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How Mathematics Happened: The First 50,000 Years
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.
316 pages, Hardcover
First published October 2, 2006
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Displaying 1 - 4 of 4 reviews
April 5, 2011
This book is a weird concoction: popular maths and archaeology? It could be unique...
It tells the (pre)(hi)story of mathematics from the days of Neolithic hunter-gatherers to the timeof the famous ancient Greek mathematicians, such as Pythagoras. This means working with very limited data and there are a lot of assumptions, suppositions and intuitions filling in the gaps between the data. Of course that is the nature of all archaeology, which is, after all, the task of reconstructing a culture from a subset of its material productions. Rudman is very good in this territory; he is very clear about what is a fact, what is a hypothesis, what is a theory, what the assumptions are, what is his personal view and what is generally accepted theory.
For me, the book got more interesting as it went along, largely because the maths itself got more interesting; at the outset there is only counting, by the end there is rigorous proof of the irrationality of root 2 and the Pythagoras Theorem (which it turns out was known but not proven well before Pythagoras came on the scene).
I do think the book has flaws, though. There are plenty of "fun questions" through out - but for me most of them weren't fun. Fortunately they are easily ignored. Rudman also "pulls a Dawkins" with several snide remarks about religion and theists that have no place in the book at all and the final chapter on maths teaching methods has some obviously fallacious arguments mixed in with sensible observations - but that chapter is just as out of place in this book as comments about the "intellectual weakness" of theists. Rudman's terminology seems a little obfuscatory on occassions, too: what's a frustrum? What's a truncated pyramid? I bet you can guess what the latter is - but a frustum is the same thing! He also uses "prism" when he means cuboid. That said, the arguments are generally clear.
Overall I feel that an intriguing topic has been poorly but not hopelessly served by this book - a better writer could have improved it greatly.
It tells the (pre)(hi)story of mathematics from the days of Neolithic hunter-gatherers to the timeof the famous ancient Greek mathematicians, such as Pythagoras. This means working with very limited data and there are a lot of assumptions, suppositions and intuitions filling in the gaps between the data. Of course that is the nature of all archaeology, which is, after all, the task of reconstructing a culture from a subset of its material productions. Rudman is very good in this territory; he is very clear about what is a fact, what is a hypothesis, what is a theory, what the assumptions are, what is his personal view and what is generally accepted theory.
For me, the book got more interesting as it went along, largely because the maths itself got more interesting; at the outset there is only counting, by the end there is rigorous proof of the irrationality of root 2 and the Pythagoras Theorem (which it turns out was known but not proven well before Pythagoras came on the scene).
I do think the book has flaws, though. There are plenty of "fun questions" through out - but for me most of them weren't fun. Fortunately they are easily ignored. Rudman also "pulls a Dawkins" with several snide remarks about religion and theists that have no place in the book at all and the final chapter on maths teaching methods has some obviously fallacious arguments mixed in with sensible observations - but that chapter is just as out of place in this book as comments about the "intellectual weakness" of theists. Rudman's terminology seems a little obfuscatory on occassions, too: what's a frustrum? What's a truncated pyramid? I bet you can guess what the latter is - but a frustum is the same thing! He also uses "prism" when he means cuboid. That said, the arguments are generally clear.
Overall I feel that an intriguing topic has been poorly but not hopelessly served by this book - a better writer could have improved it greatly.
June 2, 2019
Many books have been written on the history of mathematics, but much of what is written seems to focus almost entirely on the mathematical traditions beginning with the Greeks and extending into modernity with only passing mentions of the more ancient systems that existed in many parts of the world long before Pythagoras, Archimedes, or Euclid. It's refreshing to read a book that largely reverses this trend by devoting the majority of its attention to what I've seen described as the pre-history of mathematics. By the nature of this exercise, however, one can't simply trace the development of theorems through time but must instead focus equally on mathematical concepts and archaeological ones. Because the subjects of both mathematics and archaeology are so vast, it would be impossible to give even a cursory treatment to either topic in a single volume of only 314 pages. Instead, it's best to consider this book a very broad survey of the development of ancient mathematics, occasionally narrowing its focus to discuss a concept or result of particular interest but generally maintaining its position as a cursory overview of an entire field of inquiry.
Approximately the first half of the book is devoted to the origin of various number systems. Many readers are likely already familiar with the fact that our decimal system is not the only or even the oldest number system, but even those who already know a little bit about ancient mathematics will likely find it interesting to learn about Torres-Strait body-parts counting or the Babylonian sexagismal system. However, for the reader whose undergraduate mathematics education did not include (as mine did) significant time spent performing operations with unit fractions (sometimes written in Egyptian Hieroglyphics), this first part of the book might seem to drag on a bit longer than it needs to.
The book begins to obtain a broader general interest in the second half when it moves on from counting systems and the most basic concepts of arithmetic into the early history of algebra, spending a good amount of time discussing the algebraic problems from ancient Egypt recorded in the Rhind Papyrus. The author does an excellent job of communicating, in relatively few pages, the depth of mathematical understanding possessed by the ancient Egyptians, including concepts of geometric series, algebraic geometry, and precursors to integral calculus. The following chapter traces an alternate lineage of mathematics in Babylon covering similar but distinct mathematical territory including a rather stunning approximation of the square root of 2 and a special case of the Pythagorean theorem.
