What is the brutal truth about mathematicians

Drawbridge out of order


Mathematics beyond culture

An exterior view by Hans Magnus Enzensberger

The tones are always the same: "Stop it! You can chase me with math." - "A torture, even at school. I have no idea how I got through high school back then." - "A nightmare! Completely untalented as I am." - "I can just manage the VAT with the calculator. Everything else is too high for me." - "Mathematical formulas - that's poison for me, I just switch off."

One hears such assurances every day. Very intelligent, educated people bring them up routinely, with a strange mixture of defiance and pride. They expect an understanding audience, and there is no shortage of them. A general consensus has emerged, which tacitly but massively determines the attitude towards mathematics. Nobody seems to mind that their exclusion from the sphere of culture amounts to a kind of intellectual castration. Anyone who finds this state of affairs regrettable, who mumbles something about the charm and importance, about the scope and beauty of mathematics, will be marveled at as an expert; if he reveals himself to be an amateur, at best he is considered an eccentric who is engaged in an unusual hobby, such as breeding turtles or collecting Victorian paperweights.

Much less often one meets people who claim with similar emphasis that the thought of reading a novel, looking at a picture or going to the cinema causes them insurmountable torment; Since graduating from high school, they had painstakingly avoided any contact with the arts of any kind; they would rather not be reminded of previous experiences with literature or painting. And you hardly ever hear curses on the music. Certainly there are people who claim, possibly not wrongly, that they are unmusical. One of them sings loudly and wrongly, the other does not play an instrument, and very few listeners rush to the concert with the score under their arm. But who would seriously say they don't know any songs? It doesn't matter whether it's the Spice Girls or the national anthem, techno or Gregorian chant, nobody is completely immune to the music. And for a good reason. The ability to make and listen to music is genetically anchored; it belongs to the anthropological universals. That doesn't mean, of course, that we are all equally musically gifted. Like all other gifts and properties, this aspect of our equipment follows the Gaussian normal distribution. Just as rarely as extremely gifted people can be found in any population who are musically completely deaf; the statistical maximum is reached in the middle field.

The same goes for math skills, of course. They are also genetically created in the human brain, and they too are distributed in every population exactly according to the model of the bell curve. It is therefore a superstitious notion that mathematical thinking is a rare exception, an exotic freak of nature.

We are faced with a mystery. Why is it that mathematics has remained something of a blind spot in our civilization, an extraterritorial area in which only a few initiated ones have holed up?

A certain isolation

Anyone who wants to make the answer easy for himself will say that it is the mathematicians' own fault. This explanation has the merit of simplicity. It also confirms a stereotype that the outside world has always made of the professional representatives of the discipline. One imagines a mathematician to be a profane high priest who jealously guards his special grail. He turns his back on the ordinary things of this world. Exclusively occupied with his incomprehensible problems, he has difficulty communicating with the outside world. He lives withdrawn, regards the joys and sorrows of human society as annoying disturbances and generally indulges in a solitary activity that borders on misanthropy. With his logical pedantry he gets on the nerves of the people around him. Above all, however, he tends towards a form of arrogance that is difficult to bear. Intelligent as he is - nobody disputes this title for him - he looks at the helpless attempts of others to get one or the other thought with disdainful condescension. Therefore it would never occur to him to promote his cause.

So much for the caricature, which is often enough taken at face value. This is of course nonsense. Aside from what they do, mathematicians are probably little different from other people, and I know men and women in the field who are fun-loving, urbane, funny, and at times even unreasonable. Nevertheless, as usual, there is a real core to the cliché. Each profession has its own risks, its specific pathologies, its own deformation professional. Miners suffer from pneumonia, writers from narcissistic disorders, directors from megalomania. All of these defects can be traced back to the production conditions under which the patients work.

As far as mathematicians are concerned, their work requires extreme and prolonged concentration. There are very thick and very hard boards that you have to drill. No wonder that any external irritation is felt to be unreasonable. On the other hand, the time of the universal mathematicians such as Euler or Gauss has long expired. Nobody today has an overview of all areas of their science. But this also means that in research the circle of possible addressees is shrinking. Works that are really original are initially only understood by a few specialist colleagues; they circulate by e-mail among a dozen readers between Princeton, Bonn and Tokyo. Indeed, this creates a certain amount of isolation. Such researchers have long since given up trying to make themselves understandable to outsiders, and it may well be that this attitude rubs off on other, less advanced workers in the vineyard of mathematics.

