Connect

    mail icontwitter iconBlogspot iconrss icon

Sport 40: 2012

The fairytale of the totally symmetrical butterfly

page 274

The fairytale of the totally symmetrical butterfly

Amalie Emmy Noether1882 –1935

Once upon a time there was a little girl who wanted to know everything that can possibly be found out by asking questions and by reflecting. The child’s eyesight was poor, but her head was wide awake. While other girls were absorbed with crocheting and darning, Emmy, whose story this is, wanted to know how one finds again the stars one has seen before, in spite of the sky changing with time as it turned. Then she asked how rain comes about, then wanted to understand what causes the wind to start blowing and to stop, then she became interested in how the sun shines, and finally she tried to find out what sort of things numbers are.

Emmy was born at a time when it was comparatively safe for a girl to ask many such questions. People were then slowly becoming ‘modern’—fortunately they had finally discovered not just that there are no Easter bunnies and Santa Claus, but also that there probably never were women who were led to evil ways by their curiosity and ended up, depending on their level of education, as common witches, as ‘wiccans’, or as those particularly mysterious ‘benandanti’. The mystical knowledge possessed by these women could cause animals, fields and other women to become infertile, enabled them to fly through the air like birds, and allowed them to mate with devils. As people became modern, they were ashamed that at one time, they had tortured and killed women on that account.

On the twenty-third of March of the year eighteen hundred and eighty- two, Amalie Emmy Noether was born in Erlangen. Her mother was page 275 Ida Kaufmann, who came from a well-to-do Jewish family, and her father was Max Noether, a mathematician.

His particular interest was in trying to create by simple arithmetical means a bridge between the high peaks of geometry and algebra, i.e. between measurement science and the study of the structures of number systems. Instead of building castles, killing dragons or going to war, Max Noether fashioned equations for his inquisitive Emmy and in doing so, became a King in her eyes. He investigated algebraic curves in space, and together with his colleague Wilhelm von Brill he published a long dissertation on the ‘Development of the Theory of Algebraic Functions, in Former and in Recent Times’. This dissertation appeared when Emmy was twelve. As was fitting for her age and gender she occupied herself with housework and needlework and was beginning to get excited about piano lessons and dancing. One morning in the summer of that year she awoke to discover a small fire on her window sill.

She tossed back her blanket and walked over to the flickering flame at her window. Close-up she realised that the fire was in fact a butterfly. The creature glowed orange, not red, with darker splotches at the edge. It slowly opened and closed its wings with their regular pattern, a very proper drawing.

It definitely reminded Emmy of an ink blot pattern, of how it is formed by letting a splash of ink run on one half of a piece of paper, then folding it down the middle, pressing the two halves together and opening it up again. What kind of butterfly is it, she asked herself, a brown tiger or a garden tiger?

Quite unexpectedly the little creature began to talk: ‘So you like my pattern, do you? Ah well, that is only natural. Because—you may take this as a prophecy—it is similar to the pattern which you are going to discover, and that one you will like much more.’

Emmy never had any doubt that the voice was the butterfly’s and that it came from nowhere else. It sounded exactly the way the voice of a butterfly just had to sound.

‘I am to discover a pattern? Your pattern? You mean later on—once I am a grown-up?’

‘That is true, it will take a few more years,’ the butterfly agreed.

page 276

‘That must mean,’ said Emmy as she squatted down on the floor, because she felt it more polite to converse on the same level, ‘that I am going to be a naturalist and collect insects?’

The butterfly laughed. It continued very softly and, even though the room was brightly lit by sunshine, it made Emmy feel quite eerie. ‘No, nothing that trivial. You will look into far more general matters—into what ink-blot patterns have in common with mirror images and how both are related to the patterns on my wings.’

Emmy had often listened to her father and his guests talk about mathematics, especially geometry, and so she understood what the little fire referred to: ‘You are talking about symmetry.’

‘And about physics.’

‘How is that, about physics?’

‘I am talking about those properties of a thing that remain the same when the thing itself is changed. I am talking about what you will discover, namely what it is that determines what remains the same: I am talking about the symmetries.’

‘Does that mean that I will become a mathematician? And a good one?’

‘Not so fast. You won’t be able to accomplish that out of thin air—others have prepared what you are going to find.’ The butterfly sounded a little pretentious to Emmy, but she kept that to herself. ‘The great astronomer Christian Huygens, for example. In 1703 he realised that in the course of a collision between two bodies, the momentum remains the same after both of them have changed their positions in space. He even guessed correctly that this is a consequence of the laws of nature being invariant to a transition from one uniformly moving inertial system—one neither accelerating nor decelerating— to another.’

