Surveying the history of nineteenth-century science in his magisterial A History of European Thought in the Nineteenth Century (1904–12), John Theodore Merz concluded that one “of the principal performances of the second half of the nineteenth century has been to find … the greatest of all exact generalisations – the conception of energy.” In a similar vein, Sir Joseph Larmor, heir to the Lucasian Chair of Mathematics at Cambridge once occupied by Newton, wrote in the obituary notice of Lord Kelvin (1824–1907) for the Royal Society of London in 1908 that the doctrine of energy “has not only furnished a standard of industrial values which has enabled mechanical power … to be measured with scientific precision as a commercial asset; it has also, in its other aspect of the continual dissipation of energy, created the doctrine of inorganic evolution and changed our conceptions of the material universe.” These bold claims stand at the close of a remarkable era for European physical science, which saw, in the context of British and German industrialization, the replacement of earlier Continental (notably French) action-at-a-distance force physics with the new physics of energy.

This chapter traces the construction of the distinctively nineteenth-century sciences of energy and thermodynamics. Modern historical studies of energy physics have usually taken as their starting point Thomas Kuhn’s paper on energy conservation as a case of simultaneous discovery. Kuhn’s basic claim was that twelve European men of science and engineering, working more or less in isolation from one another, “grasped for themselves essential parts of the concept of energy and its conservation” during the period between 1830 and 1850.

“Global environmental change,” three words heard with increasing frequency in both science and policy circles, is shorthand for the inevitability of change in the geosphere-biosphere. It also expresses the realization that human activities have now reached the level of a planetary force. Since 1945, we have grown increasingly apprehensive about a number of global environmental issues, including population, energy consumption, pollution, and the health of the biosphere. At the beginning of a new millennium, instead of standing firmly on the technoscientific foundations of our “enlightened” predecessors, we find ourselves apprehensive about global environmental change, teetering on the uncertainties of a new century and unsure about the future quality and even habitability of the global environment.

Much of the concern is rightfully focused on changes in the atmosphere caused by human activities. Only a century after the discovery of the stratosphere, only five decades after the invention of chlorofluorocarbons (CFCs), and only two decades after atmospheric chemists warned of the destructive nature of chlorine and other compounds, we fear that ozone in the stratosphere is being damaged by human activity. Only a century after the first models of the carbon cycle were developed, only three decades after regular carbon dioxide (CO,) measurements began at Mauna Loa Observatory, and only two decades after climate modelers first doubled the CO, in a computerized atmosphere, we fear that the earth may experience a sudden and possibly catastrophic warming caused by industrial pollution.

Mathematical knowledge has long been regarded as essentially stable and, hence, rooted in a world of ideas only superficially affected by historical forces. This general viewpoint has profoundly influenced the historiography of mathematics, which until recently has focused primarily on internal developments and related epistemological issues. Standard historical accounts have concentrated heavily on the end products of mathematical research: theorems, solutions to problems, and the technical difficulties that had to be mastered before a well-posed question could be answered. This kind of approach inevitably suggests a cumulative picture of mathematical knowledge that tells us little about how such knowledge was gained, refined, codified, or transmitted. Moreover, the purported permanence and stability of mathematical knowledge begs some obvious questions with regard to accessibility – known to whom and by what means? Issues of this kind have seldom been addressed in historical studies of mathematics, which often treat priority disputes among mathematicians as merely a matter of “who got there first.” By implication, such studies suggest that mathematical truths reside in a Platonic realm independent of human activity, and that mathematical findings, once discovered and set down in print, can later be retrieved at will.

If this fairly pervasive view of the epistemological status of mathematical assertions were substantially correct, then presumably mathematical knowledge and the activities that lead to its acquisition ought to be sharply distinguished from their counterparts in the natural sciences. Recent research, however, has begun to undercut this once-unquestioned canon of scholarship in the history of mathematics. At the same time, mathematicians and philosophers alike have come increasingly to appreciate that, far from being immune to the vicissitudes of historical change, mathematical knowledge depends on numerous contextual factors that have dramatically affected the meanings and significance attached to it.

