Women in Mathematics — Emmy Noether and Sophie Germain

Institution Meets Mathematics

Almost every mathematician in the previous eight articles was male. Not because “men are better at mathematics,” but because women could not enter universities until the 19th century — the reason was institutional impossibility, plain and simple.

Yet some women went around the wall — by pseudonym, by direct appeal — and left world-class work:

• Sophie Germain (1776–1831) — French Revolution era; enters academic circles under the male name “M. Le Blanc.” Contributes to Fermat’s Last Theorem
• Emmy Noether (1882–1935) — Hilbert defends her with “the university is not a bathhouse.” Founder of abstract algebra; Noether’s theorem stands at the heart of physics

Through their two lives, we trace the fight between institution and individual in mathematics.

Sophie Germain — The Double Life of “M. Le Blanc”

Marie-Sophie Germain (1776–1831) was a Paris-born mathematician and physicist. She grew up in the turmoil of the French Revolution, at a time when public mathematical education was almost impossible for women, and educated herself through self-study and pseudonymous correspondence.

The Archimedes Encounter

Germain was born into a wealthy silk-merchant family. At 13, the Revolution broke out, and she spent long days at home. Reading Montucla’s History of Mathematics in her father’s study, she was moved by the story of Archimedes killed at the fall of Syracuse while lost in geometrical figures:

“If a discipline can enthrall a man like that, I must learn it myself.”

Parental Opposition

Her parents tried to stop her: “Mathematics is unsuitable for a woman.” They took away her candles and cut her heating at night. Nonetheless, she is said to have continued studying wrapped in blankets, warming her freezing inkwell.

A girl breaking through the “women should not do mathematics” norm of late-18th-century France — even when her family cut off the household’s warmth.

The Correspondence of “M. Le Blanc”

In 1794, the newly founded École Polytechnique in Paris did not admit women (the school Fermat and Cauchy of earlier articles had attended).

Germain obtained lecture notes from a former student, Antoine-Auguste Le Blanc, and submitted assignments under his name. Lagrange, impressed by “M. Le Blanc’s” talent, asked to meet him.

When Germain revealed her identity, Lagrange was astonished but encouraging, and became her mentor. “A woman leaving a record under a man’s name” — a double life reminiscent of Shakespeare’s Twelfth Night.

Correspondence with Gauss — “M. Le Blanc” Again

From 1804, inspired by Gauss’s Disquisitiones Arithmeticae, Germain began writing to Gauss with questions about number theory, again as “M. Le Blanc.”

1806 — Remembering Archimedes

In 1806, Napoleon’s army invaded Brunswick, where Gauss lived. Germain, remembering the Archimedes anecdote, arranged through French military acquaintances to have Gauss’s safety looked after.

Learning later that this had been done by “Mademoiselle Germain,” Gauss was astonished and wrote in a famous letter:

“A taste for mathematics is rare … but when a person of the sex which, according to our customs and prejudices, must encounter infinitely more difficulties than men in familiarizing themselves with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating into the most obscure parts, that person must certainly have the noblest courage, the most extraordinary talent, and superior genius.”

Praise from Gauss — Germain had been recognized by “the Prince of Mathematicians” once she revealed her sex. The moment her double life turned to respect.

Fermat’s Last Theorem — Sophie Germain Primes

On Fermat’s conjecture (no positive-integer solutions to x^n + y^n = z^n for integer n ≥ 3), Germain established:

1. For the “first case” (none of x, y, z divisible by n), the conjecture holds when p is an odd prime and 2p+1 is also prime
2. Such primes p are today called “Sophie Germain primes” (e.g., 2, 3, 5, 11, 23, 29, 41, …)
3. She proved the “first case” for primes below 100

This is the most important early-19th-century progress on Fermat’s Last Theorem. Through Dirichlet, Legendre, and Kummer, it eventually led to Wiles’s complete proof in 1995.

Foundation of Modern Cryptography

Sophie Germain primes matter in cryptography (Diffie-Hellman key exchange, choice of strong primes). Part of the mathematical foundations of 21st-century internet security lies in the work of a self-taught 19th-century woman.

