8

Legacy

“If she knew how useful her mathematics had become today, she’d probably turn over in her grave.”

In 1898, the faculty of the University of Erlangen, in the city of Emmy Noether’s birth and education, officially pronounced that the admission of female students to the university was a “measure that would overthrow all academic order.”1

Since then, the rights of German women to participate in academia have steadily improved. Just ten years later, Noether received her doctorate from Erlangen. After the end of World War II, Erlangen, along with other institutions of higher education in Germany, displayed an interest in making up for lost time. They redoubled their efforts to ensure equal treatment for women and to recognize the accomplishments of those of the present and the past.

Today, Noether is celebrated in Erlangen as one of the most important citizens in its long history. In a sense, this celebration was kicked off in 1958, when the University of Erlangen memorialized the fiftieth anniversary of Noether’s degree by bringing together many of her former students, and their students, to discuss the legacy and influence of her work.2

Noether herself, after dispensing with the mathematical problems raised in her study of general relativity in her 1918 paper, in which she proved Noether’s theorem, and after being granted her habilitation largely on the strength of the theorem, lost interest in the subject. Aside from one brief, uncredited note, she never referred to her paper in any subsequent publication, nor did she suggest related topics to any of her doctoral students.3

This peculiar preference of Noether’s stands out even in a life full of curious ironies. It’s an example of how, in a fertile career replete with fundamental results, others may attach supreme value to a piece of work of little enduring interest to its creator. After the theorem, Noether returned to her consuming passions in pure mathematics, mainly algebra. For her, the theorem was the formal culmination of her efforts as a team player; she arrived at it to help her friends and colleagues—Einstein, Hilbert, and Klein—out of their confusion. The motivation arose from collegial interests, not from an internal fire. Noether’s theorem, and the habilitation that it supported, punctuated her transition from an assistant in Hilbert’s school to a leader in her own right: she would quickly become the center of her own school of mathematics at Göttingen and take her place in the international community of mathematicians as a forger of fresh paths.

For any typical mathematician or scientist, a result of the import of Noether’s theorem would be an enviable achievement, a ticket to immortality. Yet here we are confronted with an intellect for whom this achievement was a side issue, a task of nothing more than momentary interest, completed as a favor for others—others who themselves happened to be among the giants of intellectual history. It can be a challenge to keep things in perspective, at times—or even to know what such a perspective would look like.

Nonetheless, the theorem largely defines her legacy in physics and other empirical sciences. It continues to inspire new research and find new life in a wide variety of applications that would surely have astonished its discoverer. As we will see in this chapter, Noether and her work have also had a wider cultural influence and legacy.

This book is mainly the story of an idea called Noether’s theorem and the trajectory of its fortunes from its creation around 1918 up to the present day. This chapter is chiefly about how that idea continues to inspire research and find new applications. But as I’ve just pointed out, the theorem was of little more than passing interest to its discoverer, although it is the chief source of her enduring importance in the empirical sciences. As Noether is a giant in the world of pure mathematics, I hope I can be forgiven a slight digression in the service of briefly discussing something about her legacy in that world.

A famous mathematician of our day, Ian Stewart, who has also made fundamental contributions to theoretical physics, responded this way when asked which forgotten mathematical figure everybody ought to know about: “Emmy Noether was the greatest female mathematician of her era. She started a revolution in mathematics by focusing on abstract structure rather than computational details. This led to greater generality, and greater power. Today’s mathematics—pure and applied—couldn’t exist without it.”4

Here Stewart is emphasizing a point that I’ve alluded to repeatedly in the book. She emphasized this aspect herself, in some of her rare comments about the import of her own work. Noether saw her most enduring legacy not in a collection of results but in the dissemination of methods of working through the mathematical community. This stress on the approach was connected to her emphasis on abstraction and her move away from detailed computation. The new style of research arose as she turned away from the methods of her mentor Paul Gordan and her own thesis and began to enthusiastically adopt something more in the manner of David Hilbert.

As a reminder, an example of the difference between computation and abstraction can be found in the approach to showing that a particular class of equations has solutions. The Gordan school would insist that someone prove this by calculating and displaying a particular solution. Hilbert might prefer a clever proof that solutions must exist, without ever constructing any solutions. This was the essence of the conflict between the two men, culminating in Gordan’s complaint that Hilbert’s work resembled theology more than it did mathematics. Noether’s thesis cranked through massive amounts of detailed calculations of this sort, work that she would later recoil from in distaste.

