Euler's invention of integral transforms

Euler invented integral transforms in the context of second order differential equations. He used them in a fragment published in 1763 and in a chapter of Institutiones Calculi Integralis (1769). In introducing them he made use of earlier work in which a concept akin to the integral transform is implicit. It would, however, be reading too much into that earlier work to see it as contributing to the theory of the integral transform. Other work sometimes cited in this context in fact has different concerns.     

Heaviside's operational calculus and the attempts to rigorise it

At the end of the 19^th century Oliver Heaviside developed a formal calculus of differential operators in order to solve various physical problems. The pure mathematicians of his time would not deal with this unrigorous theory, but in the 20^th century several attempts were made to rigorise Heaviside's operational calculus. These attempts can be grouped in two classes. The one leading to an explanation of the operational calculus in terms of integral transformations (Bromwich, Carson, Vander Pol, Doetsch) and the other leading to an abstract algebraic formulation (Lévy, Mikusiski). Also Schwartz's creation of the theory of distributions was very much inspired by problems in the operational calculus.     

Number crunching vs. number theory: computers and FLT,from Kummer to SWAC (1850–1960), and beyond

The present article discusses the computational tools (both conceptual and material) used in various attempts to deal with individual cases of FLT, as well as the changing historical contexts in which these tools were developed and used, and affected research. It also explores the changing conceptions about the role of computations within the overall disciplinary picture of number theory, how they influenced research on the theorem, and the kinds of general insights thus achieved. After an overview of Kummer’s contributions and its immediate influence, I present work that favored intensive computations of particular cases of FLT as a legitimate, fruitful, and worth-pursuing number-theoretical endeavor, and that were part of a coherent and active, but essentially low-profile tradition within nineteenth century number theory. This work was related to table making activity that was encouraged by institutions and individuals whose motivations came mainly from applied mathematics, astronomy, and engineering, and seldom from number theory proper. A main section of the article is devoted to the fruitful collaboration between Harry S. Vandiver and Emma and Dick Lehmer. I show how their early work led to the hesitant introduction of electronic computers for research related with FLT. Their joint work became a milestone for computer-assisted activity in number theory at large.     

Geometric division problems, quadratic equations, and recursive geometric algorithms in Mesopotamian mathematics

Most of what is told in this paper has been told before by the same author, in a number of publications of various kinds, but this is the first time that all this material has been brought together and treated in a uniform way. Smaller errors in the earlier publications are corrected here without comment. It has been known since the 1920s that quadratic equations played a prominent role in Babylonian mathematics. See, most recently, Høyrup (Hist Sci 34:1–32, 1996, and Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. Springer, New York, 2002). What has not been known, however, is how quadratic equations came to play that role, since it is difficult to think of any practical use for quadratic equations in the life and work of a Babylonian scribe. One goal of the present paper is to show how the need to find solutions to quadratic equations actually arose in Mesopotamia not later than in the second half of the third millennium BC, and probably before that in connection with certain geometric division of property problems. This issue was brought up for the first time in Friberg (Cuneiform Digit Lib J 2009:3, 2009). In this connection, it is argued that the tool used for the first exact solution of a quadratic equation was either a clever use of the “conjugate rule” or a “completion of the square,” but that both methods ultimately depend on a certain division of a square, the same in both cases. Another, closely related goal of the paper is to discuss briefly certain of the most impressive achievements of anonymous Babylonian mathematicians in the first half of the second millennium BC, namely recursive geometric algorithms for the solution of various problems related to division of figures, more specifically trapezoidal fields. For an earlier, comprehensive (but less accessible) treatment of these issues, see Friberg (Amazing traces of a Babylonian origin in Greek mathematics. WorldScientific, Singapore 2007b, Ch. 11 and App. 1).     

