Über die Autorin bzw. den Autor

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.

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HOW MATHEMATICS HAPPENED

Prometheus Books

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

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...