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- What is mathematics for Franka Brueckler
What is mathematics for Franka Brueckler
This is a very difficult question, and there is probably no one answer to which all mathematicians would subscribe. We have collected a number of partial answers. There is great variation in the ambitions of the authors and the demands placed by them on the reader
by Franka Brueckler
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.
“Sorry, but my calculator seems to be out of order, could you please add these figures up for me?” Tony hands Frank a paper with some twenty numbers written on it. Frank, not wanting to be rude, starts adding. And makes a mistake. And starts over. And crosses another obvious mistake. And this takes more than thirty seconds. “Hey, didn’t you say that you are a mathematician?” “Yes, I am, I teach mathematics at the university.” “I am not sure if I believe you, you are slower in adding these numbers than me, I would have done it quicker.” “But I’m a mathematician.” “Yes, you said that. But how come you are so slow in calculation then? Are you better in multiplication? Could you multiply 14757479 by 2334679 in your head?” “No.” “Now I’m puzzled. Isn’t mathematics the science of numbers and calculations?” “No. Mathematics may be about what numbers are and what it means to add or multiply them, but it is not about how the actual calculations are performed. And, to quote Halmos once again, You can no more expect mathematician to be able to add a column of figures rapidly and correctly than you can expect a painter to draw a straight line or a surgeon to carve a turkey.” “I must support Frank”, Joe adds, “for calculations we have calculators and computers, and accountants. Mathematics is an exact science that solves various problems. For example, if you want to know how much dye you need to paint your room, the mathematics gives you the rules how to calculate the area your walls cover. And you can calculate the area in the same way if you decide to paint another room. The rules are mathematics; the actual calculations are just a finishing touch.” “I do not agree”, Frank replies, “at least not completely. I do agree that mathematics is a science that can help solve many real world problems after transforming them into a mathematical model and often concrete real-world problems are the reason some of the mathematical theories were developed or some theorems proven. But if the definition of mathematics were that it is a tool for other sciences, this would cover only a fraction of what mathematics really is.” “I often wondered what it means: mathematics is an exact science?”, Geraldine adds, “and do you at least agree that mathematics is an exact science?” “Yes, I would venture to say even more: mathematics is THE exact science. Exact means that it can give accurate and precise answers to posed problems, and that the methods of obtaining the answers are rigorous. Often physics and chemistry, and some parts of biology or even psychology and medicine, are considered to be exact sciences also. But the rules under which mathematical conclusions are drawn are very precisely specified. In physics and chemistry (and other sciences) you gain insights through experiments. The results of experiments, since they are dealing with nature, depend on the conditions under which they have been performed and thus it is possible that the obtained results are not utterly exact and new experiments may overturn theories founded on old experiments. Mathematical results, on the contrary, are obtained by performing only mental logical operations. And once proven, they are true forever.”
“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.”
“Now I am completely confused”, Geraldine joins in, “and I think I have a new definition of mathematics: it is the art of saying complicated sentences.” “I agree that it is an art. But not of formulating complicated sentences.” “An art”, all three laugh, “mathematics is as far from art as Earth is from the end of universe, if such an end exists.” “Oh, you are so wrong. How would you define art?” “Art is something created to please my senses or tingle my emotions, to uplift my spirit. It is essentially creative, imaginative.” “If you describe a work of art, with what words would you describe it?” “I think in most cases I would say it was beautiful.” “Well, mathematicians are motivated in their work mostly by the idea of beauty. It is not only the truth of a proof that counts, the proof is the more appreciated if it has elegance, has beauty.” “An example please?” Frank turns a new page in his notebook and draws a grid of a chess board (see picture right). “Now, imagine you cut out any two white or two black squares of this chess board.” “Don’t you think we should first imagine that this is a chess board”, Joe laughs. “Good point”, Frank joins in laughing, “may I ask you all to imagine that this drawing represents a chess board, that the blue approximate squares represent the black squares of the board and that white correspond to white. Further, imagine that I gave you 31 dominoes and that the sizes of the squares are such that one domino piece fits to cover exactly two squares. Now, imagine you cut out two squares of the same colour.” “For example, two opposite corner squares?”, Geraldine asks. “Sure, lets cross them out to represent that they were cut out.” Frank crosses two opposite white corner squares. “But now we cannot play chess anymore”, Tony puts in, “but I don’t mind that. I prefer dominoes anyway.” “Oh, we won’t play either”, Frank replies, “now we have cut out the two squares, you are asked to find out if can you cover the board with the dominoes so that every domino covers exactly two squares and no dominoes overlap?” “Uh-huh”, is the sound Frank hears as a response, and he continues: “It is obvious that if you program a computer to check all the possible tilings of the board with the dominoes, you could get an answer. If the board were smaller, say 4×4, there would be not so many possibilities and you could even try all yourselves. In any case, you would get a negative answer: you cannot cover such a board completely with dominoes. You could try cutting out some other combination of two squares of the same color, but you would end up with the same answer. And sooner or later the computer program could check and draw all the possibilities of making such a mutilated chess board and covering it. This would be a proof that there is no such tiling of the board with two squares of the same colour removed. But it would be a very inelegant proof by exhaustion.” Geraldine ponders, “Yes, but what else can be done?” “Mathematics”, Franks mouth begins to stretch into a wide smile, “beautiful and elegant mathematics. It’s just about a good idea. Is it possible to cover two squares of the same colour with one domino piece?” Joe immediately joins in: “Obviously no.” “So, each domino piece will cover two squares of different colours. This means that any number of dominoes covers the same number of white and black squares of the board. But, when you removed two squares of the same colour, you got a board with different numbers of black and white squares. Consequently, you cannot cover the board with dominoes in the required fashion, no matter which two white or two black squares you remove. Proof finished!” Frank is now wearing a triumphant smile as he sees that all of three are impressed. “Wow, thatwas nice”, Geraldine says. “Beautiful”, Tony says and Frank asks: “What makes it beautiful?” All three are searching for the right words, but Frank continues: “You don’t have to answer. The mathematician Arthur Cayley once said: As for everything else, so for a mathematical theory: beauty can be perceived but not explained. And you need imagination to create mathematics. As you can learn to write correctly, but this doesn’t make you a good writer, you can learn to make correct logical conclusions, but the ideas for connecting seemingly unrelated concepts in an unexpected or at least not obvious way are what makes a good mathematician. The great mathematician David Hilbert once said regarding a former student of his: He is a writer now – he had not enough imagination. . Doing mathematics means that you discover new relations, new patterns, new properties of already known mathematical objects or that you even create new mathematical objects. Mathematics is essentially creative.” “I must agree”, Geraldine says, “there is something in it. Why has nobody told me that in school?” “The better question would be: why has nobody showed you that mathematics is creative and imaginative. Unfortunately, school maths is usually very far from the real thing.”
