An Introduction to the Early Development of Mathematics

Michael K. J. Goodman

• Publisher: Wiley
• Year: 2016
• ISBN:
  9781119105053
• Edition: 1
• Topic: Mathematics
• Subtopic: History & Philosophy of Mathematics
• Index: Mathematics

An easy-to-read presentation of the early history of mathematics

Engaging and accessible, An Introduction to the Early Development of Mathematics provides a captivating introduction to the history of ancient mathematics in early civilizations for a nontechnical audience. Written with practical applications in a variety of areas, the book utilizes the historical context of mathematics as a pedagogical tool to assist readers working through mathematical and historical topics.

The book is divided into sections on significant early civilizations including Egypt, Babylonia, China, Greece, India, and the Islamic world. Beginning each chapter with a general historical overview of the civilized area, the author highlights the civilization's mathematical techniques, number representations, accomplishments, challenges, and contributions to the mathematical world. Thoroughly class-tested, An Introduction to the Early Development of Mathematics features:

• Challenging exercises that lead readers to a deeper understanding of mathematics
• Numerous relevant examples and problem sets with detailed explanations of the processes and solutions at the end of each chapter
• Additional references on specific topics and keywords from history, archeology, religion, culture, and mathematics
• Examples of practical applications with step-by-step explanations of the mathematical concepts and equations through the lens of early mathematical problems
• A companion website that includes additional exercises

An Introduction to the Early Development of Mathematics is an ideal textbook for undergraduate courses on the history of mathematics and a supplement for elementary and secondary education majors. The book is also an appropriate reference for professional and trade audiences interested in the history of mathematics.

Michael K. J. Goodman is Adjunct Mathematics Instructor at Westchester Community College, where he teaches courses in the history of mathematics, contemporary mathematics, and algebra. He is also the owner and operator of The Learning Miracle, LLC, which provides academic tutoring and test preparation for both college and high school students.

1
INTRODUCTION

This book is not a comprehensive history of mathematics. There are excellent books, much longer than this one, that present a thorough survey of ancient math. There are also more specialized books that go into great depth about certain topics (the history of algebra, the development of numerals, etc.) or certain civilizations (math in ancient Greece, math in ancient China, etc.). I hope that you are interested to look at these other books one day, and some of them surely are in your college library right now.

When I start my course, I tell the students that this is obviously a math course. It’s offered within the Math Department, and the students will solve math problems. Some of the problems are practical, and some of the problems are exercises that students had to do in their schools in Egypt and Babylonia long, long ago. It’s a math class.

It’s also a history class because what we’re exploring here is where ancient mathematical ideas came from, what kinds of problems these old civilizations needed to solve, and how mastering math helped them develop their agriculture, engineering, government, economic, military, and social systems. So it’s a history class.

I tell my students it’s also a bit of an art class. Not everybody could read and write in the ancient world. In fact, it was a select minority who could. And a long period of training and education was required to train the scribes who did the counting and computing and administrative work that made ancient Egypt, ancient Babylonia, and ancient China run. I have my students practice making the numeric symbols that these old civilizations used, just as if they were in a scribe school, and this takes patience and accuracy.

It’s also a bit of an archeology class and an anthropology class. If we’re talking about the mathematics used by old civilizations, we need to know a little bit about these civilizations, and generally students get curious about how we know what we know about these civilizations—that’s where the archeology comes in. And if we go further back in history, to the nomadic, tribal, primitive kinds of societies that lived before the agricultural revolution, we get into the realm of anthropology—how and why did people even begin to think mathematically? Remarkably, there are groups of people around the world who were so isolated until modern times that we can use what we know about them to make reasonable guesses about our quite remote human ancestors.

I even tell my students this is a bit of an English class, because I ask them to write a short paper about a mathematician or about a bit of mathematics. Basically I’m looking for one idea that can be linked to one mathematician. Students can choose something from long ago (like Archimedes working out an approximation of π) or something modern (like Mandelbrot developing fractals) or something in between (like Omar Khayyam solving cubic equations). I look for clear writing that explains the idea, describes the world the mathematician lived in, and relates the idea to mathematical knowledge that came before it or grew out of it.

