August 2, 2016

Lessons learned from the History of Mathematics

I've read the book A History of Mathematics by Carl Boyer. It's a few years old, which is good because old books are available for free so you can download your own copy here. Just make sure to download the pdf file in which the text consists of images because the other versions are missing most of the important images. These images are important because the author has included images of old numbers from like Egypt in the main text, so you will not understand what's going on without them because they are missing in the other versions.

The history of mathematics can be characterized by several ups and downs with so-called "Golden Ages" where the development of math really accelerated, followed by a cold winter where not much happened, and then a Golden Age happened once again. But you can say the everything began around the Mediterranean in Egypt and then in Greece. This period was one of those Golden Ages, but then winter arrived when math took a detour to the Middle East.

When the Islamic State conquered Alexandria in 641, which is the city that for many years had been the mathematical center of the world. All mathematical developments so far could now have disappeared because the Islamic State were not that interested in math and were very close to burn all the books in the famous library of Alexandria. Neither the Romans were interested in math so math almost disappeared in Europe. But a few Islamic scholars were interested in trying to develop the math previously developed around the Mediterranean, not at least because they needed to apply the math. Why? Because the complicated nature of laws governing inheritance encouraged the study of math in the Islamic State. The contributions by the Islamic State to the development of mathematics may not be that great, but they saved the knowledge until Europe was once again ready to continue developing mathematics. The book includes chapters on math in China and India, but the key discoveries happened in Europe.

The word mathematics has meant different things to the peoples of the world at different periods in history. As already mentioned, the Islamic State found a purpose with math, but not all famous mathematicians had a practical purpose. A common question young students ask their teacher is: What's the point with math? When will I need this math to solve problems? A common answer is that the student will need the math when shopping. But that's the wrong answer, because math doesn't need to have a practical purpose. Math is math, and then people with problems should use the "useless" math to solve the problems. Or as Bertrand Russell said:
Math is the subject in which no one knows what he is talking about, nor whether what he says is true.   

Working as a mathematician today is far easier than it once was. Today we have computers and we connect with other mathematicians through the Internet - and we risk not to lose our heads in the French revolution, which was the fate of a few mathematicians. The history of mathematics consists of hundreds of mathematicians. You can summarize them like this:
  • Some made the key discoveries when they were young and others when they were old. So the saying that if you are older than 30, you will never be able to develop math is clearly wrong.
  • Some had math education others didn't.
  • Some made the key discoveries alone and others were part of a larger network.
  • Some thought a little too much about math and were admitted to insane asylums, while others lived happily ever after. 
  • Some managed to solve what they were working with, while others failed with the main problem, but instead they discovered something else, which turned out to be their key discovery.
  • Some published their results and others didn't want to publish. So some made key discoveries only to die without any recognition, and then someone found their notes and realized that hey this guy was really on to something. 

Of all mathematicians in the book the one I remember the most is the Frenchman √Čvariste Galois. His story goes like this:
Galois was born just outside Paris in the village of Bourg-la-Reine, where his father served as mayor. His well-educated parents had not shown any particular aptitude for mathematics, but the young Galois did acquire from them an implacable hatred of tyranny. When he first entered school at the age of twelve, he showed little interest in Latin, Greek, or algebra, but he was fascinated by Legendre's Geometry. Later he read with understanding the algebra and analysis in the works of masters like Lagrange and Abel, but his routine classwork in mathematics remained mediocre, and his teachers regarded him as eccentric. 
By the age of sixteen Galois knew what his teachers had failed to recognize - that he was a mathematical genius. He hoped, therefore, to enter the school that had nurtured so many celebrated mathematicians, the Ecole Polytechnique, but his lack of systematic preparation resulted in his rejection. This was but the first embittering failure. 
Nevertheless, Galois at the age of seventeen worked up his fundamental discoveries in a paper, which he asked Cauchy to present to the Academic. Cauchy not only misplaced the paper, as he had misplaced one of Abel's important articles; he lost the paper! Now Galois hated not only examiners but also academicians. A failure in his second attempt at admission to the Ecole Polytechnique heightened his bitterness; but the heaviest shock of all was yet to fall. 
Under attack because of clerical intrigues, his father felt himself persecuted and committed suicide. Despite the blows that he had experienced, Galois entered the Ecole Normale to prepare for teaching; he also continued his research, in 1830 submitting a memoir in competition for the Academie's prize in mathematics. Fourier, the secretary of the Academie took the paper home, died shortly afterward, and the paper was lost. 
Faced on all sides by tyranny and frustration, Galois made the cause of the 1830 revolution his own. A blistering letter criticizing the indecision of the director of the Ecole Normale resulted in Galois' expulsion; but once more he tried to submit a paper to the Academie, this time through Poisson. The paper contained important results now a part of what is known as Galois theory; but Poisson, the referee, returned it with the remark that it was "incomprehensible." 
Thoroughly disillusioned, Galois joined the National Guard. In 1831, at a gathering of republicans, he proposed a toast that was interpreted as a threat to the life of Louis Philippe, and he was arrested. Although released, he was again arrested some months later and sentenced to six months in jail. Shortly afterward he became involved with a coquette and, under a code of "honor," was unable to avoid a duel. In a letter to friends he wrote, "I have been challenged by two patriots - it was impossible for me to refuse." 
The night before the duel, with forebodings of death, Galois spent the hours jotting down, in a letter to a friend, notes for posterity concerning his discoveries. He asked that the letter be published (as it was within the year) in the Revue Encyclopedique and expressed the hope that Jacobi and Gauss might publicly give their opinion as to the importance of the theorems. 
On the morning of May 30, 1832, Galois met his adversary in a duel with pistols. He was shot through the intestines and lay where he fell until a passing peasant took him to a hospital, where he died of peritonitis the following morning. His funeral was attended by several thousand republicans. He was only twenty years old at the time, the youngest mathematician ever to make such significant discoveries.

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