Mathematician Israel Kleiner, in his History of Abstract Algebra, lends his voice to the importance of Noether’s influence in setting the direction for mathematics:

The concepts she introduced, the results she obtained, the mode of thinking she promoted, have become part of our mathematical culture.…

Emmy Noether was a towering figure in the evolution of abstract algebra. In fact, she was the moving spirit behind the abstract, axiomatic approach to algebra.5

Noether was not unique in her lack of enduring interest in her own result. Although it is the source of her fame among physicists who are aware of it, the theorem is also generally of little interest to pure mathematicians. Noether is instead a giant in this community because of the main line of her work in abstract algebra.

Sharon McGrayne puts it nicely, in her Nobel Prize Women in Science: Noether’s theorem is “both her most famous and her least famous accomplishment.”6 Among physicists, the theorem is about the only thing she is known for (although the result remains better known than the author). Among mathematicians, however, the result is relatively obscure.

Noether’s algebraic legacy includes more than an impressive host of specific results and theorems, several of which are named after her. Her influence also involves the promotion of algebra from its historical status as a calculational tool to one of the foundation stones of mathematics.7 Noether is one of perhaps three mathematicians responsible for this transformation; today algebra is rivaled in fundamental importance only by logic itself. Noether is also largely responsible for modern algebra’s abstract, “modern” nature and the methodologies surrounding work in the field.

But Noether’s legacy in mathematics proper is not limited to algebra. Her impact on mathematical culture and her encouragement of a certain style of research, through her teaching and mentorship, influenced fields far from her own usual specialties.

As mathematician Peter Roquette put it, “Her methods eventually penetrated all mathematical fields, including number theory and topology”8 and Alexandrov and Heinz Hopf’s classic text Topologie credits Noether for creating the “algebraization” of topology.9

Alexandrov also gives this assessment of her stature among mathematicians during her life:

Emmy Noether lived to see the full recognition of her ideas.… In 1932, at the International Congress of Mathematicians in Zurich, her accomplishments were lauded on all sides. The major survey talk that she gave at the Congress was a true triumph for the direction of research she represented. At that point she could look back upon the mathematical path she had traveled not only with an inner satisfaction, but with an awareness of her complete and unconditional recognition in the mathematical community.10

It’s remarkable how many scholars and historians of mathematics have noticed a similar thread in Noether’s influence on mathematical culture. According to Uta C. Merzbach, the Smithsonian’s first curator of mathematical instruments, “The remarkable change in the language and methods of mathematics [wrought by the publications of Noether and her coauthors] is apparent upon contrasting nearly any early twentieth century mathematical publication with a contemporary one.”11

This is quite a remarkable claim by a prominent historian of mathematics: Merzbach finds Noether’s influence in nearly all of the current publication in that field.

Emmy Noether was a mathematician’s mathematician, believing that mathematics should be enjoyed for its own sake, without any thought of application. In the opinion of her nephew, “If she knew how useful her mathematics had become today, she’d probably turn over in her grave.”12

The physics world was, as we have seen, slow in catching on to the value of Noether’s theorem. In the words of French mathematician Yvette Kosmann-Schwarzbach, the theorems, whose “importance remained obscure for decades, eventually acquired a considerable influence on the development of modern theoretical physics, and their history is related to numerous questions in physics, in mechanics and in mathematics.”13 She refers to “theorems”: remember that what we and most others call Noether’s theorem is really a set of two related theorems and their converses.

According to the views of some physicists and physics watchers, including Ruth Gregory, the head of the Department of Physics at King’s College in London, Noether might have won the physics Nobel Prize had she lived longer.14 This view is also implied in a chapter about Noether in Sharon McGrayne’s Nobel Prize Women in Science, which deals with women who might have or should have won the prize, as well as those who did.

However, I believe that this speculation is both a distraction from the mission of elevating Noether to her rightful place in the history of science and that it is unrealistic. The prize is not awarded posthumously, and for Noether to have been a recipient, she would have had to live long enough for the importance of her contribution to be generally recognized in the physics community. The significance of her work might not have been clear until the 1960s, when the success of the standard model was generally accepted and when its dependence on the theorem was better understood. Recipients often received their prizes decades after the cited work was accomplished. Einstein, for example, received the Nobel in 1922, chiefly for one of his 1905 papers.