The Challenge of Colour: Eighteenth-Century Botanists and the Hand-Colouring of Illustrations

Colourful plant images are often taken as the icon of natural history illustration. However, so far, little attention has been paid to the question of how this beautiful colouring was achieved. At a case study of the eighteenth-century Nuremberg doctor and botanist, Christoph Jacob Trew, the process of how illustrations were hand-coloured, who was involved in this work, and how the colouring was supervised and evaluated is reconstructed, mostly based on Trew's correspondence with the engraver and publisher of his books, Johann Jacob Haid in Augsburg. Furthermore, the question of standardizing colours, their uses and their recipes is discussed at two examples of the same time period: the colour charts of the Bauer brothers, arguably the most renowned botanical draughtsmen of the period, and the colour tables by the Regensburg naturalist, Jacob Christian Schaeffer. Hand-colouring botanical illustrations, it is argued, was far from a straightforward task but confronted botanists and their employees with a plethora of practical and methodological problems, to which different solutions were developed in the course of time. Analysing these problems and solutions reveals some new and interesting aspects of the practices of eighteenth-century botany and of the production of scientific illustrations in general.     

From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept

Historical work on the emergence of sheaf theory has mainly concentrated on the topological origins of sheaf cohomology in the period from 1945 to 1950 and on subsequent developments. However, a shift of emphasis both in time-scale and disciplinary context can help gain new insight into the emergence of the sheaf concept. This paper concentrates on Henri Cartan’s work in the theory of analytic functions of several complex variables and the strikingly different roles it played at two stages of the emergence of sheaf theory: the definition of a new structure and formulation of a new research programme in 1940–1944; the unexpected integration into sheaf cohomology in 1951–1952. In order to bring this two-stage structural transition into perspective, we will concentrate more specifically on a family of problems, the so-called Cousin problems, from Poincaré (1883) to Cartan. This medium-term narrative provides insight into two more general issues in the history of contemporary mathematics. First, we will focus on the use of problems in theory-making. Second, the history of the design of structures in geometrically flavoured contexts—such as for the sheaf and fibre-bundle structures—which will help provide a more comprehensive view of the structuralist moment, a moment whose algebraic component has so far been the main focus for historical work.     

Einstein's struggle for a Machian gravitation theory

The story of Einstein's struggle to create a general theory of relativity, and his early discontentment with the final form of the theory (1915), is well known in broad outline. Thanks to the work of John Norton and others, much of the fine detail of the story is also now known. One aspect of Einstein's work in this period has, however, been relatively neglected: Einstein's commitment to Mach's ideas on inertia, and the influence this commitment had on Einstein's work on general relativity from 1907 to 1918. In this paper published writings and archival material are examined, to try to reconstruct the details of Einstein's thinking about inertia and gravitation, and the role that Mach's ideas played in Einstein's crucial work on the general theory. By the end, a clear picture of Einstein's conceptions of Mach's ideas on inertia, and their philosophical motivations, will emerge. Several surprising conclusions also emerge: Einstein's desire for a Machian gravitation theory was the central force driving his work from 1912 to 1915, keeping him going despite numerous frustrating setbacks; Einstein's continued commitment to Mach's ideas in 1916–1917 kept him at work trying various strategies of modification of the field equations, in order to exclude anti-Machian solutions (including the addition of the cosmological constant in 1917); and as late as early 1918, Einstein was ready to call the whole General Theory a failure if no way of squaring it with Mach's ideas on inertia could be found. But by 1920 Einstein advocated a view that granted spacetime (under the name ‘ether’) independent existence with physical qualities of its own, a complete break with his earlier Machian views.     

John Dalton’s puzzles: from meteorology to chemistry

Historical research on John Dalton has been dominated by an attempt to reconstruct the origins of his so-called “chemical atomic theory”. I show that Dalton’s theory is difficult to define in any concise manner, and that there has been no consensus as to its unique content among his contemporaries, later chemists, and modern historians. I propose an approach which, instead of attempting to work backward from Dalton’s theory, works forward, by identifying the research questions that Dalton posed to himself and attempting to understand how his hypotheses served as answers to these questions. I describe Dalton’s scientific work as an evolving set of puzzles about natural phenomena. I show how an early interest in meteorology led Dalton to see the constitution of the atmosphere as a puzzle. In working on this great puzzle, he gradually turned his interest to specifically chemical questions. In the end, the web of puzzles that he worked on required him to create his own novel philosophy of chemistry for which he is known today.     

