The History of Mathematics
eBook - ePub

The History of Mathematics

A Brief Course

Roger L. Cooke

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eBook - ePub

The History of Mathematics

A Brief Course

Roger L. Cooke

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Praise for the Second Edition

"An amazing assemblage of worldwide contributions in mathematics and, in addition to use as a course book, a valuable resource... essential."

This Third Edition of The History of Mathematics examines the elementary arithmetic, geometry, and algebra of numerous cultures, tracing their usage from Mesopotamia, Egypt, Greece, India, China, and Japan all the way to Europe during the Medieval and Renaissance periods where calculus was developed.

Aimed primarily at undergraduate students studying the history of mathematics for science, engineering, and secondary education, the book focuses on three main ideas: the facts of who, what, when, and where major advances in mathematics took place; the type of mathematics involved at the time; and the integration of this information into a coherent picture of the development of mathematics. In addition, the book features carefully designed problems that guide readers to a fuller understanding of the relevant mathematics and its social and historical context. Chapter-end exercises, numerous photographs, and a listing of related websites are also included for readers who wish to pursue a specialized topic in more depth. Additional features of The History of Mathematics, Third Edition include:

  • Material arranged in a chronological and cultural context
  • Specific parts of the history of mathematics presented as individual lessons
  • New and revised exercises ranging between technical, factual, and integrative
  • Individual PowerPoint presentations for each chapter and a bank of homework and test questions (in addition to the exercises in the book)
  • An emphasis on geography, culture, and mathematics

In addition to being an ideal coursebook for undergraduate students, the book also serves as a fascinating reference for mathematically inclined individuals who are interested in learning about the history of mathematics.

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Part I
What is Mathematics?
This first part of our history is concerned with the “front end” of mathematics (to use an image from computer algebra)—its relation to the physical world and human society. It contains some general considerations about mathematics, what it consists of, and how it may have arisen. This material is intended as an orientation for the main part of the book, where we discuss how mathematics has developed in various cultures around the world. Because of the large number of cultures that exist, a considerable paring down of the available material is necessary. We are forced to choose a few sample cultures to represent the whole, and we choose those that have the best-recorded mathematical history. The general topics studied in this part involve philosophical and social questions, which are themselves specialized subjects of study, to which a large amount of scholarly literature has been devoted. Our approach here is the naive commonsense approach of an author who is not a specialist in either philosophy or sociology. Since present-day governments have to formulate policies relating to mathematics and science, it is important that such questions not be left to specialists. The rest of us, as citizens of a republic, should read as much as time permits of what the specialists have to say and make up our own minds when it comes time to judge the effects of a policy.

Contents of Part I

1. Chapter 1 (Mathematics and Its History) considers the general nature of mathematics and gives an example of the way it can help to understand the physical world. We also outline a series of questions to be kept in mind as the rest of the book is studied, questions to help the reader flesh out the bare bones in the historical documents.
2. Chapter 2 (Proto-mathematics) studies the mathematical reasoning invented by people in the course of solving the immediate and relatively simple practical problems of administering a government or managing a construction site. In this area we are dependent on archaeologists and anthropologists for the historical information available.
Chapter 1
Mathematics and its History
. . .all histories, to the extent that they contain a system, a drama, or a moral, are so much literary fiction.
Those who cannot remember the past are condemned to repeat it. (Often misquoted as “Those who do not learn from history are doomed to repeat it.”)
George Santayana (1863–1952), Spanish–American philosopher
(born Jorge Agustin Nicolás Ruiz de Santayana y Borrás)
The history of mathematics is a hybrid subject, taking its material from mathematics and history, sometimes invoking other areas such as psychology, political history, sociology, and philosophy to give a detailed picture of the development of mathematics. Obviously, no one can be an expert in all of these areas, and some compromises have to be accepted. Especially in an introductory course, it is often necessary to oversimplify both the mathematics itself and the social and historical context in which it arose so that the most significant portions can be included. No history of the subject that covers more than a narrow band of time can aim for anything like completeness.

1.1 Two Ways to Look at the History of Mathematics

One of the most distinguished historians of mathematics, Ivor Grattan-Guinness (b. 1941), has made a distinction between history and heritage. History asks the question “What happened in the past?” Heritage asks “How did things come to be the way they are?” Obviously, the first of these two questions is more general than the second. Many things happened in the past that had no influence on the current shape of things, not only in mathematics but in all areas of human endeavor, including art, music, and politics. Such events are history, but not heritage. The study of history in this sense is a purely intellectual exercise, not aimed at any applications, nor to teach a moral, nor to make people better citizens. What it does aim at is getting an accurate picture of the past for the edification of those who have a taste for such knowledge. It is difficult to write such a history, as the first epigram from George Santayana given above shows.
Even on the most impersonal, objective level, we don't want the raw, unedited past, which is a raging tsunami of sneezes and hiccups; some judgment is needed to select the events in the past that are of interest. To that extent, Santayana's implication is correct: All history is literary fiction. The danger for the historian lies in trying to frame a particular picture of the past in order to make it tell the story that one personally would like to hear. In the history of mathematics, there is a special danger because the mathematics itself fits together in a very logical way, while the routes by which it has been discovered and developed have all the illogical disorder that is inherent in any process involving human thinking. For example, it is known that there is no finite algebraic formula involving only arithmetic operations and root extractions that will yield a root of every quintic equation when the coefficients of that equation are substituted for its variables. This result follows very neatly from what we now call Galois theory, after Evariste Galois (1811–1832), who first introduced its basic ideas. It is nowadays always proved using this technique. But the theorem was first stated and given a semblance of a proof by Paolo Ruffini (1765–1822) and Niels Henrik Abel (1802–1829), neither of whom knew Galois theory. They both proceeded by counting the number of different values that such a hypothetical formula would generate if all possible values were substituted in the formula for each nth root it contains. This example is typical of many cases in the history of mathematics, where the proof of a proposition resulted not from rigorously arranged steps following in logical order from one another, but from a number of independent ideas gradually coming into focus.

