Cover

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Title Page, Copyright

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pp. i-iv

Contents

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pp. v-vi

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Preface

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pp. vii-x

...The purpose of this volume is to bring together a number of different perspectives on modern mathematics, with the aim of understanding how the work of historians, philosophers, and mathematicians can be integrated in an interdisciplinary study of...

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An Opinionated Introduction

Philip Kitcher and William Aspray

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pp. 3-58

...Each of the essays that follow could be sent forth as an orphan, with only the most perfunctory comment, to make its own solitary way in the world, and some of them would undoubtedly do well. However, we believe...

I: Logic and the Foundations of Mathematics

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Poincaré against the Logicists

Warren Goldfarb

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pp. 61-81

...Although the great French mathematician Henri Poincare wrote on topics in the philosophy of mathematics from as early as 1893, he did not come to consider the subject of modern logic until 1905. The attitude...

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Logical Truth and Analyticity in Carnap's "Logical Syntax of Language"

Michael Friedman

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pp. 82-94

...Throughout his philosophical career, Carnap places the foundations of logic and mathematics at the center of his inquiries: he is concerned above all with the Kantian question...

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The Emergence of First-Order Logic

Gregory H. Moore

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pp. 95-136

...To most mathematical logicians working in the 1980s, first-order logic is the proper and natural framework for mathematics. Yet it was not always so. In 1923, when a young Norwegian mathematician...

II: Reinterpretations in the History of Mathematics

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Kronecker's Place in History

Harold Edwards

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pp. 139-144

...At a conference on the history and philosophy of mathematics, it seems especially appropriate to talk about Kronecker's place in history. My objective is to show that the prevalence...

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Felix Klein and His "Erlanger Programm"

Garrett Birkhoff and M. K. Bennett

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pp. 145-176

...Felix Klein's "Erlanger Programm" (E.P.), listed in our references as (Klein 1872), is generally accepted as a major landmark in the mathematics of the nineteenth century. In his obituary biography...

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Abraham Robinson and Nonstandard Analysis: History, Philosophy, and Foundations of Mathematics

Joseph W Dauben

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pp. 177-200

...Historically, the dual concepts of infinitesimals and infinities have always been at the center of crises and foundations in mathematics, from the first "foundational crisis" that some, at least, have associated...

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How Can Mathematicians and Mathematical Historians Help Each Other?

Richard Askey

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pp. 201-218

...dropped, but I could argue both sides of that question. The present title presumes the optimistic answer, and while we all hope that this is the correct answer, the present time may not be the right one for this answer. The history of mathematics is not an easy field...

III: Case Studies in the History and Philosophy of Mathematics

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Fitting Numbers to the World: The Case of Probability Theory

Lorraine J. Daston

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pp. 221-237

...mathematics is a perpetual miracle. Neither formalist nor logicist nor Platonist (at least of the latter-day dilute variety) can provide a plausible explanation of why numbers should...

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Logos, Logic, and Logistiké: Some Philosophical Remarks on Nineteenth-Century Transformation of Mathematics

Howard Stein

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pp. 238-259

...so profound that it is not too much to call it a second birth of the subject— its first birth having occurred among the ancient Greeks, say from the sixth through the fourth century...

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Ten Misconceptions about Mathematics and Its History

Michael J. Crowe

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pp. 260-277

...teaching, and writing history of mathematics. During those decades, I have become convinced that ten claims I formerly accepted concerning mathematics and its development are both seriously wrong and a hindrance to the historical study of...

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Mathematics and the Sciences

Felix E. Browder

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pp. 278-292

...interaction between mathematics and the sciences and to relate the trends to the historical development of mathematics as an intellectual discipline and of the sciences as they have developed since the seventeenth century. This story is interesting...

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Mathematical Naturalism

Philip Kitcher

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pp. 293-326

...century has been concerned with the enterprise of providing a foundation for mathematics. There is no doubt that this enterprise has often been mathematically fruitful. Indeed, the growth of logic as an important field within mathematics owes much to the pioneering...

IV: The Social Context of Modern Mathematics

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Partisans and Critics of a New Science: The Case of Artificial Intelligence and Some Historical Parallels

Judith V Grabiner

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pp. 329-345

...in mathematics courses—we see the presence of the computer. We are told that we are entering a new age, that of the Computer Revolution, and that the field known as Artificial Intelligence is at the forefront of that revolution. Its practitioners have...

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The Emergence of Princeton as a World Center for Mathematical Research, 1896-1939

William Aspray

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pp. 346-366

...University, reflecting its ambitions for graduate education and research. At the time, Princeton, like other American universities, was primarily a teaching institution that made few significant contributions...

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Contributors

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pp. 367-370

...Richard Askey, Professor of Mathematics, University of Wisconsin at Madison William Aspray, Associate Director, Charles Babbage Institute for the History of Information Processing, University of Minnesota M. K. Bennett, Professor of Mathematics, University of Massachusetts at Amherst...

Index

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pp. 371-386

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About the Author

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pp. 387-387

...Currently a professor of philosophy at the University of California, San Diego, Philip Kitcher has also taught at Vassar College, the University of Vermont, and the University of Minnesota, where he served...