Mathematics

Arts of Calculation: Numerical Thought in Early Modern Europe by David Glimp (Editor), Michelle R. Warren (Palgrave Macmillan) The essays in this volume focus primarily on 16th and 17th century Europe and are broadly interdisciplinary. They answer questions such as: what kinds of cultural work do numbers do?; What roles does calculation play in colonial, imperial, and/or national projects or ideologies?; What are the relationships between aesthetic practices and bureaucratic modes of calculation (such as accounting, census taking, and demography)?; What kinds of agencies and subjectivities do numbers and numbering enable and foreclose?; How do different kinds of economic strategies (eg exchange, gift, capital, debt, interest) affect representational strategies and vice versa?; What cultural dynamics inform spatial measurements, such as cartography?; and what do analyses of counting practices tell us about the production of knowledge more generally?

Excerpt: Though the sixteenth and seventeenth centuries have long been recognized as watershed moments of scientific discovery in Western Europe, recent work in "historical epistemology" has underscored the complexity and unevenness of the transition from older to newer forms of knowing and acting upon the world. Building on these insights, the essays that make up Arts of Calculation extend our understanding of how people come to count--how numbers create forms of agency, objects of inquiry, and kinds of cultural authority. This multidisciplinary collection traces a powerful convergence of numerical thought across disciplinary and national boundaries.

There is little doubt that the act of quantifying—of representing things in numbers—is fundamental to the development of human civilization, as we know it. Scientific discovery, geographical exploration, commercial trade, architecture, and so forth, all depend on our ability to calculate "things" such as distances, directions, values, masses, spaces, stresses, and weights. The earliest evidence also suggests that we have been engaged in various forms of counting and calculating from the beginning. It seems to be essential to the way we order the world. Arguably, calculating things is part of what it means to be human.

Despite the efforts of deconstruction and poststructuralism over the last three decades, there are still many who understand the process of quantification as a fundamentally objective description of the world. Whether we talk about bookkeeping practices, engineering principles, or mapping procedures, there is a strong sense that these acts of quantification more or less reflect or embody the financial, physical, or geographical reality of the world as it is today. If these and other quantifying acts indeed were inherently neutral and objective, there would be no reason for a study on "calculation" in the early modern period to be published in a series of books dedicated to exchanges between culture and various social practices, including various forms of artistic production.

*Arts of Calculation:
Quantifying Thought in Early Modern Europe* is worthy of consideration, not
because it reexamines the mysterious study of numerology, or the meaning of
number patterns in Renaissance son-net sequences, or ponders the religious
significance of the number 3 in medieval lyrics (as earlier scholarship did),
but because editors David Glimp and Michelle R. Warren and all the essayists
gathered in this volume hold that "particular forms of quantification are
historically and culturally specific." How things are calculated or how
calculations are deployed to bolster a specific claim or determine an outcome
are not culturally neutral. Quantifying activities are interested activities,
and acts of calculation are calculating acts.

This is not to suggest there existed some kind of conspiracy of a small number of powerful individuals who skewed the numbers or misinterpreted results (although there were of course individuals like that); rather Arts of Calculation seeks to contribute to a much broader "historical epistemology" in early modern Europe by examining the role played by quantifying practices in knowledge production. The contributors to Arts of Calculation investigate how early modern rulers, philosophers, bureaucrats, military men, scholars, playwrights, and poets contributed to, thought about, resisted, or dramatized the development and proliferation of quantification. The essays here range widely, from Robert Batchelor's attempt to link Leibniz's interest in binary numbers with a form of epistemological imperialism, to Eugene Ostashevsky's discussion of Shakespeare's novel employment of Roman and Hindu-Arabic numbers in Henry V to signal a shift away from a referential use of numbers. Essays by Joel Kaye and Alina Sokol explore how the increase in use of money generated new modes of thought and perception in Europe and gave rise in seventeenth-century Spain to an "economization of value" and a crisis in the nexus between a thing and its value. Carla Mazzio counters claims for the homogenizing effect of referential numbers on people and argues instead that they disoriented people and fractured their understanding of the world. Gordon Hull explains the supportive role played by geometry in the political thought of Hobbes, while Timothy Reiss identifies a myriad ways in which "Colonial calculus" aided colonial leaders in the seizure, control, and exploitation of foreign territories. Like Reiss, Patricia Cahill is interested in the "calculable" human being of the early modern period who, in the context of the military (in Christopher Marlowe's Tamburlaine), becomes a person defined with arithmetical precision in relationship to other "calculable" humans. Christopher Johnson writes on the use of numbers to evoke an emotional response and thereby persuade the reader of a certain point of view. Benjamin Liu offers a wonderful account of a fifteenth-century Spanish tailor who accepted poems as payment for his labors in a way that signals the decline of the economy of the gift and anticipates the economy of monetary payment.

