [of mathematical economics]. I never read mathematics now: in fact I have forgotten even how to integrate a good many things." In almost every other instance where Marshall criticizes mathematics, one finds remarks restoring its value within certain guidelines. As Pigou, his successor at Cambridge, once observed: "So far as he [Marshall] was against mathematical elaborations in economics, it was only because he feared that realism might suffer. Convince him that any particular line of mathematical attack would indirecdy help realism, and he would have been its enthusiastic friend."64 If Marshall was primarily bent on empirical veracity, his misgivings came too late. When he built the Cambridge school in the last few decades of the nineteenth century, we know from various sources that he was, as Foxwell put it, "thoroughly mathematical." However much Marshall might have tried in his later years to warn against the excessive use of mathematics, it appears to have been of litde or no avail. His own analyses of such concepts as consumer surplus and the price elasticity of demand demonstrate all too clearly his allegiance to the view that mathematics, as he once declared, is "the chief means of scientific progress."65 What can we conclude from this? Evidendy, Marshall did much to promote mathematical economics in the period 1870-1900. Whether or not he acknowledged a debt to Jevons, or even had contracted one at the substantive level, Marshall's lectures and writings reinforced the radical measures set out by Jevons.

CHAPTER EIGHT

Mathematical Hegemony The science [of economics] will have become completely mathematical by 2002-3, when . . . editors will be unable to read a non-mathematical article. —George Stigler Numerical precision is the soul of science. —Jevons, Principles of Science

seen throughout that Jevons's reformulation of the content and methods of economic theory was rooted in a rich appreciation of logic and natural philosophy. It is not too far-fetched to argue that the rise of mathematical economics was as indebted to his assimilation of the philosophical insights of Boole, De Morgan, Herschel, Whewell, and Mill, as to the specific assertions on value and distribution put forth by the classical economists. Certainly Jevons's program to mathematize economic theory owed more to his broader conception of scientific reasoning than to any single problem in the theory of production or consumption. Jevons had also swallowed a hefty dose of Hume's tonic. While he did not relinquish a belief in the uniformity of nature, he constantly reminded his readers that science could never grasp more than a small fraction of the goings-on in nature. Science, in its very methods, was a matter of hit and miss and all its claims were subject to revision at one time or another. It followed from this that economics, in emulating the natural sciences, did not have as far to go as might have been supposed. Although previous economists such as Mill had argued that the methods of economics and classical physics were one and the same, they also acknowledged how far short their discipline fell when it came to handling the various frictions said to impede the wheels of the marketplace. Jevons recognized that the physical scientists were not as successful at predicting and explaining the world as had hitherto been thought and that many of their most fundamental notions were highly vulnerable. Whereas Mill's comprehension of science was more in keeping with the late Enlightenment, Jevons was au courant with Darwin and Maxwell. His specific analogies were still mostly drawn from rational mechanics, but his appraisal of the status W E HAVE

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of the natural sciences was more in step with the general retreat from apodictic schools of thought. Jevons had also witnessed both the rise of pure mathematics and classical geometry's fall from grace. Mathematics was neither a handmaiden to physics nor a bearer of certainty. In principle, then, it was just as suited to the study of economics as to any other branch of science. Certainly, theoretical explorations might utilize mathematical tools prior to the process of verification. There was thus no necessary link between the degree of exactitude, of the fit between theory and reality, and the presence of mathematical reasoning. In short, Jevons's campaign for the mathematization of economic theory did not rest on a promise of greater certainty. Mathematics had already demonstrated its value in generating new insights. The equation for exchange and articulation of the principle of the final degree of utility made this plain. Quantitative reasoning also brought greater clarity and rigor, such as the specification of the dimensions of the key economic variables. The discernment of economic generalizations was much aided by noting the appropriate algebraic functions and the various limiting cases. But most importantly, one enlisted the calculus to handle phenomena that vary continuously and to determine points of maxima and minima. In addition to these technical refinements, Jevons advanced the idea that mathematics was to logic as quantity was to quality. Hence, the presence of quantitative relationships in economics called for a mathematical treatment. However self-evident or trivial this might sound to us, it must be remembered that Jevons was one of the first to ascribe logical foundations to mathematics, and to attempt to arrive at criteria that nonetheless distinguished the one branch of knowledge from the other. And while the thesis of logicism is subject to doubt on many fronts, it has a certain appeal within the domain of economic reasoning. Jevons may have overstepped the bounds of sound judgment when he asserted that "into almost all our acts of thought number enters," but certainly within the confines of the marketplace, the mind tends to follow numerical paths. If economics was to become a mature science, it was important to lay the proper foundations. By analogy to the physical sciences, economics must develop a system of mechanics that, like its counterpart in physics, would be mathematical and universal in scope. Jevons extended these mechanical analogies quite freely, perhaps too freely. But, once again, there is a certain cogency to his program. The mechanics of the marketplace, the motion of prices and quantities exchanged, are as amenable to the calculus as the motion of planets or falling bodies. Imposing the concept of static equilibrium or aggregat-

Mathematical Hegemony · 137 ing multi-bodied systems from pairs of independent bodies strains one's credulity in either the physical or the economic realm. Much the same could be said for the concepts of virtual velocity and the center of gravity and their economic counterparts. All of these had an element of contrivance that was no more warranted in one discourse than in the other. If anything weakened Jevons's new approach, it was his appeal to psychology. One and all might agree that economic value arises from mental deliberations, but to insist that this could be exhausted by the utility calculus put more weight on the slim shoulders of psychology than it could bear.1 Although Jevons wrote at a time in which the physiological dimensions of psychology had been much advanced, his vision of reducing prices first to psychological and ultimately to physiological states is still largely uncharted territory. Jevons was not content, as economists have since become, to speak only of revealed preferences or a logic of choice.2 In many respects, the Jevonian Revolution was as profound and farreaching as the ruptures we associate with Smith and Maynard Keynes. But, in contrast to these two other major shifts in the history of economics, one is hardpressed to find any direct implications for economic policy.3 Whereas with Smith we find an obvious program for extending free trade and fostering the practice of laissez faire, and with Maynard Keynes a direct plea for government intervention, with the British marginalists there seems to have been litde attempt if any to disturb the status quo. To some, of course, that was precisely the point. Marginalism was advanced in the 1870s and 1880s as a means of discrediting socialism and thus meshed perfecdy well with current government policies. But the case for this is just not there. Not only did Wicksteed find socialism compatible with marginalism (likewise the Fabians and some of the economic historians at Oxford), but the late classicists, Fawcett and Cairnes, were more vehemently opposed to socialism than any of the early marginalists. Furthermore, as Mark Blaug has stressed, nothing could be more antithetical to working-class interests than the classical wage-fund theorem.4 The first generation of marginalists wrote extensively on questions of social reform and, in the case of Jevons, Wicksteed, and Marshall, had taken up political economy in order to address the problems of poverty and thereby better the world. This hardly squares with the simplistic view of marginalism serving as a form of apologia. Moreover, if marginalism was adopted with hidden agenda, one must explain why so few of the reforms advocated by the early neoclassicists were informed by the new theoretical insights. By and

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large, policies were worked out case by case, primarily under the guidance of Mill's general precepts. One of the few exceptions is a brilliant paper by Edgeworth, published in 1897, that uses the principles of the utility calculus to advance a system of progressive taxation.5 And Edgeworth was the most politically conservative of the group. More extensive applications of the new tools had to wait until Arthur C. Pigou's Wealth and Welfare of 1912. Only then did economists use the techniques of marginalism to equate sound economic policy with social justice.6 But this came some forty years after Jevons issued his Theory. In comparison to the policy debates engendered by the great tomes of Smith and Maynard Keynes, the Jevonian Revolution appears to have had little immediate impact on economic policy. Perhaps we have been misled by searching for an analogous sequence of events. What did follow immediately from Jevons's proclamations was a shift in the line of demarcation between theoretical and applied economics. Mill, in his 1836 essay, had painstakingly distinguished the science of political economy, to wit a set of truths, from the art of legislation. A comparable distinction was found in the connection of the theory of projectiles to the art of gunnery. If economics is to be a science, Mill argued, it must be pursued apart from practical applications. Indeed, "political economy does not of itself instruct how to make a nation rich." National advancement could be informed by political economy, but a wider knowledge of history and human nature was essential. "The mere political economist, he who has studied no science but political economy, if he attempt to apply his science to practice, will fail." The reason for this was obvious to Mill. Economists were treating only one facet of human behavior. As he remarked in 1836: "not that any political economist was ever so absurd as to suppose that mankind are really thus constituted [to act solely from the desire for wealth]."7 The theory alone, based as it was on an incomplete model of human behavior, could only serve as a partial guide for political leaders., Jevons initiated a distinct recasting of the link between economic theory and policy. His remedy of subdividing the burgeoning discipline of economics transferred much of what fell under Mill's notion of the art of legislation to the science of economics, as a type of engineering.8 Mill's caveats about the need for additional expertise in the process of policymaking were soon forgotten. The shift to a more veritable homo economicus also facilitated the possibility of encompassing policy under the purview of science. Utility was the operative notion. As Marshall insisted in 1890, economists "deal with man as he is: not with an abstract 'economic' man; but a man of flesh and blood."9 The contrast with Mill's model of economic

Mathematical Hegemony · 139 man could not be starker. By grounding their theory in a model of man "as he is," the marginalists constructed a more solid bridge between theory and policy, one which could readily transport the apparatus of economics into the houses of Parliament. Of course, this did not happen overnight. The widespread employment of economists in the government is primarily a post-World War II phenomenon.10 But Jevons's and Marshall's conception of economic engineering, as opposed to Mill's notion of an art of political economy, constituted a major step in this direction. The rise of mathematical economics in the latter part of the nineteenth century coincided with a number of profession-enhancing activities. In 1891, the British Economic Association (BEA) and the Economic Journal were founded, and in 1895 the London School of Economics. The healthy state of the discipline was such that methodological and political pluralism was officially tolerated. Indeed, the detachment from political controversies by the founders of the BEA, in marked contrast to their American counterparts, suggests once again that the debris stirred up by Jevons had setded into place.11 There is litde question that mathematics had the consequence, only pardy grasped at the time, of serving as a barrier to entry into the profession. By the turn of the century, if not before, most contributions to pure economic theory called for some mathematical analysis. Simply to survive Marshall's courses at Cambridge one had to be adept at geometry. And although his Principles was intended for the layperson, we have seen that it was far more accessible to the mathematically trained mind. Still, it strains the record to maintain that geometry or the calculus were adopted in order to exclude the amateur.12 There were much deeper currents at work that would probably have resulted in the mathematization of economics even if Jevons had not lived or written as he did. Many, including Marshall, had discerned a mathematical character in Ricardo's Principles and, while no one picked up immediately on Malthus's remark about the extremal properties of economic analysis, it was only a matter of time before economists perceived that questions of efficiency lend themselves to thinking in terms of most and least and thus the calculus. Indeed, the point of no return may have been reached as early as the mid-1880s. Of course, it was not until the mid-twentieth century that virtually all economists were required to learn the calculus. But much the same could be said about physics in Victorian Britain, where the mathematics covered on the regular degree program did not suffice to advance its theoretical frontiers. In both cases, there was a considerable lag between innovations at the level of mathematical theory and instruction in the requisite skills in the university curriculum.

140 · Chapter 8

No other economist at the time shared Jevons's conviction in the almost Pythagorean essence of the economic world. His expertise in logic and natural philosophy was also unmatched among economists in his day. As a result, Jevons imparted far more momentum to the shift to mathematical economics than any other economist, Cournot and Walras notwithstanding. Edgeworth, Wicksteed, and Marshall were all swept along on the tracks he first laid. Sociological considerations aside, there is something in the nature of the subject matter of economics that lends itself to the language of mathematics. The early neoclassicists recognized this, and used mathematics when appropriate. Since then, mathematical economics has grown to the point of overtaking the purely literary format.13 Whether contemporary theorists are as judicious in their use of mathematics, or so impartial to the professional rewards of mathematical wizardry, is a matter others will have to assess.

Notes

CHAPTER ONE MATHEMATICAL PURSUITS

1. See Theocharis, Early Developments in Mathematical Economics, for a discussion of approximately thirty contributors to mathematical economics before 1838. Say had the case of Condorcet and Nicolas-Frangois Canard in mind when he discredited moving in the direction of mathematics. Comte, despite his own education in mathematics, argued categorically against its applicability to the social sciences. 2. Cournot's Recherches sur ks principes mathernatiques de la thuorie des richesses (1838) was ignored until the 1870s, despite sustained efforts on his part to gain recognition and despite lucid analyses of the law of demand and imperfect competition. Malthus is renowned for having noted that "many of the results in political economy . . . have some resemblance to the problems de maximis et minimis." See N. B. de Marchi and R. P. Sturges, "Malthus and Ricardo's InductMst Critics: Four Letters to William Whewell," Economica (1973): 379-93. Mill had studied the differential and integral calculus at the University of Montpellier. Other examples of economists versed in mathematics were Jules Dupuit, William F. Lloyd, Mountifort Longfield, Henry Fawcett, and Jacob Waley. 3. The main locus for these debates is a special issue of History of Political Economy (1972), reprinted as Black, Coats, and Goodwin, eds., The Marginal Revolution in Economics. On scientific revolutions, see Bronfenbrenner, "The 'Structure of Revolutions' in Economic Thought," and Coats, "Is There a 'Structure of Scientific Revolutions' in Economics?" On pre-1870 economists, see Bowley, "The Predecessors of Jevons—The Revolution that Wasn't," and Stigler, "The Adoption of the Marginal Utility Theory." 4. See Schumpeter, History, p. 952; Jaffe\ "Jevons, Menger, and Walras De-Homogenized"; and Blaug, "Was There a Marginal Revolution?" 5. Hutchison, On Revolutions, p. 58. A Jevonian Revolution is also explicidy acknowledged by Dobb, Theories of Value. 6. See Erich Streissler, "To What Extent Was the Austrian School Marginalist?" History of Political Economy 4 (1972): 426-41. 7. See Herv6 Dumez, L'Economiste, la Science et Ie Pouvoir: Le cas Walras (Paris: Presses Universitaires de France, 1985). 8. Even a recent attempt to argue that our current physics need not make a commitment to the realism of numbers does not deny this claim. See Hartry Field, Science with Numbers (Princeton: Princeton University Press, 1980). 9. See, for example, Axel Leijonhufvud, "Life among the Econ," Western Economic Journal 11 (1973): 327-37, and A. S. Blinder, "The Economics of Brushing Teeth," Journal of Political Economy 82 (1974): 887-91.

