The Most Important Mathematician You've Never Heard Of

Sometimes, even the great, pivotal figures in science don't get their due. In today's very special "Ask a Physicist," I'll introduce you to Emmy Noether, one of the most important thinkers you might not have heard of.

Top image: NASA.

The Most Important Mathematician You've Never Heard Of

My column normally focuses on answering your outstanding (in both senses) questions about space, time, particles, and the nature of reality. And by all means, keep 'em coming.

But today I wanted to focus on a bit of science history: the mathematician Amalie "Emmy" Noether. If you haven't heard of Noether, don't feel too bad (and if you have, you should continue feeling smug). She is one of the most important, and most ridiculously underappreciated mathematicians of the 20th century. Despite providing an essential foundation for much of the Standard Model of Physics, she's largely forgotten, even by people in the field. I'll get to one of her most important in due course, but first, a little bit of context.

A short history of underappreciated scientists

Emmy Noether was born in 1882 in Bavaria, now part of Germany. Her father, Max, was a prominent mathematician at the University of Erlangen, and Emmy decided to follow in his footsteps. This was, perhaps, easier said than done. German Universities in 1900 didn't allow women to enroll in classes or to sit for examinations. Instead, she was no-so-subtly encouraged to do something more appropriate like teach French.

In 1898, the faculty of Erlangen went so far as to claim that admitting women would "overthrow all academic order." Noether decided to risk it, and essentially audited her entire undergraduate education. Afterwards, the ban on women having finally been lifted, Noether basically sprinted through her doctoral degree, completing it in 1908.

And then, nothing.

Even though German universities had progressed to the point of allowing women to get degrees, there was virtually no possibility of Noether getting an actual academic appointment.

In a bit of parallelism, you may recall that Einstein faced similar difficulties (albeit for decidedly different reasons). During the first few years that Noether spent as an unpaid researcher at Erlangen, Einstein was still toiling away in a Swiss patent office. But then, in 1915, Noether and Einstein's intellectual paths crossed in an important way.

Though he ran the physics gamut, it's probably fair to say that Einstein is best known for his theories of relativity. His theory of Special Relativity rocketed him to international science stardom (which, yes, is a thing) and ultimately to the university of Zurich.

But it was Einstein's Theory of General Relativity, published in 1915, that was perhaps his ultimate achievement. General Relativity was – and remains, nearly a century later – our best, most complete explanation of how gravity works. A big part of Einstein's theory was the so-called "equivalence principle," which, put simply, says that there's no measurable distinction between being in deep space – a true absence of gravity – and being in free-fall. The two feel the same and, according to Einstein's theory, essentially are the same.

Relativity was, in short, one of the most revolutionary ideas ever published and almost instantly transformed our understanding of how space, time, and gravity really work. There was something incredibly elegant and deeply symmetric about relativity, but nobody really understood how it all fit together. It was also right up Noether's alley. Her thesis and subsequent work had focused on "invariants," the mathematics of how various quantities remain unchanged as you change some of their underlying parameters. This was, in short, just the sort of thing to really understand Einstein's equivalence principle.

The eminent mathematicians David Hilbert and Felix Klein (of the eponymous bottle) invited Noether to the University of Göttingen in 1915 to help in unraveling some of the hidden symmetries of the theory.

Under normal circumstances, bringing in an expert at her level would have justified hiring her as a professor – or at the very least, you know, paying her. But just as at Erlangen, biases against her gender interfered. Hilbert was outraged. At a faculty meeting, he exclaimed:

I do not see that the sex of the candidate is an argument against her admission as a Privatdozent (roughly, an associate professor) After all, we are a university, not a bathhouse.

Hilbert and Noether skirted the rules by listing Hilbert as a course instructor and then having Noether as the perennial guest lecturer, though this didn't extend to getting Noether any sort of paycheck. It wasn't until 1922 that the Prussian Minister for Science, Art and Public Education gave her any sort of official title or pay at all, and even then only a pittance. As Hilbert described it in his memorial address for Emmy Noether:

When I was called permanently to Göttingen in 1930, I earnestly tried to obtain from the Ministerium a better position for her, because I was ashamed to occupy such a preferred position beside her whom I knew to be my superior as a mathematician in many respects. I did not succeed. . . . Tradition, prejudice, external considerations, weighted the balance against her scientific merits and scientific greatness, by that time denied by no one.

In all events, bringing her to Göttingen turned out to be an incredibly good idea. Almost immediately upon her arrival, Noether derived what's become known as Noether's 1st Theorem and by 1918 had cleaned it up enough for public consumption. And this is where we pick up the physics part of the story.

Noether's (1st) Theorem

Symmetries are at the heart of what we now understand as the Standard Model of particle physics. We know that now but it all began with Noether. It no coincidence that my book on symmetry (now available in paperback!) features Noether as its hero.

Noether recognized that there is a mathematical relationship between symmetries of the natural universe and what are known as conservation laws. While there is a fair amount of mathematics behind it, the upshot of what's known as Noether's Theorem is essentially:

Every symmetry corresponds to a conservation law.

If you are a bit underwhelmed, don't be. Conservation laws are the bread and butter of physics. For instance, one of the biggies is conservation of electric charge. In every reaction we've ever produced in a lab, positive charges and negative ones are always created or destroyed in perfect concert. If the Big Bang produced an electrically neutral universe, a very reasonable assumption, then conservation would demand that it still must be neutral today.