This is characterized by a phrase that the freshman will already hear in any lecture on function theory or vector spaces. This derivation or that assignment, it says, is "trivial", and that's that. There is no need for any further explanation; it would be beneath the dignity of the mathematician, so to speak. It is indeed tedious and tedious to unravel every single link in a chain of evidence anew every time. That is why mathematicians are trained to skip recurring intermediate steps, that is, to simply assume their validity, which has been tried and tested a thousand times. That is undoubtedly economical. But it influences communicative behavior in a very specific direction. Experts can only speak of someone who is able to speak, for whom the trivial is trivial, i.e. who understands itself. Everyone to whom this does not apply, i.e. at least 99 percent of humanity, are in this sense hopeless cases with which it is simply not worth talking to.

In addition, mathematicians not only have a peculiar technical language, like other scientists, but also their own notation, which differs from the usual script and which is indispensable for their internal communication. (Here, too, one can speak of an analogy to music, which has also developed its own code.) Now, however, most people panic as soon as they see a formula. It is difficult to say where this escape reflex comes from, which mathematicians in turn cannot understand. They are of the opinion that their notation is wonderfully clear and far superior to any natural language. So they don't see why they should bother to translate their ideas into German or English. Such an attempt would be a terrible corruption in their eyes.

So the mathematicians would be to blame for the insular situation of their science? Would you have turned your back on society yourself and willfully pulled up the drawbridge to your discipline? The answer can only be made that easy if you underestimate the problem and its scope. It is simply not plausible to pass the buck on a minority of experts as long as an overwhelming majority voluntarily refrains from appropriating a cultural capital of immense importance and of the greatest appeal.

Between utility and elegance

It is well known that ignorance is a heavenly power of invincible strength. Most people are probably convinced that it is quite easy to live without a knowledge of mathematics, and that the science is unimportant enough to be left to scientists. Many even suspect that it is a matter of a breadless art, the benefits of which are by no means obvious. In this mistake they may feel strengthened by the views of some mathematicians, who defend the purity of their work with strong words. According to the eminent English number theorist Godfrey Harold Hardy, who made the following famous confession: "I have never done anything that would have been 'useful'. For the well-being of the world, none of my discoveries - for better or for worse - have ever made the slightest difference I helped train other mathematicians, but mathematicians of the same kind I am, and their work, at least as far as I helped, was as useless as my own By all practical standards the value of my mathematical life is zero, and outside of mathematics it is trivial anyway. " - There it is again, the ominous word trivial, with which everything is stigmatized that the author despises. - "I only have one chance," continues Hardy, "to avoid the verdict of utter triviality, namely by being admitted that I have created something that was worth creating. It is not permissible that I have created something deny; the only question is whether it is worth something. " (A Mathematician's Apology, Cambridge 1967.)

Wonderfully said! A modesty that can hardly be distinguished from aristocratic arrogance. Nothing is further from a mathematician like Hardy than to court the approval of his fellow men and to appeal to the practical benefits of his work. He is right and wrong at the same time. His attitude comes close to that of an artist. From a purely business point of view, it would not only have been difficult for Ovid and Bach, but also for Pythagoras and Cantor. Your work would hardly have yielded the fifteen percent instant return that is under the banner of today shareholder value apply as a guide. Of course, the vast majority of human activities would be obsolete from this point of view. (Incidentally, mathematical research is one of the cheapest cultural achievements. While the new particle accelerator at Geneva’s CERN is estimated at four to five billion, the Max Planck Institute for Pure Mathematics in Bonn, a research center of international renown, only takes 0.3 percent of the cost Household of the Max Planck Society. Great mathematicians like Galois or Abel were impoverished all their lives. Cheaper geniuses are likely to be hard to find.)

The autonomy that Hardy demands for his fundamental research finds its counterpart in the arts, and it is by no means a coincidence that most mathematicians are not alien to aesthetic criteria; it is not enough for them that a proof is stringent; their ambition is aimed at "elegance". This expresses a very specific sense of beauty that has characterized mathematical work from its earliest beginnings. This, of course, again raises the puzzling question of why the public may appreciate Gothic cathedrals, Mozart's operas and Kafka's stories, but not the method of infinite descent or Fourier analysis.

But when it comes to social benefits, it is easy to refute Hardy's claims. An engineer who has to calculate a completely normal electric motor makes use of complex numbers with the greatest possible ease. Wessel and Argand, Euler and Gauss could not have suspected anything of this when they created the theoretical basis for this extension of the number system at the turn of the 19th century. Our computers would be inconceivable without the binary number code that Leibniz developed. Einstein would not have been able to formulate his theory of relativity without Riemann's preliminary work, and quantum mechanics, crystallographers, and communications engineers would be left pretty empty-handed without group theory. The study of prime numbers, a branch of number theory of inexhaustible charm, has always been an esoteric specialty. For a few millennia, and not just since Eratosthenes and Euclid, the best minds have dealt with these highly capricious figures without being able to say what it was for - until in our century suddenly secret services, programmers, military and bankers realized that factoring and trapdoor codes can be used to wage wars and do business.