142 · Notes to Pages 7-10 10. See Joel Franklin, "Mathematical Methods in Economics," American Mathematical Monthly 90 (April 1983): 229-44. 11. Thomas S. Kuhn, "Mathematical versus Experimental Traditions in the Development of Physical Science," in The Essential Tension (Chicago: University of Chicago Press, 1977). 12. See Senior, Political Economy, p. 6, and James R. McCulloch, Principles of Political Economy (London, 1825), p. 15. In the Encyclopaedia Metropolitan, Senior's article on political economy was part of the first division of pure sciences. 13. In his Preliminary Discourse on the Study of Natural Philosophy, Herschel used examples from political economy to illustrate more general points about science. See pp. 12-13. Whewell wrote several papers on economics and helped to secure its place in the Cambridge curriculum. 14. See Silvan S. Schweber, "Darwin and the Political Economists: Divergence of Character," Journal of the History of Biology 13 (1980): 195-289, and Porter, Rise of Statistical Thinking. Lyell also utilized concepts drawn from the social sciences, including political economy. See Martin J. S. Rudwick, "Transposed Concepts from the Human Sciences in the Early Work of Charles Lyell," in L. J. Jordanova and Roy S. Porter, eds., Images of the Earth: Essays in the History of the Environmental Sciences (Chalfont St. Giles, Buckshire: British Society for the History of Science, 1979). 15. See Robert Bud and Gerrylynn K. Roberts, Science versus Practice: Chemistry in Victorian Britain (Manchester: Manchester University Press, 1984). Much the same story holds for geology. See Salim Rashid, "Political Economy and Geology in the Early Nineteenth Century: Similarities and Contrasts," History of Political Economy 13 (1981): 726-44. It is worth recollecting that the three founding fathers of these sciences, Adam Smith, Joseph Black, and James Hutton, were close friends at Glasgow. 16. The Senior quote is from Hutchison, On Revolutions, p. 58. Bagehot, "The Postulates of Political Economy," p. 1. 17. Hutchison, "Decline and Fall of English Classical Political Economy," pp. 450-51. 18. Jevons, Theory, p. xliv. 19. See Edwin W. Eckard, Economics of W. S. Jevons (Washington, D.C.: American Council on Public Affairs, 1940). 20. Rosamond Konekamp, "Biographical Introduction" to Jevons, Papers and Correspondence, 1:1-51, and Keynes, "William Stanley Jevons, 18351882," in Essays in Biography. Black has written frequently for commemorative issues of the Manchester School (1962, 1972, 1982). 21. I had the good fortune to help locate about one hundred letters that are now part of the Seton-Jevons Papers at Seton Hall University. See Black, "Supplementary Note," and McNiven, "Hand-List of the Jevons Archives." A virtually complete list of Jevons's published writings has been compiled by Takutoshi Inoue, "Bibliographical List of W. Stanley Jevons' Writings," Journal of Economics ofKwansei Gakuin University 39 (January 1986): 1-33. 22. He described the work to Macmillan as "the most important book that

Notes to Pages 10-15 · 143 I can ever hope to produce." See Jevons, Papers and Correspondence, 3:250. Some of the nine books counted above are collections of essays published posthumously. See Appendix Four of Jevons, Theory, for a complete list of his books. 23. Whitaker, "Mill's Methodology," p. 1033. In the Principles there are only a few references to the social sciences. See pp. xxvii, 334, 501, 759-61. 24. According to one scholar, "the most striking instance of the impact of Jevons' results in scientific method on his economics is to be found in the introduction to his paper on "The Solar Period and the Price of Corn' (1875)." MacLennan, "Jevons's Philosophy of Science," pp. 62-63. Wolfe Mays has examined Jevons's logic in more detail, but only suggested the possibility of intersections with his economics. See Mays, "Jevons's Conception of Scientific Method." 25. On Jevons's contributions in these areas, see the articles by David Laidler, Stephen Stigler, and Terence Hutchison in a commemorative issue of the Manchester School 50 (December 1982). CHAPTER TWO JEVONS'S LIFE

1. The name Jevons is Welsh in origin, as our protagonist was the first to document. He undertook this research in connection with an unsuccessful attempt to win the Celtic Literature Prize at University College in 1861. See Jevons, Papers and Correspondence, l:3n.3. 2. See George Chandler, William Roscoe of Liverpool (London: B.T. Batsford, Ltd., 1953). 3. See Howard Channon, Portrait of Liverpool (London: Robert Hale & Co., 1976), pp. 75-76. 4. The principal of the Mechanics Institute, William B. Hodgson, was later to become Professor of Political Economy at Edinburgh. Shortly after Hodgson died, in 1880, Jevons wrote: "his teaching made a great impression upon me and I have never forgotten it." In 1876, Jevons was awarded an LL.D. from the University of Edinburgh (Hodgson claimed he had no hand in the award). See Jevons, Papers and Correspondence, 4:17; 5:107. 5. Professor Black has made this point in "W. S. Jevons and the Foundation of Modern Economics," p. 370. Harry helped Jevons to get the positions in Australia and at Owens College, to publish some of his first scientific papers, and to be elected to the Royal Society. 6. This included trigonometry, conic sections, analytical geometry, probability theory, and the differential and integral calculus. On Jevons's regard for De Morgan, see Black, "Jevons, Bentham, and De Morgan." 7. Jevons, Papers and Correspondence, 1:185. 8. Jevons made extensive visits to the gold fields, at which he took some of the first photographs of that region. He published his findings in the Memoirs of the Literary and Philosophical Society of Manchester. Weekly summaries of his weather reporting were published in the Sydney Empire and various articles on meteorological and geological topics were published in the Sydney

144 · Notes to Pages 15-18 Magazine of Science and Arts and the London Philosophical Magazine. During his last year in Sydney, Jevons published a long essay on the Australian climate for the 1859 Almanac. This meteorological work, which utilized various statistical techniques, was later viewed by professional meteorologists as quite ingenious given the limited range of data available at the time. See John La Nauze, "Jevons in Sydney," pp. 28-30. In 1861, Jevons sent some of his studies on clouds to John Herschel, who responded with praise. See Jevons, Papers and Correspondence, 2:432-33. 9. The first work is at the Mitchell Library, Sydney; the other is purportedly missing, though there are essays on music in the Jevons Archives, including one that, not unexpectedly, explores applications of mathematics. 10. Jevons, Papers and Correspondence, 2:280. 11. Ibid., 2:159, 244. Jevons was once counseled by his brother Herbert that "it is a mistake for you to compare yourself to [Harry Roscoe]." Jevons's envy, however, was hard to contain. In 1862 he confessed: "If I had even a small percentage of HER's ways, I should by this time have been soaring up in the regions of fame ease & fortune, instead of labouring away in obscurity & without a halfpenny of profit." Ibid., 2:424, 452. 12. Ibid., 2:334. 13. Ibid., 2:296, 326. Jevons also wrote, "I judge of a religion by the abstract morality it contains, and by its effects on the character, condition & happiness of men." Ibid., 2:296. He rejected natural theology because of the pervasiveness of evil and claimed that "pure science is necessarily atheistical." See his letter to his brother Tom, Seton-Jevons Papers 14/1, September 10, 1858. Jevons eventually regained his faith. In the last year of his life he set out to write a sequel to the Bridgewater Treatises, & series of texts that explicitly sought to demonstrate God's design in nature. 14. Jevons, Papers and Correspondence, 2:322, 321. 15. Jevons, for example, had been elected to the Philosophical Society (Australia) in June 1856. By the time of his deparature, the editors of the Sydney Magazine of Science and Art wrote of Jevons, "We much regret to learn that Australia is about to lose this labourious and unassuming yet most promising natural philosopher. We fear it will be long ere we shall find another observer so industrious, so talented, and so modest." Quoted in La Nauze, "Jevons in Sydney," p. 38. 16. Jevons, Papers and Correspondence, 1:185; 2:293, 307. 17. Ibid., 2:355, 66, 291. In June 1857, he wrote to Henrietta: "I have never had the courage to open the many mathematical books I have with me, but what do you think I would do if I had the opportunity over again, attend College & De Morgans [sic] Mathematical lectures. The utility of Mathematics is one of the most incomprehensible things about it, but though I was never bright or successful in his class, in spite of working hard, I feel the greatest benefit from it." 18. Jevons, Theory, p. xviii. Bostaph and Shieh, "Lardner's Railway Economy," maintain that Lardner's more empirical bent did not have as direct an

Notes to Pages 18-21 · 145 impact on Jevons's more theoretical treatment of prices as has often been assumed. 19. Jevons, Papers and Correspondence, 2:321. 20. Jevons, tetters and Journal, pp. 118-19. 21. Jevons, Papers and Correspondence, 1:133; 2:361-62. 22. Ibid., 7:118-19; 2:362, 321. 23. Ibid., 1:184. 24. Ibid., 2:406. His studies included Greek, Latin, French, German, Greek History, Roman History, British History, Animal Physiology, Logic, Moral Philosophy, and Natural Philosophy. Jevons had already passed the examinations in Chemistry, Mathematics, and Experimental Philosophy in 1852. 25. On the University College Examination for 1859-1860, Waley asked: "What are the conditions as to the tenure and cultivation of land which must exist in order that the Ricardian theory of rent may be applied?" and "Give instances of social arrangements, now or formerly existing, under which the Ricardian theory of rent is inapplicable." University College, London, Calendar for the Session 1859-60 (London: Walton and Maberly), p. 163. A similar question can be found on the Joseph Hume Examination in Political Economy for 1863-1864. In the 1820s, Richard Jones had launched an attack on the Ricardian theory for its pretensions at universality. 26. Jevons, Papers and Correspondence, 2:415-16. Jevons never did ask Waley for a letter of recommendation. 27. Jevons wrote articles on the "Balance," "Clouds," "Gold Assay," "Hydrometer," "Hygrometer," "Thermometer," and "Volumenometer" for a dictionary edited by Henry Watts. He also published two articles on Kirchhoff s work on the spectrum, for the London Quarterly Review (April 1862) and the Philosophical Magazine (July 1862), as well as an article on "Light and Sunlight" for the National Review (July 1861). The British Association paper was entided "On the Deficiency of Rain in an Elevated Rain-gauge as caused by Wind," Philosophical Magazine (1861). 28. Jevons, Papers and Correspondence, 1:185. 29. Quoted in Black, "Jevons, Bentham, and De Morgan," pp. 123-24. 30. Martineau had previously served as a Unitarian minister in Liverpool. In 1859, Jevons's older sister, Lucy, had broken her engagement to Martineau's eldest son, Russell. Jevons wrote: "We shall perhaps have to live rather quiedy in London [in] order that Lucy may not meet the Martineaus too much." Jevons, Papers and Correspondence, 2:392. These remarks are contained in a letter to his brother Tom. Seton-Jevons Papers 2/14, January 4, 1861. 31. Jevons, Papers and Correspondence, 1:179, 186; 3:52. The full tide is Pure Logic, or the Logic of Quality apart from Quantity, with Remarks on Boole's System and the Relation of Logic and Mathematics. The work was reprinted as Pure Logic and Other Minor Works (London: Macmillan, 1890). De Morgan offered the most encouragement: "I find great difficulty in forming an opinion. I admit the ingenuity of your views, and the truth of the bulk of them. The questions of difficulty are there between you and George Boole."

146 · Notes to Pages 22-25 Letter from De Morgan to Jevons, September 15, 1863. The Jevons Archives (JA6/2/114). 32. Jevons, Papers and Correspondence, 7:120. 33. The first view is argued by La Nauze, "The Conception of Jevons' Utility Theory." White, "Jevons in Australia," has suggested that Jevons had formulated the basic premises of his Theory before he returned to England. With such little evidence to go by, however, neither case is persuasive. 34. Jevons, Papers and Correspondence, 2:410-11. 35. Ibid., 2:422. 36. See Hutchison, Cm Revolutions and Progress, pp. 86-87, for the claim about Bentham. For a recent statement of Jevons's purported solution to the paradox, see Robert Cooter and Peter Rappoport, "Were the Ordinalists Wrong about Welfare Economics?" Journal of Economic Literature 22 (June 1984): 507-30, especially 510. On Jevons's acknowledgment of the problem, see his Theory, p. 79. The anomaly, clearly formulated by both Smith and Ricardo, derived from the recognition that certain goods, such as water, that have the greatest use-value command the lowest or even no exchange value; diamonds, on the other hand, have little use-value but high exchange value. The formal solution lies in the realization that only incremental, not total, utility is appropriate for exchange value; the final degree of utility for water is low relative to that for diamonds in normal market situations. Incidentally, there is an interesting parallel to the purported influence of the Michelson-Morley experiment on Einstein's theory of special relativity. In both cases, what appears in hindsight to have been the key anomaly of the previous paradigm figured only indirecdy in the conceptual genesis of the leading instigator of the revolution. 37. Jevons, Papers and Correspondence, 1:179. 38. Ibid., 2:452; 1:188. A brief abstract was listed in the Transactions of the British Association and a title appeared in the Times. On the publication history of this paper, see Howey, Marginal Utility School, pp. 17-20. 39. Jevons tried several publishers in Paternoster Row before finding a map publisher who was willing to undertake the venture, provided he printed them at his own expense (about twenty-five pounds). Eventually the diagrams did sell, almost to the point of retrieving the initial expense. See Jevons, Papers and Correspondence, 1:180-81; 3:33. 40. Ibid., 3:6. Henry Enfield Roscoe was Professor of Chemistry at Owens College, one of the few colleges at the time that would employ nonconformists. Owens College became the Victoria University of Manchester in 1879. 41. Cairnes was Mill's heir apparent, and Fawcett, also a Millian, had recendy been appointed Professor of Political Economy at Cambridge. For their responses to the book, see Jevons, Papers and Correspondence, 3:16-18, 4 3 45. Cairnes informed Jevons immediately that he had reached similar conclusions in several papers of 1859-1860, but Jevons pleaded ignorance of them. On his Ricardo Scholarship Examination, however, there was a question that referred specifically to Cairnes's studies of the effects of the recent gold discoveries in Australia and California. Although it would have been possible to

Notes to Pages 25-27 · 147 have answered the question without having read Cairnes, it is odd that Jevons had not remembered the reference when it came time to write the book a couple of years later, particularly given his firsthand experience with the Australian case. See University College, London, Calendar for the Session 186162, pp. 90-91. 42. Jevons, Papers and Correspondence, 1:192. Professor Black has included an informative note on the reception of the pamphlet. According to Maynard Keynes, "it is scarcely an exaggeration to say that he [Jevons] made as much progress in this brief pamphlet as has been made by all succeeding authors put together." Keynes, Essays, p. 269. 43. Jevons, Papers and Correspondence, 3:42. To his brother he wrote: "I am also about undertaking the Subject of the exhaustion of Coal in England which I believe is a very serious matter. A good publication on the subject would draw a good deal of attention. I am convinced that it is necessary for the present at any rate to write on popular subjects." Ibid., 3:52. 44. Ibid., 3:103. Jevons also wrote to his sister: "excuse my being a litde too full of myself at present. It is hard even for me to feel the full meaning of such sudden and complete success. If I had worked ten or twenty years longer, I might have been glad to have got the result I already have got." Ibid., 3:97. 45. When Jevons took the post in London, he accepted a meager annual salary of seventy pounds. But, either through marriage or royalties, he seems to have lived very comfortably. In 1875, he turned down an offer of a professorship at the University of Edinburgh with an income of six hundred pounds. 46. Jevons, Papers and Correspondence, 1:204, 208-9. This led him to write another short book, The Substitution of Similars, the True Principle of Reasoning, Derived from a Modification of Aristotle's Dictum (London: Macmillan and Co., 1869), which seems to have fared no better than the first. 47. See Konekamp, "Harriet Ann Jevons." 48. Jevons actually believed that he might not live to finish the work, though he was only in his mid-thirties. Numerous entries in his private papers attest to a mild case of hypochondria, an affliction that often beset Victorian intellectuals. See Jevons, ^ters and Journal, p. 257. 49. See Jevons, Papers and Correspondence, 3:166-78, and Jenkin, "The Graphical Representation of the Laws of Supply and Demand," in his Papers. 50. On the details and subsequent revisions of this hypothesis, see Mirowski, "Macroeconomic Instability and the 'Natural' Processes in Early Neoclassical Economics." 51. The first edition, some 1,250 copies, sold in about two years. A second edition was published in 1877 and, by 1878, Jevons wrote enthusiastically to his brother Tom, "the Principles is soon to be in the third edition, and is adopted as a text-book at two or more universities." Although the third edition never materialized, the book was used in courses at Oxford and Dublin. Jevons, Papers and Correspondence, 4:286, 291. 52. The talk was on "The Progress of the Mathematical Theory of Political Economy" and was published in the Transactions of the Manchester Statis-