There is also another, perhaps more fundamental reason that the theorem would not be likely to be the basis for a Nobel Prize in physics, despite its central importance for that science. A prize in physics is unlikely because Noether’s theorem is difficult to categorize as a part of physics. It’s not a theory of physics, such as general relativity or quantum mechanics. To repeat a phrase I’ve used before, it’s rather in the nature of a theory construction kit. Noether’s theorem is a mathematical result that sits underneath all of physics and provides structure and interpretation. It tells physicists what theories are possible and where to look for new physics. It’s so indispensable that entire edifices of physics could not exist without it. But it itself is not a physics theory. Noether’s theorem is not the kind of thing that the Nobel committee is used to dealing with.

Obsessions about Nobel Prizes are best left to journalists, who take these awards more seriously than do scientists, who sometimes find having won an annoyance despite the welcome cash.15 There is little doubt that several female physicists and astronomers were unfairly passed over for the physics prize, but Noether’s name should not be among them.

The importance of Noether’s theorem for physics is the main theme of this book, of course. Its legacy seems finally secure, as is the credit for its creator. Certainly, many physicists still have at best a vague idea of what the theorem is about and an even cloudier awareness of the story behind it. But as many people, including Nobel laureates and science writers who bring cutting-edge ideas to a wider audience, attest, the theorem’s importance as a cornerstone of physics is enthusiastically recognized, to the benefit of Noether’s place in history.

In the opinion of astrophysicist Brian Koberlein, “Emmy Noether should probably be put in the same group as Isaac Newton and Albert Einstein as one of the greatest physicists who ever lived.…[She] single-handedly revolutionized the way we understand physical theories.”16 Koberlein also believes that Noether was “the most brilliant mathematical physicist of all time,” and her theorem’s “power is hard to overstate.… Everything we study within physics depends upon Noether’s theorem, from dark energy to the Higgs boson. It has transformed the way we view the cosmos, and it demonstrates the real power of mathematics when it comes to understanding the Universe.”17

We increasingly find direct mention of Noether’s theorem in research papers and textbooks, often with an aside about its immense value. This situation is a welcome contrast to the way things were near the middle of the last century, where, as described briefly in Chapter 7, an emerging field of physics depended critically on the theorem but it was rarely mentioned in any of the relevant papers.

The physicists Leon Lederman and Christopher Hill, quoted in the introduction, have more to say about the import of Noether’s work for physics:

Noether’s theorem is a profound statement, perhaps running as deeply into the fabric of our psyche as the famous theorem of Pythagoras. Noether’s theorem directly connects symmetry to physics, and vice versa. It frames our modern concepts about nature and rules modern scientific methodology. It tells us directly how symmetries govern the physical processes that define our world. For scientists, it is the guiding light to unraveling nature’s mysteries, as they delve into the innermost fabric of matter, exploring the most minuscule distances of space and shortest instants of time.18

Symmetry has been interesting to scientists, philosophers, and artists since the beginning of recorded history. Many popularizers of science have written about how the general concept of symmetry is somehow crucial to understanding, discovery, or appreciation. But the preceding quote brings out how Noether’s theorem makes the importance of symmetry precise and actionable. It explains “how symmetries govern the physical processes.” After Noether’s theorem, symmetry doesn’t simply describe how something appears. It participates actively in the dynamics of a system, defining what behaviors are possible.

Noether’s theorem is firmly embedded at the center of high-energy physics, classical mechanics, and gravitation, among other fields of physics. But physicists continue to discover new and sometimes unexpected ways to apply the theorem in areas where it has not traditionally occupied center stage. Recently, physicists studying the role of fluctuations in fluids and related systems discovered that Noether’s theorem provided, as in particle physics, a guide to the construction of theories in statistical mechanics.19 This discovery is important for physics because much of it could not exist without statistical mechanics. It is through that specialty that we deduce the large-scale properties of matter from the behavior of its constituent atoms or molecules. Aside from his theories of relativity, most of Einstein’s important contributions to physics rest in large part on statistical arguments; this was the branch of physics that he turned to repeatedly to explain matter, radiation, and quantum phenomena.

