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\"This book presents reverse mathematics to a general mathematical audience for the first time. Reverse mathematics is a new field that answers some old questions. In the two thousand years that mathematicians have been deriving theorems from axioms, it has often been asked: which axioms are needed to prove a given theorem? Only in the last two hundred years have some of these questions been answered, and only in the last forty years has a systematic approach been developed. In Reverse Mathematics, John Stillwell gives a representative view of this field, emphasizing basic analysis--finding the \"right axioms\" to prove fundamental theorems--and giving a novel approach to logic. Stillwell introduces reverse mathematics historically, describing the two developments that made reverse mathematics possible, both involving the idea of arithmetization. The first was the nineteenth-century project of arithmetizing analysis, which aimed to define all concepts of analysis in terms of natural numbers and sets of natural numbers. The second was the twentieth-century arithmetization of logic and computation. Thus arithmetic in some sense underlies analysis, logic, and computation. Reverse mathematics exploits this insight by viewing analysis as arithmetic extended by axioms about the existence of infinite sets. Remarkably, only a small number of axioms are needed for reverse mathematics, and, for each basic theorem of analysis, Stillwell finds the \"right axiom\" to prove it. By using a minimum of mathematical logic in a well-motivated way, Reverse Mathematics will engage advanced undergraduates and all mathematicians interested in the foundations of mathematics. \"-- Provided by publisher.

Before Palm Pilots and iPods, PCs and laptops, the term \"computer\" referred to the people who did scientific calculations by hand. These workers were neither calculating geniuses nor idiot savants but knowledgeable people who, in other circumstances, might have become scientists in their own right. When Computers Were Human represents the first in-depth account of this little-known, 200-year epoch in the history of science and technology. Beginning with the story of his own grandmother, who was trained as a human computer, David Alan Grier provides a poignant introduction to the wider world of women and men who did the hard computational labor of science. His grandmother's casual remark, \"I wish I'd used my calculus,\" hinted at a career deferred and an education forgotten, a secret life unappreciated; like many highly educated women of her generation, she studied to become a human computer because nothing else would offer her a place in the scientific world. The book begins with the return of Halley's comet in 1758 and the effort of three French astronomers to compute its orbit. It ends four cycles later, with a UNIVAC electronic computer projecting the 1986 orbit. In between, Grier tells us about the surveyors of the French Revolution, describes the calculating machines of Charles Babbage, and guides the reader through the Great Depression to marvel at the giant computing room of the Works Progress Administration. When Computers Were Human is the sad but lyrical story of workers who gladly did the hard labor of research calculation in the hope that they might be part of the scientific community. In the end, they were rewarded by a new electronic machine that took the place and the name of those who were, once, the computers.

Running in something approaching chronological order, this book shows that every breakthrough in math represents a single step forward, resting on the work of others, and brings to life the importance of numbers, shapes, and patterns in the world around us.

01 02 Navigational Enterprises in Europe and its Empires, 1730–1850 brings together the work of international authors to explore European experiences in the development of new navigational techniques and instruments in the eighteenth and nineteenth centuries. This is the period in which the 'longitude problem' has been presented as being solved in an unproblematic way. Challenging this narrative, the book looks beyond just the British story to examine the role of governments, institutions, men of science, practitioners and navigators across Europe, and the use of the new and old techniques and instruments in practice. As the different chapters show, the methods available, including long-established navigational techniques such as dead reckoning and the newer astronomical and timekeeping methods of longitude determination, were complementary rather than exclusive. When and how they were used depended on local, national and other circumstances, although their development must be seen as the result of international and transnational exchanges. 04 02 1. Introduction; Rebekah Higgitt and Richard Dunn 2. A Southern Meridian: Astronomical Undertakings in the Eighteenth-Century Spanish Empire; Juan Pimentel 3. The Longitude Committee and the Practice of Navigation in the Netherlands, c.1750–1850; Karel Davids 4. From Lacaille to Lalande: French Work on Lunar Distances, Nautical Ephemerides and Lunar Tables, 1742–85; Guy Boistel 5. The Bureau des Longitudes: An Institutional Study; Martina Schiavon 6. Patriotic and Cosmopolitan Patchworks: Following a Swedish Astronomer into London's Communities of Maritime Longitude, 1759–60; Jacob Orrje 7. 'Perfectly Correct': Russian Navigators and the Royal Navy; Simon Werrett 8. A Different Kind of Longitude: The Metrology and Conventions of Location by Geodesy; Michael Kershaw 9. Testing Longitude Methods in Mid-Eighteenth Century France; Danielle M. E. Fauque 10. Navigating the Pacific from Bougainville to Dumont d'Urville: French Approaches to Determining Longitude, 1766–1840; John Gascoigne 11. Navigation and Mathematics: A Match Made in the Heavens?; Jane Wess 12 . Longitude Networks on Land and Sea: The East India Company and Longitude Measurement 'in the Wild', 1770–1840; David Philip Miller 02 02 This book explores the development of navigation in the eighteenth and nineteenth centuries. It examines the role of men of science, seamen and practitioners across Europe, and the realities of navigational practice, showing that old and new methods were complementary not exclusive, their use dependent on many competing factors. 08 02 \"Higgitt, Dunn and their learned authors present a fascinating alternative history of longitude, latitude and navigation ... Historians of science and empire, maritime and physical histories will want it on shelves as soon as possible.\" - Alison Bashford, Jesus College, University of Cambridge, UK 13 02 Richard Dunn is Senior Curator of the History of Science at the National Maritime Museum in Greenwich, UK, where he has worked since 2004. Rebekah Higgitt is Lecturer in History of Science at the University of Kent, UK, and formerly Curator of History of Science at the National Maritime Museum, Greenwich. Dunn and Higgitt are co-authors of Finding Longitude: How Clocks and Stars Helped Solve the Longitude Problem (2014).