148 · Notes to Pages 27-32 tical Society (1874-75), pp. 1-19, the Journal of the Royal Statistical Society 35 (1874): 478-88, and has been reprinted in Jevons, Payers and Correspondence, 7:75-85. 53. Ibid., 4:50, 104. 54. Ibid., 4:272, 278; 5:24. Walras reached much the same conclusion. See ibid., 4:64-65. 55. Jevons published the list in several prominent European journals, including the Journal des Economistes. See Jevons, Papers and Correspondence, 4:262-63. 56. See Black, "Jevons, Marginalism and Manchester," pp. 6-8. 57. Jevons, Principles, p. 467. 58. According to Ashton, Economic and Social Investigations, p. 92, Jevons was "the authentic voice of the Manchester school." His cousin Harry informed him that he was at the top or very near the top of the ballot for that year's membership to the Royal Society; Jevons, Papers and Correspondence, 3:249. 59. The closest Jevons came to questioning the status quo is the following remark: "The laws of property are a purely human institution, and are just so far defensible as they conduce to the good of society; yet we maintain them to the extent of wasting and misusing no inconsiderable fraction of the land and wealth of the country." Jevons, "Economic Policy," in Smyth, Essays, pp. 2829. 60. See Hutchison, "The Politics and Philosophy in Jevons's Political Economy," and Stigler, "The Politics of Political Economists," in his Essays. 61. Jevons, Papers and Correspondence, 1:191; 5:144. 62. Jevons was a good swimmer but the waters in Devon happened to be much colder that year, and the shock most likely induced a heart attack. Although he had plans for several books, it is unclear if he would have been able to complete them. According to his wife, "his health was an increasing anxiety and his mind so encumbered with all he wanted to do that he seemed to feel life grow more and more hard to him every year." Harriet Jevons to Tom, August 19, 1882, Seton-Jevons Papers 15/9. CHAPTER THREE THE MATHEMATICAL THEORY OF ECONOMICS

1. Jevons, Theory, pp. xlii; xviii. References throughout are to the fifth edition, unless otherwise noted. 2. Jevons, Theory, pp. xxv, 148-52. On his acquaintance with Whewell, see Jevons Archives, JA6/6/159. Whewell's papers were published in the Cambridge Philosophical Transactions (1829, 1831, 1850). See James L. Cochrane, "The First Mathematical Model," History of Political Economy 2 (1970): 419-31; and James P. Henderson, "William Whewell's Mathematical Statements of Price Flexibility, Demand Elasticity, and the Giffen Paradox," Manchester School 41 (1973): 329-42. 3. Jevons, Theory, pp. xxvii-xxviii. Even his qualified praise for Macleod vexed Foxwell. See Jevons, Papers and Correspondence, 5:76-77. Jevons de-

Notes to Pages 32-37 · 149 fended himself as follows: "I have said the most civil things I can of his [Macleod's] books, and I see no need to dwell upon his errors because they are not likely to do any harm." Ibid., 5:79. 4. Tozer was a Fellow at Cambridge. The papers (1838) have been reprinted; see David A. Collard, ed., Mathematical Investigation of the Effect of Machinery (New York: Augustus M. Kelly, 1968). 5. Jevons, Theory, p. xxvi. Jevons had a penchant for promoting neglected figures, such as James Anderson and Richard CantiUon. His praise for Hutcheson, at Bentham's expense, may be another instance of this. Interestingly, Jevons was more laudatory of Bentham in the second edition of his Theory than in the first. Compare the same passage, pp. 11-12 of the 1871 edition, with p. 10 of the 1879 version of the Theory. 6. See Jevons, Papers and Correspondence, 3:168n. For a more detailed assessment of the exchange between Jevons and Jenkin, see Michael V. White, "Why Are There No Supply and Demand Curves in Jevons?" 7. Schumpeter, History, p. 826. 8. See Porter, Rise of Statistical Thinking. This distinction was also drawn by Neville Keynes. See Keynes, Scope and Method, pp. 342—43. 9. See Becher, "William Whewell and Cambridge Mathematics," and Joan L. Richards, Mathematical Visions: The Pursuit of Geometry in Victorian England (San Diego: Academic Press, 1989). 10. Jevons, Theory, pp. 29-30. The other dimensions to pleasure and pain identified by Bentham were certainty, propinquity, fecundity, purity, and extent. 11. See ibid., pp. 14, 26-27. This position is reasserted in his Principles, p. 12. "A revolver is a means of attaining both good and evil, but we have only to consider whether it is wanted, and, if wanted, how it may be obtained with the least cost of labour." 12. Jevons, Theory, pp. 12-13. Jevons may have decided to limit his comparisons to two contiguous feelings because of the problems raised in assessing the role the memory plays in the retention of sensations. These problems were addressed by both Hume and Hartley, whom he had read in some detail. 13. Ibid., pp. 30-31. In the Principles of Science, Jevons claims that "time is the magnitude which seems to be capable of the most exact estimation," though he also recognizes that there is no absolute measure of time. See Jevons, Principles of Science, pp. 303, 309. 14. "A definition of time must not be looked for; . . . Time is undoubtedly one of those primary notions which can only be defined physically, or by observation of phenomena which proceed in time." Ibid., pp. 307-8. 15. Jevons, Theory, p. 32. 16. Ibid, pp. 37-39. 17. We know that he admired the psychological texts of Bain and Spencer, but not of Mill. According to Blaug, Jevons was also acquainted with Theodor Fechner's seminal work in psycho-physiology, but I can find no confirming evidence of this. Blaug, Economic Theory, p. 349. 18. Jevons, Theory, p. 45. Jevons tends to move back and forth indiscrimi-

150

· Notes to Pages 37-40

nately between the state of consumption and mere possession. Yet, if all eco­ nomic actions are said to stem from considerations of pleasure and pain, it would seem more appropriate to regard consumption as the state in which to ascertain the final degree of utility. See ibid., pp. 51, 138. To the best of my knowledge, the extant theory of value (the theory of revealed preferences) has yet to address the distinction. 19. Ibid., p. 48. As Stigler first noted, Jevons has committed the error of extending his analysis of the individual mind to a competitive group. See Stig­ ler, Production and Distribution Theories, p. 17n4. Be this as it may, the as­ sumption of the divisibility and homogeneity of commodities is one that econ­ omists have yet to setde satisfactorily. 20. Jevons, Theory, p. 51. 21. Ibid., pp. 52-53. Jevons draws the curve as downward sloping through­ out. But his actual definition of the principle allows for an initial horizontal section. He also wisely refrains from drawing the curve so as to intersect the axes. As others have noted, Jevons's final degree of utility is not precisely the same as the modern concept of marginal utility, although operationally they serve the same analytical function (and will be used interchangeably in this work). Alfred Marshall was the first to note the distinction between the amount of utility acquired from an additional increment of a good, and the marginal utility, the change in utility at the margin, at the final increment of the good just acquired. "If u be the total utility of an amount of χ of a com­ modity to a given person at a given time, then marginal utility is measured by (du/dx)dx, while duldx measures the marginal degree of utility." Marshall, Principles, p. 690. 22. See Jevons, Theory, pp. 58-61. 23. "The general result is that commodity, if consumed by a perfectly wise being, must be consumed with a maximum production of utility." Ibid., pp. 59-60, 71. 24. Jevons, "Utilitarianism," in Pure Logic, p. 290. 25. Jevons's racism is quite pronounced on this point. "If the moral, not to speak of the physical characteristics of the lower animals, are so distinct, why should there not be moral and mental differences among ourselves, descend­ ing, as we obviously do, from different stocks with different physical charac­ teristics?" Ibid., p. 292. Already at the age of twenty-one, Jevons regarded benevolence and sympathy as driven by hedonistic motives. See Jevons, Pa­ pers and Correspondence, 1:133. 26. Jevons, Theory, pp. 86-87. 27. Ibid., pp. 89-90. 28. Ibid., pp. 85-86. Jevons later learned that Cournot had preceded him in defining an economic market. Stigler has suggested that Jevons's "merging of the concepts of competition and the market was unfortunate, for each de­ served a full and separate treatment." Stigler, "Perfect Competition, Histori­ cally Contemplated," in Essays, p. 245. 29. Jevons, Theory, p. 91. He regarded it as a self-evident principle. 30. "This law of indifference, in fact, is but another name for the principle

Notes to Pages 40-44 · 151 of competition which underlies the whole mechanism of society." Jevons, Principles of Economics, p. 60. There is a little more to the story. Jevons only introduced the name in the second edition and subsequendy maintained that the law of indifference is a direct analogue to the principle of the substitution of similars. 31. Jevons, Theory, pp. 94, 100. Jevons admits in the preface to the first edition of the Theory (p. viii) that this principle is the "cardinal difficulty of the whole theory." As Alfred Marshall remarked in his review of the Theory, "some amusement has been derived from the absurd result which is obtained by integrating the [above] equation." Jevons, Papers and Correspondence, 7:145. 32. Jevons, Theory, p. 95. 33. Ibid., pp. 133, 119. 34. Ibid., p. 102. Jevons wrote to George Darwin: "I wish you would consider the mathematical nature of the equations. I have a standing difference with my friend Barker, who says they are (or at any rate ought to be) differential equations demanding integration, whereas I hold that, though deduced by the use of differentials, they are simple algebraic equations." Jevons, Papers and Correspondence, 4:87. By the second edition, Jevons acknowledged his limited command of the techniques. A correspondent, Harald Westergaard, had "pointed out that a litde manipulation of the symbols, in accordance with the simple rules of the differential calculus, would often give results which I labouriously argued out. The whole question is one of maxima and minima, the mathematical conditions of which are familiar to mathematicians." Jevons, Theory, p. xiii. 35. Jevons, Theory, pp. 140, 141, 145. 36. Blaug, Economic Theory, p. 328. 37. Bostaph and Shieh, "Jevons's Demand Curve." 38. Jevons, Principles of Economics, pp. 57-58. Jevons, Theory, p. xxxvi. For this reason, he dismissed Gossen's geometric illustrations as "misplaced ingenuity." 39. Jevons, Theory, p. 168. This definition purposely excludes activities such as games and sports that are undertaken solely for the pleasure of the moment. In his later work, he admits that it is not possible to arrive at "a single definition [to] solve the perplexities of a complicated subject [like labor]." Jevons, Principles of Economics, p. 71. 40. See Jevons, Principles of Economics, pp. 85-89, and his Lectures on Political Economy, reprinted in Jevons, Papers and Correspondence, 6:52, for explicit denunciations of this classical distinction. 41. Jevons, Theory, pp. 168-69, 173. The solution offered is misguided. The relevant components are not the lengths qm and dm, but the slopes of the two curves. 42. Ibid., p. 177. Jevons later notes that if dl is measured by the same scale as du, it is "intrinsically negative." But he does not bother to adjust his equation to meet this condition. See ibid., p. 185. 43. Ibid., p. 186.

152 · Notes to Pages 45-49 44. Ibid., p. 187. 45. Ibid., pp. 165-66. He stated this view in his 1862 essay as well. "I assume, as obviously true, that the abilities of men are infinitely varied, whether by nature or by education, so that both the same person may vary in his power of producing different objects, and any two persons may vary in respect of the same object. This, indeed, is in direct opposition to the erroneous simplification of the science effected by Ricardo, when he assumed that all labourers have a certain uniform power." Ibid., pp. 307-8. 46. See ibid., pp. 204-9. Jevons cites Charles Babbage, On the Economy of Machinery and Manufactures; S. Haughton, Principles of Animal Mechanics; and W. S. Jevons, "On the Natural Laws of Muscular Exertion," Nature (30 June and 18 August 1870). 47. Sandra Peart, however, has recently argued that Jevons's theory of long-run economic growth upheld the Malthusian account of reproductive rates as a function of the real wage. See Sandra Peart, "The Population Mechanism in W. S. Jevons's Applied Economics." 48. See Jevons, Theory, pp. 179-83. 49. Ibid., pp. 163, 165. 50. Ibid., pp. 165, 167. 51. Ibid., p. xiv. The same proposal is made in his later work: "labour forms a subject of almost equal importance with utility; it constitutes the main mass of disutility, and it is really the algebraic sum of utility and disutility which we should endeavour to maximise." Jevons, Principles of Economics, p. 71. 52. Pigou, Memorials, p. 100. Black reiterates this view: "there is no doubt that Jevons did present his work in such a way as to stress the break he was trying to make with classical doctrine." Black, "Introduction," p. 30. 53. Jevons, Theory, p. vi. In the preface to the second edition, this isomorphism is underscored: "we are forced . .. to admit that rates of wages are governed by the same formal laws as rents." Ibid., p. xlvii. 54. Ibid., pp. 215-16. He has one simplification, however. He prefers to hold capital constant and vary the increments of labor, claiming that it is a very different problem to consider rent as the result of a fixed quantity of labor "assisted by different quantities of capital." 55. Ibid., pp. xlix, 218,1. 56. As Stigler has shown, Jevons mistakenly conflates this instantaneous rate of interest with the annual rate. See Stigler, Production and Distribution Theories, pp. 26-29. 57. Jevons, Theory, pp. 314, 270, 273. Again, "Capitalists can never exact from labourers the whole advantages which their capital confers—they can exact only a rate determined by the advantage of the last increment." Ibid., p. 257. 58. Jevons, Papers and Correspondence, 6:38. 59. Black, "Introduction," pp. 27-28. 60. Jevons, Theory, p. 164. 61. See Jevons, Principles of Economics, pp. 126-33, and Jevons, Papers and Correspondence, 6:48-54.

Notes to Pages 49-55 · 153 62. Jevons, Principles of Economics, pp. 131-32. 63. Jevons, Theory, p. 295. 64. Ibid., p. 243. Jevons, Principles of Economics, p. 68. This insight on human capital only appears in his posthumously published "Fragment." See Jevons, Theory, pp. 300-302. 65. Jevons, Papers and Correspondence, 6:39. Jevons harks back to the Aristotelian notion of household management: "Housekeeping is an occupation involving wages, capital and interest, like any other business, except that the owner consumes the whole result." Jevons, Theory, p. 263. 66. Ibid., pp. 229, 270. In Australia, Jevons observed that "it was quite common for companies of miners to be formed who were thus the miners and yet found all their own capital." Jevons, Papers and Correspondence, 6:76. 67. Jevons, Theory, p. 262. 68. Ibid., p. 264. 69. Ibid., pp. 264, 312. 70. Ibid., p. 312; Jevons, Papers and Correspondence, 6: 52-53; Marshall, Principles, pp. 53-54. 71. Young, "Jevons's Theory of Political Economy," p. 230; Marshall, "Review of the Theory of Political Economy," reprinted in Jevons, Papers and Correspondence, 7:145. 72. Jevons, Theory, pp. vii, xiii-xiv. CHAPTER FOUR LOGIC AND SCIENTIFIC METHOD

1. The initial break with Aristotelian logic had begun a decade or two before, with the work of Richard Whately, George Bentham, Sir William Hamilton, and H. L. Mansel. 2. Jevons, Principles of Science, p. 399 (hereinafter cited as Principles). Sense impressions were granted absolute certainty: "If I see blue sky, I may be quite sure that I do experience the sensation of blueness. Whatever I do feel, I do feel beyond all doubt." Ibid., p. 236. A contemporary observer, George Croom Robertson (founding editor of the journal Mind ), wrote in his review of Jevons's Principles that "as a methodologist he [Jevons] has fairly outstripped predecessors as great as Herschel, Whewell, and Mill.... [But] Jevons does not equal either Whewell or Mill in philosophical grasp.. .. He appears least at ease when he touches on questions properly philosophical." See Robertson, "Jevons's Formal Logic," p. 207. Much the same could be said of Herschel. See Curt J. Ducasse, "Herschel's Methods of Experimental Inquiry," in Blake, Theories of Scientific Method, p. 181. 3. For a discussion of this debate, see Gerd Buchdahl, "Inductivist versus Deductivist Approaches in the Philosophy of Science as Illustrated by Some Controversies between Whewell and Mill," The Monist 55 (1971): 343-67; H. T. Walsh, "Whewell and Mill on Induction," Philosophy of Science 29 (1962): 279-84; E. W. Strong, "William Whewell and J. S. Mill: Their Controversy about Scientific Knowledge," Journal for the History of Ideas 16 (1955): 209-32; and C. J. Ducasse, "Whewell's Philosophy of Scientific Dis-

154 · Notes to Pages 56-59 covery," Philosophical Review 60 (1951): 56-69, 213-34. On the broader context, see Laudan, "Induction and Probability in the Nineteenth Century." 4. See Whitaker, "Mill's Methodology," p. 1034n. Mill had some acquaintance with the natural sciences, botany in particular, but nothing that could compare with Herschel, Whewell, or Jevons. 5. Jevons, Principles, p. xxix. 6. Ibid., p. xxvii. 7. Jevons, Theory, p. 16. Mill's position is expounded in his System of Logic. 8. Jevons, Principles, pp. xxvii-xxviii. 9. "Here, as throughout this work, I shall devote my attention to truths which I can exhibit in a clear and formal manner, believing that in the present condition of logical sciences, this course will lead to greater advantage than discussion upon the metaphysical questions which may underlie any part of the subject." Jevons, Principles, p. 14. 10. Hempel praised the quality of Jevons's analysis of the problems of classification. See Hempel, Aspects of Scientific Explanation, p. 147n. 11. Jevons, Principles, pp. 7-8. 12. Jevons asks in this passage: "Are not the Laws of Identity and Difference the prior conditions of all consciousness and all existence? Must they not hold true, alike of things material and immaterial? . . . I am inclined, in short, to regard them as true both 'in the nature of thought and things,' . . . and I hold that they belong to the common basis of all existence." Ibid., p. 8. 13. Ibid., p. 7. 14. Ibid., p. 9. Recall that Jevons arrived at this insight in 1866 and published the results in an 1869 essay, "The Substitution of Similars." But because it made no mark, it was not until the publication of the Principles in 1874 that scholars brought to Jevons's attention the fact that the idea had already been anticipated by German logicians, notably Leibniz, Reusch, and Beneke. In the introduction to the second edition of the Principles, Jevons acknowledges these precursors and added that the idea could be extracted from Aristotle; ibid., p. 21. When Jevons made his first "discoveries" in logic, he was immersed in the works of both Aristotle and Leibniz. But Jevons maintained, even in the second edition, that the idea of substitution had come from his studies of mathematics rather than from a reading of philosophy. Ibid., p. xvi. 15. Ibid., p. 163. Jevons recognized that a problem was raised by the use of inequalities in algebra, such as the statement "three is greater than two." He responds with the rather unconvincing argument that in order to manipulate an inequality, one still relies upon the concept of equality: If b > c, then b = c + q, where q is some positive integer. Thus "even when an inference seems to be obtained without the use of an equation, a disguised substitution is really performed by the mind, exactiy such as represented in the equations." Jevons, "The Substitution of Similars," in Pure Logic, pp. 95-96. 16. Jevons, Papers and Correspondence, 3:13. 17. Jevons, Pure Logic, p. 86.