Head and universe

There is something amazing about the unexpected usefulness of mathematical models. It is by no means clear why highly precise fantasies, which are far from all empiricism, as it were l'art pour l'art, have been conceived, are capable of explaining and manipulating the real world as it is given to us. More than one has wondered about "the unreasonable effectiveness of mathematics". For more religious times this pre-established harmony was not a problem: Leibniz could calmly claim that with the help of mathematics we could "gain a pleasant insight into the divine ideas", simply because the Almighty was personally the first mathematician. Today philosophers find it much more difficult. The old quarrel between Platonists, Formalists and Constructivists seems to be petered out in a dull tie. Mathematicians hardly care about such questions in their practice. An obvious explanation, though not very popular with the custodians of tradition, could be that it is one and the same evolutionary processes that have produced the universe and our brain, so that a weak anthropic principle ensures that we are find the same rules of the game in physical reality and in our thinking.

In his inaugural speech in Tübingen in 1927, Konrad Knopp was able to triumphantly declare that mathematics is "the basis of all knowledge and the carrier of all higher culture". Exaggerated and formulated pathetically, but not wrongly. The only difference is that the tangible benefits, the technical application usually only appear afterwards, to a certain extent behind the backs of the mathematical pioneers who, like Hardy, ruthlessly go their own way, from which nobody can predict in advance where they will lead. The mediation between pure and applied mathematics is often difficult to see through; That too may be a reason why the importance of mathematical research in today's societies is fantastically underestimated. Besides, there should be no other field in which the cultural time lag is so enormous. The general consciousness has lagged behind research by centuries, yes, one can cold-bloodedly state that large parts of the population never got beyond the level of Greek mathematics. A comparable deficit in other fields, such as medicine or physics, would probably be life-saving. In a less direct way, the same is likely to apply to mathematics; for there has never been a civilization that has been so permeated and dependent on mathematical methods into everyday life as ours.

The cultural paradox we are dealing with could be further exaggerated. There is good reason to believe that we are living in a golden age of mathematics. In any case, the contemporary achievements in this field are sensational. The visual arts, literature, and theater, I fear, would fare rather poorly on a comparison.

I do not trust myself to justify such an assertion more precisely. As a hopeless layman, I can only roughly follow the arguments of mathematicians. Often I have to be happy when I understand what they are actually about. For me, too, the drawbridge to her island remains pulled up. However, that does not prevent me from taking a look at the other bank. What I can see there, at least, enables me to make my thesis plausible with a few examples.

Most people probably never heard of the class number problem.It is one of the most difficult puzzles in number theory. Formulated by Gauß in 1801, it was finally solved by Zagier and Gross in 1983 after lengthy preparatory work. It took just as long to prove the so-called classification theorem. The aim was to organize the infinite variety of simple groups, which are completely wrongly named because they are damned complicated in nature. Aschbacher and Solomon found the keystone only one hundred and eighty years after group theory was founded. I can save myself further receipts. Gödel's two incompleteness theorems, who was probably the most brilliant mathematician of the century, are well known. It is also likely that word got around that Fermat's last theorem, on which three centuries failed, was proven in 1995 by Andrew Wiles. I would like to see the soccer championship, which could compete with such triumphs - not to mention the documenta exhibitions and theater meetings of recent years.

Nevertheless, there is no storm of enthusiasm from the audience, which brings us back to the initial question of my considerations. And at this point there is only one scapegoat left, namely our intellectual socialization, more precisely: the school. It is not just about the acute excessive demands that this institution suffers from today. The failures are deeper and have older roots. One may wonder whether there is such a thing as mathematics teaching at all in the first five years of the curriculum. What is taught there was rightly called "arithmetic" in the past. Even today, children are tortured almost exclusively with boring routines for years, a procedure that goes back to the era of industrialization and is now completely out of date. Until the middle of the twentieth century, the labor market only required three rudimentary skills of the majority of employees: reading, writing and arithmetic. The elementary school was there to provide a makeshift literate workforce. That is probably the explanation for the fact that a purely instrumental relationship to mathematics has established itself in school. Now I do not want to deny that it makes sense to master the multiplication table and to know how to do simple calculations of three sets or fractions. But none of this has anything to do with mathematical thinking. It's like introducing people to music by letting them practice scales for years. The result would probably be lifelong hatred of this art.

Childish fascination

In the higher school classes it is usually not much different. Analytical geometry is mainly treated as a collection of recipes, as is infinitesimal calculus. As a result, you can get good grades without actually understanding what you are doing. Good results are granted to every high school graduate, all the more so since he has not the slightest influence on the curriculum and method. But one should not be surprised that such teaching ultimately promotes mathematical illiteracy. It has long since lost its functional meaning because the standards of the labor market and technology have changed significantly in the last few decades. No sixteen-year-old will understand why he should bother with boring calculations that any department store calculator can do faster and better.