As mentioned in the previous chapter, the unfinished business of modern fundamental physics, its unrealized dream, is to unify its two great models of reality—general relativity and the standard model—into one unified theory that explains everything. It’s a difficult task both conceptually and mathematically, because the standard model is based on quantum mechanics, while no quanta exist in the geometry of gravitation. The two pictures of reality must be reconciled somehow, with a future theory of “quantum gravity”: some way of encompassing quantum particle interactions with the non-Euclidean geometry of general relativity.

No one has yet succeeded in constructing such a theory, but recent experience suggests that Noether’s theorem will remain an essential tool in the effort. The most well known attempt to create a theory of quantum gravity is string theory, along with its various offshoots. This activity remains controversial, as it’s vulnerable to criticism that it has failed to produce specific predictions amenable to verification by experiment. However, it’s worth noting that the mathematics of these approaches makes heavy use of Noether’s theorem.

We know that despite its beauty and power, general relativity can’t be the whole story of gravity. As we follow the evolution of the cosmos backward in time and approach the Big Bang, we eventually enter a realm where conditions are so extreme that they can never be reproduced in our laboratories. The physics of the early—very early—universe, the first tiny fraction of a second, will always be beyond experiment. As physics, despite the flights of imagination brought to bear on it by theorists, is at its core an empirical science, a more encompassing, unified theory, one that applied from the birth of the universe to the present, will always contain some arbitrary ingredients beyond the possibility of verification. Some things we will never know.20

But we do know that in the very early universe, general relativity is not sufficient. The only way to cover those highly dense conditions would be with a theory that unifies gravity with the other forces of nature, which are now described by quantum field theory. And such a theory has still eluded our most brilliant physicists.

Noether’s theorem is an important guide to constructing possible alternative theories of gravity that make general relativity even more general and that may be extended to the quantum realm.21 While groping in the dark, beyond the reach of the light of experiment, the theorem illuminates the outlines of what such theories should look like. It can do this because of its status as a theory construction kit, not tied to any particular physical hypothesis.

Particle physicists have so far not enjoyed much success in going beyond the standard model. The persistent dream is to unify all its forces into a grand unified theory (which would be grand, and unified, but which would still not encompass gravity). The program for unifying the forces of the standard model involves searching for grander symmetries. Through Noether’s theorem, such symmetries would imply new conservation laws, which would need to be mediated by new particles. Some of these predicted entities are beyond the reach of current, or even foreseeable, particle accelerators—the energies required are just too big. Others are not outside our reach, and the search for these new particles can be undertaken. This was the program followed during the rise of the standard model, and it was a succession of exciting successes. But so far, the quest to enlarge the model has failed.

Physicist Sabine Hossenfelder explains the problem with contemporary particle physics in an engaging and illuminating video.22 She points out that in their attempts to enlarge the standard model, particle physicists have made a series of predictions of new particles since the 1970s and that essentially all these predictions have failed. The basic strategy has been as described above: to find larger, more complicated symmetries that encompass the symmetries of the standard model but, in some sense, complete it and, through Noether’s theorem–type arguments, to calculate the properties of the still-unobserved particles that the more elaborate symmetries require. Why has the power of Noether’s theorem let them down?

Hossenfelder has a good answer, and I believe that she’s correct. The history of physics presents examples of successful theories that began as more complicated elaborations of existing theories. What the successful theories all have in common is that the complications were needed to explain observations or to remedy an inconsistency in existing theory. Sometimes the theories took care of both, as with Einstein’s theory of general relativity. It explained the precession of the orbit of Mercury, and it healed the inconsistencies in Newtonian gravity. General relativity explained the long-standing mystery of the identity of gravitational and inertial mass and eliminated Newton’s problematic instantaneous action at a distance. The mathematics is immensely more complicated than what’s needed for classical gravity, but this complication is not gratuitous. It was the natural mathematical language for the job at hand. General relativity was, and is, spectacularly successful, making a series of predictions that continue to be borne out up to the present day.

The history of the quantum theory follows the same pattern, from the early atomic theory through quantum field theory and the standard model. At each step, theories were enlarged and made more mathematically complex when needed to explain observations or to resolve inconsistencies. Each new model made new predictions, and these were often confirmed. The long list of triumphs bears out this pattern: Paul Dirac’s prediction of antimatter, the standard model’s prediction of new particles, the calculation of the fine structure constant, and other achievements were consequences of theories’ grappling with observed physical reality.