What Noether proposed sounds quite simple, almost content-free, until you get into the nitty-gritty. The conceptual problem for us is that Noether was a mathematician, which means that the details involve a lot of equations. And because we're not going to derive equations, it couldn't hurt to start off by giving you a few results, just so you know what to expect. Noether's Theorem predicts:

  • Time Invariance → Conservation of Energy
  • Spatial Invariance → Conservation of Momentum
  • Rotational Invariance → Conservation of Angular Momentum

What I really mean is that the laws of physics don't change if you adjust the clock of the universe or move to a different place or point in a different direction.

The laws of physics, for instance, seem to be unchanging over time. This is more than just an assumption. In 1971, geologists discovered something remarkable in the village of Oklo in Gabon. For some time previously, the French had been mining uranium in Gabon when researchers found what could be described only as a 2 billion year-old nuclear reactor.

Just to anticipate the obvious: I'm not advocating some sort of Chariots of the Gods scenario here. It was just a small region where the leeching of minerals from ancient rocks, the flow of rivers, and some hungry, hungry bacteria conspired to deposit a far higher concentration of uranium than is typical anywhere outside a nuclear reactor.

Though it's all fairly nasty stuff, there are many different isotopes of uranium, and they don't all behave the same. The most common isotope is uranium-238, but the isotope that does the heavy lifting in nuclear fission is uranium-235 (U-235). For nuclear fission reactors to work, one of the crucial steps is enriching the uranium with a series of centrifuges, until U-235 makes up a few percent. In natural samples, U-235 makes up only about 0.7 percent of the total uranium on earth, but 2 billion years ago, it made up a much healthier 3.7 percent, the levels used in modern lightwater reactors. This is because uranium is in a constant state of decay, and U-235 decays six times faster than U-238.

This combination of natural enrichment and a big pile of uranium produced a critical mass. The uranium fissed into isotopes of palladium and iodine, as well as a lot of energy, perpetuating the process. The Oklo site was a natural nuclear reactor that burned for millions of years. It's amazing that such a site exists at all. But more amazing is that the detritus left behind from the nuclear fission is in exactly the proportions that we would find in a nuclear reactor today. Nuclear reactions are a tricky business. If the nuclear forces had changed over that time—and remember, we're talking about 2 billion years here—we'd be able to see it more or less directly.

Noether's great contribution to physics is the mathematical proof that so long as the laws of physics don't change with time, then energy can't be created or destroyed. It can change form, from mass to thermal energy, but not built out of whole cloth.

And again, given the laws of physics really do seem to be the same everywhere, Noether's Theorem immediately tells us that there's a conservation of momentum. If you're flying through deep space, you can't expect to glide to a halt; you'll just keep drifting at the same speed forever. You may also know this as Newton's First Law of Motion.

Noether didn't invent the idea of energy conservation. It is embodied in the First Law of Thermodynamics, but she did show that the first law was simply a consequence of the immutability of physical laws.

Likewise, conservation of momentum (which drops out of the laws of physics being the same everywhere) also wasn't new. It was discovered by Isaac Newton in the seventeenth century, and all three of his famous Laws of Motion are various ways of describing the conservation of momentum for an isolated system.

Noether's Theorem describes much, much more – things that we didn't already know. It describes and explains conservation of spin, electrical charge, of "color," (the equivalent of charge in the strong nuclear force), and on and on, ultimately providing the mathematical foundation for much of the standard model of particle physics. She explained, ultimately, why symmetry is such a big deal.


Epilogue

Given all of that, it's a bit puzzling that Noether has, to some degree, been forgotten. Part of it, I think, stems from the fact that there are only a handful of calculations that one can do with Noether's Theorem. Once you've established that there is a conservation of energy or charge or maybe half a dozen other quantities, that's it. You don't need to or get to do any more useful calculations. Some of these conserved quantities might even have popped out of the equations using brute force, but who wants to do that?

We saw how Noether's story intersected that of Einstein's. Her life parallels Einstein's in other, sadder, ways. Like Einstein, she fled to the United States in 1933. Einstein settled in Princeton, at the newly built Institute for Advanced Study. Noether went to nearby Bryn Mawr College. And then only 2 years after coming to America, Emmy Noether was diagnosed with a cancerous tumor, and in the aftermath of a surgery, she died from infection. She was only fifty-three. In Einstein's words:

In the judgment of the most competent living mathematicians, Fräulein Noether was the most significant creative mathematical genius thus far produced since the higher education of women began.

Most important for our purposes, Noether finally explained why it is that symmetry shows up just about everywhere in the physical laws governing the universe. Symmetry isn't just something beautiful or elegant. The mere existence of symmetry ultimately gives rise to new laws! In a sense, she turned symmetry into order.

Dave Goldberg is a Physics Professor at Drexel University, your friendly neighborhood "Ask a Physicist" columnist, and, most recently, author of The Universe in the Rearview Mirror: How Hidden Symmetries Shape Reality, which is now out in paperback (and from which today's article is adapted). You should absolutely send him all of your questions about the universe. You can also follow him on Facebook.