‘Does invariant mean the same as unchanging?’

‘You see,’ her strange guest exclaimed with pleasure, ‘it’s beginning already! That is exactly what this is all about. And there are many more laws about such invariance, about conservation, than just the one regarding momentum. You will be the first to generalise what Huygens had noticed: Every conservation law corresponds to such an invariance with respect to changes, namely to a particular symmetry.’

‘I don’t know what might be so clever about finding out something page 277 like that. This is, after all, the meaning of laws of nature, that they are as valid in Berlin as in Erlangen.’

‘Looking at it that way, it seems self-evident. But nothing is less self-evident than that certain properties of a thing remain the same when the thing is somehow changed. If I were, for example, larger than I am, I would collapse, because while my surface area increases as the square of my size, my volume increases as the third power— but what stabilises me is my skeleton. That is why there are no giant insects, for the larger they would become, the heavier they would be and the more strength they would require to keep from collapsing under their own weight. With respect to size, therefore, there is no such invariance as there is with respect to a change in location. But as we have seen, there is invariance with regard to position. And that action and reaction are equal to each other, that is something one can depend on—as one can depend on the constancy of the speed of electro-magnetic waves.’

‘And what is the connection between conservation and symmetry?’

‘The symmetry of space corresponds to the conservation of momentum which Huygens had already noticed—momentum is written as a vector because it corresponds to a spatial transformation. What is so wonderful about it is that deriving conservation laws from symmetry, which you will accomplish, makes it possible to draw the inverse conclusion. Look at it this way: Momentum is conserved because space is the same everywhere, is homogeneous. But turning this around, the more confidently scientists can conclude from their experiments that momentum is conserved, the more confidently they can assume space to be homogeneous. And space is just the beginning. There is also the symmetry of time, the invariance as to whether an interaction takes place sooner or later, it also corresponds to a conservation law, namely energy conservation—for nothing is ever lost. And besides that, there is the symmetry of rotation, which is going to be very important in particle physics (which, to be sure, does not even exist yet). Otherwise I could reveal to you some astonishing stuff: For rotational symmetry corresponds to the conservation of angular momentum, which is of great importance for small particles. It is just as in skating: If you fold your arms against your chest, you rotate faster, so that the product of mass and rotational velocity the page 278 same. Symmetry of rotation therefore means that there is no preferred direction in the Universe.’

Emmy scoffed: ‘Most of what you keep on telling me, I don’t understand. I am not even sure that my father would understand it.’

‘Well, how could he?’ the butterfly mocked, as it briefly beat its flaming wings twice and added very softly, almost inaudibly. ‘It is best we remain silent on what your father would make of what all this will lead to in less than a hundred years. Symmetries everywhere, including even the super-symmetry between bosons and fermions, gauge invariance theory, and the dream of unifying in a single model all four fundamental forces of nature: gravity, electro-magnetism and the strong and weak nuclear forces . . .’

‘You keep prattling on wildly about all kinds of clever stuff,’ scolded Emmy.

The butterfly responded with a sound that Emmy was at first unable to interpret, but when it began to speak again, in a less mocking and peculiar voice than before, it became clear to her that it had been a sigh. ‘Yes, indeed, clever stuff . . . what can I say? I am just happy that the times are as modern as they are slowly getting to be. You, at any rate, consider me to be merely prattling on, and don’t think of some occult stuff or other. My species, the Lepidoptera—we have had to put up with the silliest stories about us throughout the ages and in all cultures. In Malaysia they say that vampires can turn themselves into butterflies, the Japanese take us to be messengers of the kami (meaning their ancestors); and in the Balkans people believe that the souls of the dead rise to heaven in the guise of butterflies.’

‘I don’t believe in ghosts,’ Emmy reassured it.

The butterfly responded in a business-like tone: ‘Well, then. That is, after all, the reason that I stopped by in the first place: because I know that all you will remember of our meeting are the parts you can believe in. And that will amount to little more than an encouragement to learn. We will only meet twice more during your lifetime and so it is very important that you will explain to me as soon as possible—’

From somewhere in the house there came a rumbling sound. It startled Emmy.

She turned her head toward the door involuntarily. But when it page 279 failed to open and Emmy saw nothing that could have explained the rumbling, she turned again to the butterfly. It had vanished however.

The butterfly had been right, of course. In the days, weeks, months, and years that followed Emmy’s adventure with that capricious little flame she forgot almost everything about it, except for the encouragement to continue asking questions. The first steps towards forgetting, a growing conviction that it had all been a dream, she had already taken while their meeting was still going on. But that detracted nothing.