But Germain’s own proof was mentioned only in a footnote of Legendre’s 1825 paper and never published under her own name — a classic case of gender-based erasure in the attribution of credit.

Chladni Figures and 1816 — First Woman to Win the Academy Prize

In 1808, the German physicist Ernst Chladni showed that sand sprinkled on a glass plate and set to vibrating with a bow forms beautiful geometric patterns (Chladni figures). The French Academy posed a prize question: derive the mathematical law governing vibrations of elastic surfaces.

• Rejected in 1811 and 1813
• Won on her third attempt in 1815 (formally awarded 1816)
• First woman to win a grand prize of the French Academy

Her fourth-order PDE was not perfect; Kirchhoff and others later completed elasticity theory. Still, a pioneering physical–mathematical treatment of vibration.

Germain’s Death and Erased Legacy

She died of breast cancer in 1831, aged 55. The year before, Gauss had proposed that Göttingen award her an honorary doctorate in 1837 — but she died six years before it could be conferred, and never received a degree during her lifetime.

The city of Paris recorded her occupation on her death certificate as “rentière” (rentier, “person of independent means”) — not “mathematician.” Though her work resonated worldwide, women were not officially recognized as “mathematicians.”

Today near the Luxembourg Gardens are Rue Sophie-Germain and the Lycée Sophie-Germain. The Prix Sophie-Germain was created by the French Academy in 2003. Two centuries later, her name was written into institutions.

Emmy Noether — The Greatest Woman Mathematician of the 20th Century

Amalie Emmy Noether (1882–1935) was a Jewish mathematician born in Erlangen, Germany. Widely regarded as the greatest woman mathematician of the 20th century, she is one of the founders of abstract algebra, and gave physics the immortal Noether’s theorem — the correspondence of symmetries and conservation laws.

Under the banner of “modern algebra” she systematized rings, ideals, and representation theory, and through van der Waerden’s Moderne Algebra (1930–31) she established the standard of 20th-century algebra.

In his obituary of her, Einstein wrote:

“The most significant creative mathematical genius thus far produced since the higher education of women began.”

Noether’s Early Life

• 23 March 1882 — Born in Erlangen, Bavaria. Her father Max Noether was professor of mathematics at Erlangen (algebraic curves)
• Grew up when women could not attend universities; from 1900, women were allowed to attend as auditors, and she studied at Erlangen
• 1907 doctorate (advisor: Paul Gordan) on “the complete system of invariants of ternary cubic forms” (she later called it “garbage”)
• In the 1910s she shifted style, toward structural and abstract methods

To Göttingen — Hilbert’s Invitation and “The University Is Not a Bathhouse”

In 1915, Hilbert and Felix Klein invited Noether to Göttingen. But at that time German universities had laws forbidding women to hold formal professorships.

Against opposition that “Habilitation (the qualification to lecture) must not be granted to a woman,” Hilbert famously said:

“I do not see that the sex of the candidate is an argument against her admission as Privatdozent. After all, the Senate is not a bathhouse.”

Even so, the Habilitation was not granted, and for years Noether taught “under Hilbert’s name” — the lectures officially Hilbert’s, though in fact given by her. Structurally identical to Germain’s “M. Le Blanc” a hundred years earlier.

• 1919 — Habilitation finally granted
• 1922 — Named “unofficial extraordinary professor” — but with a negligible salary

Noether’s Theorem (1918) — At the Heart of Physics

While discussing conservation laws in general relativity with Hilbert, Noether proved in general the correspondence between continuous symmetries of a Lagrangian and conservation laws.

Noether’s First Theorem

“If the action of a physical system is invariant under a continuous group of transformations (a Lie group action), there is a corresponding conserved quantity.”

Examples

• Time translation → conservation of energy
• Space translation → conservation of momentum
• Rotation → conservation of angular momentum
• Gauge symmetry → conservation of charge

Modern particle physics (the Standard Model), quantum field theory, and cosmology all stand on Noether’s theorem. The modern physical principle “symmetry = conservation law” was born from one woman mathematician’s paper.

“Had she confined herself to pure mathematics, the history of physics might have gone another way.”

