How Mathematics Happened: The First 50,000 Years - Hardcover

Peter S. Rudman

Hardcover

ISBN 10: 1591024773 ISBN 13: 9781591024774

Publisher: Prometheus, 2006

Synopsis

In this fascinating discussion of ancient mathematics, author Peter Rudman does not just chronicle the archeological record of what mathematics was done; he digs deeper into the more important question of why it was done in a particular way. Why did the Egyptians use a bizarre method of expressing fractions? Why did the Babylonians use an awkward number system based on multiples of 60? Rudman answers such intriguing questions, arguing that some mathematical thinking is universal and timeless. The similarity of the Babylonian and Mayan number systems, two cultures widely separated in time and space, illustrates the argument. He then traces the evolution of number systems from finger counting in hunter-gatherer cultures to pebble counting in herder-farmer cultures of the Nile and Tigris-Euphrates valleys, which defined the number systems that continued to be used even after the invention of writing. With separate chapters devoted to the remarkable Egyptian and Babylonian mathematics of the era from about 3500 to 2000 BCE, when all of the basic arithmetic operations and even quadratic algebra became doable, Rudman concludes his interpretation of the archeological record. Since some of the mathematics formerly credited to the Greeks is now known to be a prior Babylonian invention, Rudman adds a chapter that discusses the math used by Pythagoras, Eratosthenes, and Hippasus, which has Babylonian roots, illustrating the watershed difference in abstraction and rigor that the Greeks introduced. He also suggests that we might improve present-day teaching by taking note of how the Greeks taught math. Complete with sidebars offering recreational math brainteasers, this engrossing discussion of the evolution of mathematics will appeal to both scholars and lay readers with an interest in mathematics and its history.

About this title

About the Author

Peter S. Rudman (Haifa, Israel) is professor (ret.) of solid-state physics at the Technion-Israel Institute of Technology and the author of more than 100 articles in physics.

Excerpt. © Reprinted by permission. All rights reserved.

HOW MATHEMATICS HAPPENED

THE FIRST 50,000 YEARSBy PETER S. RUDMAN

Prometheus Books

Copyright © 2007 Peter S. Rudman
All right reserved.
ISBN: 978-1-59102-477-4

Contents

List of Figures.............................................................................7List of Tables..............................................................................11Preface.....................................................................................151. Introduction.............................................................................211.1 Mathematical Darwinism..................................................................211.2 The Replacement Concept.................................................................281.3 Number Systems..........................................................................382. The Birth of Arithmetic..................................................................492.1 Pattern Recognition Evolves into Counting...............................................492.2 Counting in Hunter-Gatherer Cultures....................................................533. Pebble Counting Evolves into Written Numbers.............................................673.1 Herder-Farmer and Urban Cultures in the Valley of the Nile..............................673.2 Herder-Farmer and Urban Cultures by the Waters of Babylon...............................823.3 In the Jungles of the Maya..............................................................1144. Mathematics in the Valley of the Nile....................................................1314.1 Egyptian Multiplication.................................................................1314.2 Egyptian Fractions......................................................................1414.3 Egyptian Algebra........................................................................1584.4 Pyramidiots.............................................................................1755. Mathematics by the Waters of Babylon.....................................................1875.1 Babylonian Multiplication...............................................................1875.2 Babylonian Fractions....................................................................2085.3 Plimpton 322-The Enigma.................................................................2155.4 Babylonian Algebra......................................................................2315.5 Babylonian Calculation of Square Root of 2..............................................2406. Mathematics Attains Maturity: Rigorous Proof.............................................2496.1 Pythagoras..............................................................................2496.2 Eratosthenes............................................................................2546.3 Hippasus................................................................................2587. We Learn History to Be Able to Repeat It.................................................2637.1 Teaching Mathematics in Ancient Greece and How We Should but Do Not.....................263Appendix: Answers to Fun Questions..........................................................273Notes and References........................................................................295Index.......................................................................................309

Chapter One

INTRODUCTION

1.1 MATHEMATICAL DARWINISM

As far as most people know or care, our number system and the arithmetic we do with it descended from Mount Sinai inscribed on the reverse side of the Ten Commandments: "Thou shalt count with the decimal system using the ten symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9; and do arithmetic thus and so."