Notes to Pages 59-61

·

155

18. Jevons, Principles, pp. 14-15. 19. Ibid., pp. 49, 22. Jevons believed that he had to fight against as much dogmatism in logic as in economics. Mill's criticism of the syllogism had, in Jevons's eyes, done little to advance the situation. His reforms were apparendy successful. George Croom Robertson wrote in his review of the Princi­ ples that "we may consider Mr. Jevons's doctrine as the best outcome of the modern revolt against the Aristotelian system, sure that nothing has been urged in opposition more strongly than he urges it." Robertson, "Jevons's For­ mal Logic," p. 208. 20. According to Nagel, Jevons's "text-books on elementary logic, if judged by the size of their sales, are among the most successful ever written." His Elementary Lessons in Logic underwent almost forty printings and his Primer in Logic was translated into Italian, Hindustani, and Polish. See Nagel, Intro­ duction, p. xlvii. 21. The main difference between the three systems lay in the representa­ tion of the quantifier. De Morgan used parentheses, indicating by the direc­ tion of the parenthesis whether or not the subject and/or predicate were quan­ tified. For example, "All X is Y" was written as X))Y and "Some X is Y" as X()Y. The latter statement was equivalent, in the double negative form, to X(..(Y. The system, as these examples demonstrate, was visually very confus­ ing and did not find many followers. Boole used additional symbols, ν and w, to represent quantification. As with Jevons, he did not use an additional sym­ bol to represent an unquantified term, i.e., "AD X" was simply X, as opposed to De Morgan's X). Boole gained something by using extra symbols; subtrac­ tion and division could be performed because the equation sign maintained a full equality on both sides. Jevons did not retain this and, as a result, ruled out the possibility of division. If one were to divide A = AB by A, to obtain 1 = B, it shortly became clear that one was led to absurd results. For this reason as well, Jevons did not adopt the symbol " 1 " to represent the universe of discourse, a concept that Boole used to much advantage. Jevons also used a new symbol for disjunction, ·|·, rather than the addition sign, + (though he used this in his earlier work, Pure Logic) because, in his view, the analogy was not sufncientfy strong. See Jevons, Principles, p. 68. For the more re­ stricted operations (essentially addition and multiplication), the Jevonian sys­ tem offered the simplest means for representing the quantified predicate or subject. See Lewis, Symbolic Logic, pp. 37-78, and Kneale and Kneale, De­ velopment of Logic, p. 422. 22. Jevons did not use the terms distribution and transitivity in his work. Also implicit in his system are the laws AO = 0 and Α·|·0 = 0, a point drawn to my attention by Professor Alisdair Urquhart of the University of Toronto. 23. Jevons, however, claims that they are analogous. See Jevons, Princi­ ples, pp. 161-62. Boole adopted AA = A but not A + A = A, primarily be­ cause the latter does not hold for the value 1. Jevons, however, maintains that these two laws are obeyed by all numbers and for this reason, retains them both in his system. The reason, as we will see, stems from his definition of a number, as a form of "logical discrimination." Although, according to Lewis,

156 · Notes to Pages 61-63 Jevons was right to insist, in opposition to Boole, that A + A = A holds in logic, his attempt to interpret this law numerically (he substitutes the disjunctive with the addition sign) is not very convincing. See Lewis, Symbolic Logic, p. 73. 24. In an earlier work, he accepted the orthodox view that "What is notnot-A is A." Jevons, Pure Logic, p. 30. And there is one reference, in passing, to this problem in his Principles, p. 45: "Thus 'Iron is solid' might be expressed as 'Iron is not not-solid'." But there the matter rests. 25. Jevons seems to have been confused over the ways in which his logic paralleled the operations of arithmetic. In his 1863 work, although he adopted the addition sign for the disjunctive, he also states categorically that "there are no such operations as addition and subtraction in pure logic. The operations of logic are the combination and separation of terms, or their meanings, corresponding to multiplication and division in mathematics." Jevons, Pure Logic, pp. 70-71. But in the Principles, where he objects to Boole's use of subtraction and division, he endorses notions of addition and multiplication. See Jevons, Principles, pp. 160-62. 26. This example is from Jevons, "The Substitution of Similars," in Pure Logic, pp. 112-13. 27. Jevons, Principles, pp. 95-96. 28. One of these is on permanent display at Oxford University. See Jevons, "On a Logical Abacus," presented to the Manchester Literary and Philosophical Society (April 1866); and Jevons, "On the Mechanical Performance of Logical Inference," Philosophical Transactions of the Royal Society (1870), pp. 497-518, reprinted in Pure Logic. Also see Mays and Henry, "Jevons and Logic"; Martin Gardner, Logic Machines, Diagrams and Boolean Algebra (New York: Dover Press, 1968), pp. 91-104; Herman Goldstine, The Computer from Pascal to von Neumann (Princeton: Princeton University Press, 1972). 29. Jevons, Pure Logic, p. 95. 30. Jevons, Pure Logic, p. 66. For an informal discussion of hypotheticals, see Jevons, Studies in Deductive Logic, pp. 137-48. Nagel claims that "Jevons does not even suggest the possibility of a calculus of relations," but Jevons did acknowledge this limitation. See Nagel, "Introduction," p. xlviii, and Jevons, Principles, pp. 22—23. At the time, De Morgan had made considerable progress on this branch of logic. 31. In both his notebooks and the Principles, Jevons discusses the possibility of using 0 and 1. See Mays and Henry, p. 488. But as early as 1863, Jevons remarked that "Boole spoils his system by introducing 1 and 0." Jevons, Papers and Correspondence, 3:13. On the spread of the Jevonian system, see Kneale and Kneale, Development of Logic, p. 422, and Encyclopedia of Philosophy, s.v. "Logic, History of (Modern Logic: The Boolean Period)," by P. L. Heath. 32. Jevons, Pure Logic, pp. 4, 7. 33. See Lewis, Symbolic Logic, p. 73. Jevons showed some hesitation on this point. See Principles, p. 48.

Notes to Pages 64-69 · 157 34. Jevons, Principles, p. 156. "Boole inverted the true order of proof when he proposed to infer logical truths by algebraic processes." Ibid., p. 113. 35. Ibid., p. 156. 36. Frege found Jevons's work suggestive, and superior to Mill's inductive approach, but also noted many remaining difficulties. See Gottlob Frege, The Foundations of Arithmetic: A Logico-Mathematical Enquiry into the Concept of Number, J. L. Austin, trans. (Evanston, 111.: Northwestern University Press, 1980), pp. 46-58. 37. Jevons, Principles, p. 155. Jevons also speculates "that many unexpected points of connection will in time be disclosed between the sciences of logic and mathematics." Ibid., p. 145. 38. See Robert F. McRae, "Introduction" to Mill, System of Logic, 7: xxxv. McRae argues that "since for Mill there is no such thing as deductive inference . . . it is not necessary that inductions or inferences in order to be sound be warranted [by deduction]. It is evidence from the particular facts alone, and not they together with a general warrant, which makes an induction or inference valid.... Contrary to a common misunderstanding there is no circle in Mill's account of 'the ground of induction.' " This is a just interpretation of Mill's intentions, but he did not succeed in making a sufficiently convincing case to win the approval of his contemporaries or successors. Mill's logic is regarded as an attempt to solve the "problem of induction," but not as a solution. 39. Jevons, Pure Logic, p. 122; Jevons, Principles, pp. 228, 595. 40. Jevons, Principles, pp. 149, 218. 41. An example is the following: given the combinations ABC, aBC, abC, and abc, the only premises consistent with these are A = AB and B = BC. To see this, expand A, using the law of duality, substitute for A and B and eliminate those with contradictions. Jevons provides ten problems with up to six terms per proposition. In the second edition of the Principles, he reports that readers sent him a number of correct answers. 42. Ibid., pp. 1, 190, 150. 43. Ibid, pp. 198, 2. 44. Of the over 700 pages of the Principles, less than ten address the problem of causation. See pp. 220-26 for the most extensive discussion. 45. Ibid., pp. 222, 226. 46. Ibid., pp. 149-50. 47. Ibid., pp. 197, 200. De Morgan and Boole were also responsible for reviving the study of probability theory and applying it to the problem of induction. Jevons adopted a number of De Morgan's conclusions, for example, that "the probability of the hypothesis will increase very rapidly with the number of accordances" during the initial testing period, but will then level off. See ibid., p. 516. 48. Ibid., p. 200. Jevons appears to have taken this dictum to heart. While happening upon a snake in his bedroom one night, he wrote: "this is the first snake of any size that any of us have met this summer, and in all probability I may go to bed every day of my life and not meet a second." Jevons, Papers

158 · Notes to Pages 69-73 and Correspondence, 2:208. Although the incident never occurred again, he was known to suffer from insomnia. 49. It is difficult to know precisely what Jevons meant by partial knowledge, but this is indeed what he claims: "Perfect knowledge alone can give certainty, and in nature perfect knowledge would be infinite knowledge, which is clearly beyond our capacities. We have, therefore, to content ourselves with partial knowledge—knowledge mingled with ignorance, producing doubt." Jevons, Principles, p. 197. 50. Nevertheless, Jevons insists that observed outcomes of random events could feed back onto the estimated values. See ibid., pp. 260-62. 51. Ibid., pp. 206-7, 198, 261. Jevons recognizes that this "long run" is really "an infinity of cases." 52. Ibid., p. 210. See John Maynard Keynes, A Treatise on Probability (London: Macmillan and Co., 1921), pp. 41-64, for a discussion and critique of the subject. Keynes refers to Jevons as a well-known exponent of the principle. Contemporary critics of Jevons were Boole, von Kries, and Venn. See John Venn, The Logic of Chance (London: Macmillan and Co., 1888), p. 136. The main objection centered on the claim that it was wrong to assume that every proposition must have some numerical value of probability. 53. Jevons, Principles, pp. 216-17. 54. Ibid., p. 217. 55. Ibid., p. 228. Jevons suggested that scientists are usually "assisted by the whole body of science previously accumulated" in their search for hypotheses. Ibid., p. 509. 56. Ibid., pp. 267, 243. According to Peter Medawar, "the account of scientific method which became recognized as the official alternative and rival to Mill's was not Whewell's but Stanley Jevons." His sources, however, are not identified. See Peter B. Medawar, The Art of the Soluble (London: Methuen and Co., Ltd., 1967), p. 149. 57. See Jevons, Principles, pp. 253-55. 58. The best discussion of Jevons on the inverse method is by Strong, "De Morgan, Boole, and Jevons." Also see Laudan, "Induction and Probability in the Nineteenth Century," and Edward H. Madden, "Jevons on Induction and Probability," in Blake, Theories of Scientific Method. Keynes, after Boole, laid the most decisive blows on the inverse method. 59. See Strong, "De Morgan, Boole, and Jevons," p. 204; and Jevons, Principles, p. 255. Although this in itself was sufficient to have undermined Mill's methods, it appears that Jevons was unaware of the implications of his position. His attack on Mill, though always restrained in public, came more from his revival of the Humean position on causation and prediction, and from his own emphasis on the role of deduction and the theory of probability in the scientific method. In a later essay, Jevons also took Mill to task on his empiricist interpretation of Eucbdean geometry, which, Jevons claims, Mill "sacrificed" to accord with his philosophical goals. See the Jevons Archives (JA6/5/ 44). 60. Jevons, Principles, pp. 507, 647, 509.

Notes to Pages 73-85 · 159 61. See ibid., pp. 525, 547-50. Jevons distinguishes four classes of scientific observations, of varying degrees of theoreticity. Indeed, one class is generated solely by theory, such as the measurement of the vapor density of gases. 62. Ibid., pp. 510, 238. 63. Ibid, pp. 516-17. 64. Ibid., pp. 558, 592, 623. 65. Jevons, Theory, p. 276. Jevons, Principles, p. 737. 66. Jevons, Principles, p. 400. 67. Ibid., pp. 573, 587. 68. Ibid., pp. 629, 631. 69. Ibid., pp. 270, 273, 526. 70. Ibid., pp. 154, 457. 71. Jevons, "Helmholtz on the Axioms of Geometry," Nature 4 (April-October 1871): 481-82. For some delightful polemics on the mental scrutiny of geometric propositions, see Jevons, "John Stuart Mill's Philosophy Tested," reprinted in Pure Logic. 72. Jevons, Principles, pp. 271, 457-58. 73. Ibid., pp. 471, 483, 486. 74. Ibid., p. 463. CHAPTER FIVE MARKETS AND MECHANICS

1. Jevons, Theory, p. 7. 2. Jevons, Principles, pp. 752-53. 3. Jevons, Theory, p. 276. 4. Neil de Marchi has done a convincing job of discrediting part of Jevons's accusations. Although he agrees that Mill's writings were used to the exclusion of others, he does not find Mill and his disciples in control of the academic community at the time. See de Marchi, "Noxious Influence of Authority: A Correction of Jevons's Charge." 5. Jevons, Theory, pp. 276-77. 6. Ibid., p. 6. This perspective ties in well with Jevons's great admiration for Archimedes, whom he declared to have been "the greatest of ancient philosophers." Jevons, Principles, p. 581. 7. Jevons, Theory, p. 5. In his 1862 paper, Jevons points out this distinction from the very start; see ibid., pp. 303-4. We will see in the next chapter that one of these prejudiced persons was J. S. Mill. 8. Jevons, Theory, p. 8. 9. Ibid., pp. 22, 10-11. 10. Ibid., p. 266. 11. Ibid., pp. 17, 19. 12. Jevons, Theory (1st ed.), p. 24. 13. Jevons, Theory, p. 18. 14. Jevons, Principles, pp. 309, 307. 15. Jevons, Theory, p. 21.