Emmy wasted no time on dream interpretation, but learned to learn instead. She not only added to her knowledge in mathematical- scientific areas, but also learnt several languages and after attending the ‘School for Young Ladies’ she received a certificate for teaching French and English in schools for girls. But she longed to study at a university, as her older brother was already doing. The conditions for Emmy’s entry into the academic life were not the most favorable, however. A pronouncement of the University of Erlangen from as late as 1898 stated in so many words that admitting women to university was tantamount to sounding the first trumpet blast of the Apocalypse.

But armed with letters of agreement from respected professors, it was nevertheless possible for a woman to attend lectures. Emmy had little trouble obtaining such documents, passed her required matriculation exam in 1903, and attended lectures in history, philology and mathematics at the University of Erlangen for the next two years.

She began her actual work in mathematics under Paul Albert Gordan who was not satisfied with Hilbert’s existence proof, according to which the invariants of certain systems have a finite basis. It is not that Gordan doubted that a finite basis existed, but just like Kronecker and Brouwer, he had a preference for constructive methods, and so his student Noether also worked with them. Later on she disparaged that approach and called it ‘number crunching’, but she was also ready with stronger expressions, ‘garbage’ being a particularly unambiguous one.

She visited Göttingen for the first time in the winter semester of page 280 1903/1904. David Hilbert, who ruled there like Prospero, arranged for her visit. The outside world was then gradually realising what a remarkable place that university was and soon the inhabitants of this island received the most outlandish requests. For example, after the insane mass murderer Haarmann was executed in Berlin, his brain was put at the disposal of science, so that scientists could find its evil ingredient by rummaging through it. And it was taken for granted that the requisite science would be available in Göttingen.

At first Emmy attended the lectures of Karl Schwarzschild and Hermann Minkowski, two of the people who were soon to clean up the mathematics in Einstein’s special relativity theory. Later on she attended Hilbert’s lectures. Three years later she received her doctorate with highest honours from Erlangen University and immediately afterwards she began to teach—without pay and unofficially, and mostly as substitute for her father. It got around among mathematicians that this woman had something to contribute to their field, and only two years after her graduation Emmy Noether was admitted to the German Mathematical Association.

In the time between her graduation and the end of the First World War she managed to find her voice. She continued the work done by Hilbert, Gordan, her father, and by others, employing an increasingly abstract and more generalised approach. Impressed by her ideas, Klein and Hilbert invited her to Göttingen. There she developed, as others there were doing, precise theoretical tools for a clearer formulation of Einstein’s theory, while Hilbert fought to gain for her an academic position that was commensurate with her scientific standing.

The pettiness of the regulations opposing his efforts was indeed discouraging. An academic career was then far from assured even for highly gifted scholars who had been certified to lecture. Hilbert’s exhortation to his colleagues that the university faculty was, after all, not a bathing institution was to no avail. Since she was not licensed to lecture, Emmy gave lectures in Hilbert’s name. Occasionally she accepted guest professorships in Vienna, and later also in the Soviet Union.

After the death of her father she missed the modest financial support that he had provided. Starting in 1922 she held at least an unofficial, page 281 even though initially unpaid, position as university assistant; later on she was paid, though hardly magnificently, for teaching algebra to undergraduates.

Apart from the financial constraints, these career worries bothered her hardly at all. For while wandering around Prospero’s island she had discovered something that seemed to have been waiting for her: rings and ideals. In the year 1921 Emmy Noether’s most famous work was published: Theory of Ideals in Ring Domains. Just as the twelve-year-old had been told by the little flame on her windowsill, the rules she developed in that paper for the elements of specific abstract sets are also applicable to the elementary particles of contemporary physics. The symmetries that correspond to the physical conservation laws had now been discovered and were as pleasing aesthetically as they were valuable for gaining greater conceptual economy.

Rings and Ideals: An ideal is a subset of a number ring. A ring is a set of numbers for which the following holds:

1. Addition is associative and commutative:

(a + b) + c = a + (b + c) (associativity); and a + b = b + a (commutativity)

2. Additive identity applies, meaning that there exists an element 0, such that for all elements a of the set:

0 + a = a + 0 = a.

3. Multiplication is associative: (a * b) * c = a * (b * c).

4. For every a there exists an inverse element (–a), such that:

a + (–a) = (–a) + a = 0.

5. Distributivity applies on the left as well as on the right:

for all a, b, c in the ring,

a * (b + c) = (a * b) + (a * c) and (b + c) * a = (b * a) + (c * a).

