## ‘The most significant creative mathematical genius thus far produced since the higher education of women began’

**By Jessica Hanzlik**

The story of Amalie “Emmy” Noether parallels that of countless women, who, throughout generations and across cultures, were denied access to knowledge solely on the basis of their sex. Noether, however, leaves behind her not just a story of overcoming inequality and struggling for recognition, but also a beautiful mathematical theorem whose power becomes ever clearer with the passage of time.

Born in Erlangen, Germany in 1882 to a professor of mathematics Max Noether and his wife Ida, Emmy Noether did not show indications as a child of mathematical genius to come. Like many girls of her era, she attended a school for languages and arts, and upon completion of these studies, prepared to become a teacher of French and English at a local girls’ school, one of the few professions open to academically-inclined women in her region and era.

Abruptly, however, she changed plans for reasons unknown, and she expressed a desire to attend graduate level mathematics courses at the university. It was an accomplishment in and of itself for her to audit university classes, as women (who were certainly not permitted to enroll as full students) had to obtain the express permission of the lecturing professor to sit in on his class. Despite her requests to audit mathematics lectures being frequently declined, as more than a few of the professors viewed the higher education of women as having the potential to “overthrow all academic order,” she ultimately succeeded in earning a doctorate in mathematics.

Emmy Noether had help, as well as hindrance, on her path toward mathematical fame. While she was attending lectures at the University of Erlangen, many of her peers and colleagues advocated for her. David Hilbert, well on his way to becoming a renowned theoretical mathematician whose work would become instrumental to the as yet undeveloped field of quantum mechanics, commented that he didn’t “see why the sex of the candidate is relevant – this is after all an academic institution not a bath house.”

Being surrounded by and working with many of the great mathematical minds of her era acted as a formative influence on Noether’s approach to mathematical proofs. This generation of scholars was a part of the new standard of developing mathematical thoughts in an abstract, philosophical way, which was in opposition to the current trend of rigorous computation and detailed examples. Noether was truly at the forefront of this movement.

She had a remarkable and almost uncanny ability to abstract mathematical concepts from numbers and computation to the deeper idea. She thought in objects, not numbers. In practice, this new way of thinking meant that she would begin working on a particular, specific question and would rapidly see the underlying metaphysical concepts. As one of her students described it, “Any relationships between numbers, functions and operations only become transparent, generally applicable, and fully productive after they have been isolated from their particular objects and been formulated as universally valid concepts. Her originality lay in the fundamental structure of her creative mind, in the mode of her thinking and in the aim of her endeavours.”

Not only is Emmy Noether known in the mathematical world as one of the founders of the field of abstract algebra, her work also provides the theoretical underpinnings of modern particle physics. Her insight into symmetries and her ability to see the most general form of a question, however specific an example presented itself, would ultimately result in Noether’s Theorem, a proof which was unparalleled in its power and prescience.

Noether’s Theorem proves that there is a direct relationship between conservation laws in physics and symmetries of nature. For example, her theorem explains that momentum is conserved in all physical processes because the laws of physics are the same anywhere in space. The physics law that momentum is *conserved* corresponds to the spatial *symmetry* of the laws of physics. Conversely, her theorem also proves that if physicists observe a symmetry of nature, such as the fact that the equations of motion which describe a particle’s action in the universe do not change over time, then it must be the case that there is a corresponding entity which is conserved in those types of interactions, which for this example is energy.

This energy and momentum conservation following from the spatial and temporal symmetries of the laws of physics is a concrete example of Noether’s theorem at work. Incidentally, as Albert Einstein was writing down his equations for general relativity, it was not yet clear that his new understanding of the universe would guarantee that energy and momentum were conserved. Of course, it would certainly be possible that they not be conserved, but physicists at the time were uncomfortable with the idea of throwing out such a fundamental aspect of their understanding of science. Noether clearly and elegantly demonstrated that general relativity was just one particular instance of a class of theories in which energy and momentum are naturally conserved. Her work calmed some of the fears scientists had about accepting Einstein’s theory. But, her theorem does much more than prove that the universe conserves energy and momentum. Other elegant, more subtle symmetries that appear in formulations of quantum mechanics have without fail always resulted in the observation of a new conserved quantity in the field of particle physics.

It wasn’t until 40 years later that mathematicians and physicists began to take her work seriously. This neglect probably wasn’t malicious; it just took those intervening decades for the potential applicability of her discovery to become clear. Today, so often has Noether’s Theorem held true, even in the face of seemingly esoteric artifacts of a theory, that it drives much of the field of theoretical physics.

Noether’s story is remarkable. Had she merely “kept up with” the giants of her generation– Einstein, Hilbert, Weyl, Klein– that would have been success enough, as women’s intellectual capabilities, particularly in the “masculine” fields of mathematics, philosophy, and logic, were considered less-than. Had she merely demonstrated to her colleagues that she was just as capable of comprehending esoteric ideas and mathematical proofs as their other, male, students that would have been a large step forward for the intellectual status of women. But, she did not merely rise to this level; she superseded it in an incredible way.

As a Jewish woman, lecturing at a university, Emmy Noether fled from Nazi Germany in the early 1930s. She was offered the position of visiting professor at Bryn Mawr College in the United States. There, she continued to collaborate with mathematicians and physicists at the nearby Princeton, but regrettably she died unexpectedly following complications from surgery after only two years.

Albert Einstein submitted an obituary for her to the New York Times in which he described “Fraulein Noether” as “the most significant creative mathematical genius thus far produced since the higher education of women began.” It may indeed be true that Noether’s genius surpasses that of all other female mathematicians, but a case could be made for the brilliance of Noether’s work shines among her peers even without the implied modification of “genius” with “female”.

**References**

Byers, Nina. “E. Noether’s Discovery of the Deep Connection Between Symmetries and Conservation Laws.” July 16, 1998. *Published in the Proceedings of a Symposium on the Heritage of Emmy Noether, held in Bar-Ilan University, Israel on 2-4 December, 1996.*

Byers, Nina. “The Life and Times of Emmy Noether.” November 11, 1994. *Presented at the International Conference on “The History of Original Ideas and Basic Discoveries in Particle Physics” and published in the Proceedings of the Conference. *

New York Times. 5 May 1935. “Professor Einstein Writes in Appreciation of a Fellow Mathematician.”

Mathematical Monthly 1972. Vol. 79, No 2. “Emmy Noether.” Clark H. Kimberling. *Published by the Mathematical Association of America. *

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