160 · Notes to Pages 85-91 16. Ibid., pp. xvi-xviii. 17. For a more detailed account that has withstood the test of time, see Marie Boas, "The Establishment of the Mechanical Philosophy," Osiris 10 (1952): 412-541. 18. He published some papers on his experiments on Brownian Motion (or "pedesis" as he called it). Einstein's analysis of this phenomenon removed virtually every doubt regarding the existence of atomic particles. Jevons had thus fortuitously placed himself in the mainstream. See Jevons, "On the Movement of Microscopic Particles Suspended in Liquid," Quarterly Journal of Science 8 (April 1878): 167-86, and Jevons, "On the So-Called Molecular Movements of Microscopic Particles," Proceedings of the Manchester Literary and Philosophical Society (1870). Also see Stephen G. Brush, "A History of Random Processes: Brownian Movement," Archives for History of Exact Science 5 (1968): 1-36. 19. Jevons, Principles, pp. 735-36 (my emphasis). At an earlier point in the same work, Jevons writes: "Ultimately, as we can hardly doubt, all phenomena will be recognised as so many manifestations of energy." Ibid., p. 323. But recall that he was in other places quite skeptical about how much chemistry could solve. 20. Ibid., p. 769. 21. Jevons, Theory, pp. 304, 13-15. 22. Ibid., pp. 11-12. 23. Jevons, Papers and Correspondence, 3:246-47. 24. Jevons, Theory, pp. 146-47. 25. Ibid., p. 147. As early as 1861, Jevons attempted to collect such data. He decided to examine the price and quantity movements of a few imported goods, such as sugar and tea, but found the project too difficult, primarily because of complications introduced by substitute and complementary commodities. See Jevons, Principies of Economics, pp. 146-48. 26. Jevons, Papers and Correspondence, 7:81. 27. Jevons, Theory, pp. 62, 65. 28. See Gavin Reid's paper, "Jevons's Treatment of Dimensionality," for a more thorough treatment of the subject. 29. Jevons, Theory, pp. 92-93. 30. Ibid., p. 92. The incremental nature of price movements is explained by the fact that "a change of price is always occasioned by the overbalancing of the inclinations of those who will or will not sell just about the point at which prices stand." Ibid., p. 110. 31. Ibid., p. 93. Note that he has severed all ties with an annual agrarian cycle, which figured so largely in the classical literature. 32. Ibid., pp. 93-94. 33. Ibid., pp. 94, 111, 109. 34. Jevons, Theory, pp. 106, 102. Jevons defines the principle as follows: "if any number of forces be in equilibrium at one or more points of a rigid body, and if this body receive an infinitely small displacement, the algebraic sum of the products of each force into its displacement is equal to zero." Ibid.,

Notes to Pages 91-94

·

161

p. 104. In mathematical terms, this is expressed as: Σ m,v,dvt - 0. The prin­ ciple is now known as the principle of virtual work. See Louis Lagrange, Μέchanique Analytique (Paris: 1788), p. 12. 35. Jevons's course on natural philosophy and astronomy, which he took from Richard Potter in 1852-1853, covered statics (including the center of gravity and the principle of virtual velocities), dynamics, hydrostatics and dy­ namics, acoustics, optics, electricity, and astronomy. See University College, London, Calendar for the Session 1852-53, p. 24. 36. Jevons, "The Future of Political Economy," p. 199. 37. Jevons, Theory, p. xiii. In the first edition, Jevons is somewhat more cautious. He points to a resemblance rather than to an exact analogy. "The Theory of Economy thus treated presents a close analogy to the science of Statical Mechanics, and the Laws of Exchange are found to resemble the Laws of Equilibrium of a lever as determined by the principle of virtual veloc­ ities." See ibid., p. vii. 38. Jevons, Papers and Correspondence, 4:101. 39. Jevons, Theory, p. xxxi. In an 1873 letter to W.H.B. Brewer, Jevons remarked that Cournot's Recherches is "a very able and mosdy sound work, though it hardly gets anywhere to the bottom of the matter." Jevons, Papers and Correspondence, 4:5. 40. Jevons, Principles, pp. 459, 475. 41. Jevons, Theory, pp. 114—17. 42. Jevons, Principles, p. 760. 43. Jevons, Theory, p. 89. 44. Jevons, Principles, p. 363. In the Theory, Jevons refers to the "single average individual," but he does not adopt Quetelet's phrase of "the average man." See Jevons, Theory, pp. 64, 90. Nor does he make a specific reference to Quetelet's Treatise, which contains the exact same analogy: "The average man, indeed, is in a nation what the centre of gravity is in a body." Lambert A. J. Quetelet, A Treatise on Man, trans. Solomon Diamond (Gainesville, Fl.: Scholars' Facsimiles and Reprints, 1969; reproduction of 1842 English trans­ lation), p. 96. But we know that Jevons had read Quetelet and had admired his work gready. In fact, he specifically adopts Quetelet's use of the word average to denote a fictitious mean in the same section of the Principles (The Method of Means) in which he discusses the concept of the center of gravity. Jevons no doubt had borrowed the idea from Quetelet and forgotten or failed to acknowledge the source. 45. Jevons, Principles, p. 363. 46. Jevons, Theory, pp. 90, 16. This analogy between the average man and the center of gravity first appeared in the second edition. 47. Jevons, "The Future of Political Economy," p. 198. 48. Jevons added "Dogs certainly have a strong though perhaps limited idea of property, as you will soon discover if you interfere between a dog and his bone." Ibid., p. 197. We know from a letter to his wife that the claim about animals was partly intended as a joke. See Jevons, Papers and Correspon­ dence, 4:182.

162 · Notes to Pages 95-98 49. See Jevons, "The Future of Political Economy," pp. 197, 201-6. Jevons comments on this renaming of the discipline in the preface to the second edition of the Theory, and makes a point of using economics consistently in place of political economy. He would have changed the tide of his book as well, but did not wish to cause any confusion. See Jevons, Theory, pp. xiv-xv. 50. Jevons, Principles, pp. 670-71. 51. Jevons, Theory, p. 163. 52. Jevons, Principles, pp. 681, 691, 674. 53. Jevons, Theory, p. 267. 54. Dobb, Theory, p. 176. 55. Schumpeter, History, p. 826. CHAPTER SIX RESPONSE TO THE THEORY

1. Jevons, Papers and Correspondence, 4:39. 2. Jevons's wife made a similar observation about an earlier period in his life. She claimed that her husband's overwork had made him "more inclined to exaggerate the chances against his appointment [to Owens College] than he would otherwise have been." She also noted that her husband's concentrated work on the Theory ruined his health. Jevons, Letters and Journal, pp. 157, 221. 3. Jevons, Papers and Correspondence, 2:415-16; 3:240. For an appraisal of Mill's work, see de Marchi, "Success of Mill's Principles." By 1936, some 7,000 copies of the Theory had been sold. This certainly exceeded Jevons's original estimation of a little more than a thousand (one edition). The book not only reached four editions, but there have been two subsequent reprints. See Keynes, Essays in Biography, p. 302. 4. The only review of Walras's work, by Charles LeTort, was a two-page notice in the journal Economiste Frangais, 31 October 1874, pp. 540-41, the French equivalent of the British business weekly, the Economist. See Howey, Marginal Utility School, pp. 61-69, 200,256. Walras's original memoir, "Principe d'une Theorie Mathematique de l'Echange," Journal des Economistes, ser. 3, 34 (1874): 5-21, was the paper he first sent to Jevons. See Jevons, Papers and Correspondence, 4:36-37. 5. Jevons received notices in the Manchester Daily Examiner and Times, November 15, 1871, p. 7, and the Manchester Guardian, November 22, 1871, p. 7, both of which gave him half a page of coverage. He also received reviews in the Glasgow Daily Herald, December 16,1871, p. 3, and the Evening Standard, December 1871. Both the Athenaeum (November 4, 1871), pp. 589-90, and the Westminster Review 97 (January 1872): 102-3, reviewed his Theory along with John Macdonnell's A Survey of Political Economy, giving each author a little over one column of print. Both reviews emphasized Jevons's stricdy mathematical treatment of economics and his claim that a science need not be exact to be mathematized. The Athenaeum described the Theory as "an admirable work" and the Westminster Review began by stating that Jevons had "done invaluable service" to the science of political economy. The

Notes to Pages 98-100 · 163 contributor to the British Quarterly Review 5 (January 1872): 243-44, not only endorsed Jevons's use of mathematics and analysis of exchange in terms of the theory of utility, but noted Mill's "tyranny" and impending downfall. 6. Newcomb wrote: "Just now, when the claims of political economy to a place among the sciences are hotly disputed, . . . it requires no little courage to come before the world expressing its truths by mathematical formulae. Holding that this mode of expression is now what political economy stands most in need of, we welcome every attempt to introduce it, as showing that there certainly are truths of this science which admit of exact mathematical expression and reasoning.... The theory of utility and exchange here described we regard as entirely correct in its results." Newcomb also made some criticisms, but the general tenor of the review was quite enthusiastic. North American Review 114 (April 1872): 435-^10. 7. These reviews have been reprinted in Jevons, Papers and Correspondence, 7:141-52. British periodicals that did not notice the publication of the Theory were the Contemporary Review, the Quarterly Review, and the magazines Blackwood's, Fraser's, and Macmillan's. 8. It has been suggested that George Wirgman Hemming wrote the review for Saturday Review (November 11, 1871), reprinted in Jevons, Papers and Correspondence, 7:152-57. 9. Kgou, Memorials, pp. 99-100. 10. Jevons, Papers and Correspondence, 2:453. It is appropriate that several of the reviews spoke of Jevons's "courage" in writing the book. 11. William Ashley, "A Survey of the Past History and Present Position of Political Economy" (1907), in Smyth, Essays, p. 232. 12. Some of the economists of the time that will be passed over are L. L. Price, James Bonar, G. J. Goschen, Edward C. Lunt, Charles F. Dunbar, David Syme, John Macdonnell, and John L. Shadwell. 13. See Black, "Jevons and Cairnes," for a lengthy discussion of their correspondence and defense of their position on the gold question in the Times. On Cairnes's theory on the value of gold and its conformity with his methodological prescriptions, see Hirsch, "Cairnes' Methodology." 14. Jevons wrote to Luigi Bodio in 1874: "This work [Theory] was very unfavourably received in this country, and almost the only English economist of importance who noticed it, namely Professor J. E. Cairnes, repudiated it altogether" Jevons, Papers and Correspondence, 4:80. 15. For a study of the state of economics in Britain at the time, see Checkland, "Economic Opinion in England as Jevons Found It," pp. 154,161. Mill's high regard for Cairnes is assessed in O'Brien, "Mill and Cairnes." 16. Jevons, Papers and Correspondence, 4:81. Shortly after the review appeared, Jevons wrote to Cairnes defending his position and apologizing in a very diplomatic manner for having "inadequately explained" his views in the Theory. Ibid., 3:246. 17. See ibid., 4:62, and J. E. Cairnes, Some Leading Principles of Political Economy (New York: Harper, 1878), p. 21. 18. See Jevons, Theory, pp. 276-77.

164 · Notes to Pages 100-103 19. Jevons, Papers and Correspondence, 7:150; 3:246. 20. Cairnes, Leading Principles, p. 21. 21. Jevons, Papers and Correspondence, 4:102. Jevons remarked, upon receiving a preliminary draft of Darwin's article, "that it can hardly be called, as it stands, a review of Cairnes' book, but rather a defence of mine." Ibid., 4:87. Johan D'Aulnis and R. O. Williams noted Cairnes's unjust criticisms, as did Cliffe Leslie; see ibid., 4:62, 132; 7:157-62. Darwin's review will be discussed in the next chapter. 22. Ibid., 7:84; 4:62. 23. In his review, Cairnes excused himself from discussing the matter. "Besides the doctrine of value, there is much in Mr. Jevons's volume that will attract the attention of economists. In particular his view as to the amenability of economic problems to mathematical treatment is well deserving of consideration. Into this question, however, I do not now enter." Ibid., 7:151. 24. Quoted in de Marchi, "Mill and Cairnes," p. 35On. 25. Jevons, Papers and Correspondence, 7:152. 26. In February 1875, Marshall wrote to Jevons with an endorsement of Darwin's attack on Cairnes. Ibid., 4:100. 27. Cairnes, Logical Method, p. v. 28. Ibid., p. 129. 29. In the preface to the second edition of the Theory, Jevons made this very point: "I contend that all economic writers must be mathematical so far as they are scientific at all, because they treat of economic quantities.... Even those who have most strongly and clearly protested against the recognition of their own method, continually betray in their language the quantitative character of their reasonings. What, for instance, can be more clearly mathematical in matter than the following quotation from Cairnes' chief work: [Leading Principles]." Jevons, Theory, p. xxi. 30. Jevons, Papers and Correspondence, 4:99. Either Jevons had forgotten that he had read the first edition or he had not told the whole truth when he wrote Cairnes to that effect in 1863, for the two editions, aside from the issue of mathematical economics, are not significandy different. Perhaps, upon skimming the second edition, Jevons only then realized that Cairnes's work could be put to good effect. Jevons made a point of acknowledging Cairnes's contribution in the 1879 edition of the Theory. 31. Jevons, Papers and Correspondence, 5:80. 32. Cairnes, Logical Method, pp. 34, 62, 83, 92, 93, 130. 33. "As regards origin and mode of development, therefore, the parallel between Political Economy and the physical sciences is complete." J. E. Cairnes, "M. Comte and Political Economy," in Essays in Political Economy (London: 1873). 34. These similarities were recognized by both Jevons and Cairnes. Jevons, Papers and Correspondence, 4:113-14. In 1875, Jevons set an exam question on Cairnes's views on method for the Cambridge moral sciences tripos, a sign, perhaps, of his general endorsement of the work; see ibid., 7:137; 3:19. 35. Black, "Jevons and Cairnes," p. 214.