交通流宏微观参数关系方程组实证分析

本文以James H．Banks关于平均时间间隔的研究为基础，提出交通流宏微观参数关系方程组，理论上较全面地描述两类交通参数的关系；应用实测交通数据对该方程组进行数据实证。分析表明，使用关系方程组产生的偏差不仅与统计时间间隔有关，而且与检测点的位置、交通状态相关。通过分析关系方程组的推导可知，偏差源于两个等式在理论上与实际运用中的不同。     

Genetic epistemology,equilibration and the rationality of scientific change

Genetic epistemology is concerned not only with the development of knowledge in the individual person, but also with the epistemological development of scientific thought. If so, problems and solutions in the one area should be roughly isomorphic to problems and solutions in the other area. To illustrate this, I consider Piaget's theory of equilibration and show how this theory has implications for issues concerning scientific progress, change and rationality. Piaget's views about these issues have much in common with claims of contemporary philosophers of science (such as Popper and Lakatos) concerning the rational growth of science, but in addition purport to be grounded in empirical psychology. This suggests thatt it may be fruitful to investigate the possibility of integrating cognitive psychology and philosophy of science in a new way.     

The analytic geometry of genetics: part I: the structure, function, and early evolution of Punnett squares

A square tabular array was introduced by R. C. Punnett in (1907) to visualize systematically and economically the combination of gametes to make genotypes according to Mendel’s theory. This mode of representation evolved and rapidly became standardized as the canonical way of representing like problems in genetics. Its advantages over other contemporary methods are discussed, as are ways in which it evolved to increase its power and efficiency, and responded to changing theoretical perspectives. It provided a natural visual decomposition of a complex problem into a number of inter-related stages. This explains its computational and conceptual power, for one could simply “read off” answers to a wide variety of questions simply from the “right” visual representation of the problem, and represent multiple problems, and multiple layers of problems in the same diagram. I relate it to prior work on the evolution of Weismann diagrams by Griesemer and Wimsatt (What Philosophy of Biology Is, Martinus-Nijhoff, the Hague, 1989), and discuss a crucial change in how it was interpreted that midwifed its success.     

“The soul of the fact”—Poincaré and proof

Henri Poincaré acquired a reputation in his lifetime for being difficult to read. It was said that he missed out important steps in his arguments, assumed the truth of claims that would be difficult if not impossible to prove, and in short that he lacked rigour. In the years after his death this view coalesced into an exaggerated claim that his work was simply too vague, and has become a cliché. This paper argues that Poincaré was far from indifferent to rigour, and that what characterises his work is an attempt to convey a particular sense of what it is to understand a topic. Throughout his working life Poincaré was concerned to promote the understanding of many domains of mathematics and physics. This is as apparent in his views about geometry, his conventionalism, and his theory of knowledge, as it is in his work on electricity and optics, on number theory, and function theory. It is one of the ways Poincaré discharged his responsibilities as a scientist, and that it accounts not only for a surprising degree of unity in his work but also gives it its distinctive character—at once profound and elusive.     

Einstein and Singularities

Except for a few brief periods, Einstein was uninterested in analysing the nature of the spacetime singularities that appeared in solutions to his gravitational field equations for general relativity. The existence of such monstrosities reinforced his conviction that general relativity was an incomplete theory which would be superseded by a singularity-free unified field theory. Nevertheless, on a number of occasions between 1916 and the end of his life, Einstein was forced to confront singularities. His reactions show a strange asymmetry: he tended to be more disturbed by (what today we would call) merely apparent singularities and less disturbed by (what we would call) real singularities. Einstein had strong a priori ideas about what results a correct physical theory should deliver. In the process of searching through theoretical possibilities, he tended to push aside technical problems and jump over essential difficulties. Sometimes this method of working produced brilliant new ideas—such as the Einstein–Rosen bridge—and sometimes it lead him to miss important implications of his theory of gravity—such as gravitational collapse.     