Mathematics is not just the product of lone geniuses dreaming up ideas remote from everyday life: real social, political and cultural problems drive developments in the subject, and the work of mathematical practitioners surrounds us everywhere we look. This lavishly illustrated book accompanies the Science Museum's landmark new Mathematics: The Winton Gallery. It includes an absorbing series of essays by world-leading experts in the history and modern practice of mathematics, alongside vivid accounts of mathematical work underpinning some of our most fundamental human concerns, from life and death to war and peace, money, trade, beauty and our attempts to control nature's most elemental forces. 'Mathematics' uses the world-class collections of the Science Museum to offer a vivid vision of the people behind 400 years of mathematical practice, driven by problems that affect us all.

\"Galileo famously wrote that the book of nature is written in mathematical language. The Language of Mathematics is a wide-ranging and beautifully illustrated collection of short, colorful histories of the most commonly used symbols in mathematics, providing readers with an engaging introduction to the origins, evolution, and conceptual meaning of each one. In dozens of lively and informative entries, Raúl Rojas shows how today's mathematics stands on the shoulders of giants, mathematicians from around the world who developed mathematical notation through centuries of collective effort. He tells the stories of such figures as al-Khwarizmi, René Descartes, Joseph-Louis Lagrange, Carl Friedrich Gauss, Augustin-Louis Cauchy, Karl Weierstrass, Sofia Kovalevskaya, David Hilbert, and Kenneth Iverson. Topics range from numbers and variables to sets and functions, constants, and combinatorics. Rojas describes the mathematical problems associated with different symbols and reveals how mathematical notation has sometimes been an accidental process. The entries are self-contained and can be read in any order, each one examining one or two symbols, their history, and the variants they may have had over time. An essential companion for math enthusiasts, The Language of Mathematics shows how mathematics is a living and evolving entity, forever searching for the best symbolism to express relationships between abstract concepts and to convey meaning\"-- Provided by publisher.

This essay argues that the diversity of the history of mathematics community in the United Kingdom has influenced the development of the subject and is a significant factor behind the different concerns often evident in work on the history of mathematics when compared with that of historians of science. The heterogeneous nature of the community, which includes many who are not specialist historians, and the limited opportunities for academic careers open to practitioners have had a profound effect on the discipline, leading to a focus on elite mathematics and great mathematicians. More recently, reflecting earlier developments in the history of science, an increased interest in the context and culture of the practice of mathematics has become evident.

\"This book follows the historical trail by which humanity has determined the shape and internal structure of the Earth. It is a story that bears on aspects of the history of science, the history of philosophy and the history of mathematics. At the heart of the narrative is the important philosophical practice of performing thought experiments -- that is, the art of considering an idealized experiment in the mind. This powerful technique has been used by all the great historical practitioners of science and mathematics, and this book looks specifically at the long history of considering what would happen if an object could be dropped into a tunnel that cuts all the way through the Earth's interior. Indeed, the story begins with a historical whodunit, tracing back through the historical literature the origins of what is now a classic, textbook problem in simple harmonic motion\"-- Provided by publisher.

During the 1970s and 1980s, a team of Automated Theorem Proving researchers at the Argonne National Laboratory near Chicago developed the Automated Reasoning Assistant, or AURA, to assist human users in the search for mathematical proofs. The resulting hybridhumans+AURAsystem developed the capacity to make novel contributions to pure mathematics by very untraditional means. This essay traces how these unconventional contributions were made and made possible through negotiations between the humans and the AURA at Argonne and the transformation in mathematical intuition they produced. At play in these negotiations were experimental practices, nonhumans, and nonmathematical modes of knowing. This story invites an earnest engagement between historians of mathematics and scholars in the history of science and science studies interested in experimental practice, material culture, and the roles of nonhumans in knowledge making.