Notes to Pages 103-105 · 165 36. Cairnes wrote in 1874, "I should be sorry if my dissent from Mr. Jevons on this point should convey an impression that I undervalue the work.... I am far from being insensible to the lucid statements of economic doctrine, and to the numerous original and suggestive remarks, with which the volume [the Theory] abounds." Cairnes, Leading Principles, p. 21n. To Jevons, Cairnes once wrote "there are few persons living for whose judgment on a question of logical method I have a higher esteem than for yours." Jevons, Papers and Correspondence, 4:113. 37. De Marchi, "Mill and Cairnes," pp. 343-47. Mill had studied the calculus for at least a year, including the works of Lacroix, Lagrange, Laplace, Whewell, and De Morgan. 38. De Marchi, "Success of Mill's Principles." Portions of the Principles have been recognized to be highly original. See Stigler, "The Nature and Role of Originality in Scientific Progress," Essays, pp. 1-15. 39. De Marchi, "Mill and Cairnes," p. 350. 40. See Mill, Essays on Economics and Society, 4:318. 41. De Marchi, "Mill and Cairnes," pp. 352, 349. De Marchi also suggests, quite plausibly, that Mill would not allow demand to be measured subjectively, that associationist psychology did not lend itself to the same treatment of sensations as Jevons's hedonic calculus, and that, on methodological grounds, the law of diminishing marginal utility could not explain the shape of the demand curve. 42. Mill, Later Letters, 17:1862. 43. In May 1872, Mill wrote to the editor of the Fortnightly Review, John Morley, "I am very desirous that the F. shd continue, & increase rather than diminish in importance." In the last years of his life, Mill corresponded frequendy with Cairnes. Mill was in London in January 1872 and, judging from his next letter to Cairnes, written in April 1872, the two had definitely met then. Ibid., pp. 1892, 1862. 44. I can find no occurrence, in the Mill correspondence, of additional disparaging remarks on Jevons. Perhaps Mill had not recorded such statements. Cliffe Leslie maintained, in a letter to Walras, that from his personal acquaintance with Mill he knew that Mill did not support the mathematical treatment of economics. But Leslie's other claim in the same letter, namely that Mill did not endorse the deductive method, suggests that he was not such a reliable witness. See JafK, Walras Correspondence, 1:395. 45. Mill wrote of Jevons in 1866, "Such of Mr. Jevons's writings as I am acquainted with give evidence of decided originality, much knowledge and mental vigour, and an unusual degree of precision of thought and investigation." Jevons, Papers and Correspondence, 3:119-20. Mill's address is reprinted in ibid., 7:119. 46. See Schabas, "Mill to Jevons." 47. Mill, System of Logic, 7:255 (hereinafter cited as Logic). 48. Ibid., pp. 477-78. This view was developed in a later work: "Applied mathematics is not the measurement of extension and number. It is the measurement by means of extension and number, of other quantities which ex-

166 · Notes to Pages 105-110 tension and number are marks of; and the ascertainment by means of quantities of all sorts, of those qualities of things which quantities are marks of." Mill, Hamilton's Phibsophy, 9:477 ff. 49. According to Mill, "what is commonly called mathematical certainty, therefore, which comprises the twofold conception of unconditional truth and perfect accuracy, is not an attribute of all mathematical truths, but of those only which relate to pure Number, as distinguished from Quantity. . . . The certainty usually ascribed to the conclusions of geometry, and even to those of mechanics, is nothing whatever but certainty of [deductive] inference." Mill, Logic, 7:259. 50. Mill, Hamilton's Philosophy, 9:481. Mill's remarks stem from the fact that Hamilton discredited altogether the importance of studying mathematics. 51. Mill, Logic, 8:845-46, 864. Mill made this point in order to justify the approximate nature of the "empirical laws of human nature." 52. Mill, Hamilton's Philosophy, 9:480-81. Also see his Logic, 8:621. 53. Comte, Positive Philosophy, 2:99. 54. Mill, Logic, 8:620. This section is only two pages long. Mill excuses his superficial treatment of the problem by "referring the reader who desires a more thorough acquaintance with the subject, to the first two volumes of M. Comte's systematic work." Elsewhere, Mill claimed: "I do not know that the logical value of mathematics has ever been more finely and discriminately appreciated than by M. Comte in his latest work [1856]." Mill, Hamilton's Philosophy, 9:472n. 55. MiU, Logic, 8:620. 56. Mill, Hamilton's Philosophy, 9:480. Jevons, Theory, p. 6, made the same point. 57. Jevons, Theory, p. 5. 58. See Mill, "On the Definition of Political Economy and on the Method of Investigation Proper to It," in Essays on Economics, 4:309-39; and Mill, Logic, 8:895-904. 59. Mill, Logic, 7:620. 60. Mill, Essays on Ethics, Religion and Society, 10:258. 61. Mill, Logic, 8:904. 62. Mill, Hamilton's Philosophy, 9:485. 63. Mill, Earlier Letters, 12:36. 64. Mill had dissented from Comte on the very crucial question of the right of political economy to exist as a separate discipline. Mill's famous recantation of the wage-fund theory also suggests a flexible mind. 65. Although Jevons complained to Walras in 1875 that it would be necessary to fight "the disciples of J. S. Mill, . . . [who are] bitterly opposed to any innovation upon his doctrines," he could cite only one example of such opposition (apart from Cairnes)—an anonymous article in the London Examiner (28 November 1874, pp. 1294-95), that was written in response to an 1874 address by Jevons. According to Professor Black, the article "combined unqualified praise of Mill with unqualified condemnation of Jevons." Perhaps

Notes to Pages 110-113 · 167 because the bias of the author was so pronounced, Jevons did not take the trouble to reply. See Jevons, Papers and Correspondence, 4:91-92, 104-5. 66. See Koot, "Origins of the English Historical School of Economics," and Coats, "Historist Reaction." 67. J. K. Ingram, "The Present Position and Prospects of Political Economy," Presidential Address to Section F of the British Association (1878), in Smyth, Essays, p. 65. 68. Ingram's address refers to Lowe and Cairnes as culprits but not to Jevons. On this issue, I differ with the judgment of Professor Black, who has claimed that "on the whole it seems correct to regard Jevons as assailed rather than supported by the writings of the 'English Historical School.' " See Black, "Jevons and the Economists of His Time," pp. 214-15. 69. See J. K. Ingram, History of Political Economy (Edinburgh: 1893), p. 232. 70. For Leslie on Mill, see Koot, "Origins," pp. 319-22. For Leslie on the wage-fund controversy, see ibid., p. 329ff. Leslie's other criticisms of Mill can be found in his essay "On the Philosophical Method of Political Economy" (1876), in T. E. Cliffe Leslie, Essays in Political Economy, 2nd ed. (London: 1888), pp. 171-80, and Ingram, "Prospects of Political Economy," p. 57. 71. Jevons, Papers and Correspondence, 7:77. Jevons, Theory, p. xlv. 72. There were also personal affections for one another. In the early 1880s, Jevons signed a memorial to Gladstone to request a pension for Leslie, who was very ill, but Leslie died before receiving the fund. Jevons wrote the obituary on Leslie for the Economist, 4 February 1882, p. 133. See Jevons, Papers and Correspondence, 5:169-71. 73. T. E. Cliffe Leslie, "Untitled Review of the Second Edition," in Jevons, Papers and Correspondence, 7:157-62. 74. Ingram, History, pp. 232-34. 75. This view is implicit in Leslie's 1879 review of the Theory, in which he suggests that it would have been better to treat Ricardo in mathematical terms and thereby render him all the more obscure: "We regret that so much of Mr. Jevons' own reasoning is put into a mathematical form. . . . On the other hand, we not only concede that a mathematical shape might have been given to a great part of Ricardo's system, but we regret that it ever received any other . . . since the basis is false, the more the superstructure is hidden the better, and we should be glad to see it obscured in every treatise in which it is put forward by a liberal use of the calculus." Jevons, Papers and Correspondence, 7:160. 76. Quoted in Kadish, Oxford Economists, p. 55. See also pp. 59, 67-70. 77. The belief that economics was a universal science was canonized by Lionel Robbins, An Essay on the Nature and Significance of Economic Science (1932). For some of the anthropological debates, see Edward LeClair and Harold Schneider, Economic Anthropology: Readings in Theory and Analysis (New York: Holt, Rinehart, & Winston, Inc., 1968). 78. See Coats, "Sociological Aspects," and Maloney, Marshall. According to

168 · Notes to Pages 113-119 Kadish, the Cambridge group formed a far more cohesive unit than the one at Oxford. Kadish, Oxford Economists, p. 153. 79. Jevons, Papers and Correspondence, 3:114. Bagehot was referring to Jevons's study on the value of gold. 80. Sidgwick, Principles, p. v. Sidgwick was a leading moral philosopher. 81. Foxwell also edited Jevons's papers on finance. See Jevons, Investigations, p. xxxiv. 82. Keynes, Scope and Method, p. 89. 83. Jevons, Papers and Correspondence, 5:78. 84. Walter Bagehot, Economic Studies, ed. R. H. Hutton (London: 1911), p. 20. 85. Herbert S. Foxwell, "The Economic Movement in England," Quarterly Journal of Economics 2 (1888): 90. 86. Jaffe, Walras Correspondence, 2:161. 87. Schumpeter claims that Keynes's book "setded most of these methodological issues in a spirit of judicial reasonableness to the satisfaction of the profession. For two decades this book held a well-earned position of authority." Schumpeter, History, p. 824. 88. Keynes, Scope and Method, p. 29. 89. Ibid., p. 199. 90. Henry Sidgwick, "The Scope and Method of Economic Science" (1885), in Smyth, Essays, p. 83. 91. Henry Sidgwick, "Economic Method," Fortnightly Review 31 (January 1879):310-11. 92. Foxwell wrote to Walras in 1886: "I went to see Dr. Sidgwick myself, in order to be quite sure about his relation to Jevons. He authorizes me to say that he is 'quite content to be described in general terms as a follower of Jevons.' So am I." Jaffe, Walras Correspondence, 2:160. 93. Bagehot, Economic Studies, pp. 29, 97. 94. Ibid., p. 17. 95. Foxwell, "Economic Movement in England," p. 88. 96. See, for example, the examinations in political economy in the University College Calendars for the years 1880 to 1900. 97. Jaffe, Walras Correspondence, 2:160. 98. We learn, from an 1883 letter to Walras, that Keynes had "long taken an interest in the mathematical theory of Political Economy." Ibid., 1:777. 99. Keynes, Scope and Method, pp. 237, 241-42. 100. Ibid., pp. 237, 251. 101. Bagehot, Economic Studies, p. 209. 102. See Keynes, Scope and Method, pp. 293-96. 103. Schumpeter, History, p. 953. CHAPTER SEVEN MATHEMATICAL ECONOMICS TAKES HOLD

1. Jevons, Papers and Correspondence, 4:99. In his correspondence to his Continental enthusiasts, Jevons most frequendy cited Darwin as his closest compatriot; see ibid., 4:63, 66, 102.

Notes to Pages 120-123 · 169 2. George Darwin, "The Theory of Exchange Value," Fortnightly Review 23 (1875): 243-53, p. 250. 3. Jevons, Papers and Correspondence, 4:81; and Darwin, "Theory of Exchange Value," p. 249. He also refers to "the mathematical form into which Mr. Jevons was unavoidably driven to express his views." Ibid., p. 252. 4. George Darwin, "Commodities versus Labour," Contemporary Review 22 (1873): 689-98. 5. See Hennings, "George Darwin, Jevons, and the Rate of Interest," p. 201, and Jevons, Papers and Correspondence, 4:63. 6. These sentiments are expressed in a letter to Walras, written in 1876. See Jaffe\ Walras Correspondence, 1:503. 7. Darwin was elected to the Royal Society in 1879, in part due to Jevons's support, and was eventually knighted in 1905. 8. On Wicksteed as a disciple, see Schumpeter, History, p. 832, and Stigler, Production and Distribution Theories, p. 38. Apparendy, an economist at the turn of the century, Clara Collet, informed J. M. Keynes that Jevons had met Wicksteed but the evidence is not conclusive. See Black, "Jevons and the Economists of His Time," p. 215. Wicksteed's only biographer states that there is no record of an acquaintance. See C. H. Herford, Philip Henry Wicksteed (London: J. M. Dent & Sons, 1931), p. 23n. 9. See J. H. Burrows, "Teaching of Economics 1876-1902," History of Economic Thought Newletter, 20: 8-14. 10. George Bernard Shaw, "The Jevonian Criticism of Marx," To-Day 3 (January 1885): 22-26; reprinted in Wicksteed, Common Sense, 2:724-30. Philip Henry Wicksteed, "The Jevonian Criticism of Marx: A Rejoinder," Today 3 (April 1885): 177-79; reprinted in Wicksteed, Common Sense, 2:73133. 11. Marxist historians are reputed to view the marginalist theory as "one of retreat and evasion." But neither Jevons nor Wicksteed adopted the new theory in order to defend the capitalist system. Rather, because of such debates as the one between Wicksteed and Shaw, the neoclassical theory came to be associated with those who were openly hostile to the Marxist position. See Mark Blaug, Economic Theory, p. 317; and Donald Winch, "Marginalism," p. 326. 12. Shaw, "Marx," p. 725. 13. Wicksteed, Common Sense, 2:808. 14. Both are reprinted in ibid., 2:734-54, 809-13. Wicksteed's first book, The Alphabet of Economic Science (London: 1888), is noted for its use of the word "marginal" throughout. His other major work, "An Essay on the Coordination of the Laws of Distribution" (London: 1894), has been reprinted by the London School of Economics. 15. Wicksteed, Common Sense, 2:811, 765. 16. Ibid., 2:771, 779, 769. 17. Stigler, Production and Distribution Theories, p. 108. 18. Edgeworth may have studied mathematics at Trinity College, Dublin, or at Oxford, or consulted W. K. Clifford, whom he befriended at the Savile Club. See John Creedy, "F. Y. Edgeworth's Mathematical Training," Journal of the Royal Statistical Society 146 (1983): 158-62.

170 · Notes to Pages 123-126 19. For a list of the many societies to which Edgeworth belonged, see Keynes, Essays in Biography, p. 223. 20. Schumpeter, History, p. 831. 21. Jevons, Papers and Correspondence, 5:201-2, 98, 145. Edgeworth had applied for the chair in philosophy at King's College, London (1880), and for the professorship in Logic, Mental and Moral Philosophy and Political Economy at University College, Liverpool (1881). For a brief account of the Edgeworth-Jevons relationship by their mutual friend James Sully, see Howey, Marginal Utility School, p. 99. 22. Creedy, Edgeworth, p. 51. 23. Edgeworth, Mathematical Psychics, pp. 9-15. Fechner and the other German psycho-physiologists, Weber and Helmholtz, were preoccupied with possible links between energy conservation and the inner workings of the mind. 24. Arthur Bowley, "Obituary of Francis Ysidro Edgeworth," Econometrica 2 (1934): 112—24. For a discussion of these and other analytical breakthroughs, see A. J. Nichol, "Edgeworth's Theory of Duopoly Price," Economic Journal 45 (March 1935): 51-66, and Donald A. Walker, "Edgeworth's Theory of Recontract," Economic Journal 83 (March 1973): 138-^19. 25. William Jaffe\ "Edgeworth's Contract Curve, Part I: A Propaedeutic Essay in Clarification," History of Political Economy 6 (Fall 1974): 343-59. 26. William Stanley Jevons, "Critical Notice of Mathematical Psychics," Mind, 6 (1881): 581. 27. Marshall also recognized many of the analytical achievements in the work, such as the contract curve, as well as Edgeworth's attempt to demonstrate the value of mathematics independently of the availability of exact data. The review has been reprinted in Whitaker, Writings of Alfred Marshall, 1: 265-68. 28. See Creedy, Edgeworth, Chs. 3-6. 29. Edgeworth, Mathematical Psychics, p. 138. Like Jevons, he found much in Marshall to support the cause. 30. Edgeworth, Papers, 2:275, 281, 287. 31. Ibid., 1:4-7, 6. Also see 2:274. 32. See, for example, H. Cunynghame, A Geometrical Political Economy (Oxford: Clarendon Press, 1904). 33. See Joseph Shield Nicholson, Political Economy as a Branch of Education (Edinburgh: David Douglas, 1881), pp. 20-21. 34. George Stigler, from a quantitative survey of leading American periodicals, has dated the rise of mathematical techniques from the 1920s, but by this, he meant the calculus or more elaborate methods. See Stigler, Essays, p. 48. 35. See the oft-cited G. F. Shove, "The Place of Marshall's Principles in the Development of Economic Theory," Economic Journal 52 (1942): 294-329. 36. On the issue of inconsistency, see H. M. Robertson, "Alfred Marshall's Aims and Methods Illustrated from His Treatment of Distribution," History of Political Economy 2 (1970): 1-65. For the next two accusations, see Howey,

Notes to Pages 127-128 · 171 Marginal Utility School, pp. 85-87n, and Schumpeter, History, p. 839. Maynard Keynes has perhaps the best account of Marshall's character flaws; see "Alfred Marshall," in Keynes, Essays in Biography, pp. 174-76. For Marshall's tendency "of being less than frank," see Whitaker, Writings of Alfred Marshall, 1:103; and for the last characteristic, see Maloney, Marshall, p. 65. 37. See Schumpeter, History, p. 830. The two pamphlets, available as London School of Economics Reprints, are entided "Pure Theory of Foreign Trade" and "Pure Theory of Domestic Values." Despite the immense popularity of The Economics of Industry, Marshall was in later years to deplore its existence and even went to the trouble in 1892 of reissuing, though as sole author, a quite different version with the same tide. Prior to 1879 Marshall had published only four essays, perhaps the most important of which was "On Mr. Mill's Theory of Value," Fortnightly Review, April 1876, reprinted in Pigou, Memorials, pp. 119-33. It defended Mill against the recent criticisms of Cairnes. On Mrs. Marshall's interesting career, see Keynes, "Mary Paley Marshall," in Essays in Biography, pp. 324-47. 38. See Jevons, Papers and Correspondence, 4:204-5. Jevons chose Foxwell as his substitute instead. 39. This seems to have been their only encounter, unless their paths crossed at British Association meetings. Jevons had attempted unsuccessfully to meet Marshall on his first visit to Cambridge. 40. Jevons, Papers and Correspondence, 4:100; 5:66. 41. Keynes, "William Stanley Jevons," in Essays in Biography, p. 287. From the extant correspondence, which consists of five letters in all, it is apparent that there must have been at least two additional letters. Of the three exchanges, in 1875, 1877, and 1879, not one of the letters exceeds a printed page. On Marshall's low regard for Walras, see Whitaker, Writings of Alfred Marshall, 1:103-7. 42. Alfred Marshall, "Mr. Jevons's Theory of Political Economy" (and undated comments on the review), reprinted in Pigou, Memorials, pp. 93-100, quoting from pp. 93, 99, 95, 99-100. Similar uncharitable comments appear in his later work; see Marshall, Principles, pp. 673-75. As far as Marshall was concerned, the fact that Ricardo did not use mathematics was of little consequence: "There is a class of economic problems which cannot be safely treated by anyone of less genius than Ricardo without the aid of some apparatus, either of mathematics or of diagrams"; ibid., p. 688. 43. In 1908, Marshall recollected the following: "My main position as to the theory of value and distribution was practically completed in the years 1867 to 1870, when I translated Mill's version of Ricardo's or Smith's doctrines into mathematics; and that, when Jevons's book appeared, I knew at once how far I agreed with him and how far I did not": Marshall to John Bates Clark, 24 March 1908, in Pigou, MemoriaL·, p. 416. In his early work, Marshall also shared with Jevons an appreciation for drawing analogies to classical mechanics, though he shifted over time toward a preference for biological motifs. See Gordon, "Alfred Marshall."

