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# What is mathematics for Franka Brueckler

By on 01/02/2016

A mathematician walks into in a coffee-shop. You may, but are not obliged to, imagine that he looks just like the Mathematician painted by Diego  Rivera in 1918: grave, absorbed, accountant-like. Many people imagine that professional mathematicians look or at least have an air of Rivera’s mathematician, so let’s take it from here. If you know any mathematician personally you are free to substitute his or hers appearance instead of the one suggested. As already said, the mathematician enters the coffee-shop. He has never visited it before. It is not really crowded, but he has a reference for sitting on bar stools and takes a place at the bar, orders a strong coffee and a big glass of water, takes a notebook out of his pocket and starts a quasi-periodic process of taking a sip of coffee, thinking for some tens of second and writing something down.

So far, so good. But… Next to the mathematician sits a couple. They are obviously friends of the waiter and converse much with him. While the waiter is busy and the female part of the couple dissapears to freshen up, the male part looks arround, and finds interest in the activities of our mathematician. In a moment when the mathematician is in the phase of taking a sip of coffee the man asks: “I apologize if I’m disturbing you, but I’m curious. You are a mathematician, right?” After a few seconds needed to exit the quasi-periodic routine, the mathematician answers: “Yes, I am. Why do you ask?” “Oh, I am a chemist. I accidentally saw your notes and noticed that you are solving a differential equation, are you not?” “Err… Yes.” Our mathematician lightens a little bit up: it is not often that somebody shows interest in his work. “So, you are a chemist? Of what kind?” “Oh, I’m a physical chemist and work at the institute for physical and chemical resarch. May I introduce myself: I’m Joe Brand.” “I’m Frank Bridger, nice to meet you.” At this moment the girl returns. “Oh, Geraldine, may I introduce you to Mr. Frank Bridger, he is a mathematician. This is my girlfriend Geraldine Stillins, she is a history teacher.” While shaking hands the conversation continues: “A mathematician? I’m afraid I must say I was never good at maths. In fact, I was bad. What is it good for anyway?” “How can you ask that”, Joe intervenes, “mathematics is an indispensable tool for scientists and engineers.” Before Frank can answer, the fourth drammatis persona enters the picture (again): the waiter joins the conversation. We skip the introduction part because the only new information transferred in it is that the waiter’s name is Tony. Ah, yes, and that Frank, Joe and Geraldine ordered more to drink. Tony is soon off to attend to the other guests, and Frank takes up the conversation with Joe.

“You said that you consider mathematics a tool?” Joe is surprised: “Sure. I wonder you of all people should ask me that. After all, mathematics developed as a tool to solve real world problems. I personally could not do my research without extensive usage of mathematics, particularly statistics. If I have some data, I need statistics to conclude something from them.” “Oh, but you are not doing mathematics if you calculate the mean and the standard deviation of your data. You are just using mathematics.” “And may I ask what is the difference?” “Doing mathematics is creating new mathematics. Proving theorems, discovering interrelations and structures.” Geraldine is puzzled: “I don’t understand you. If I ever understood anything about mathematics, it was that it is about juggling with formulas.” “Oh, no. For example, the discovery and proof of the formula for the solution of a quadratic equation, that was doing mathematics. After it was known, everybody is free to use it. When somebody uses it to solve a problem, this has as much to do with mathematics as the work of a Western Union messenger boy has to do with Marconi’s genius. A great mathematician, Paul Halmos, said that once, and I must say I fully agree with that.” “But I always thought that you professional mathematicians earn your living by solving particularly complicated equations. And if I understand you correctly, you just said that you do not solve equations but find the ways of solving them?” “Well, yes and no. In most cases it is not about equations at all. But I must apologize myself for a minute, I’ll be back soon.” Frank leaves to you know what little room, and Joe and Geraldine have a chance to comment something you would probably see if you were with them. But you are not, so let them tell you: “This Frank, he somehow is not like a mathematician”, Geraldine says, “I mean, he looks like a mathematician, but he is much more of an alive person than he looks it, if you know what I mean.” “Maybe we were wrong about mathematicians”, Joe muses, “take a better look at him. He looks much more like a person you ordinary meet in coffee-shops and bars, than it seemed in the first moment. Could it be that because of our prejudices we saw him not as he is, but as we thought he should be?” “Hey, who is more into humanities, you or me?” Geraldine laughs. And then Frank is back. And Tony also.

“I apologize for interrupting,” Tony says, “I am not sure anymore if I know what is a theorem. Wasn’t there the Pythagorean theorem we had to learn? I think it was $a^2 + b^2 = c^2$ or something like that? “A theorem is a proven mathematical statement. And the theorem of Pythagoras is not $a^2 + b^2 = c^2$ “Oh, sorry, it was really so long a time ago that I guess I forgot the formula, but it was something like it, wasn’t it?” “No, the formula is all right. But the formula is not the theorem. The theorem is: If a, b and c are the lengths of the sides of a right triangle, and if c is the length of the hypotenuse, then the $a^2 + b^2 = c^2$ holds. The theorem does not say that the equation is, or is not, true for any other meanings given to a, b and c. It does not even say that right triangles exist. But, if there is a right triangle, then the lengths of its sides are sure to be related in the described way.”

“I have a question”, Joe says, “what if you remove two squares of different colours?” “Oh, this is obvious”, Tony replies, “if you have proven that you cannot cover the board when two squares of the same colour are removed, you can do it when they are of different colours.” Frank corrects him: “True conclusion. But obtained incorrectly. If you have proven the theorem that it is impossible to cover the board when two squares of the same colour are removed, you do not know anything about the situation when two squares of different colours were cut out. The previously idea for proof obviously does not work since on such a board there remain the same number of squares in both colours. Maybe there could be some cases when for such a board the required tiling were also not possible. After all, you want the theorem to be as general as possible. Not: this particular board can or cannot be tiled with dominoes. But: any chess board with two squares of different colours removed can be covered with dominoes. This is known as Gomory’s theorem, and the proof is also beautiful: he draws a labyrinth like this (see picture on the left). You cut any two squares of different colours. Now you imagine a caterpillar with its only crawler overlaid with 31 dominoes. You let the caterpillar drive along the the labyrint, placing the dominoes. If it skips the two holes, it will exactly cover the board with the dominoes.” All three have just one comment (and wide smiles): “Wow.”