Dismissed as inconsequential before the 1970s, the history of the contributions of women to the physical sciences has become a topic of considerable research in the last two decades. Best known of the women physical scientists are the three “great exceptions” from central Europe – Sonya Kovalevsky, Marie Sklodowska Curie, and Lise Meitner– but in recent years, other women and other countries and areas have been receiving attention, and more is to be expected in the future. The overall pattern for most women in these fields, the nonexceptions, has been one of ghettoization and subsequent attempts to overcome barriers.

PRECEDENTS

Before 1800 there were several self-taught and privately-tutored “learned ladies” in the physical sciences. Included were the English self-styled “natural philosopher” Margaret Cavendish (1623–1673), who wrote books and in the 1660s visited the Royal Society of London, which had not elected her to membership; the German astronomer Maria Winkelmann Kirch (1670– 1720), who worked for the then-new Berlin Academy of Sciences in the early 1700s; the Frenchwoman Emilie du Chatelet (1706–1749), who translated Newton’s Principia into French before her premature death in childbirth in 1749; the Italians Laura Bassi (1711–1778), famed professor of physics at the University of Bologna, and Maria Agnesi (1718–1799), a mathematician in Bologna; Ekaterina Romanovna Dashkova (1743–1810), the director of the Imperial Academy of Sciences in Russia; and Marie Anne Lavoisier (1758– 1836), who helped her husband Antoine with his work in the Chemical Revolution.

During the nineteenth and twentieth centuries, astronomy has changed from a relatively homogeneous discipline to one of tremendous diversity. Before this period, the main business of astronomy had been the measurement and prediction of planetary motion and stellar position. Earlier astronomers depended on a limited range of observational equipment – the optical telescope and various instruments for measuring angles and positions against the sky – in order to map the locations of the stars and to track the motions of the planets as they wandered against this fixed background. By the early nineteenth century, astronomers had, as theoretical tools, not only Newtonian gravitation but also the fruits of a century’s further refinement of celestial mechanics. Not only could astronomers calculate the orbits of individual planets around the Sun; they could also investigate the mutual perturbations of the various bodies and the stability of the solar system as a whole, far into the future. Within their well-defined realm, early-nineteenth-century astronomers congratulated themselves on possessing a predictive power exceeding that of all other fields of natural science.

Yet astronomers were eventually to trade their sure grasp of their traditional portion of the world for a much less certain hold on broad, new domains: the study not just of position and motion but also of the physical nature of celestial objects of all kinds, from the Sun, stars, and planets to nebulae and galaxies. This expansion of subject was, in significant part, technology driven, and many new observational technologies contributed to making it possible, including the building of telescopes with tremendously increased light-gathering power and finer resolution, and the introduction of photography as an astronomical tool.

Until the mid-1950s, the word “computer” commonly referred to a woman employed in operating a calculating machine in a business office or a scientific calculating laboratory. With the invention in 1945 of the stored-program computer, several months after the Second World War ended, and with the publicity surrounding the introduction in 1952 of the first commercial computer (the Universal Automatic Computer, or UNIVAC), the word computer became associated with a machine, rather than a human.

This machine had three attributes that rendered prior calculating technologies obsolete in less than two decades. The electronic switching of its components eventually made the computer billions of times faster than its mechanical ancestors. The digital storage of information enhanced precision to practically unrestricted levels. The stored program capability, that is, the ability to store instructions as well as data inside the machine and to have the machine process those instructions during the course of a computation without human intervention, had two advantages: First, it enabled almost any computer to be used as a universal machine, in other words, to carry out virtually any computation possible by a machine. Second, stored programming was critical to the automation of the computational process, so that the overall speed of computation could reflect the electronic speed of the components.