An ideal I for a ring defined in this way is then an additive group for which the following holds: If x belongs to the ring and y belongs to the ideal, then xy and yx belong to the ideal. For example, if a ring is all the integers, then the set of all even numbers is its ideal.

Shivering as always after one of those nights under three blankets and without a reasonable heating system, Emmy Noether sat in a room in Moscow in the clammy spring of 1925. The tiles on the wall were page 282 supposed to be heated from inside the wall by a wood fire, but instead of being warm, they were merely not completely cold.

She poured herself a cup of steaming hot tea from a valuable pre- revolutionary teapot and enjoyed the hot vapour rising towards her. Suddenly the blue butterfly baked on the porcelain teapot changed its colour, beat its wings once, once more, three times, freed itself from the pottery and finally sat trembling softly on the teapot lid. And as if several decades had not passed, Emmy Noether spoke as in a dream:

‘Well, here you are again. In Russia, of all places. I really cannot say that I expected it.’

‘Go ahead, put on your glasses and take a good look if you don’t believe it!’ spoke the same voice as that time in the children’s room, slight but distinct, and in the same snippy tone as before.

‘You know,’ threatened Emmy Noether not too seriously, ‘that ghosts like you are less welcome here in this country than anywhere in the world? The people here have decided that they want to believe in as little as possible, and this new disbelief is defended with the gun, if necessary.’

‘What’s all this about ghosts? I am a butterfly and nothing else. All right, a caterpillar I never was. I hail from another place altogether and will remain forever what I am: axially symmetric, prettily coloured, very well informed. In any case, as long as the world does not collapse, if this ambiguous allusion is permitted.’

At the end of this sentence it seemed to Emmy Noether as if the butterfly had taken an almost inaudible breath in order to give the brief pause a more provocative effect. Then it said, ‘A lot has been happening with you, that is certain. You have become even more short- sighted, also quite dignified, an imposing personage—you know that your students refer to you as “Mr Noether”? If you ask me, they are an ungrateful lot. You go swimming with them, serve them pudding in your home, and some of them even publish your ideas under their names. Has it, in fact, ever been acknowledged who you are and what you have accomplished? There still are quite a few colleagues who, upon getting a paper authored by E. Noether in their hands for the first time, think that an Emil Noether must be behind it.’

‘If you cannot think of anything more entertaining than to slander my students, I may scald you straight away with my tea here. Nobody page 283 could find a more loyal group of people. The Dutchman van der Waerden, for instance, never misses an opportunity to sing my praises so loudly that it almost goes to my head.’

The butterfly fluttered. ‘Ah yes, dear Lord, your Noether-boys as they are called in Göttingen. And leading them all, Bartel Leendert van der Waerden—that’s correct, his book, which he titled modestly enough Modern Algebra, will in a few decades make you and your ideas more famous than most of your colleagues who are faculty members. The second volume will be chock full of almost exclusively your work, and in America, the work of Noether will simply be discussed as the

‘New Algebra’, thanks to van der Waerden’s propaganda. But what good does that do you, and most of all, what good does it do me? You really ought to work more in physics. Or at least solve problems that are close enough to what physics will soon become.’

Noether shook her head. ‘You really are extremely well-informed and hold many and very succinct opinions. But you can no more prescribe to me what to investigate or to teach than the gentlemen could lock me out of science for good. Just what emboldens you to be so impudent?’

The answer was much less boastful than everything the butterfly had said before: ‘I would be happy to explain everything if we had more time. But that is the real issue, that I know so much. For I consist much less of porcelain or of any other tangible material, than of the information that I carry in me. The experience we are now going through is axially symmetric, you know—the axis is the present meeting here. The first time we met you hardly knew anything and I knew just about everything. Today at our second meeting we both know some things, but far from everything. But when we see each other for the last time, I will hardly know anything, and you—’

There was a knock at the door. A voice with a Russian accent inquired politely, ‘Are you already awake, Professor?’

‘Just a moment!’ she answered and looked instinctively toward the door. As she had secretly expected, the talkative butterfly had changed back into a picture on the teapot. ‘Until the next time, then,’ she sighed, and admitted her far more prosaic visitor.

page 284

Time, asymmetrically enough, kept marching on, always in the same direction. As before, Emmy retained hardly any recollection of the strange butterfly and what it had said, once it was gone. Busy working years followed, as before. Occasionally a caterpillar would crawl on the desk, something white would hover near the pond where she was bathing, or among the violets of March, and tug at forgotten memories. And once towards the end of the 1920s, a fat green elephant hawk- moth sat on the lecture hall drapes and Emmy, lost in thought, gazed at it for several minutes before continuing her lecture.