The largely fruitless course followed by particle physics for the last half century has disregarded the lessons of history. Untethered from the demands of observation and experiment, brilliant theoreticians have given free range to their imaginations, creating a variety of lovely fantasies such as supersymmetry and string theory. These surmises have either produced predictions that turned out to be false, can’t be tested, or produce nothing testable at all.

Instead, these theory creators are motivated by a philosophical or aesthetic desire to unify the several forces of the standard model and sometimes gravity as well. Certainly, aesthetics and metaphysics can be a guide for the scientist. Einstein’s rapture about the beauty of his new theory helped convince him that it must be true. But in the absence of the constraints of observation, aesthetic concerns lead somewhere else than to successful science. Without these constraints, we have a profusion of approaches to unification, searching for a solution to a problem that doesn’t exist and leading, as we’ve repeatedly seen, to science that doesn’t work.

When we are searching for a solution to a problem that does exist, Noether’s theorem helps show us where to look. As a theory construction tool, it can provide the scaffolding, something to stand on while you build your tower. But it has no opinions about the physics that you think might describe the world. It lies beneath the physics. To use a loose analogy from today’s pervasive technology, the theorem is a key component of the operating system, which lies beneath any number of application programs. Therefore the theorem cannot tell us what the sequel to the standard model is, but it can show us where to look.

According to astrophysicist Katie Mack, “Noether’s theorem is to theoretical physics what natural selection is to biology.”23 And some biologists are beginning to find insight in the theorem for their science.

I admire Mack’s formulation. It perfectly captures how Noether’s theorem sits underneath all of physics as an organizing principle that can inform us about what a reasonable theory might look like, no matter the details of the phenomenology under study.

One application of the theorem to biology is in the concept of scale symmetry. A system has scale symmetry if it looks the same at different scales. As you zoom in, the system remains unchanged.

Imagine sitting far above a planet and observing its continents with their jagged coastlines separated by seas, oceans, land bridges, canals, and so forth. Now imagine diving down for a closer look. You zero in on the edge of one of the continents, and after a while, you see a pattern that looks familiar: a set of land masses with jagged coastlines, separated by various bodies of water, land bridges, and so forth. You might think of these land masses as islands rather than continents. But at least on this planet, the pattern is eerily familiar. You’ve gotten closer, but everything looks the same.

Now imagine diving down to take a closer look at one of these islands and discovering the same pattern again: the island is made up of smaller islands, with bodies of water and land bridges separating them.

This example is an illustration of a type of scale symmetry. Some aspects of some systems, surprisingly, remain unchanged as you magnify your view. This kind of symmetry is different from the space-time and other symmetries we usually talk about in physics and not one of those considered by Noether. However, as her theorem deals with symmetry at an abstract level and is not restricted to symmetries of time and space (for example, the preceding examples of gauge symmetries), any newly discovered symmetry is potentially important. If it can be fit into the mathematical framework of Noether’s theorem, then it can be shown to be equivalent to a conservation law. As in the case of the standard model discussed earlier, new conservation laws can lead to new predictions about nature.

It turns out that scale symmetry is more than an idea in our imaginations. Some biologists believe that some natural neural networks, including brains, exhibit some form of scale symmetry. Researchers have already connected this property with other properties, such as efficient signal transfer within the network.

The possibility of scale symmetry in the brain’s neural network has now reached the level of detailed experimental investigation. Several years ago, a group of neuroscientists and physicists from London and Zürich carried out a combined experimental and theoretical study of scale invariance in animal brains.24 Studying the question mathematically, they used Noether’s theorem to derive a set of conserved quantities related to the scale invariance symmetry. These conserved quantities aren’t the familiar ones, such as energy or momentum, but the scientists derived specific formulas for the new conserved quantities. They then discovered that some of the properties of animal brain activity that they measured followed the conservation laws they had derived.

This was the first study of conservation laws for scale invariance from Noether’s theorem. These findings may open up avenues for research in physics and may guide future research into biological neural networks.

Of course, no one was thinking about computers in 1917. I won’t try to speculate on how Noether would feel about these machines had she lived to see them. She might have found a use for them in the initial stage of her mathematical journey, which was full of tedious calculations. But after her definitive transition to more abstract forms of mathematical thought, which created in her such a distaste for her earlier work that she purportedly referred to her doctoral thesis with a mild obscenity, she probably would have had little use for computers.