172 · Notes to Pages 128-131 44. Alfred Marshall, "Fragments," in Pigou, Memorials, p. 359, and Marshall, Principles, pp. 670; 76. 45. Marshall, "Jevons's Theory," in Pigou, Memorials, p. 99. Marshall also wrote: "there are few writers of modern times who have approached as near to the brilliant originality of Ricardo as Jevons has done": Marshall, Principles, p. 673. 46. See Whitaker, Writings of Alfred Marshall, 1:103, and Howey, Marginal Utility School, pp. 79-80. 47. Marshall to John Bates Clark, in Pigou, Memorials, pp. 416-17 (on the Principles of Economics), and Marshall, "Fragments," ibid., p. 358. Mary Paley Marshall maintained that Jevons's Principies of Economics was used in the Cambridge lectures, but Whitaker has noted, quite soundly, that she must have meant Jevons's Theory, since the other work was not published until 1905. See Whitaker, Writings of Alfred Marshall, 1:11. 48. Jevons, Papers and Correspondence, 5:78. Foxwell also mentions that Henry Sidgwick "thinks that Marshall's theory of Distribution is founded on your [Jevons's] theory of Final Utility." Although Foxwell was a colleague of Marshall's, he was also prone to playing the ambassador and may have embellished things in order to improve relations between Jevons and Marshall. 49. Whitaker, Writings of Alfred Marshall, 2:286. 50. See Howey, Marginal Utility School, pp. 79-80, and Whitaker, Writings of Alfred Marshall, 1:44. Although Marshall used Jevons's phrase "final utility" in his 1879 papers, he made a point, in his Principles, of switching to "marginal utility." In the preface to the first edition he maintained that he had known of the latter phrase all along from the work of von Thunen, that is, from 1869 or 1870. But it was pointed out to him that von Thunen never used the phrase in German, and Marshall corrected this error in the second edition. However, in a letter to J. B. Clark, dated 1900, Marshall forgot his error and restated his illegitimate debt to von Thunen, another instance, perhaps, of his refusal to acknowledge his links to Jevons. See Howey, Marginal Utility School, pp. 83-84, for this interesting detective work. 51. Marshall, "Undated Comments," in Pigou, Memorials, p. 100. Marshall also once claimed in a letter that "I make it a point of honour to acknowledge my obligations—whenever I contract them, and when they are not obvious," quoted by Whitaker, Writings of Alfred Marshall, 1:49. 52. Marshall to Edgeworth, 8 February 1880, quoted in Creedy, Edgeworth, p. 51, and Whitaker, Writings of Alfred Marshall, 1:5. Marshall may have been restricted by the more traditional Cambridge curriculum. See Becher, "William Whewell and Cambridge Mathematics." Keynes has suggested that an important source of inspiration was W. K. Clifford, who was friendly with Jevons, Edgeworth, and Marshall, but I have been unable to substantiate this point. Keynes, Essays in Biography, p. 159n. 53. Marshall, "Fragments," in Pigou, Memorials, p. 359, and Marshall to J. B. Clark, 2 July 1900, in ibid., p. 412. A perusal of Marshall's early papers suggests that he derived a number of insights by playing around with demand

Notes to Pages 131-133 · 173 and supply curves, rather than with the differential calculus. See, for example, Whitaker, Writings of Alfred Marshall, 1:117-64. 54. Marshall, Principles, pp. viii-ix (preface to first edition); see also Marshall to Clement Colson, 1907, printed in Econometrica 1 (1933): 221-22. 55. Marshall, Principles, p. ix. In a draft of a letter to his publisher, 1878, Marshall describes a potential "book on the method of diagrams distinct from but allied to the methods of analytical mathematics applied to economic theory including Foreign trade curves." Whitaker, Writings of Alfred Marshall, 1:63. Marshall may also have avoided mathematics out of fear of offending his mentor, Benjamin Jowett: See Marshall, "Reminiscences," in Pigou, Memorials, p. 66. According to Jowett, "Political Economy is human & concrete & should always be set forth in the best literary form: the language of symbols may be relegated to notes & appendices": Jowett to Marshall, 25 December 1884, quoted in Whitaker, Writings of Alfred Marshall, 1:28. 56. Keynes, Essays in Biography, pp. 16On, 164n, 196. E. A. Benians also mentions the high attrition in Marshall's course: see his "Reminiscences," in Pigou, Memorials, p. 78. 57. Marshall, Principles, p. ix. On the math beneath Marshall's analysis, see A. C. Pigou, "In Memoriam: Alfred Marshall," in Pigou, Memorials* p. 86; Keynes, Essays in Biography, pp. 159-60; and, most recently, Mirowski, "Physics and the 'Marginalist Revolution.'" For keeping math in the background, see Marshall to Walras, 19 September 1889, in Jaffe, Walras Correspondence, 2:355. 58. Marshall to F. Macmillan, 12 April 1887, as quoted in C. W. Guillebaud, "The Marshall-Macmillan Correspondence Over the Net Book System," Economic Journal 75 (1965): 519, and Marshall, Principles, p. 85. On Marshall's oversight of the printing see Keynes, Essays in Biography, p. 180. Marshall also wanted to keep the price of the book low, and helped Macmillan to experiment with a new pricing system: see Guillebaud, "Marshall-Macmillan Correspondence." Marshall again owns up to the need for mathematics in the Principles, p. 688, and in his and his wife's Economics of Industry a problem is passed over because "an exact treatment... requires the aid of mathematics": quoted in J. K. Whitaker, "The Marshallian System in 1881: Distribution and Growth," Economic Journal 84 (1974): 1-17. 59. Marshall, Principles, pp. xiv-xv. 60. Ibid., p. 644. 61. Quoting from Maloney, Marshall, p. 59; on the value of translations, see A. W. Coats, "Alfred Marshall and the Early Development of the London School of Economics: Some Unpublished Letters," Economica 36 (1967): 410-11. On the use of diagrams, see Keynes, Essays in Biography, pp. 190, 252. For the Cambridge address, see Alfred Marshall, "The Old Generation of Economists and the New," in Pigou, Memorials, p. 301. 62. Keynes, Essays in Biography, p. 170. This view may also explain his preference for diagrams, since he and others realized the possibility, in the not-too-distant future, of statistics, particularly the method of least squares, lending empirical credibility to the theoretical analysis of markets. On the de-

174 · Notes to Pages 133-139 velopment of econometrics see Mary S. Morgan, The History of Econometric Ideas (Cambridge: Cambridge University Press, 1990). 63. Marshall to A. Bowley, 27 February 1906, in Pigou, Memorials, p. 427. The passage in full is as follows: "But I know I had a growing feeling in the later years of my work at the subject that a good mathematical theorem dealing with economic hypotheses was very unlikely to be good economics: and I went more and more on the rules—(1) Use mathematics as a shorthand language, rather than as an engine of inquiry. (2) Keep to them till you have done. (3) Translate into English. (4) Then illustrate by examples that are important in real life. (5) Burn the mathematics. (6) If you can't succeed in 4, burn 3. This last I did often." 64. Maloney, Marshall, pp. 233-34. Pigou, quoting from Bladen, From Adam Smith to Maynard Keynes, p. 361. Jacob Viner maintained that Marshall's distrust of mathematical economics "was probably a lurking puritan suspicion of the morality of any highly pleasurable activity" and that he must have delighted in mathematical manipulations. Viner, "Marshall's Economics in Relation to the Man and His Times," American Economic Review 31 (1941): 231. 65. Jevons, Papers and Correspondence, 5:78, and Marshall, Principles, p. 700. CHAPTER EIGHT MATHEMATICAL HEGEMONY

1. This point is effectively made by Amartya K. Sen, "Rational Fools," in Frank Hahn and Martin Hollis, Philosophy and Economic Theory (Oxford: Oxford University Press, 1979). 2. See Robbins, "Place of Jevons," pp. 6-7. 3. This point is made by Stigler, "Adoption of the Marginal Utility Theory," p. 578, and by Black, Introduction, p. 37. 4. Blaug, Economic Theory, p. 317. 5. See "The Pure Theory of Taxation," in Edgeworth, Papers, 2:63-125. 6. See Maloney, Marshall, Chs. 8-10. 7. John Stuart Mill, "On the Definition of Political Economy; and on the Method of Investigation Proper to It," in Mill, Essays on Economics and Society, 4: 312-15, 331. 8. Jevons, Theory, p. xvii. 9. Marshall, Principles, p. 22. 10. A. W. Coats, "Introduction to Economists in Government," History of Political Economy 13 (1981): 341-64. 11. See Coats, "Origins of the Royal Economic Society," and Mary O. Furner, Advocacy and Objectivity (Lexington: University Press of Kentucky, 1975). 12. While an excellent study of the period, Maloney goes a little too far when he remarks that: "economic orthodoxy, accused on all sides in the 1870s and 80s of both theoretical inadequacy and social irrelevance, resolved this position not by a successful answer to these criticisms, but by capturing

Note to Page 140 · 175 a dominant position in which it could largely ignore its critics." Maloney, Marshall, p. 4. 13. Stigler found, even in 1964, that two-thirds of the articles in the leading American economics journals used some mathematics. See "Statistical Studies in the History of Economic Thought," in Stigler, Essays, pp. 47-48.

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Index

algebra, 33, 59, 6 1 , 104-5, 108, 154n.l5; applied to economics, 4, 23, 34-35, 38, 41, 47-48, 52, 130, 132 analogy, 59, 96; to mechanics, 84-85, 90-91, 93, 124, 135-36, 171n.43; to science in general, 67, 75, 80 Anderson, James, 149n.5 Arago, Frangois, 74 Archimedes, 94 Aristotle, 60, 82, 100, 153n.l, 154n.l4, 155n.l9 Ashley, William, 99, 112, 163n.ll Ashton, T. S., 28, 148n.58 Austrian School, 5, 141n.6 Babbage, Charles, 8, 46, 152n.46 Bacon, Francis, 54 Baconian induction, 71-72 Bagehot, Walter, 9, 100, 113-17, 142n.l6 Bain, Alexander, 21, 35, 58, 149n.l7 ballot-box of nature, 67, 69, 71, 73-74 Becher, Harvey, 149n.9, 172n.52 Benians, E. A., 173n.56 Bentham, George, 153n.l Bentham, Jeremy, 18, 2 1 , 23, 34, 36, 38, 54, 104, 110, 146n.36, 149nn. Berkeley, George, 54 Black, Joseph, 142n.l5 Black, R.D.C., 10, 28, 33, 49, 103, 141n.3, 142n.21, 143nn., 145n.29, 147n.42, 148n.56, 152n.52, 163n.l3, 164n.35, 169n.8, 174n.3 Bladen, Vincent, 174n.64 Blaug, Mark, 5, 42, 137, 141n.4, 149n.l7, 151n.36, 169n.ll, 174n.4 Blinder, A. S., 141n.9 Boas, Marie, 16On. 17 Bodlo, Luigi, 163n.l4 Bonar, James, 163n.l2 Boole, George, 6, 21, 54, 58, 60-65, 72, 75, 96, 104, 135, 155nn., 156n.25, 157n.47, 158n.52 Bostaph, Samuel, 43, 144n.l8, 151n.37 Bowley, Arthur, 126, 133-34, 170n.24, 174n.63

Bowley, Marian, 141n.3 Brewer, W.H.B., 161n.39 Bridgewater Treatises, 144n. 13 British Association for the Advancement of Science, 8, 20, 24, 28-29, 31-32, 99, 125, 145n.27, 146n.38, 171n.39 Bronfenbrenner, Martin, 141n.l3 Brush, Stephen G., 160n. 18 Buchdahl, Gerd, 153n.3 Bud, Robert, 142n.l5 Burrows, J. H., 169n.9 Cairnes, John Elliott, 25, 87, 98-105, 109-12, 115-16, 118, 119-22, 137, 146nn., 163-65nn., 167n.68 calculus: mathematical, 3, 20, 23, 33, 42, 44, 47, 52, 67, 80, 84, 91-92, 96, 121, 125, 130, 133, 136, 139, 151n.34, 165n.37; utilitarian, 34, 130, 137-38, 165n.41 Cambridge University, 8, 33, 110, 113, 119, 121, 127, 129-34, 139, 142n.l3, 146n.41, 164n.34, 168n.78, 171n.39, 173n.61 Canard, Nicolas-Frangois, 141n.l Cannon, Edwin, 112 Cantillon, Richard, 149n.5 Checkland, S. G., 163n.l5 Clark, John Bates, 126, 131, 171n.43, 172nn. classical theory of political economy, 4-5, 9, 39, 42-43, 47, 50-52, 82, 95, 99, 117, 122, 137 Clifford, W. K., 119, 169n.l8 Coats, A. W., 141n.3, 167nn., 173n.61, 174n.l0 Cochrane, James L., 148n.2 Collard, David A., 149n.4 Collet, Clara, 169n.8 Colson, Clement, 173n.54 Comte, Auguste, 4, 8 1 , 103, 106-7, 10910, 141n.l, 166nn. Condorcet, Marquis de, 4, 70, 141n.l Cooter, Robert, 146n.36

188 · Index Coumot, A. Α., 4, 28, 33, 92, 114, 116, 122, 128-30, 140, 141n.2, 150n.28, 161n.39 Courtney, Leonard, 126 Creedy, John, 169n.l8, 170nn. Cunningham, William, 112 Cunynghame, H., 126, 170n.32

Fisher, Irving, 119, 126 FoxweU, H. S., 27, 29, 92, 100, 103, 11314, 116, 122, 127, 131, 134, 148n.3, 171n.38, 172n.48 Franklin, Joel, 142n.l0 Frege, Gottlob, 157n.36 Furner, Mary O., 174n.ll

Darwin, Charles, 8, 39, 101, 135 Darwin, George, 28, 100-103, 119-21, 151n.34, 164nn., 169nn. D'Aulnis de Bourouill, Johan, 101, 164n.21 Davy, Humphrey, 8, 81 demand and supply curves. See geome­ try, applied to economics De Marchi, N. B., 104, 141n.2, 159n.4, 162n.3, 164n.24, 165nn. De Morgan, Augustus, 135, 165n.37; on logic, 54, 59-60, 63, 104, 155n.21, 157n.47; relation to Jevons, 14, 19-22, 25, 33, 65, 69-70, 121, 143n.6, 144n.l7, 145n.31 Descartes, Rene, 72, 75, 85, 108 diminishing marginal utility, 4-5, 104, 128, 165n.41 Dobb, Maurice, 97, 141n.5, 162n.54 Ducasse, Curt J., 153nn. Duhem, Pierre, 78 Duhem-Quine problem, 73 Dumez, Herve, 141n.7 Dunbar, Charles F., 163n.l2 Dupuit, Jules, 33, 141n.2