Whatever it was that occupied her mind at that moment vanished without a trace. There was plenty to do. Together with Helmut Hasse and Richard Brauer, she turned to generalised studies in the growing field of non-commutative algebra in 1927. She was increasingly at home in the scientific community where she was now accepted as a full-fledged and indispensable member. She was co-editor of the important Annals of Mathematics, she addressed major congresses, received honours, and was less and less often taken for an Emil. And everything might well have continued that way.

Instead of that, the National Socialists came to power and began to hound Jews out of Germany’s academic life. Some students at Göttingen, and not just the Noether boys, composed letters and petitions requesting that Emmy Noether be kept at the University.

Although she was herself in danger and terror loomed everywhere, she nevertheless found time to organise, together with Hermann Weyl, an emergency fund for the mathematicians who lost their positions after being designated Jewish according to the Nuremberg laws. As a committed pacifist and one-time member of the German Socialist Party, she took an interest in politics only to the extent of one who could not believe, as she did not, that one could lead a good and productive scientific life without paying attention to such things.

To save herself she had to leave the country to whose most intelligent inhabitants she had belonged. She fled to America. There she found a position at the women’s college of Bryn Mawr near Philadelphia, which secured her an income, but one that again did not correspond to her academic standing.

The American mathematician Anna Johnson Pell Wheeler stood by her side there as a good friend. She had at one time studied in page 285Göttingen after winning a fellowship and had herself come to Bryn Mawr in 1918 where she taught until 1948. Wheeler had been the first woman to present the very prestigious Colloquium Lectures of the American Mathematical Society.

But Emmy’s hopes of being able to work in peace once again were dashed when the recent emigrant had to face serious health problems. On the 10th of April 1935 she was operated on to remove some cysts and four days later she suddenly lost consciousness and developed a high fever. The physicians’ attempts to save her were in vain.

In the garden beneath the window of the room where Emmy Noether lay unconscious on her sickbed, spring flowers were in bloom, some of them native species, others recently domesticated from South America or Europe. A butterfly sat on a low wall at the garden’s edge.

It was talking to itself: ‘It is always the same, never changes, I come whenever I happen to come. But this time I am too late. What happens to all the knowledge it is possible to acquire—she had known enough to be able to envisage that. I would have liked to find out from her where all that information ends up. Whether it can ever be completely destroyed, or if there is a conservation law for information preserved in ordinary matter. What happens when a book is swallowed up by a black hole? Is everything written in it subtracted from the total information content of the Universe? What is the fate of memory if no one remembers? One really ought to discover the conservation law for that. She could have told me how that stands. She would have known what symmetry it corresponds to.’

When she was born, it seemed that people had got over witch burnings. When she died, crimes were again being committed in Europe that could easily be compared to those ancient cruelties. There is an adage of the ‘benandanti’ that says: ‘When we, who have resolved to live between sunset and moonrise and to search for the truth, want to find protection in the hour of the dead, we must take care that in the Great Struggle we stand on the side of the innocents.’

And though she has died, on the side of the innocents is where Emmy Noether lives.

page 286
From the author’s Introduction:

Intoxicating Heights contains sample episodes from the lives of great twentieth- century mathematicians, some of them re-told (in the essays), others newly combined (in some of the stories), and some, I hope, happy inventions (in the remaining stories). Much of what is written here is not true: Georg Cantor was not, as the first story suggests, identical to two numbers, Poincaré has not been resurrected up to now, Emmy Noether never came across a talking butterfly, Gödel never haunted anyone, Dirac had no quarrel with bohemians, Turing is not godfather to a secret artistic project, Kolmogorov had no truck with werewolves, John von Neumann’s reincarnation has remained unnoticed, Gregory Chaitin is not an artist, Julia Robinson never practiced medicine, Benoît Mandelbrot was not lost anywhere, Edward Witten did not crash-land in Colorado, and the writings of Lena Dieringshofen cannot be read anywhere outside this book (not yet).

These diversions based on people who are still alive or have actually lived are relevant to the book’s theme of twentieth-century mathematics because while fantasy was needed to invent and reconstruct them, a precise fantasy was called for; not any old ideas, but meaningful ideas.

This all makes sense and should serve to encourage the reader to follow the paths indicated here and to delve a little more deeply into the domain of the precise, which is often no less fantastical.

From Dietmar Dath, Höhenrausch. Die Mathematik des 20. Jahrhunderts in zwanzig Gehirnen © AB —Die Andere Bibliothek GmbH & Co. KG, Berlin, 2003, 2011. English translation © Josef Eisinger, 2003.