The prince's deputy

About the life and work of the 19th century mathematician Geord Friedrich Bernhard Riemann 

Georg Friedrich Bernhard Riemann was born in 1826 in a small village near the city of Hanover, the son of a poor and not very young Lutheran priest. In the 19th century Europe was very strong in matters of urbanization and the industrial revolution. The poor continued to be poor, just lived in more densely populated areas. The trendiest disease in the slums of Europe was tuberculosis, all of Europe's poverty aristocracy contracted tuberculosis. If you didn't lose a child or years to tuberculosis, you weren't poor. One in five patients in Britain, one in six in France, literally hyperinflation of tuberculosis. By these standards the Riemann family was in the top decile.

Georg Friedrich Bernhard's mother and three sisters died of tuberculosis when he was still young, leaving a father and three children, boys, with no one who knows how to cook, do laundry or manage life. Accordingly, don't trust Father Rieman to know when to enroll his son in school. He just taught the son at home. The problem was that the son excelled in mathematics. In about a minute and a half, Bernhard learned everything his father could teach. Schultz, the principal of the nearby school, was next in line. This was enough until the age of 10. Instead of sending the son to work as a plumber's apprentice and put some food on the family table, Father Rieman sent him to Hanover to live with his grandmother and study at a middle school - Lissaom. Perhaps Friedrich Bernhard would have preferred to work. He was an introverted, quiet child, with abnormal stage fright, and the children at school did not make it easy for him. He studied a classical system of hours that contained lessons in Hebrew, Greek, the Bible and such things. Registered as a good student, but not particularly outstanding. Years later, grandmother dies, young Ryman goes to high school - gymnasium.

A quiet, introverted student, doesn't talk, doesn't excel, isn't problematic. One has to make an effort to distinguish its existence among the 381 registered students. The headmaster of the school, Constantine Schmalfus did notice. By an amazing coincidence he was the only principal ever in the history of the school who came from the field of mathematics and studied it at the university and not at the monastery, he bothered to ask Riemann about his interests and when he heard, he let him play with rather difficult mathematical books. Riemann ate them without salt and then asked for some more. The most famous legend about Riemann's appetite for mathematics concerns his fat book by Legendre on number theory and the fact that Riemann finished the 900 pages of the book in less than a week. Schmalfus also describes the great pleasure he had in his mathematical conversations with the student Riemann. He notes his pleasant manners, his pleasant voice and his happy mood. The introverted Riemann became something else when spoken to properly. Schmalfus also noted in his diaries that he learned much more from Riemann than Riemann learned from him.

Now it was already clear that there was talent here and that it was concentrated in mathematics. Unfortunately for Schmalfuss, Riemann's insane perfectionism made him not submit works unless they were perfect. This caused him to be months late in handing in assignments in the other subjects and he was about to leave high school without a matriculation certificate. This is where Dr. Gustav Heinrich Schaffer came into the picture. The young Hebrew teacher. Shafer, in agreement with Shmalfus, took Rieman to live in his house, sat with him until the wee hours of the night and watched over him until he finished his work. Rieman even helped improve the writing of the Hebrew textbook that will be taught in all the Lutheran schools in the area. Their deep acquaintance over the years led to Shaffer's amazing diagnosis that this guy is not going to be successful with women and that he is simply charming. Riemann, therefore, leaves with an excellent matriculation certificate solely thanks to Konstantin Schmalfus. From here to university.

Mathematics doesn't put food on anyone's table unless you have some rich patron. The only way Riemann can continue to study is to become a clergyman. Father Riemann, it is not clear from where, perhaps from the gray market, manages to raise enough money to send his son to the University of Göttingen to study theology.

The year is 1846 and Göttingen is still just a place. Then the prince came to Göttingen and turned it overnight into one of the leading universities in the world. who is the prince It's a Gaussian.