What then does it say that her theorem has turned out to have a direct bearing on today’s cutting-edge investigations into several areas of computer applications in science? It reminds us of the immense power of mathematical abstraction. Deep thought that uncovers unsuspected relationships between ideas has a timeless value. A theorem is true forever. We still use the Pythagorean theorem all over the world, countless times each day, millennia after it was first inscribed in the sand in ancient Greece. Noether’s theorem will never become obsolete. Those who grasp it centuries from now will find new areas that it illuminates, just as today’s scientists, working a century after it was proven, are applying it to fields that Noether and her friends could not possibly have foreseen.

Noether’s theorem has found recent applications in quantum computing.25 Research into quantum computers is motivated by the theoretical possibility that they could perform certain types of calculations thousands of times faster than any computer designed along familiar (classical) lines. The way to speed up computation for any type of computer is to perform as much of the work as possible in parallel, because the rate of improvement in the speed of single processors is slowing down due to the existence of ineluctable physical limits. This is why your laptop has multiple cores, and why large simulations of complex systems, such as the atmosphere, are performed on supercomputers with thousands of processors.

In a quantum computer, the parallelism is achieved by exploiting the physics of the quantum realm, where many potentialities coexist until a single one is selected by a measurement. While the computation is underway, it’s essential that the computer, and its internal quantum states, remain isolated from the environment. That is, except at the beginning, when the operators define the problem, and at the end, when they read the result—during the computation, the computer must remain isolated. Interactions between these two moments can disturb or destroy the superposition of states essential for maintaining parallelism. In the context of quantum computing, these undesirable environmental interactions are referred to as noise. Elimination of noise is one of the central problems in the design of quantum computers.

Physicist Evan Fortunato and his coworkers have published a series of papers, beginning with Fortunato’s 2002 PhD thesis, in the general area of error control in quantum computers. Fortunato’s techniques are based on results derived from Noether’s theorem.26 Briefly, he has discovered that the relevant components of the noise-system interaction are characterized by certain symmetry groups. Through Noether’s theorem, he can relate these symmetries with subspaces of conserved quantum states and use that relationship as a theoretical basis for control of the computation.

In his conversation with me about his work, it was clear that Fortunato considered Noether’s theorem a key to the success of his research program. He suggested that others laboring without its insights sometimes had to take a long way around, while he was able to take a shortcut.

Noether’s theorem has also been used to clarify problems in ordinary, nonquantum computing. Scientists and engineers routinely use computer simulations to predict the weather and climate, design airplanes, study the gravitational interactions of galaxies, and much more. In many of the algorithms underlying these simulations, the homogeneous, isotropic space in which we live is represented on some kind of grid, something that resembles the lines and boxes on graph paper.

In this computational universe, an object (an atom, an airplane, or a planet) can’t glide smoothly in any direction, as it can in the real (classical) world. Instead, it’s forced to jump from square to square on the predefined grid, like a piece in a game of checkers.

This grid universe doesn’t have the symmetries of real space: it looks different when you rotate or translate it, just as a piece of graph paper looks different when rotated—the lines are on a slant. Therefore, the (classical) conservation laws that Noether’s theorem shows us are equivalent to these spatial symmetries cannot hold within these computer simulations. Since the grid universe doesn’t have, for example, spatial isotropy, it can’t have conservation of angular momentum, as Noether’s theorem tells us that the real (pre-Einstein) universe must have.

However, we need our simulations to be accurate. They should respect conservation of momentum and angular momentum, even if they don’t fully respect the symmetries of real life. Here Noether’s theorem has been useful in analyzing the effects of grids on conservation in computations and showing how to get around the absence of symmetry in certain cases.

I was once part of an undergraduate seminar whose leader, a professor of physics, confidently mentioned in passing that all of economics could be reduced to a single physics equation. Certainly this remark, which I hope was not meant to be taken entirely seriously, was just an example of that hubris of which our kind is often accused, usually by scientists in other disciplines.