Emergent evolutionism,determinism and unpredictability

The fact that there exist in nature thoroughly deterministic systems whose future behavior cannot be predicted, no matter how advanced or fined-tune our cognitive and technical abilities turn out to be, has been well established over the last decades or so, essentially in the light of two different theoretical frameworks, namely chaos theory and (some deterministic interpretation of) quantum mechanics. The prime objective of this paper is to show that there actually exists an alternative strategy to ground the divorce between determinism and predictability, a way that is older than—and conceptually independent from—chaos theory and quantum mechanics, and which has not received much attention in the recent philosophical literature about determinism. This forgotten strategy—embedded in the doctrine called “emergent evolutionism”—is nonetheless far from being a mere historical curiosity that should only draw the attention of philosophers out of their concern for comprehensiveness. It has been indeed recently revived in the works of respected scientists.     

Phenomenotechnique: Bachelard's critical inheritance of conventionalism

The concept of phenomenotechnique has been regarded as Bachelard's most original contribution to the philosophy of science. Innovative as this neologism may seem, it benefited from a generation of debates on the nature and status of scientific facts, among conventionalist thinkers and their opponents. Granting that Bachelard stood among the opponents to conventionalism, this article nonetheless reveals deep similarities between his work and that of two conventionalist thinkers who insisted on what we call today the theory-ladenness of scientific experiment: Pierre Duhem and Édouard Le Roy. This article, therefore, compares Bachelard's notion of phenomenotechnique with Duhem's developments on the double character of scientific instruments, and with Le Roy's claim that scientific facts are fabricated to meet the requirements of theory. It shows how Bachelard retained Duhem and Le Roy's views on the interplay between theory and experiment but rejected their sceptical conclusions on the limitations of experimental control. It claims that this critical inheritance of conventionalism was made possible by a reflection on technology, which led Bachelard to re-evaluate the artificiality of scientific facts: instead of regarding this artificiality as a limitation of science, as Le Roy did, he presented it as a condition for objective knowledge.     

Causation,information, and physics

This work outlines the novel application of the empirical analysis of causation, presented by Kutach, to the study of information theory and its role in physics. The central thesis of this paper is that causation and information are identical functional tools for distinguishing controllable correlations, and that this leads to a consistent view, not only of information theory, but also of statistical physics and quantum information. This approach comes without the metaphysical baggage of declaring information a fundamental ingredient in physical reality and exorcises many of the otherwise puzzling problems that arise from this view-point, particularly obviating the problem of ‘excess baggage’ in quantum mechanics. This solution is achieved via a separation between information carrying causal correlations of a single qubit and the bulk of its state space.     

Entanglement and Open Systems in Algebraic Quantum Field Theory

Entanglement has long been the subject of discussion by philosophers of quantum theory, and has recently come to play an essential role for physicists in their development of quantum information theory. In this paper we show how the formalism of algebraic quantum field theory (AQFT) provides a rigorous framework within which to analyse entanglement in the context of a fully relativistic formulation of quantum theory. What emerges from the analysis are new practical and theoretical limitations on an experimenter's ability to perform operations on a field in one spacetime region that can disentangle its state from the state of the field in other spacelike-separated regions. These limitations show just how deeply entrenched entanglement is in relativistic quantum field theory, and yield a fresh perspective on the ways in which the theory differs conceptually from both standard non-relativistic quantum theory and classical relativistic field theory.     

Ptolemy and the meta-helikôn

In his Harmonics, Ptolemy constructs a complex set of theoretically ‘correct’ forms of musical scale, represented as sequences of ratios, on the basis of mathematical principles and reasoning. But he insists that their credentials will not have been established until they have been submitted to the judgement of the ear. They cannot be audibly instantiated with the necessary accuracy without the help of specially designed instruments, which Ptolemy describes in detail, discussing the uses to which each can be put and cataloguing its limitations. The best known of these instruments is the monochord, but there are several more complex devices. This paper discusses one such instrument which is known from no other source, ancient or modern, whose design was prompted by the geometrical construction known as the helikôn. It has several remarkable peculiarities. I examine its design, its purposes, and the merits and shortcomings which Ptolemy attributes to it. An appendix describes an instrument I have built to Ptolemy’s specifications (possibly the first of its kind since the second century bc), in an attempt to find out how satisfactorily such a bizarre contraption will work; and it explains how various practical problems can be resolved.