In one sentence he was said to be Riemann's evil twin. Like Riemann, he was introverted, did not like to speak in public, came from a poor family and was gifted like a demon in mathematics. This is where the imagination ends. Instead of a supportive and loving father, Gauss got a father who insisted that he be a construction worker despite repeated requests from Gauss's teachers who begged him to send the boy to study mathematics. Gauss was not a shy type in the conventional sense of the word, a better description of him would be a man-hater. While he wasn't hating Adam he was chasing dresses. In contrast to Riemann's true modesty, Gauss was arrogant and conceited, to Gauss's credit he had receipts from the earth to the moon to cover his intolerable arrogance. These parallels, in the fields of set theory, composite analysis and physics, rightfully earned him the nickname the prince of mathematicians.

Riemann, therefore, arrives in Göttingen in 1846 to study theology and hears several lectures from the prince, who was a particularly horrible lecturer. It doesn't bother Riemann, he catches everything without a problem. Riemann Sr. agrees, after one semester, to release his son from the theology course and Riemann goes full force in mathematics. Gauss, didn't see him for a while. The one who did notice him was Moritz Abraham Stern, who was the first Jewish professor at a German university. He paid close attention to the talent he had in his hands, and also realized that the guy was very shy and would not speak of his own initiative about the development of his career. He sent young Riemann to Berlin. In Berlin Riemann develops relationships with several colleagues, and forms a true friendship with Johann Peter Gustav Drichkel.

Drichla shares his working method with Riemann. The first thing is to make it intuitively clear to yourself what you want to prove, then do a simple and precise logical analysis of the fundamental questions of your problem and most importantly avoid entering into lengthy calculations at all costs. The result is elegant, short and simple math to test.

1849, Riemann returns to Göttingen to do his doctorate with the prince. The doctorate is wonderful, Riemann takes the concepts of complex functions developed by Koshy and Gauss before him, gets rid of the complicated calculations and power series and infuses this branch of mathematics with a wonderful elegance. The doctoral thesis is nice, but what about academic coolness? In order to be accepted as a free lecturer, the lowest rank at Riemann University, one is required to lecture to an audience on a topic chosen by the moderator - Gauss. And Gauss had serious suspicions about geometry. Some definitions, five axioms from which everything is derived.

Gauss wrote to his friend Olvers, "I am becoming more and more convinced that there is no way to prove the physical necessity of Euclidean geometry, at least not by human reason. Perhaps in another incarnation we will be able to see the nature of the space, which today we cannot achieve. Until then, we must place geometry not in the same class with arithmetic, which is completely a priori, but with mechanics"

Well, I think you deserve a brief explanation of what Gauss wants from our lives: let's start with the concept of "a priori knowledge". A priori knowledge as knowledge that does not depend on our experience, knowledge that derives solely from definitions and logical conclusions from these definitions or knowledge that derives from axioms. For example "every size is equal to itself" we do not test it in reality before we declare it as true, this axiom does not depend on reality, it is a product of pure thought. Arithmetic, according to Gauss, is a priori. The number systems, addition and multiplication laws do not depend on reality. They are the creations of human thought and can be developed even if we do not see or hear anything. Geometry, Gauss suspects, is not a priori, even though it is built from definitional axioms and sentences derived from them. No observations, no predictions, nothing physical here. So why does Gauss think geometry is part of physics?

Well, in 1826, when Riemann was just born, Gauss gave one of the most significant flashes of his coolness. Gauss introduced the concept of curvature of a surface into the world of mathematics, proposed a way to measure curvature at any point on the surface, and most importantly, proved his amazing theorem (that's not me, that's what Gauss called it: the amazing theorem). What the amazing theorem says is that the curvature at any point on a surface is invariant under isometry.

As my younger brothers of the Internet generation say, WTF? I'll just explain the sentence like this: suppose I'm a two-dimensional creature living on a two-dimensional plane. In order to know the curvature of my two-dimensional world, all I need is to draw a triangle and measure the sum of the angles. More than 180 - positive curvature, less than 180, negative curvature, exactly 180 - zero curvature. This. I don't need to know anything about the 180D universe outside to calculate curvature and frankly, if tomorrow the shape of this XNUMXD world changes in a way that doesn't distort the distances, I won't notice any difference in curvature in mine. If so, Euclidean geometry proves that the sum of angles in a triangle is XNUMX degrees, any triangle, anywhere and anytime. This is not a mathematical statement. Gauss's amazing theorem proves that this claim is physical. It is only true if the curvature of our universe is zero!. The proof of the geometric theorem about the sum of angles in a triangle relies on the axiom of parallels, so we discovered here that the axiom of parallels is not an axiom at all. is a physical assumption about the universe.