The nineteenth century saw enormous advances in electrical science, culminating in the formulation of Maxwellian field theory and the discovery of the electron. It also witnessed the emergence of electrical power and communications technologies that have transformed modern life. That these developments in both science and technology occurred in the same period and often in the same places was no coincidence, nor was it just a matter of purely scientific discoveries being applied, after some delay, to practical purposes. Influences ran both ways, and several important scientific advances, including the adoption of a unified system of units and of Maxwellian field theory itself, were deeply shaped by the demands and opportunities presented by electrical technologies. As we shall see, electrical theory and practice were tightly intertwined throughout the century.

EARLY CURRENTS

Before the nineteenth century, electrical science was limited to electrostatics; magnetism was regarded as fundamentally distinct. In the 1780s, careful measurements by the French engineer Charles Coulomb established an inverse-square law of attraction and repulsion for electric charges, and electrostatics occupied a prominent place in the Laplacian program, based on laws of force between hypothetical particles, then beginning to take hold in France. The situation was soon complicated, however, by Alessandro Volta’s invention in 1799 of his “pile,” particularly as attention shifted from the pile itself to the electric currents it produced. Much of the history of electrical science in the nineteenth century can be read as a series of attempts to come to terms with the puzzles posed, and the opportunities presented, by currents like those generated by Volta’s pile.

Throughout the late eighteenth and nineteenth centuries, there were two distinctly different ways of thinking about the earth – two different evidentiary and epistemic traditions. Such men as Comte Georges de Buffon and Léonce Elie de Beaumont in France, William Hopkins and William Thomson (Lord Kelvin) in the United Kingdom, and James Dwight Dana in the United States tried to understand the history of the earth primarily in terms of the laws of physics and chemistry. Their science was mathematical and deductive, and it was closely aligned with physics, astronomy, mathematics, and, later, chemistry. With some exceptions, they spent little time in the field; to the degree that they made empirical observations, they were likely to be indoors rather than out. In hindsight, this work has come to be known as the geophysical tradition. In contrast, such men as Abraham Gottlob Werner in Germany, Georges Cuvier in France, and Charles Lyell in England tried to elucidate earth history primarily from physical evidence contained in the rock record. Their science was observational and inductive, and it was, to a far greater degree than that of their counterparts, intellectually and institutionally autonomous from physics and chemistry. With some exceptions, they spent little time in the laboratory or at the blackboard; the rock record was to be found outside. By the early nineteenth century, students of the rock record called themselves geologists. These two traditions – geophysical and geological – together defined the agenda for what would become the modern earth sciences. Geophysicists and geologists addressed themselves to common questions, such as the age and internal structure of the earth, the differentiation of continents and oceans, the formation of mountain belts, and the history of the earth’s climate.

Until about 1840, the theory of probability was used almost exclusively to describe and to manage the imperfections of human observation and reasoning. The introduction of statistical methods to physics, which began in the late 1850s, was part of the process through which the mathematics of chance and variation was deployed to represent objects and processes in the world. If this was a “probabilistic revolution,” it was a multifarious and gradual one, the vast scope of which went largely unremarked. Yet it challenged some basic scientific assumptions about explanation, metaphysics, and even morality. For this reason, it sometimes provoked searching reflection and debate within particular fields, including physics, over what, in retrospect, appears as an important new direction in science.

At the most basic level, statistical method meant replacing fundamental laws whose action was universal and deterministic with broad characterizations of heterogeneous collectives. Statistics, whether of human societies or of molecular systems, involved a shift from the individual to the population and from direct causality to mass regularity. In social writings, it was linked to bold claims for scientific naturalism. Statisticians claimed to have uncovered a lawlike social order governing human acts and decisions that had so far been comprehended by Christian moral philosophy in terms of divine intentionality and human will. Their science seemed to devalue moral agency, perhaps even to deny human freedom. In other contexts, and especially in physics, statistical principles appeared, rather, to limit the domain of scientific certainty. They directed attention to merely probabilistic regularities, the truth of which was uncertain and approximate.