So, don’t expect everything from this slender book. It is a starting point. I have my own favorite websites about the history of math, but your professor may have his (or hers) and there are many good ones. I show my students videos of people making cuneiform tablets and of people building models of Platonic solids. I show the Mayan codex books that survived and I also show the Hindu derivation of the square root of 2. I read out loud to the class the view of the Crow Indians that honest people don’t need numbers larger than 1000.

So I urge you to read, think, and be curious. You’ve got an electronic calculator at your fingertips and the advantage of a clever base-10 system with 10 simple digits. But put yourself in the position of someone thousands of years ago, who needed to solve an arithmetic problem and didn’t have these wonderful and convenient things. Put yourself in that position often enough this semester, and you’ll have a profound understanding of the history of math and the development of mathematical thought.

2
MATHEMATICAL ANTHROPOLOGY

In the chapters to come, this book will focus on major ancient civilizations and the development of math in those civilizations. You probably already anticipate chapters on Egypt, Babylonia, and China. But it is natural to wonder: what came before that? Where did the very earliest mathematical ideas come from?

It turns out we have some information. Anthropologists have found, during the past few centuries, dozens of isolated societies that live (or lived) with Stone Age technology. These societies were found in remote areas, like Pacific islands and tropical jungles. These societies got along perfectly well, for thousands of years, without contact with the rest of the world.

From the study of these societies, anthropologists are able to make an informed guess about the lives of very early humans. We might think of them as cavemen or hunter-gatherers, or primitive people, but whatever label we give them we have to recognize a few key things about them:

• They were around for a very long time. The paleontological evidence shows humans with our brains and our capacities were living for tens of thousands of years before recorded history.
• They included an enormous range of cultures, all different from each other. The aborigines of Australia included people who spoke very different and mutually unintelligible languages. These aborigines were quite unlike their contemporaries in New Guinea and Tasmania, and even more unlike their contemporaries in Indonesia, Micronesia, and southeast Asia. All these people had decidedly different lifestyles and cultures than the people who lived in the rest of Asia, in Africa, Europe, the Americas, and the Arctic.
• They knew enough to thrive in their varied environments. They did not starve, die from diseases, get completely killed off by other tribes, or go extinct because of any of the hazards they faced. They survived.

Part of what kept them alive is that they could count.

There were a lot of things to count. Are we all here? is an important question. Do we have all our stuff? is another. How many fish do we need to catch? How many days will it take to walk to the next place? How many people are in that band of strangers you saw? (If we outnumber them, they probably are not a threat to us.)

The ability to count and communicate numbers gave a group a big advantage over a group that couldn’t count and communicate.

The earliest evidence of counting is notched bones and sticks. One famous bone, from Africa about 20,000 years ago, appears to have been cut so that the person holding it could use the notches to count something. Was he counting animals, people, days, or something else? That, we don’t know. But the regular, evenly spaced notches suggest that this tool was used again and again for the purpose of counting. The clusters of notches come in intriguing quantities (like 11, 13, 17, and 19) and have engendered much analysis and speculation about what early people may have known about numbers.

It also makes sense that some fairly large number had to be counted. All those primitive societies anthropologists found on islands and in jungles counted on their fingers (and sometimes toes and other body parts if fingers weren’t enough). There would be no need to cut dozens of notches into a bone (and perhaps carry the bone around from place to place) if the number of things that had to be counted was only a small number. So the counting bone was probably a very important possession.

Since we are speculating about prehistory, we cannot know just how and when certain mathematical ideas arose, and whether they came all together or one by one. We can’t be certain how long it took to develop an idea, and the order in which the ideas came. What we can say with some confidence is that these ideas were discovered again and again, in different bands and tribes and societies, because of the isolation of people. It is likely that the world’s population 10,000–30,000 years ago consisted of a very large number of very small groups who knew on...