1.1.1 History, but not Heritage

During the fifteenth and sixteenth centuries, tables of sines were used to simplify multiplication and reduce it to addition and subtraction. This procedure was called prosthaphæresis, from the Greek words prosthairesis (πρoσθα ´ιρεσις), meaning taking toward, and aphairesis (α,ϕα ´ιρεσις), meaning taking away. This technique disappeared almost without a trace after the discovery of logarithms in the early seventeenth century, and it is nowadays unknown even to most professional mathematicians. Nevertheless, it was an important idea in its time and deserves to be remembered. We shall take the time to discuss it and practice it a bit. As we shall see, it is actually more efficient than logarithms for computing the formulas of spherical trigonometry.

1.1.2 Our Mathematical Heritage

The appeal of history is to a person of a particular “antiquarian” bent of mind. Heritage, which is parasitic upon history, has a somewhat more practical aim: to help us understand the world that we ourselves live in. This is the “useful” part of history that historians advertise to the public to gain support, and it is the point of view expressed in the second of Santayana's epigrams at the beginning of this lecture. (Notice that the two epigrams taken together imply that the human race needs a variety of history that is actually literary fiction.)
If you have taken a course called “modern algebra,” for example, you found yourself confronted with a collection of abstract objects—groups, rings, fields, vector spaces—that seemed to have nothing in common with high-school algebra except that they required the use of letters. How did these abstract subjects come to be referred to as algebra? By tracing the story of the unsolvability of the general equation of degree five, we can answer this question.
After algebraic formulas were found for solving equations of degree 3 and 4 in the sixteenth century, two centuries were spent in the quest for a mathematical “Holy Grail,” an algebraic formula to solve the general equation of degree 5. Some people thought they had succeeded; but in the late eighteenth and early nineteenth centuries, Ruffini, Abel (one of those who for a time thought he had succeeded in finding the formula), and William Rowan Hamilton (1805–1865) were able to show that no such formula could exist. The question then arose of determining which equations could be solved by algebraic operations (the operations of arithmetic, together with the extraction of roots) and which could not. The answer to this question, as shown by Galois, depends on the abstract nature of a certain set of permutations of the roots. This was the beginning of the study of groups, a word first used by Galois. The concept of an abstract group arose some decades later, along with the rest of these abstract creations, all of which found numerous applications in other areas of mathematics. The original problem that gave rise to much of this modern algebra was, in the end, only one part of the vast edifice of modern algebra.

1.2 The Origin of Mathematics

The farther we delve into the past, the more we find mathematics entangled with accounting, surveying, astronomy, and the general administration of empires. Mathematics arises wherever people think about the physical world or about the world of ideas embodied in laws and even theology. It grows like a plant, from a seed that germinates and later ramifies to produce roots, branches, leaves, flowers, and fruit. It is constantly growing.

1.2.1 Number

It seems nearly certain that the small positive integers, the kinds of numbers that are intuitively known to everyone, are the “seed” of mathematics. Essentially all mathematical concepts can be traced ultimately to the use of numbers to explain the world. Numbers seem to be a universal mode of human thought. They were probably used originally in a kind of informal accounting, when it was necessary to keep track of objects that could be regarded as interchangeable, such as the cattle in a herd. Through anthropology, archaeology, and written texts, we can trace a general picture of arithmetical progress in handling such discrete collections, from counting, through computation, and finally to abstract number theory. Many different cultures have shown a convergent development in this area, although in the final stage there is considerable variety in the choice of topics developed. Through this history, we shall gradually introduce the properties of numbers in the chapters that follow. At the moment, we take note of just one important property that they have, namely that discrete collections can be exactly equal: If I have $9845.63 in my checking account, and you have $9845.63 in your checking account, then we have exactly the same amount of money, for all financial purposes whatsoever. When you count—votes, pennies, or attendance at a football game—it is at least theoretically possible to get the outcome exactly right, with no error at all.

1.2.2 Space

While discrete collections are naturally handled through counting, nature presents us with the need to measure quantities that are continuous rather than discrete, quantities such as length, area, volume, weight, time, and speed. Number is invoked to solve such problems; but in each case, it is necessary to choose a unit and regard the continuous quantity as if it were a discrete collection of units. Doing so adds a layer of complication, since the unit is arbitrary and culturally dependent. When people from different groups meet and talk about such quantities, they need to reconcile their units.
The essence of continuous quantities is that they can be divided into pieces of arbitrarily small size. A continuous measurement therefore always has precision limited by the size of the unit chosen. Equality of two continuous objects of the same kind is always approximate, only up to the standard unit of measurement in which their sizes are...

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