For over three millennia, cosmology had closer connections to myth, religion, and philosophy than to science. Cosmology as a branch of science has essentially been an invention of the twentieth century. Because modern cosmology is such a diverse field and has ties with so many adjacent scientific disciplines and communities (mathematics, physics, chemistry, and astronomy), it is not possible to write its history in a single chapter. Although there is no complete history of modern cosmology, there exist several partial histories that describe and analyze the main developments. The following account draws on these histories and presents some major contributions to the knowledge of the universe that emerged during the twentieth century. The chapter focuses on the scientific aspects of cosmology, rather than on those related to philosophy and theology.

THE NINETEENTH-CENTURY HERITAGE

Cosmology, the study of the structure and evolution of the world at large, scarcely existed as a recognized branch of science in the nineteenth century; and cosmogony, the study of the origin of the world, did even less. Yet there was, throughout the century, an interest, often of a speculative and philosophical kind, in these grand questions. According to the nebular hypothesis of Pierre-Simon de Laplace and William Herschel, some of the observed nebulae were protostellar clouds that would eventually condense and form stars and planets in a manner similar to the way in which the solar system was believed to have been formed. This widely accepted view implied that the world was not a fixed entity, but in a state of evolution.

The modern historical period from the Enlightenment to the mid-twentieth century has often been called an age of science, an age of progress or, using Auguste Comte’s term, an age of positivism.

Volume 5 in The Cambridge History of Science is largely a history of the nineteenth- and twentieth-century period in which mathematicians and scientists optimistically aimed to establish conceptual foundations and empirical knowledge for a rational, rigorous scientific understanding that is accurate, dependable, and universal. These scientists criticized, enlarged, and transformed what they already knew, and they expected their successors to do the same. Most mathematicians and scientists still adhere to these traditional aims and expectations and to the optimism identified with modern science.

By way of contrast, some writers and critics in the late twentieth century characterized the waning years of the twentieth century as a postmodern and postpositivist age. By this they meant, in part, that there is no acceptable master narrative for history as a story of progress and improvement grounded on scientific methods and values. They also meant, in part, that subjectivity and relativism are to be taken seriously both cognitively and culturally, thereby undermining claims for scientific knowledge as dependable and privileged knowledge.

When medical technology met computers in the last third of the twentieth century, the conjoining triggered changes almost as radical as the ones that followed the discovery of x rays in 1895. As in that earlier revolution, the greatest change was in the realm of vision. Whereas x rays and fluoroscopy allowed physicians to peer into the living body to see foreign objects, or tumors and lungs disfigured by tuberculosis (TB), the new digitized images locate dysfunction deep inside organs, like the brain, that are opaque to x rays. The initial medical impact of these new devices, like the x ray before it, was in diagnosis.

Wilhelm Conrad Röntgen’s (1845–1923) announcement of the discovery of x rays in 1896 was probably the first scientific media event. Within months, x-ray apparatus was hauled into department stores, and slot machine versions were installed in the palaces of kings and tsars, and in railroad stations for the titillation of the masses. Although the phenomenon had been discovered by a physicist who had no interest in either personal profit or any practical application, it was obvious to physicians and surgeons, as well as to those who sold them instruments, how the discovery could help make diagnoses.

The advantages seemed so great that, for the most part, purveyors of x- ray machines were either oblivious to the dangers of radiation or able to find alternative explanations for burns and ulcerating sores that kept appearing. Even so, with the exception of military medicine, exemplified in the United States by the use of x rays during the Spanish-American war, the machines were not employed routinely in American hospitals for at least a decade after their discovery.

Scientists have always expressed a strong urge to think in visual images, especially today with our new and exciting possibilities for the visual display of information. We can “see” elementary particles in bubble chamber photographs. But what is the deep structure of these images? A basic problem in modern science has always been how to represent nature, both visible and invisible, with mathematics, and how to understand what these representations mean. This line of inquiry throws fresh light on the connection between common sense intuition and scientific intuition, the nature of scientific creativity, and the role played by metaphors in scientific research.

