22/04/2025

BBC article chronicles the career of German mathematician Emmy Noether.

Reproduction of the BBC article shared in Folha de S. Paulo.

When German student Emmy Noether wanted to study mathematics , women were not yet allowed to enroll in university.

Years later, when he finally got permission to teach university students, he wasn't receiving a salary.

Nevertheless, for Albert Einstein , "Miss Noether was the most important creative mathematical genius that has ever existed since women began entering higher education."

She is considered the mother of modern algebra for her theories on rings and fields, but her contribution to science is not limited to mathematics.

His work is fundamental to understanding the theory of general relativity . And it's not limited to that. Noether is key to understanding all theories of physics.

"When you learn about Noether's story, you wonder: what other contributions would a person labeled a mathematical genius have made if all doors had been open to them from day one?" asked physicist Mayly Sánchez, currently a professor at Florida State University in the United States , in 2017 (when this report was originally written).

No salary

Emmy Noether was born in 1882, and her father, the mathematician Max Noether, was a professor at the University of Erlangen in Bavaria, Germany .

The faculty at this same university even went so far as to say that allowing women to enroll would "overthrow the entire academic order."

But, according to the American Physical Society (APS), years later, Noether was one of the women authorized to enroll at the institution.

But without the same rights as other students.

She could only attend classes as an auditor, and only if the teachers gave her explicit permission to enter the room.

"That was enough for her to pass the graduation exam in 1903 and obtain a degree equivalent to a bachelor's degree," explains Michael Lucibella, author of Noether's biography published by APS.

"She spent the following year studying at the University of Göttingen [also in Germany], but returned to Erlangen when the university finally lifted the restrictions against female students to complete her dissertation on invariants for ternary biquadratic forms in 1907," the writer points out.

Although the university had taken an important step toward allowing women to enroll, it still continued to exclude them from faculty positions.

"Noether taught in Erlangen for seven consecutive years without receiving a salary, on some occasions substituting for his father," Lucibella recounts.

'We are a university, not a sauna'

In 1915, the renowned German mathematician David Hilbert tried to bring Noether to teach at the University of Göttingen, but was rejected by his colleagues in the mathematics department.

"What will our soldiers think when they return to university and discover that they will have to learn from a woman?" asked one of the professors.

Hilbert replied: "I don't see why the sex of the candidates should be an argument against their admission. We are a university, not a sauna."

Noether had to teach under the name Hilbert for the next four years without receiving a salary.

Lucibella explains that Hilbert wanted Noether at the University of Göttingen because his knowledge and experience regarding "the theory of invariants—numbers remain constant even when manipulated in different ways—could be applied to Albert Einstein 's nascent theory of general relativity, which seemed to challenge the law of conservation of energy."

Noether's theorem

Noether developed a fundamental theorem for understanding elementary particle physics and quantum field theory .

In short, "to understand all the most sophisticated physics," explained Manuel Lozano Leyva, currently an honorary researcher at the Department of Atomic, Molecular and Nuclear Physics at the University of Seville, Spain, in 2017 to BBC Mundo, the BBC's Spanish-language service.

"When Einstein saw Noether's work on invariants, he wrote to Hilbert: 'I am impressed that these things can be understood in such a general way. The old guard of Göttingen should learn some lessons from Miss Noether. It is clear that she knows what she is doing,'" indicates the biography published by APS.

What does the theorem consist of?

Lozano explained: "The theorem, conceptually, is very simple and mathematically very complicated. It's about relating symmetry to conserved quantities."

We asked Professor Lozano, who taught his students about this in Spain for 30 years, for an explanation.

What is symmetry?

Imagine I'm holding a glass of wine and I ask you to close your eyes. While your eyes are closed, I rotate the glass on its axis and then ask you to open your eyes. You probably won't notice if the glass moved or not.

But if I rotate the glass perpendicular to its axis, that is, if I turn it upside down, and ask you to open your eyes, you will realize that something has happened to the glass.

This means that the cup is symmetrical with respect to rotations around one axis, but not symmetrical with respect to rotations around another axis.

Now consider physical quantities that everyone is familiar with, such as energy, which is neither created nor destroyed, but rather transformed. This is what we call a conserved quantity.

What Emmy Noether did was relate the symmetry of a system to the conserved physical quantities, and these quantities are a fundamental tool when formulating and solving problems in physics.

And this applies to all physical systems, from the planetary system to a crystal, or metals. "Everything!", said the professor enthusiastically.

'The most beautiful theorem in the world'

The theorem created by the German scientist has already received numerous adjectives.

"They call it the most beautiful theorem in the world, but it's not just because it's beautiful from the point of view of symmetry, but because of its tremendous mathematical power and fantastic computational capacity," Lozano stated.

"My students were amazed when I taught them this because, although it's something mathematically difficult to formulate, its consequences are enormous," he said.

And he emphasized: "All physicists owe a great deal to this woman."

This opinion is shared by Professor Sánchez from the United States.

"It's an extremely elegant theorem. It brings the beauty of the concept of symmetry to the principles of physics," he said in an interview with BBC Mundo.

"Noether is one of those figures in the history of physics who were hidden, and then you discover them," he stated.

"When I first learned about the theorem, I fell in love with the concept. My professor gave us a beautiful lesson on how this was one of the most elegant principles of physics. Now that I teach this same subject at the undergraduate level, I get emotional. It's one of the points where physics and mathematics connect in a very beautiful way."

"What my professor didn't tell me in that class is that Noether's theorem was written by Emmy Noether. He never mentioned that she was a woman, and it was only years later, during my doctoral studies, that I discovered that it was a woman who created it," she says.

'The Boys of Noether'

After the end of the First World War, there were some advances in women's rights in Germany.

"Noether began receiving a small salary at the University of Göttingen in 1923," Lucibella recounts. "But she was never given the title of full professor."

Most of the mathematics students were men, and they became known as "Noether's boys," the biography indicates.

With the rise of Nazism in Germany, Noether had to abandon academic life in the country due to a law that removed Jews from government and university positions, Lucibella recalls.

Noether was then dismissed from the University of Göttingen.

"Initially, she received the students at home, but later she was forced to leave Germany, along with other Jewish academics," says Lucibella.

Noether moved to the United States, where he continued his academic career at Bryn Mawr College and the Institute for Advanced Study in Princeton.

In 1935, she was diagnosed with a tumor in her pelvis. She underwent surgery, and although the surgery was successful, a series of complications led to her death four days later.

She was 53 years old.

Read the full article on the Folha de S. Paulo website.