Actually, the currently used decimal system, also called the base-10 system, evolved in India about two thousand years ago. It was introduced into Europe about one thousand years later in Latin translations of Arabic translations of Hindu texts. Because the Latin translations credited Arabic sources, these symbols became known as Arabic numerals. The symbols have changed so much over the years that neither the original Hindu, nor their Arabic transcriptions, nor the early European transcriptions are easily recognizable in the present symbols. Only after the invention of printing with movable type by Johannes Gutenberg in 1442 did numbers eventually standardize into the symbols almost universally used today. In our politically correct world, it is better to call the symbols Hindu-Arabic, and I shall refer to them so.

In different times and places, many different number systems and arithmetics evolved. The currently used decimal system and arithmetic are just the fittest that have survived by natural selection, but that does not mean that current practice will necessarily be the fittest in the future. The decimal system, which fortuitously exists because we have ten fingers, is not always the best in our electronic calculator/computer age, and other number systems are now widely used. But the decimal system is so deeply embedded in our culture that it will probably never be replaced for everyday use. However, the ubiquitous electronic calculator is rendering obsolete much of the arithmetic currently taught. When was the last time you actually did a pencil/paper multiplication of a multi-digit number by a multi-digit number?

By about 2000 BCE, at least in Egypt and Babylon, humans had learned to do all the basic arithmetic operations, although not with the same symbols or methods used today; had learned some algebra to generalize arithmetic; and had learned some geometry/trigonometry. We learn this mathematics nowadays as children, and so I call the eras up to about 2000 BCE the childhood of mathematics.

When development of the individual mirrors development of the species, this is eruditely expressed in biology jargon as "ontogeny recapitulates phylogeny." The mathematical education of a child nowadays does indeed mirror ancient historical progress in mathematics. However, rather than being analogous to any biological process, the mathematics of a child or a culture naturally starts with the easiest and most frequently used operation-counting. The next step is more generalized addition. The other arithmetic operations are just variations of addition, and hence come later.

To relate the other arithmetic operations to addition, and to introduce algebraic notation and jargon, let a and b be givens and x be the unknown to be calculated.

Addition: x = a + b. In current base-10, pencil/paper arithmetic using Hindu-Arabic numerals, we use a memorized addition table.

Subtraction: x = b - a, but in practice we usually do a + x = b, which asks what must we add to a to obtain b, and we use the same addition table.

Multiplication is just an efficient way of doing successive additions of the same number: x = ab = ba = b + b ... + b (a terms) = a + a ... + a (b terms). In current pencil/paper arithmetic, we multiply using a memorized multiplication table and tend to think of it as a unique operation, forgetting that the multiplication table had been calculated by successive additions. Some four thousand years ago, Babylonians also did multiplication using multiplication tables, but the Egyptians used a completely different method that obviated the need for multiplication tables. Somewhat ironically, an electronic calculator multiplies the oldest way, by successive additions.

Division: x = b/a, but in practice we usually do xa = b, which asks what must we multiply a by to obtain b, and we use the same multiplication table. Since multiplication is successive additions of the same number, it is sometimes convenient to think analogously of division as successive subtractions of the same number: b - xa = 0, which asks how many successive subtractions of a from b produce zero.

FUN QUESTION 1.1.1: Prove that ab = ba.

ANSWER: One way to prove this is diagrammatically:

Nowadays we tend to think abstractly about multiplication, not relating it to a geometrical diagram, but ancient cultures visualized multiplication in terms of the area of a rectangle. The visualization of generating an area as addition of rows followed naturally from the universal experience of generating a plowed field by addition of furrows. In chapter 3, we shall see how this visualization influenced the evolution of measurement units and even a number system. It is interesting and relevant to note that early Greek writing also followed the path of the plow: the first line was written right to left, the second line from left to right, the third line from right to left, and so forth.

FUN QUESTION 1.1.2: What is the length of one side of a square that has the same area as a rectangle of sides 2 and 4? This simple but not trivial visualization explains important Egyptian (see section 4.3) and Babylonian (see section 5.3) mathematical inventions.

ANSWER: See the appendix for answers not given contiguously. If you do not enjoy solving these problems, you can skip Fun Questions without contiguous answers with no loss of story continuity or essential concepts.