But the usual math lessons are not only boring, they also undermine the students' intelligence. It seems to be an obsession in education that children are incapable of thinking abstractly. This is of course a pure belief in coalfish. Rather, the opposite is true. The concept of the infinitely large and the infinitely small, for example, is intuitively and directly accessible to every nine or ten year old. Many children are extremely fascinated by the discovery of zero. What a limit is can be explained to them, and the difference between convergent and divergent sequences is easy to understand. Many children show a spontaneous interest in topological problems. Even with elementary group theory or combinatorial questions, they can be amused by taking advantage of their innate sense of symmetry, and so on and so forth. Their capacity for mathematical ideas is probably even greater than that of most adults; they have already passed the usual course of education. In most cases, they will never have recovered from the damage they suffered.

It would be unfair, however, to hold the math teachers solely responsible for the disaster. These unfortunate people are not only beaten by the instructions of the didactic experts and their fashions, they also have to operate on the string of the ministerial bureaucracy, which dictates very brutal curricula and learning goals for them. Perhaps the civil servant status is to blame for the fact that the teaching body, as can be shown in the example of the so-called spelling reform, tends to be obedient in advance. A certain fearfulness prevents many from using the freedom that the factual non-terminability gives them. However, there are teachers who defy the obsolete routines that are placed on them and who manage to introduce their students to the beauties, riches and challenges of mathematics. Your successes speak for themselves.

There are also isolated symptoms outside the educational system that give hope that the low point of mathematical ignorance has been reached and perhaps even passed. Initially, some things seem to be changing in the attitude of scientists. Today's generation of mathematicians corresponds less than ever to the cliché picture of the introverted, worldly remote loner. This is especially true for the Anglo-Saxon world. Not only obvious external motives such as the struggle for research funds speak in favor of such a change in mentality. It has mainly internal mathematical reasons. The so-called fundamental crisis of the first half of the century may have contributed to the fact that a less rigid habitus began to take hold. The traditional gap between pure and applied research has also shrunk since the client and beneficiary were convinced that profits can be made faster than ever from basic research. Experimental, computer-aided mathematics has also opened up completely new possibilities, although their methods have long been suspected of lacking rigor. As for the traditional arrogance of the discipline, it seems to me that nowadays it has been broken with a touch of irony. Mathematicians are more aware of their fallibility than before; they realize that their cathedral will never be completed and that there cannot even be a complete blueprint for this work. Many are even willing to talk to non-mathematicians.

Semantic approximation

It is no wonder that this must lead to communication difficulties. It is a good sign that over the last few decades more and more interpreters have been found who specialize in translating the formal language of the subject into natural language. This is an extremely delicate but also extremely worthwhile endeavor. Anglo-Saxon authors are also leaders in this area. Famous bridge masters like Martin Gardner, Keith Devlin, John Conway and Philip Davis pioneered this; in Germany magazines such as "Spektrum der Wissenschaft" and publicists such as Thomas von Randow owe important intermediary services. Occasionally, even the mass media have taken hold of mathematical subjects, as in 1976, when Appel and Haken solved the four-color problem, perhaps less relevant than notorious. The risk of fashionable distortions, as in the case of chaos and catastrophe theories, must be accepted. It is not only semantic misunderstandings that play a role here. The Sokal affair has shown the embarrassment it can lead to when amateurs incorporate scientific terms into their gibberish without knowing what they are talking about. On the other hand, it is a promising indication that "Fermat's Last Movement", a thoroughly serious scientific thriller by Simon Singh, has become an international bestseller.

It takes a certain boldness to attempt such translation in a culture characterized by profound mathematical ignorance. I cannot resist the temptation to quote from a dialogue that Ian Stewart, a brilliantly writing professional mathematician, preceded his book The Problems of Mathematics. An expert is talking to an imaginary layman.

"The mathematician: It is one of the most important discoveries of the last decade.

Layman: Can you explain this to me in words that common mortals can understand?

Mathematician: That won't work. You can't get a sense of it if you don't understand the technical details. How am I supposed to talk about manifolds without mentioning that the propositions in question only work if these manifolds are finite-dimensional, paracompact, and Hausdorff-like, and if they have an empty margin?

The layman: Then you lie a little.

Mathematician: But that doesn't suit me.

The layman: why not? Everyone else is lying too.

The mathematician (on the verge of giving in to temptation, but at odds with a lifelong habit): But I have to stick to the truth!

The layman: Sure. But you could bend it a little if it makes it easier to understand what you're actually up to.

The mathematician (skeptical, but inspired by his own daring): All right. It depends on a try. "

It is the attempt at literacy that matters; a protracted but promising project that should begin at a young age and that could give our much too lazy brains a certain fitness training and very unfamiliar feelings of pleasure.