Arguably the most interesting mathematics is found in relatively few pages toward the end of the book where the author describes the development of rigorous proof. Illustrative examples include one of my own favorite stories from mathematical history: Eratosthenes, whose sieve can be used to locate prime numbers and whose knowledge of geometry allowed for a remarkably accurate calculation of the Earth's circumference as early as some 250 BCE. While this chapter does include a few of the "greatest hits" of early (Greek) mathematics, as well as a brief description of our own mathematical lineage (it can, after all, be argued that we are all the mathematical descendants of the Greeks), it's not the strongest point of the book because the description is, frankly, far too brief to allow for any real depth of understanding.
The book concludes with a brief but important plea for improved education in mathematics. While this certainly isn't the message you buy this book to hear, it remains a message that more people interested in mathematics need to start thinking about. The author's nine pages on the subject won't solve any problems, but I applaud him for devoting even a few pages to an attempt to start a conversation we desperately need to have.
The bottom line is that this book is a fascinating if somewhat superficial look at the early history of mathematics, culminating in a brief discussion of early "rigorous" mathematics. I recommend it for anyone with an interest in mathematics, OR anyone with an interest in history. The reader needn't be an expert mathematician to understand the vast majority of the book's material (a bit of high school algebra and geometry should easily suffice). If you're looking for a book that traces some of the famous proofs in mathematics, however, you'll be disappointed. That subject is taken up only at the very end of the book in a chapter I consider to be a bridge between this book and the many other works that have been written on Greek mathematics. If you find yourself looking for more of that kind of material after finishing this book, William Dunham's Journey Through Genius provides an excellent follow-up (conversely, if you read a book like Dunham's and want some more of the earlier history, this book would be perfect for you).
Approximately the first half of the book is devoted to the origin of various number systems. Many readers are likely already familiar with the fact that our decimal system is not the only or even the oldest number system, but even those who already know a little bit about ancient mathematics will likely find it interesting to learn about Torres-Strait body-parts counting or the Babylonian sexagismal system. However, for the reader whose undergraduate mathematics education did not include (as mine did) significant time spent performing operations with unit fractions (sometimes written in Egyptian Hieroglyphics), this first part of the book might seem to drag on a bit longer than it needs to.
The book begins to obtain a broader general interest in the second half when it moves on from counting systems and the most basic concepts of arithmetic into the early history of algebra, spending a good amount of time discussing the algebraic problems from ancient Egypt recorded in the Rhind Papyrus. The author does an excellent job of communicating, in relatively few pages, the depth of mathematical understanding possessed by the ancient Egyptians, including concepts of geometric series, algebraic geometry, and precursors to integral calculus. The following chapter traces an alternate lineage of mathematics in Babylon covering similar but distinct mathematical territory including a rather stunning approximation of the square root of 2 and a special case of the Pythagorean theorem.
Arguably the most interesting mathematics is found in relatively few pages toward the end of the book where the author describes the development of rigorous proof. Illustrative examples include one of my own favorite stories from mathematical history: Eratosthenes, whose sieve can be used to locate prime numbers and whose knowledge of geometry allowed for a remarkably accurate calculation of the Earth's circumference as early as some 250 BCE. While this chapter does include a few of the "greatest hits" of early (Greek) mathematics, as well as a brief description of our own mathematical lineage (it can, after all, be argued that we are all the mathematical descendants of the Greeks), it's not the strongest point of the book because the description is, frankly, far too brief to allow for any real depth of understanding.
The book concludes with a brief but important plea for improved education in mathematics. While this certainly isn't the message you buy this book to hear, it remains a message that more people interested in mathematics need to start thinking about. The author's nine pages on the subject won't solve any problems, but I applaud him for devoting even a few pages to an attempt to start a conversation we desperately need to have.
The bottom line is that this book is a fascinating if somewhat superficial look at the early history of mathematics, culminating in a brief discussion of early "rigorous" mathematics. I recommend it for anyone with an interest in mathematics, OR anyone with an interest in history. The reader needn't be an expert mathematician to understand the vast majority of the book's material (a bit of high school algebra and geometry should easily suffice). If you're looking for a book that traces some of the famous proofs in mathematics, however, you'll be disappointed. That subject is taken up only at the very end of the book in a chapter I consider to be a bridge between this book and the many other works that have been written on Greek mathematics. If you find yourself looking for more of that kind of material after finishing this book, William Dunham's Journey Through Genius provides an excellent follow-up (conversely, if you read a book like Dunham's and want some more of the earlier history, this book would be perfect for you).
July 5, 2017
I enjoyed reading about the history of math from Egypt, Babylonian. Its a shame I have trouble figuring out math from that long ago in history
September 13, 2010
Pretty good dive into the the history of mathematics. Entertaining at times and easy to follow.
Displaying 1 - 4 of 4 reviews