How Mathematics Happened: The First 50,000 Years Review

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(More customer reviews)Written by Peter S. Rudman (Professor Physics, Technion-Israel Institute of Technology), How Mathematics Happened is explores the history and prehistory of human mathematical discovery. From the different ways Egyptians and Babylonians expressed fractions so that they would almost never be nonterminating fractions, to the evolution of pattern recognition to finger counting to pebble counting, to the underappreciated Mayan mathematics system, to concepts of abstraction and rigor invented by Greek mathematicians such as Pythagoras, Eratosthenes, and Hippasus. An amazing tour of human discovery, illustrated with black-and-white diagrams and punctuated with "fun questions" (with answers) for the reader to solve as well as references and an index, How Mathematics Happened is an engaging read for students, scholars, and laypeople alike.

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In this fascinating discussion of ancient mathematics, author Peter Rudman does not just chronicle the archaeological 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 archaeological 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.

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The World of Mathematics Review

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(More customer reviews)I pencil in the date that I finish reading each article in James R. Newman's four volume, "The World of Mathematics" After a good many years, I now find that I am more than halfway through Newman's remarkable collection that spans 2500 pages.
Newman described his work as "a small library of the literature of mathematics form A'hmose the Scribe to Albert Einstein, presented with commentaries and notes". The topics have been chosen with care. Newman preceded each article with a thoughtful commentary.
The individual articles are not abridgements, but are reprinted in their entirety. Some articles are short, some quite long, some are easy reading, some are difficult, but few are overwhelming.
I have not systematically read section by section. I find that I skip around. Often, after Newman introduces me to some mathematical topic, I find myself sidetracked, exploring other books and authors. But eventually I return to Newman, select another article, and begin the cycle again.
The Newman collection was published in 1956 as a boxed set that occasionally shows up in used bookstores. More recently, the four volumes have become available in soft cover (a Dover reprint) and can be purchased individually.
What makes Newman collection so remarkable? The answer is great original papers, great authors, and wide ranging topics.
Imagine reading Descartes on Cartesian coordinates, Whitehead on mathematical logic, Weyl on symmetry, Dedekind on irrational numbers, Russell on number theory, Heisenberg on the uncertainty principle, Turing on computer intelligence, Boole on set theory, and Eddington on group theory.
I enjoy the biographical and historical articles scattered throughout the four volumes. I especially liked Bell's article "Invariant Twins, Cayley and Sylvester", The Great Mathematicians" by Turnball, and G. H. Hardy's "A Mathematician's Apology".
Mathematicians try to define just what is mathematical thought and how a mathematician creates mathematics. Clifford writes about "The Exactness of Mathematical Laws", Von Neumann on "The Mathematician", Weyl on "Mathematical Way of Thinking", Poincare on "Mathematical Creation", Newman on "Godel's Proof", and Russell and Whitehead separately offer their thoughts.
This is the "World" of mathematics. Newman's assemblage also includes a fascinating, eclectic mix of articles that I have not encountered elsewhere like "How to Hunt a Submarine", "Durer as a Mathematician", "A Mathematical Approach to Ethics", "Geometry in the South Pacific", and "The Vice of Gambling and the Virtue of Insurance".
I have had great fun wandering through this four volume set from section to section, article to article. I assume that someday I will finally read the last article. I expect that I will simply begin again. It would be hard to say good-bye to Newman's collection.

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Vol. 2 of a monumental 4-volume set covers mathematics and the physical world, mathematics and social science, and the laws of chance, with non-technical essays by and about scores of eminent mathematicians, economists, scientists, and others. Individual articles by Galileo Galilei, Gregor Mendel, Thomas Robert Malthus, and many more. Includes numerous figures.

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