Galileo, 82, 85 Gardner, Martin, 156n.28 geometry, 33, 85, 136, 139; applied to economics, 4, 7, 18, 22-23, 35, 47, 52, 122, 125, 127, 130, 132 Gibbs, J. Willard, 126 Gladstone, W. E., 25, 167n.72 Goldstine, Herman, 156n.28 Goodwin, Craufurd, 141n.3 Gordon, H. S., 171n.43 Gossen, H. H., 28, 33, 116, 151n.38, 163n.l2 Graham, Thomas, 14, 20 Great Exhibition, 9, 14 Guillebaud, C. W., 173n.58

Eckard, Edwin W., 142n.l9 Economist, 24-25, 100, 113 Edgeworth, Francis Ysidro, 10, 29, 41, 100, 116, 119, 122-27, 130-31, 138, 140, 174n.5 Einstein, Albert, 146n.36 ethology, 124 Euclidean geometry, 33, 59, 63, 66, 74, 76, 158n.59 Euler, L., 78 Fabians, 137 Faraday, Michael, 14, 75, 83 Fawcett, Henry, 25, 110, 118, 137, 141n.2, 146n.41 Fechner, Gustav Theodor, 124, 170n.23 Field, Hartry, 141n.8

Hahn, Frank, 174n.l Hamilton, William, 58, 104, 153n.l Hamilton, William Rowan, 91 Hartley, David, 149n.l2 Haughton, S., 46, 152n.46 Heath, P. L., 156n.31 Hegel, G.W.F., 55 Helmholtz, H.L.F. von, 6-7, 76, 170n.23 Hemming, G. W., 163n.8 Hempel, Carl, 154n.l0 Henderson, James P., 148n.2 Hennings, K. H., 120, 169n.5 Henry, D. P., 156nn. Herford, C. H., 169n.8 Herschel, John, 7-8, 14, 20, 25, 54-55, 78, 135, 142n.l3, 144n.8, 153n.2, 154n.4 Hewins, William, 112 Hirsch, Abraham, 163n. 13 Hobbes, Thomas, 3 Hodgson, William, 143n.4 Hollis, Martin, 174n.l Hoppus, John, 20 Howey, Richard S., 130, 146n.38, 162n.4, 170n.21, 172nn. Hume, David, 54-55, 78, 135, 149n.l2 Hutcheson, Francis, 32, 149nn.

Index Hutchison, T. W., 5, 9, 23, 141n.5, 142nn., 143n.25, 146n.36, 148n.60 Hutton, James, 24, 142n. 15 Huygens, Christian, 72, 85 hypothetico-deductive method, 71, 79-80 Ingram, John Kells, 99, 110-12, 167n.67 Inoue, Takutoshi, 142n.21 Jaffe, William, 5, 141n.4, 165n.44, 168n.86, 169n.6, 170n.25 Jenkin, Fleeming, 26, 32, 41, 43, 147n.49, 149n.6 Jennings, Richard, 32 Jevonian Revolution, 5-6, 9, 33, 96, 118, 137-38 Jevons, Harriet Ann, 26, 123, 148n.62, 161n.48, 162n.2 Jevons, Henrietta, 13, 16, 18, 144n.l7 Jevons, Herbert (brother), 13, 19, 22-23, 144n.ll Jevons, Herbert Stanley (son), 26 Jevons, Lucy, 13, 145n.30 Jevons, Mary Anne, 12 Jevons, Roscoe, 13 Jevons, Thomas (father), 12-14, 21 Jevons, Thomas (brother), 13, 17, 19, 27, 144n.l3, 147n.51 Jevons, William (grandfather), 12 Jevons, William Stanley: in Australia, 14-19, 22, 54, 144n.8, 153n.66; on capital, 22-23, 31, 47-51; The Coal Question, 25-26, 105, 147n.43; on dimensions, 34, 51, 53, 89, 122, 136; education, 9-10, 13-15, 17, 19-24, 33, 145n.24, 154n.l4; on exchange, 4 0 42, 53, 87, 91, 93, 119, 122; on faffibilism, 6, 67-69, 73-74, 76-79, 82, 158n.49; on final degree of utility, 2 2 24, 27, 38, 40-46, 50, 87-88, 111, 119, 136, 150n.21; on human nature, 1819, 38, 40, 94; on induction, 65-72; on labor, 43-47; in Liverpool, 12-14; on logic, 4, 10, 2 0 - 2 1 , 26, 28, 54-72, 7 5 76, 80, 96; logic machine, 62, 80, 156n.28; logical alphabet, 62, 64, 6 6 67; in London, 13-14, 26-30, 123, 147n.45; in Manchester, 25-29; on mathematical economics, 10—11, 1719, 21-24, 26-29, 31-52, 80-85, 97, 109, 111, 120, 136, 147n.52; on math-

·

189

ematics, 76-78, 80-81, 83, 96, 107-9, 136; on meteorology, 15-16, 14344n.8; on money, 26; predecessors, 27-28, 32-33, 128; Principles of Economics, 27, 40, 43, 122; Principles of Science, 6, 10-11, 2 1 , 26-28, 54, 57, 59, 63, 65, 76, 78, 81-82, 86, 88, 90, 95, 98-99; on rent, 47-50; on scientific methodology, 4, 10, 54-57, 93, 143n.24; on social reform, 27, 29, 13839; on sunspots and business cycles, 26, 80, 86, 143n.24, 147n.50; Theory of Political Economy, 6, 11, 24, 26-28, 31-34, 42, 46-47, 52, 57, 78, 82-83, 92-95, 98-100, 102, 104-5, 109, 111, 113, 116, 119, 121, 125, 128-29, 131; on utility, 36-38, 41-47, 92, 95, 100102, 119-20; on value, 22-23, 25, 3 4 39,84 Jones, Richard, 110, 145n.25 Jordanova, L. J., 142n.l4 Jowett, Benjamin, 173n.55 Kadish, Alon, 167n.76 Kant, Immanuel, 55 Kelvin, Lord, 119 Kepler, Johannes, 74, 77, 106 Keynes, John Maynard, 10, 54, 78, 117— 18, 128, 132-33, 137-38, 142n.20, 147n.42, 158n.52, 162n.3, 169n.8, 170n.l9, 171-73nn. Keynes, John Neville, 100, 113-14, 11617, 131, 149n.8 Kneale, William and Martha, 155n.21, 156n.31 Konekamp, Rosamond, 10, 142n.20, 147n.47 Koot, Gerard, 167n.66 Kuhn, Thomas S., 7, 23, 142n.ll Lagrange, Joseph Louis, 23, 91, 161n.34, 165n.37 Laidler, David, 143n.25 La Nauze, John, 144nn., 146n.33 Laplace, Pierre Simon, 69-70, 72, 165n.37 Lardner, Dionysius, 18, 31-32, 144n.l8 Laudan, Larry, 154n.3, 158n.58 Lavoisier, A. L., 8, 81-82 law of indifference, 39, 45, 150n.30 LeClair, Edward, 167n.77 Leibniz, G. W., 21, 72, 154n.l4

190 · Index Leijonhufvud, Axel, 14 In.9 Leslie, Thomas Edward Cliffe, 29, 99, 110-12, 164n.21, 165n.44, 167nn. LeTort, Charles, 162n.4 lever of exchange, 91-92 Lewis, C. L, 155-56nn. Lloyd, William F., 141n.2 Locke, John, 5, 54 logicism, 64-65, 136, 157n.37 Longfleld, Mountifort, 141n.2 Lowe, Robert, 110, 167n.68 Lunt, Edward C , 163n.l2 Lyell, Charles, 14, 142n.l4 Macdonnell, John, 162n.5, 163n. 12 Mach, Ernst, 6, 78, 95 MacLennan, Barbara, 143n.24 Macleod, Henry Dunning, 32, 148n.3 Macmillan, Frederick, 98, 132, 142n.22, 173n.58 Madden, E. H., 158n.58 Maine, Henry, 110 Maloney, John, 133, 167n.78, 171n.36, 173-74nn. Malthus, Thomas Robert, 4, 8-9, 15, 139, 141n.2 Malthusian doctrine, 9, 25, 46 Manchester Guardian, 26, 98, 162n.5 Mansel, H. L., 153n.l Marginal Revolution, 4, 141nn., 146n.36 Marshall, Alfred, 10, 51, 100, 113, 12223, 125-34, 137-40, 150n.21, 153nn., 164n.26, 170-74nn.; on mathematical economics, 6, 41, 109, 116, 119, 12634, 139; Principles of Economics, 117, 129-30, 132, 139; relation to Jevons, 31, 47, 52, 98-99, 102, 126-34 Marshall, Mary Paley, 127, 171n.37, 172n.47 Martineau, Harriet, 15 Martineau, James, 21, 145n.30 Marx, Karl, 24, 121-22 mathematical economics, 3-4, 6-7, 97, 102, 112, 114-19, 122-28, 130-31, 133-34, 139-40 Maxwell, James Clerk, 8, 83, 119, 135 Mays, Wolfe, 143n.24, 156nn. McCulloch, J. R., 8, 142n.l2 McNiven, Peter, 142n.21 McRae, Robert F., 157n.38 mechanical philosophy, 85

mechanics, 3, 19-20, 78, 84-85, 87-94, 96, 102, 105, 107, 115, 124, 135, 171n.43 Medawar, Peter, 158n.56 Menger, Carl, 4 - 5 Michelson-Morley experiment, 146n.36 Mill, James, 54 MiU, John Stuart, 5, 99, 100-101, 11012, 114, 117-18, 120-22, 139, 146n.41, 149n.l7, 154nn., 159n.7, 163n.5, 174n.7; on capital and rent, 47-49; on human nature, 138; Jevons's views on, 15, 25, 29, 31, 38, 8 1 82, 84, 103-10; on logic, 6, 10, 20-21, 54-58, 65, 71-72, 99, 105-7, 135, 155n.l9, 157n.38, 158n.59; Marshall's regard for, 128-29, 131; on mathematics, 4, 76, 105-9, 141n.2; Principles of Political Economy, 9, 98, 113, 131-32; on value, 4, 9, 82, 108 Mirowski, Philip, 147n.50, 173n.57 Morgan, Mary S., 174n.62 Nagel, Ernst, 155n.20, 156n.30 Newcomb, Simon, 98, 126, 163n.6 Newton, Isaac, 23, 33, 52, 72, 81, 85 Newtonian, 8, 55 Nichol, A. J., 170n.24 Nicholson, Joseph Shield, 126, 170n.33 non-Euclidean geometry, 6, 29, 76 O'Brien, Dennis, 163n.l5 Owens College, 24-25, 43, 99, 105, 113, 143n.5, 146n.40, 162n.2 Oxford University, 112, 119, 123, 125, 137, 147n.51, 156n.28, 168n.78, 169n.l8 Paley, William, 54 Pasteur, Louis, 7 Pearson, Karl, 78 Peart, Sandra, 152n.47 pedesis, 29, 16On. 18 Peirce, C. S., 6, 78 Physiocrats, 51 Pigou, A. C , 133-34, 138, 152n.52, 163n.9, 171-73nn. Plato, 21 Poincare, Henri, 6 Political Economy Club, 28 Porter, Roy, 142n.l4

Index Porter, Theodore M„ 33, 142n.l4, 149n.8 Potter, Richard, 161n.35 Price, L. L., 112, 163n.l2 prices, 33, 39-40, 43, 45-46, 86-90, 94, 136, 160n.30 probability, 4, 33, 67, 69-72, 79, 157n.47 psychology, 32, 36, 52-53, 58, 86-87, 104, 123-25, 137, 165n.41 Pythagoras, 140 Quetelet, L.A.J., 8, 18, 161n.44 Rappaport, Peter, 146n.36 Rashid, Salim, 142n.l5 Reid, Gavin, 160n.28 Ricardian economics, 9, 20, 32, 103-4, 110, 145n.25 Ricardo, David, 4, 8-9, 31, 45, 47, 57, 82, 99, 110-12, 122, 128-29, 139, 146n.36, 152n.45 Ricardo-Mill doctrine. See classical theory of political economy Richards, Joan L., 149n.9 Robbins, Lionel, 167n.77, 174n.2 Robert, Gerrylyn, 142n.l5 Robertson, George Croom, 153n.2, 155n.l9 Robertson, H. M., 170n.36 Rogers, Thorold, 112 Roscoe, Henry Enfield (Harry), 13-18, 143n.5, 144n.ll, 146n.40, 148n.58 Roscoe, William, 12 Royal Society, 99, 143n,5, 148n.58, 169n.7 Rudwick, Martin J. S., 142n.l4 Say, J.-B., 4 - 5 , 9, 141n.l Schneider, Harold, 167n.77 Schumpeter, Joseph, 5, 33, 97-98, 118, 123, 141n.4, 149n.7, 162n,55, 168nn., 170n.20, 171nn. Schweber, S. S., 142n.l4 self-interest, principle of, 18, 39-40, 110 Sen, Amartya K., 174n.l Senior, Nassau, 9, 50, 142n. 12, 142n. 16 Shadwell, John L., 28, 163n.l2 Shaw, G. B., 121-22, 169nn. Shieh, Yeung-Nan, 43, 144n.l8, 151n.37 Shove, G. F., 170n. 35 Sidgwick, Henry, 28-29, 100, 113-15, 131, 172n.48

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191

Smith, Adam, 4-5, 8-9, 15, 57, 82-83, 110-11, 114, 118, 122, 129, 137-38, 142n.l5, 146n.36 Smith, H. L., 112 Smyth, R. L., 148n.59, 163n.ll, 167n.67, 168n.90 Spencer, Herbert, 25, 28, 39, 74, 149n.l7 statistics, 33, 83-84, 88, 94, 125-26, 131 Steedman, Ian, 31 Stigler, George, 3, 31, 135, 150nn., 152n.56, 160n.34, 165n.38, 169nn., 174n.3, 175n.l3 Stigler, Stephen, 143n.25 Streissler, Erich, 5, 141n.6 Strong, E. W., 153n.3 Strong, John V., 72, 158nn. Sturges, R. P., 141n.2 SuUy, James, 170n.21 Syme, David, 163n.l2 Tait, P. G., 83 Theocharis, R., 141n.l Thomson, William, 83 Times (London), 25, 100, 111, 146n,38, 163n.l3 Toynbee, Arnold, 112 Tozer, John Edward, 32, 149n.4 University College, London, 13, 19, 24, 26, 29, 99, 121, 127, 143n.l, 145n.25, 147n.41, 161n.35, 168n.96 utilitarianism, 5, 38, 54, 57, 120, 124 Venn, John, 72, 158n.52 Viner, Jacob, 174n.64 virtual velocities, principle of, 42, 91-92, 137, 161n.35 von Thiinen, J. H., 28, 33, 128-30, 172n.50 wage fund theorem, 9, 49, 131, 137, 166n.64 Waley, Jacob, 19-20, 141n.2 Walker, Donald A., 170n.24 Walras, Leon, 4-6, 27-29, 101, 114-16, 122, 128, 148n.54, 162n.4, 166n.65, 169n.6, 171n.41 Walsh, H. T , 153n.3 water-diamond paradox, 23, 146n.36 Webb, Sidney, 122 Weber, E. H., 170n.23

192

·

Index

Westergaard, Harald, 151n.34 Whately, Richard, 7, 15, 20, 153n.l Whewell, WiUiam, 8, 20, 31-32, 54-56, 58-59, 71-72, 79, 135, 142n.l3, 154n.4, 165n.37 Whitaker, John, 10, 129-30, 143n.23, 154n.4, 170n.27, 171-73nn.

White, Michael V., 146n.33, 149n.6 Wicksteed, P. H., 80, 100, 121-24, 131, 137, 140, 169nn. Williams, R. O., 164n.21 Winch, Donald, 169n.ll Young, A. A., 52, 153n.71