* * *

Despite its simplicity and obvious advantages, algebraic notation using a single letter to represent a generalized number is only of rather recent origin, in the sixteenth century. It may appear surprising that neither Euclid (325-265 BCE), nor Archimedes (287-212 BCE), nor any of the other great Greek mathematicians of antiquity invented algebraic notation. A proposed explanation is that the ancient Greeks used alphabetic symbols to write numbers, thereby usurping their use as generalized numbers.

Let us use this example of why some Greek mathematician did not invent modern algebraic notation to consider the reliability of whys of ancient mathematics. The given explanation appears to be consistent with what is known, and that is always the minimum requirement. Other than that, it is pure speculation. Its reliability depends on the proposer's mathematical intuition. I do not know who proposed this explanation or how good his mathematical intuition was, but my guess is that this why is not very reliable because there are other simple alternatives to the usurpation of the Greek alphabet by numbers. For example, Greek mathematicians were certainly aware of other alphabets, such as Latin and Hebrew, that did not use the Greek symbols. Today when mathematicians run out of Latin letters, they use Greek or Hebrew letters, or other symbols such as upside-down or backward letters. Therefore, a guess that appears to me just as probable is that Greek mathematicians were simply preoccupied with other mathematical problems, and their clumsy algebraic notation did not sufficiently bother them to motivate invention of a new notation.

Women's intuition is more hindsight than foresight, as expressed by a husband's whimsical wish, "If I only knew yesterday, what my wife knows today." But mathematical intuition is the starting point of much if not all mathematics. Consider the sums 1 + 3 = 4, 1 + 5 = 6, 3 + 5 = 8, 3 + 7 = 10.... Mathematical intuition leads us to guess that there is some rule operating here. One guess is that there is a rule: the sum of two odd numbers is an even number. In math jargon, such a guess is a conjecture. A proof turns a conjecture into a rule. It is easy to prove this conjecture and turn it into a rule.

FUN QUESTION 1.1.3: Prove that the sum of two odd numbers is an even number.

ANSWER: Let m and n be any integers so that 2m is an even number. (An even number has a divisor of 2, which 2m has, whatever the value of m.) An odd number is then 2m + 1. The sum of two odd numbers can thus be written as (2m + 1) + (2n + 1) = 2[m + n + 1], an even number, QED. QED is the acronym for quod erat demonstrandum, Latin for "which was to be demonstrated." It denotes the end of a successful proof.

However, if you have exceptional mathematical intuition, you may have also noted that the examples, 1 + 3 = 4, 1 + 5 = 6, 3 + 5 = 8, 3 + 7 = 10 ..., can be described by another conjecture: every even number is the sum of two prime numbers. (A prime number has only the number 1 and the number itself as divisors.) This intuitive guess is Goldbach's (1690-1764) conjecture. Since every prime number other than two is an odd number, it might be thought that this would also be easy to prove. No exceptions to this conjecture have ever been found, but neither has anybody been able to prove it. (See sections 5.2 and 6.2 for more on prime numbers.)

The purpose of this discussion of conjectures is to show that there is an essential role for judicious guessing in mathematics, and sometimes that is the best we can do. No document will ever be unearthed in which Archimedes explains why he did not invent algebraic notation. The archeological record only shows what was done, never why it was done. Answering the whys is the primary reason for historical study. I, or anyone else who tries to interpret the archeological record in terms of not just what was done, but also in terms of why it was so done, must rely on conjecture.

My conjectures are products of my mathematical intuition. I have required them to be consistent with archeological evidence, but also to be consistent with natural and intuitive mathematical thinking that I show is universal and timeless. I believe that I have given not just plausible whys, but probable whys. Others, with new archeological data, and/or better versed in archeological interpretation, and/or with better mathematical intuition, may be able to produce whys that are more probable. But whether my whys are right or wrong, it is in the search for them that the fun and mathematical insights reside.

* * *

The similarity of the learning sequence of a child to the sequence of historical progress in mathematics is only superficially analogous to any biological process, but the evolution of number systems and arithmetic is truly mathematical Darwinism. An invention that improves a number or arithmetic system is analogous to an enhancing genetic variation in a biological system. In both cases chances are better that the system will survive and propagate.