Here, so close to the end of the road, Gauss stopped. He did not dare to draw the obvious conclusion. In the same letter of Gauss in which he questions the a priori of geometry, he also says that he will never publish his results. Why actually? You did all the work, the chance for world fame is ahead of you, got cold feet? It can be said that Gauss was conservative, and was careful not to make revolutions, and this was a piece of a philosophical, mathematical and physical revolution. It can also be said that Gauss feared the status he already had, he feared the challenge that other mathematicians would call upon him, he feared that they would prove him wrong - he, the prince. Perhaps the perfectionist Gauss didn't want to publish anything that was beyond rebuttal.

The modest Riemann had no such worries. He didn't have too big of a reputation, and he didn't have a problem turning things around, Dr. Shaffer taught him to submit things even if he wasn't sure they were perfect.

10.6.1854. Riemann delivers the best PhD lecture ever given. The title of Riemann's lecture was "On the hypotheses underlying geometry".

In the first part of the lecture Riemann defines his mathematical framework. He defines the concept of size, and the concept of sheet as an ordered collection of sizes. Sounds vague to you? Well, in the two-dimensional world, every point can be described by two coordinates X and Y, that is, an ordered collection of two sizes - which makes the plane a sheet of dimension 2. In a similar way, the space can be described as a sheet of dimension 3 and there is no objection to talking about a sheet of dimension 28 as well - it is simply a collection A series of 28 sizes. Now that we have canvases we can define lengths of lines on those canvases. Here we come across the first hypothesis that is at the foundations of geometry and it is: length is not affected by position. If we measure the length of a line in a certain place on our canvas, and then move the line, its length will not change. Riemann shows that there exist sheets of any dimension greater than 2 in which this assumption about the independence of size on position does not hold. Sheets where the assumption of independence is fulfilled Riemann calls "flatness". He demonstrates how you can build a 3D canvas where the length of a line depends on its location and very elegantly shows that he is actually talking about curvature - the curvature at the point where you started measuring, and the direction in which we moved are the ones that will determine the length of the line. If the curvature is constant and the same in all directions - there really is no connection between length and position. If our canvas is flat, Euclidean geometry works fine. The first assumption, then, of geometry on the structure of the universe: the universe is a canvas with constant curvature.

Now Riemann comes to the second assumption: geometry assumes that direction also does not depend on position. That is, if we take a line pointing in a certain direction and move it without rotating, the direction will remain the same. It is easy to see how on the surface of a sphere - which is a 3-dimensional sheet with constant curvature, you can take a line pointing north and move it without rotating it so that we get a line pointing south. Riemann shows that this assumption is valid only if the curvature of the sheet is zero. That is, the hypothesis of Euclidean geometry about the structure of the universe is that it is a three-dimensional sheet with zero constant curvature. Riemann shows that this is like saying that the universe is cylindrical.

The lecture is over. Three or four short formulas were on the board. No complicated calculations and no unnecessary complications. Geometry was no longer a priori. The self-evident definitions contained hypotheses about the structure of the universe. Hypotheses that no one guaranteed to be true. In order to verify or disprove them, we need to go outside, make measurements and understand that geometry is a branch of physics.

Gauss stood up and said "understood". This is the highest compliment any human being could receive from the Prince of Mathematicians, it means he was listening.

If you scan the lists of the greatest mathematicians ever you will find Gauss always in the top five. Riemann will always push to the second or third tenth. Nothing, he didn't like fame and honor anyway. He died at the age of 39 from tuberculosis, poor like his father, shy and modest, but appreciated by everyone who knew him not only as a great mathematician, but also as a pleasant and gentle person. Thank you Constantine Schmalfus.

The above transcript is part of an episode of the "Biocast" podcast on our website www.biocast.co.il You can listen to this episode and also another episode that deals entirely with the Riemann hypothesis.