We understand, and represent, the world about us not merely through perception but with the complex interplay between perception and cognition. Representing phenomena means literally re-presenting them as either text or visual image, or a combination of the two. But what exactly are we re-presenting? What sort of visual imagery should we use to represent phenomena? Should we worry that visual imagery can be misleading?

Consider Figure 10.1, which shows the visual image offered by Aristotelian physics for a cannonball’s trajectory. It is drawn with a commonsensical Aristotelian intuition in mind. On the other hand, Galileo Galilei (1564–1642) realized that specific motions should not be imposed on nature. Rather, they should emerge from the theory’s mathematics – in this way should the book of nature be read. Figure 10.2 is Galileo’s own drawing of the parabolic fall of an object pushed horizontally off a table. It contains the noncommonsensical axiom of his new physics that all objects fall with the same acceleration, regardless of their weight, in a vacuum.

Hephaestus, arms maker to the gods, was the only deity with a physical disability. Lame and deformed, he caricatured what his own handiwork could do to the human body. Not until the later twentieth century, however, did his heirs and successors attain the power to inflict such damage on the whole human race. Nuclear weapons lent salience to the long history of military technology. The Cold War contest between the United States and the Soviet Union attracted the most attention and concern, but in the second half of the twentieth century, science and technology transformed conventional warfare as well. Even small states with comparatively modest arsenals found themselves stressed by the growing ties and tensions between science and war.

The relationship between science, technology, and war can be said to have a set of defining characteristics: (1) State funding or patronage of arms makers has flowed through (2) institutions ranging from state arsenals to private contracts. This patronage purchased (3) qualitative improvements in military arms and equipment, as well as (4) large-scale, dependable, standardized production. To guarantee an adequate supply of scientists and engineers, the state also underwrote (5) education and training. As knowledge replaced skill in the production of superior arms and equipment, a cloak of (6) secrecy fell over military technology. The scale of activity, especially in peacetime, could give rise to (7) political coalitions; in the United States these took the form of the military-industrial complex. The scale also imposed upon states significant (8) opportunity costs in science and engineering that were often addressed by pursuit of (9) dual-use technologies. For some scientists and engineers, participation in this work posed serious (10) moral questions.

In this chapter I shall illustrate some of the general trends in the development of mathematical analysis by considering its most basic element: the concept of function. I shall show that its development was shaped both by applications in various domains, such as mechanics, electrical engineering, and quantum mechanics, and by foundational issues in pure mathematics, such as the striving for rigor in nineteenth-century analysis and the structural movement of the twentieth century. In particular, I shall concentrate on two great changes in the concept of function: first, the change from analytic-algebraic expressions to Dirichlet’s concept of a variable depending on another variable in an arbitrary way, and second, the invention of the theory of distributions. We shall see that it is characteristic of both of the new concepts that they were initiated in a nonrigorous way in connection with various applications, and that they were generally accepted and widely used only after a new basic trend in the foundation of mathematics had made them natural and rigorous. However, the two conceptual transformations differ in one important respect: The first change had a revolutionary character in that Dirichlet’s concept of function completely replaced the earlier one. Furthermore, some of the analytic expressions, such as divergent power series, which eighteenth-century mathematicians considered as functions, were considered as meaningless by their nineteenth-century successors. The concept of distributions, on the other hand, is a generalization of the concept of function in the sense that most functions (the locally integrable functions) can be considered distributions. Moreover, the theory of distributions builds upon the ordinary theory of functions, so that the theory of functions is neither superfluous nor meaningless.

In 1967, Per-Olov Löwdin introduced the new International Journal of Quantum Chemistry in the following manner:

In this chapter I address the emergence and establishment of a scientific discipline that has been called at times quantum chemistry, chemical physics, or theoretical chemistry. Understanding why and how atoms combine to form molecules is an intrinsically chemical problem, but it is also a many-body problem, which is handled by means of the integration of Schrödinger’s equation. The heart of the difficulty is that the equation cannot be integrated exactly for even the simplest of all molecules. Devising semiempirical approximate methods became, therefore, a constitutive feature of quantum chemistry, at least in its formative years.