Nevertheless, one approach to economics makes heavy use of the mathematics and concepts of modern physics.27 This school of econometrics attempts to calculate invariances and conservation laws in economic systems and refers explicitly to Noether’s theorem, employing it as a central tool.28 While several “conservation laws,” or invariances, such as the income-wealth conservation law, have been recognized in economics for some time, the use of Noether’s theorem has revealed some of these to be special cases of more general invariances. The theorem had allowed the discovery of what the proponents of its use in economics call hidden conservation laws. The application of Noether’s theorem in economics, and the related apparatus of group theory, variational calculus, and the machinery of analytical mechanics borrowed from physics, has led to the blossoming of a new, highly sophisticated and theoretical branch of the “dismal science.” The literature of this branch of economics resembles, at a glance, treatments of the field theories of physics more than it does the familiar expositions of classical economics.

Whether these new approaches will lend economics the reliable predictive power that the discipline has long yearned for, and whether they will lead to its taking its place beside what are generally recognized as the real sciences, remains to be seen. But even if it turns out to be an extended academic exercise, the translation of the language of modern (and advanced classical) physics into the field of economic modeling is a curious and fascinating development.

Of course, things can get out of hand. One social science article that appears to be entirely serious attempts to explain something about Taiwanese politics by referring to the symmetrical behavior of a political party and its conserved objectives through application of Noether’s theorem.29

There is clearly something infectious about the high degree of generality inherent in the way that the theorem relates symmetries with constant properties. People can and obviously will interpret symmetry and conservation in increasingly imaginative ways.

Just as quantum mechanics and relativity theory (both versions) have been subject to countless popular treatments trafficking in unforgivable vagueness and philosophical extrapolation, Noether’s theorem, as it becomes increasingly better known in the general culture, will have to suffer alongside them. It’s a minor annoyance compared with the gratification that would come with Noether’s taking her rightful place alongside the other greats in histories of science.

Noether’s struggles to participate in academic life, and the many occasions where she was thwarted or suppressed by men who were comparative mediocrities (in contrast with the conspicuously brilliant men who exerted themselves to advance her interests) recall a scene from Oscar Wilde’s An Ideal Husband, which also happens to be a choice representation of Wilde’s effortless employment of breezy irony:

Lady Markby.… Really, this horrid House of Commons quite ruins our husbands for us. I think the Lower House by far the greatest blow to a happy married life that there has been since that terrible thing called the Higher Education of Women was invented.

Lady Chiltern. Ah! it is heresy to say that in this house, Lady Markby. Robert is a great champion of the Higher Education of Women, and so, I am afraid, am I.

Mrs. Cheveley. The higher education of men is what I should like to see. Men need it so sadly.

Lady Markby. They do, dear. But I am afraid such a scheme would be quite unpractical. I don’t think man has much capacity for development. He has got as far as he can, and that is not far, is it? With regard to women, well, dear Gertrude, you belong to the younger generation, and I am sure it is all right if you approve of it. In my time, of course, we were taught not to understand anything. That was the old system, and wonderfully interesting it was. I assure you that the amount of things I and my poor dear sister were taught not to understand was quite extraordinary. But modern women understand everything, I am told.

As we’ve seen, Noether became the center of mathematical life during the 1920s and early 1930s in Göttingen, which was then the center of mathematics in the Western world. She was one of the ten or so most important mathematicians alive in the world at the time, by any measure. Yet even in recent years, historians seem capable of a perplexing disregard not only of her importance and accomplishments but also of her very presence when they describe a milieu where she was actually dominant. I hesitate to form facile theories about the reasons for this blatant oversight, whether the causes be quirks of scholarship, psychology, or just a recurring accident, especially as this Noether-blindness is not limited to male historians. But when faced with an otherwise fine and interesting book (a collection of essays specifically about Göttingen and its importance in the development of the natural sciences, including several studies about the years during which Noether was active) and finding within its pages not a single mention of her name, one can be forgiven, I think, a certain measure of exasperation.30

An interesting paper in that very collection actually discusses Hilbert’s battles with the Göttingen faculty over the hiring of foreigners, Jews, and, specifically, women. It names many professors who were the subjects of these skirmishes—but omits any mention of Emmy Noether.31

In addition to these oversights, several mathematicians who have studied the history of Noether’s theorem have pointed out the curious fact that special cases of it are periodically “discovered” by modern mathematicians who believe that they have derived novel results, unaware that the problem was dispensed with, in full generality, over a century ago.

The legacy of Emmy Noether as a mathematician is obviously secure and permanent. She possesses the immortality of having her name attached to objects and results central to algebra and having indelibly influenced several other areas of mathematics. In addition, as many mathematicians have pointed out, even without these specific results, her legacy is permanently etched into the fabric of how mathematical research is carried out and discussed.

