By Shing-Tung Yau, Steve Nadis

Within the 20th century, American mathematicians started to make severe advances in a box formerly ruled via Europeans. Harvard's arithmetic division was once on the heart of those advancements. A heritage in Sum is an inviting account of the pioneers who trailblazed a particularly American culture of mathematics--in algebraic geometry and topology, advanced research, quantity concept, and a bunch of esoteric subdisciplines that experience infrequently been written approximately open air of magazine articles or complex textbooks. The heady mathematical recommendations that emerged, and the boys and girls who formed them, are defined right here in vigorous, obtainable prose.

The tale starts in 1825, while a precocious sixteen-year-old freshman, Benjamin Peirce, arrived on the university. He could turn into the 1st American to provide unique mathematics--an ambition frowned upon in an period while professors principally restricted themselves to instructing. Peirce's successors--William Fogg Osgood and Maxime Bôcher--undertook the duty of remodeling the maths division right into a world-class learn heart, attracting to the college such luminaries as George David Birkhoff. Birkhoff produced a blinding physique of labor, whereas education a iteration of innovators--students like Marston Morse and Hassler Whitney, who cast novel pathways in topology and different parts. Influential figures from all over the world quickly flocked to Harvard, a few overcoming nice demanding situations to pursue their elected calling.

A background in Sum elucidates the contributions of those striking minds and makes transparent why the heritage of the Harvard arithmetic division is an important a part of the heritage of arithmetic in the US and beyond.

Editorial Reviews

Review

This publication tells the story of the way arithmetic constructed at Harvard--and by means of extension within the United States--since early days. it truly is packed with interesting tales approximately a few of the mythical names of contemporary arithmetic. either fanatics of arithmetic and readers interested in the historical past of Harvard will take pleasure in it. (Edward Witten, Professor of Physics, Institute for complicated Study)

A historical past in Sum is a gorgeous tribute to a stunning topic, one who illuminates arithmetic in the course of the lens of a few of its such a lot amazing practitioners. The authors' love of arithmetic shines via each bankruptcy, as they use available and lively language to explain a wealth of heady insights and the all-too-human tales of the minds that stumbled on them. there's maybe no higher e-book for immersion into the curious and compelling background of mathematical notion. (Brian Greene, Professor of arithmetic & Physics, Columbia University)

The publication is written in a leisurely type, the scope is remarkably wide, and the subjects coated are defined astonishingly good. as soon as i began the publication, I easily couldn't placed it down and that i used to be ecstatic to simply comprehend vital arithmetic faraway from my very own examine pursuits. (Joel Smoller, Professor of arithmetic, college of Michigan)

A heritage in Sum encompasses a wealth of fine tales, tales that visit the center of the advance of arithmetic during this kingdom. The authors reach humanizing and enlivening what may well rather be a dry remedy of the topic. (Ron Irving, Professor of arithmetic, collage of Washington)

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**Extra info for A History in Sum: 150 Years of Mathematics at Harvard, 1825-1975**

**Example text**

7 Peirce had, of course, an even deeper reverence for his mentor than he did for Laplace. 9 Before long, a similar term, “father of American mathematics,” was applied to Peirce (by the British mathematician Arthur Cayley, among others). 10 On that score, Peirce faced little competition. Before he entered the scene, no one thought that “mathematical research was one of the things for which a mathematical department existed,” Harvard mathematician Julian Coolidge wrote in 1924. It was certainly not a job prerequisite since there were not nearly as many people qualified to conduct high- Benjamin Peirce and the Science of “Necessary Conclusions” level research, or inclined to do so, as there were available teaching slots.

Complex numbers assume the form of a + bi, where a and b are real numbers and i, the square root of −1, is an imaginary number. Quaternions are four-dimensional representations of the form (a,b,c,d) or a + bi + cj + dk, where a, b, c, and d are real and i, j, and k are imaginary. These numbers obey various rules, such as i2 = j2 = k2 = −1, and ij = −ji. Whereas Peacock held that the same rules applied to both symbolic and arithmetic algebra, this was not the case in Hamilton’s system: in arithmetic algebra, a × b is always equal to b × a, in adherence to the commutative law of multiplication, but the commutative law does not always apply to quaternions, since i × j, by definition, does not equal j × i.

Hamilton believed that algebraists were not bound to set rules but were instead free to write their own rules as they Benjamin Peirce and the Science of “Necessary Conclusions” saw fit. 58 Fascinated by quaternions, Peirce discussed them in a course in 1848 (the year of Hamilton’s first lectures on the topic), and he often called them his favorite subject. ”59 But Benjamin’s preoccupation eventually paid off. He identified and provided multiplication tables for 163 different algebras up to the “sixth order,” that is, containing six or fewer terms.

### A History in Sum: 150 Years of Mathematics at Harvard, 1825-1975 by Shing-Tung Yau, Steve Nadis

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