What makes Arts of Calculation
such a fascinating volume is the sheer variety of the essays. Like Linda
Woodbridge's *Money and the Age of Shakespeare*, *Arts of Calculation*
employs new methods of analysis and asks new questions that should help us
arrive at a richer and more nuanced understanding of the "early modern culture"
that we so often refer to casually as if its meaning is clear, stable, and
widely agreed upon. Mirroring the complexity of early modern culture, this
volume's variety amply underscores not only the fragmented nature of that
culture but also articulates how conflicting ways of calculation that gave rise
to social shifts and divisions could stem from a single, potentially unifying,
impulse or set of conditions. -Ivo Kamps, series editor

*
History of Mathematics *by Jeff Suzuki (Prentice Hall) The author of a text on the
history of mathematics is faced with a difficult question: how to handle the
mathematics? There are several good choices. One is to give concise descriptions
of the mathematics, which allows many topics to be covered. Another is to
present the mathematics in modern terms, which makes clear the connection
between the past and the present. There are many excellent texts that use either
or both of these strategies.

This book offers a third choice, based on a simple philosophy: the best way to understand history is to experience it. To understand why mathematics developed the way it did, why certain discoveries were made and others missed, and why a mathematician chose a particular line of investigation, we should use the tools they used, see mathematics as they saw it, and above all think about mathematics as they did.

Thus to provide the best understanding of the history of mathematics, this book is a mathematics text, first and foremost. The diligent reader will be classmate to Archimedes, al Khwarizmi, and Gauss. He or she will be looking over Newton's shoulders as he discovers the binomial theorem, and will read Euler's latest discoveries in number theory as they arrive from St. Petersburg. Above all, the reader will experience the mathematical creative process firsthand to answer the key question of the history of mathematics: how is mathematics created?

In this text I have emphasized:

1. Numeration, computation, and notation. Notation both limits and guides: Limits, because it is difficult to think "outside the notation"; guides, because a good system of notation can suggest relationships worthy of further study. As much as possible, I avoid the temptation to "translate" a mathematical result into modern (mathematical) language or notation, for modern notation brings modern ways of thinking. In a similar vein, the means of computing and expressing numbers guides what discoveries one may make, and ultimately influences the direction taken by mathematicians.

2. Mathematical results in their original form with their original arguments. The most dramatic change in mathematics over the past four millennia has been the standard of proof. What was acceptable to Pythagoras is no longer be acceptable today. The binomial theorem was, when Newton proposed it, nothing more than a conjecture, and while most of Euler's results on series summation are accepted today, his methods are not. But to provide a modern proof of a result would be historically inaccurate, while to omit how these results were supported would be to neglect a vital part of the history of mathematics. To protect modern sensibilities, I will distinguish between proofs, where the term is used in the modern sense, and demonstrations, which contain some elements no longer acceptable in a modern context, and if a non‑obvious result is presented without a proof, it will be labeled a conjecture.

3. Mathematics as an evolving science. The most important thing we can learn from a history of mathematics is that mathematics is created by human beings and not by semimythical demigods. The ideas of Newton, Euler, Gauss, and others originated from the mathematics they knew and the problems they saw around them; they made great contributions, but they also made mistakes, which were faithfully replicated (or caustically reviewed) by fellow mathematicians. Finally, no mathematical idea is born, fully mature and in the modern form: we will follow, where space and time permit, the development of mathematical ideas, through their birth pains, their early, formative years, and onwards to their early maturity and final, modern form.

To fully enter into all of the above for all of mathematics would take a much larger book, or even a multi‑volume series: a project for another day. Thus, to keep the book to manageable length, I have restricted its scope to what I deem "elementary" mathematics: the fundamental mathematics every mathematics major and every mathematics teacher should know. This includes numeration, arithmetic, geometry, algebra, calculus, real analysis, and the elementary aspects of abstract algebra, probability, statistics, number theory, complex analysis, differential equations, and some other topics that can be introduced easily as a direct application of these "elementary" topics. Advanced topics that would be incomprehensible without a long explanation have been omitted, as have been some very interesting topics that, through cultural and historical circumstances, had no discernible impact on the development of modern mathematics.

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