Her legacy as a role model, in light of the exemplary and courageous way she led her life in the midst of an endless cascade of cruel obstacles, will also be a permanent inspiration to all who learn her story. Her infinite generosity, impossible optimism, and utter devotion to scientific truth and the progress of mathematics define her as a unique personality in the entire human history of the exact sciences.

The moral issues that arise in Noether’s treatment, first as a woman and second as a Jew at the hands of the Nazis, hardly need to be belabored. Her life is the perfect example of the potential loss that can be suffered by a civilization that withholds opportunities to flourish from segments of its population. Without Noether’s work, our science would most likely not have progressed as far as it has. The tool that Noether provided future scientists with which to probe nature—from the inner workings of the atom to the structure of the cosmos—would not exist. It’s exceedingly rare to encounter a figure that combines Noether’s unique talents with the fortitude to push ahead in the face of a culturally entrenched disregard. Are there people, even today, with the ability to put a dent in the serious problems facing us but whose contribution we are denied because they have quite understandably decided not to pursue an exhausting existence, constantly asserting their right to be taken seriously?

My view of the history of ideas is somewhat unfashionable, as I attach what some would consider exaggerated significance to the innovations of individual minds. Lately, it’s more usual to encounter the assumption that, well, if A hadn’t come up with it, then B would have done so sooner or later. The idea was “in the air.” This is true of some, perhaps many, of the important turning points in the twisting course of culture through time. But there also exist unique individuals with unique abilities, and there have been ideas that would simply never have existed without these people. Some of these ideas have changed everything—we would be in a different place from where we are now had it not been for these thinkers.

In no sense could we say that Noether’s theorem was in the air. Without Noether, we would most likely never possess an equivalent, equally powerful and general result. Without this theorem—this theory construction kit—much of modern physics would not exist or would be in relative disarray. Our understanding of our physics heritage would be more logically uncertain; we would still labor under confused and ad hoc explanations of even basic concepts such as energy. The path forward in many areas, especially in attempts at unification, would be on even shakier ground.

I think Hilbert was right about Einstein. Others may have had a better grasp on the mathematics than he did, and predecessors such as Poincaré had many of the details at hand, and as I have pointed out, perhaps more than once, Einstein could not have reached the endpoint of general relativity without a lot of help. But without his singular and profound insights into how the universe was made, there would be no physics of the cosmos. Black holes, the Big Bang, gravitational waves, gravitational lensing, the orbit of Mercury, even the operation of GPS satellites—we would not know or understand any of it.

In Chapter 4, I told the story of Grace Chisholm Young, who received, under Felix Klein, the first PhD earned by a woman in Germany. Her son, Laurence Young, also became a mathematician. He had this to say about how the world often overlooks the importance of mathematicians to the world: “I shall say something of the significance of mathematics, and the value of mathematicians, in altering the course of history. I hope to convince you of this, although you do not find much mention of it in most history books. The great question is then, how do we produce more of these admirable beings?”32

So at least I have one person in my corner.

And what of the subject of this book, Noether’s theorem? Clearly, any danger that this powerful result will be lost to science is well past us. It will increasingly be taught to generations of physics students at progressively earlier phases in their education. Some teachers have already told me that introducing Noether’s ideas early on makes it easier to present physics as a unified body of thought and to relate it to other sciences. We can, and probably should, emphasize symmetry arguments earlier, and with particular emphasis on how Noether’s theorem shows how to relate symmetries to dynamical behavior.

I’ve offered a minuscule survey in this chapter to give a feel for how Noether’s theorem is sending tendrils of thought and method into several disparate areas of research. The scientists who grasp its power have a chance to achieve insights and see connections that others are not equipped to see, not only within their specialties but potentially across what Hilbert would insist are the artificial chasms separating the various kingdoms of science.

After many decades of neglect, Emmy Noether is finally beginning to appear occasionally in popularizations and histories of physics, alongside Einstein and the other giants of our science. My deepest aspiration for this book is that I’ve nudged this process along. I trust I have convinced you that she deserves a central place in the history of science and have helped you appreciate something of the grandeur of her mathematical thought.

Above all, I hope you’ve joined me in absorbing the most important lesson from Emmy Noether and her life: don’t forget to laugh.