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Journey Through Genius

William Dunham

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Nonfiction | Autobiography / Memoir | Adult | Published in 1990

When Dunham searched for theorems for his book, he did not only select ones that are already exalted in academia or in popular science. He chose his twelve theorems by also asking whether they are effective tools in conveying what intellectual life was like at the time they were formed. He starts the book early in mathematical history – with the ancient Greeks and the Egyptian school of math, based in Alexandria, and works up to modernity. Many of the later theorems owe partial credit to the discoveries of mathematicians who had already been dead for millennia.

For each period, Dunham introduces the mathematicians relevant to the development of new theorems, exploring their lives, before examining the specific thinking behind their contribution to mathematical thought. He begins with Hippocrates, a founder of geometry who hailed from Chios, and Euclid, whose contribution – Euclidean mathematics – is essential to modern number theory, linear algebra, and calculus. Today, ideas from Euclidean math are taught to children as early as kindergarten. Next, Dunham profiles Archimedes, known as the inventor of the irrational number, π (pi). Archimedes developed the notion of π while trying to imagine how one would compute the area of a circle using past geometric models, which relied on straight lines and therefore could not easily accommodate the curved forms we see in reality. Euclid, who deepened our understanding of triangular geometry, came after Archimedes.

After the groundbreaking work of these three classical mathematicians, few mathematicians proved as important until a millennium later. Dunham resumes his survey in the 1500s, with the superstitious figure Gerolamo Cardano. Just as Newton was jailed for heresy following his revelations about the heliocentric model of the solar system, Cardano was jailed for solving math equations that challenged the Catholic dogma that exerted control over intellectual life. Then, Dunham profiles Isaac Newton, who improved upon Archimedes’s estimation of π. After Newton came the Bernoulli brothers, Jakob and Johann. The Bernoullis, an argumentative pair, made important discoveries about the physical nature of the universe, including the functions of different states of matter: solid, liquid, and gas.

After the physicists of the seventeenth and eighteenth centuries came the advent of modern mathematics as we know it now. Georg Cantor and Leonhard Euler are indispensable figures during this era. Both created important abstractions about the nature of math, some of which have only recently been proven true. Euler and Cantor are now known as having pushed the boundaries of math using the sheer power of their brains.

After his survey of mathematical history, Dunham turns to trends in the present. He explains that for many contributing reasons, mathematics is now highly theoretical. One is math’s reduced reliance on the approval of social and economic structures, like the church and economic industry. Fields such as “pure math” have thus emerged. Dunham argues that many of these forms of math will one day serve practical purposes, once other fields catch up.

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