The game of survival-of-the-fittest mathematics that has been going on for millennia has taken a new twist recently. A new mathematical technique called genetic programming generates new solutions by mathematical Darwinism. Rather than limiting solutions to methods that our brains are capable of inventing, random variations in existing solutions generate new solutions. The fittest are selected for survival by defining criteria for what is a better solution. The analog of sexual reproduction is the generation of an offspring solution that combines properties of parent solutions. The analog of mutation is random changes in parts of solutions. Of course, a computer generates this mathematical evolution.

The success of genetic programming is an interesting validation of Darwin's (1809-1882) theory of evolution, and the evolution of the brain in particular. Mathematician Alan Turing (1913-1954), one of the seminal thinkers in computer science, invented the genetic programming concept in 1950, although it was only implemented in the 1990s. In considering the question of whether a computer could ever duplicate a human's brain, Turing conjectured that computer capability would eventually have to increase by the same mechanism that the capability of the human brain has increased, by genetic variation and survival of the fittest.

1.2 The Replacement Concept

In order to understand the nondecimal number systems that play an important role in ancient and modern arithmetic, I briefly introduce some simple mathematical concepts in this and the following section.

The finger counting we all first used as children assigns a value of one to each finger, and the count is the sum of extended fingers. This simple additive number system, limited to counting up to ten, is possibly the natural and intuitive way humans first started to count.

FUN QUESTION 1.2.1: In additive finger counting where all fingers have a value of one, represent an extended finger by the symbol 1 and a nonextended finger by the symbol 0. How many ways can the number one be exhibited? How many ways can the number nine be exhibited? How many ways can the number two be exhibited?

ANSWER: There are ten ways of exhibiting the number one:

10000 00000 01000 00000 00100 00000 00010 00000 00001 00000

00000 10000 00000 01000 00000 00100 00000 00010 00000 00001

See the appendix for remaining answers.

This is a very redundant counting system, which means that many recognizably different finger arrangements, or equivalently symbolic arrangements, have not been exploited to count higher. Clearly, it must be possible to count higher than ten with just ten fingers.

One method of counting higher without adding more fingers is to let each finger on the right hand still have a value of one, but to let each finger on the left hand have a value of five. Each time you count five on the right hand, replace the five extended fingers on the right hand by just one extended finger on the left hand (a 1-for-5 replacement). Now you can count to thirty (five on the right hand plus twenty-five on the left hand) with just ten fingers. Whether a finger is on the right or left hand determines its value, and so such finger counting is positional. A positional system is also called a place-value system. It is implicit in the definition of positional systems that the values at different positions are added. In this example, whatever its position on a given hand, a finger's value is the same, and the value of the hand is calculated additively. To be more precise, this counting system is both additive and positional.

FUN QUESTION 1.2.2: Consider the additive-positional finger-counting system where all fingers on the right hand have a value of one but all fingers on the left hand have a value of five. Again, represent an extended finger by the symbol 1, and a nonextended finger by the symbol 0. How many ways can the number six be exhibited? How many ways can the number twenty-four be exhibited?

ANSWER: There are twenty-five ways of exhibiting the number six. For each exhibition of the left hand, there are five ways of exhibiting the right hand: 10000, 01000, 00100, 00010, 00001. Since there are the same five possible exhibitions of the left hand, the total number of exhibitions is 5 ?5 = 25. See the appendix for the other answer.

This solution is an example of the multiplication rule of combinatorial mathematics, which states that if one arrangement can be done in M ways and another arrangement can be done independently in N ways, then both together can be done in MN ways. This is a very useful rule that I shall use again to solve other problems.

FUN QUESTION 1.2.3: Give a value of one to each finger on the right hand and a value of five only to the first finger on the left hand; assign values to the four remaining fingers on the left hand to maximize the counting limit. What is the maximized counting limit?

ANSWER: The maximized counting limit is 160. If you do not get this answer, see the appendix.

(Continues...)

Excerpted from HOW MATHEMATICS HAPPENEDby PETER S. RUDMAN Copyright © 2007 by Peter S. Rudman. Excerpted by permission.
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Edition details

• Publisher: Prometheus
• Publication date: 2006
• Language: English
• ISBN 10: 1591024773
• ISBN 13: 9781591024774
• Binding: Hardcover
• Number of pages: 316