“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.”
“So, what is a proof?”, Joe wants to know. “As I already said, the basic law of mathematics is that any new mathematical insight must be proven by a sequence of strictly logical conclusions. If it is so, the new insight is called a theorem. In the logical reasoning of the proof you are allowed to use only previously proven statements and axioms. Axioms are general assumptions taken to be true and after choosing your axioms, theorems are their logical consequences.” “And what if the axioms are not true?” “They are true by definition. It is another thing if they do or do not fit our everyday experience.” “Can you give us an example?” “The axioms of Euclidean geometry were chosen to fit our everyday experience of the space around us. And they seemed obvious. Particularly, it seemed obvious that at most one line can be drawn through any point not on a given line parallel to the given line in a plane. The whole of geometry you have all learned in school is founded on this and a few other axioms. But in the 19th century mathematicians found out that you can get as sensible a system of geometry if you postulate that you cannot draw any parallel line through a point not on the chosen line, or if you postulate that you can have more than one parallel. The first was more easily conceivable, by imagining the geometry on the sphere where lines are in fact great circles and any two intersect, so there are no parallels. The other possibility with more than one parallel line seemed just an exotic mathematical theory.” “Seemed? Surely, this is impossible!”, both Geraldine and Tony cry out and Frank replies: “Firstly, because you have no experience with something it does not mean that it does not exist. Secondly, mathematics is not about things you meet in your experience. It can be, but it can be concerned only with the ideas. The principal questions a mathematician asks himself is: what if?”. “Yes, but who cares about something that is only an idea?”, Joe asks. “Mathematicians do. And to make you rest easy, until the beginning of the 20th century it did seem that such non-Euclidean geometries are just a mathematical curiosity, but then Einstein’s theory of relativity was founded on the use of non-Euclidean geometry. Still, the mathematics of non-Euclidean geometries would be as relevant to mathematicians if the applications were not found. There are many mathematical results that have not found an application, and many of these probably never will.”
“So, you say that mathematics is in fact a logical game, a juggling of ideas?”, Geraldine asks. “No, this would reduce mathematics to a game. Although most mathematicians consider their job, the proving of new theorems and creating of new mathematics, as much fun as any game. There are similarities to, for example, to chess problems found in newspapers, but although most mathematicians like such problems, and usually a talent for solving such problems indicates a talent for mathematics, these are not the problems a mathematician is really interested in. The mathematician solves problems he or she considers important. The level of importance is usually related to the impact the result would have on other conjectures or how an answer to the posed question would relate to known results.” “For example?”, Joe wants to know, but Tony cries out: “Hey, you have to go. I have to close up.” “Wow, how the time flies”, Joe comments, and Geraldine adds: “When it’s fun. I must say I would like to hear more about your curious world of mathematics.” “Me to” both Joe and Tony agree, and Frank has nothing more to say than: “Tomorrow, same time, same place?” And as Frank walks away in one, and Joe and Geraldine in another direction from the coffee-shop, Tony runs out shouting after them: “Hey, you forgot to pay your drinks!”
P.S. Frank has repeatedly quoted P. R. Halmos’s “Mathematics as a creative art”. The whole text can be found on this link. Two books were also a helpful inspiration: “The Art of Mathematics” by Jerry P. King and “In Mathe war ich immer schlecht” by Albrecht Beutelspacher. Thanks also to professor Šime Ungar for the idea about the chessboard, and for helpful conversations and heated discussions to my chemist friends Vladimir Stilinovic, Josip Pozar, Kresimir Molcanov and Tomislav Portada, and to the boss Josip, the waiter Igor and most of the regular guests in my favourite coffee-shop “Bug” in Zagreb.
Franka Miriam Brückler, Dept. of Mathematics, University of Zagreb, Croatia