Systematizing Abstract Algebra

In the 1920s Noether fundamentally reconstructed commutative ring theory:

• Noetherian rings — rings whose ideals satisfy the ascending chain condition (finitely generated ideals). Polynomial rings and rings of integers are examples
• Noether’s basis theorem — a polynomial ring over a Noetherian ring is Noetherian (a generalization of Hilbert’s basis theorem)
• Ideal decomposition — the Lasker–Noether primary decomposition theorem
• Dedekind domains — generalizations of the ring of integers in number fields

Bartel Leendert van der Waerden systematized these in Moderne Algebra (1930), which became the standard 20th-century algebra text. Its preface explicitly states, “This is based on the lectures of Emmy Noether and Emil Artin.”

Recognition “in the preface of a textbook.” Her formal position was informal, but her influence defined 20th-century mathematics as a whole.

The “Noether Boys”

• Van der Waerden — author of Moderne Algebra
• Alexandrov — Russian topologist; stayed at Göttingen in 1928–29 and, with Noether, laid the foundations of algebraic topology together
• Hopf — topologist

These disciples were called the “Noether boys” — an extremely rare configuration at the time: a woman teacher training a cohort of male disciples. “She was the unofficial rector of Göttingen” (Hermann Weyl).

Noether’s Personality

• Passionate lectures, often running out of time before reaching a conclusion. She would tell students “to be continued next time” and leave
• Careless of dress; often forgot meals
• Large in build, loud in laughter, bright and cheerful
• A presence that transcended the prejudices of a male-dominated field

Concentration on the discipline coexisted with cheer. A completed form of “the archetypal abstract thinker.”

Nazi Exile and Early Death

In 1933, the Nazi regime enacted its Law for the Restoration of the Professional Civil Service, purging Jewish civil servants. Noether was four times a target: Jewish, female, socialist, pacifist.

Expelled from Göttingen, she went to Bryn Mawr College in Pennsylvania in 1933 (a women’s college), where she held a professorship and lectured regularly at the Institute for Advanced Study in Princeton.

In April 1935 she had surgery for an ovarian tumor and died of postoperative infection, aged 53. “Exile came in time” — but she died soon after.

Einstein’s Tribute

Hermann Weyl spoke of her work in a eulogy, and Einstein contributed to the New York Times the tribute (4 May 1935):

“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.”

The greatest physicist of the 20th century, in a public newspaper, formally honored the greatest woman mathematician of the 20th century after her death — a symbolic moment.

Modern Impact

Germain → today

• Sophie Germain primes — foundation of modern cryptography
• Prix Sophie-Germain — French Academy, 2003
• Rue Sophie-Germain, Lycée Sophie-Germain — her name preserved in Paris

Noether → today

• Physics — the foundation of the Standard Model and quantum field theory
• Algebraic geometry — scheme theory and coherent sheaves all presuppose Noetherian rings
• Chemistry — group-theoretic symmetry (crystallography, molecular structure)
• The Emmy Noether Lecture (AWM), the Emmy Noether-Programm (Germany) — programs supporting women researchers worldwide bear her name

“The whole foundation of 20th-century physics and algebra came from the papers and lectures of one woman” — a fact that, alone, changes the place of women in mathematical history.

Germain to Noether — A Century of Women in Mathematics

Germain (1776–1831) to Noether (1882–1935): a century apart. The situation shifted slightly, but the essential structure of discrimination did not:

• In common — institutional walls (no entry, no professorship), work under male names, delayed posthumous evaluation
• What changed — Germain’s death certificate reads “rentière”; Noether’s obituary appeared in the New York Times

In a century, “the invisible woman becomes officially recognized by the world after her death.” But the treatment of professorships in her lifetime remained in the domain of “unofficial” and “unpaid.”

From This Article to the Next

We have traced the lineage of women in mathematics. The next article turns to the 20th-century giants of computation and proof — Alan Turing (1912–1954) and Andrew Wiles. Turing, founder of computer science, dead at 41; Wiles, who proved Fermat’s Last Theorem in 1994 after seven years of solitary work — mathematicians who solved concrete problems in the 20th century.