Wednesday, April 5, 2017

The Music of the Primes by Marcus du Sautoy

Might the entire universe be "written" in the language of equations? British mathematician G.H. Hardy pointed out that while languages die, mathematical ideas live on.

Certainly, mathematical problems and solutions have long horizons. At the Sorbonne in 1900, David Hilbert, one of the great mathematicians of his age, set out a list of 23 problems he thought should be explored in the coming century. By 2000, all were solved but the Reimann Hypothesis, which "seeks to understand the most fundamental objects in mathematics -- prime numbers."

Princeton mathematician Enrico Bombieri played an April Fool prank on his colleagues in 1997. On the website of the International Congress of Mathematicians, he announced that "Holy Grail" of mathematics had been proved.

However, in 2000, along with six new problems for the 21st century, the Reimann Hypothesis was offered once more to challenge mathematicians. This time the solver would earn both glory and money -- a million per problem.

Solving the puzzle of the primes would change mathematics, and the world. It is thanks to the inscrutable unpredictability of prime numbers that e-business thrives. Carl Friedrich Gauss died in 1855, but the calculator clocks he invented, with their faces bearing "more hours than there are atoms in the observable universe," are vital to the security of online transactions today. In the age of internet commerce, "advances in the most obscure or abstract corners of the mathematical world now have to potential to bring business to its knees."

Prime numbers remain deeply mysterious in spite of all the work that has gone into unmasking them. One early effort was made by Erastothenes, who developed a mathematical "sieve" to eliminate numbers that could not be primes.

Marcus du Sautoy's book is full of astonishing and fascinating details. Where else could you learn about the sex of numbers in ancient Chinese thought? The 22,000-year-old Ishango bone from equatorial Africa, with its ancient markings of prime numbers? The fact that the ancient Greek mathematician Euclid was the father of mathematical proof?

Math history also includes such stories as the miraculous WWII escape of French mathematician Andre Weil. Leaving France in 1939 for Finland, he hoped to avoid the war and go on to America, where he could continue doing mathematics. Unfortunately, the Finns took him for a spy when they found that his letters to Russian mathematicians were full of equations. On the eve of his planned execution, the chief of police happened to mention his presence in prison to a Finnish mathematician, who pleaded successfully for deportation rather than death.

Returning to France, Weil was jailed for desertion. However, during this period of incarceration, he produced a promising new line of thought that promised progress on the Reimann Hypothesis. He published from prison, and fellow-mathematician Cartan testified at his trial. Weil's 5-year sentence was commuted when he agreed to go into the army. Luckily, as it turned out. Shortly after, the Germans advanced on Rouen and all prisoners were shot. A second lucky escape. He did eventually get to to Princeton, the Mecca for fleeing mathematical Europeans during WWII.

New mathematical ideas begin with hunches and intuitions: the first sketch of the idea is called a conjecture. Later it becomes a hypothesis. Only when the new idea can be proven is it promoted to the status of a theorem.

Then there's the old question of pure versus applied mathematics. While academics remained true to the pure abstract beauty of the science they studied for its own sake, the industrial revolution pushed mathematical development into applied forms to serve industry. In 1789, Napoleon founded the Ecole Polytechnique in Paris, to focus on the "needs of the state," hydraulics and ballistics for the war machine.

On the other hand, pure mathematics "has the ability to unite people across political and historic boundaries." The author quotes Julia Robinson's description of the unifying bond between mathematicians, 'a nation of our own without distinction of race creed, sex, age or even time (the mathematicians of the past and of the future are our colleagues too) -- all dedicated to the most beautiful of the arts and sciences.'

In this fascinating romp through centuries of mathematics, author Marcus du Sautoy says that "The primary drive of the mathematician's existence is to find patterns, to discover and explain the rules underlying nature, and to predict what will happen next." When it comes to prime numbers, though, those patterns and predictions have yet to be uncovered or explained.

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