A mathematician is a machine for turning coffee into theorems. – Paul Erdos
This is a wonderful world we live in: with its variety of changes, new things happening every day and creativity all around. In all of this there is transience: nothing seems to be permanent. Yet, undergirding all this are the immutable laws of nature that don’t change with time or place. The laws that govern processes here today must have been present eons ago: apples fell down from tree branches in days of yore as of now. These same laws must be operating in the stars and in the farther-most galaxies too. Or, as physicists would put it, the laws of physics are invariant in space and time. When one probes into the fundamental features of the physical world with the microscope of mathematics, one uncovers a few other principles of invariance also.
It turns out that invariance is intimately related to the symmetry aspects of the world. For example, a circle is perfectly symmetrical. No matter how you rotate it, its form is the same. A square and an equilateral triangle are also symmetrical, but in different ways: only rotation through certain angles will bring them to their original forms. If we move along a straight line in empty space, everything remains the same from any point on it. This is called translational symmetry. If we turn around in space from one direction to another, the world appears the same. This, as with the circle and the sphere, is known as rotational symmetry. Reflection and scaling (altering the size) are other symmetry operations.
Then again, processes in the world are constrained by rules of conservation. There is only so much matter-energy in the universe, so much momentum, so much electric charge, etc. Irrespective of what happens, the totality of these remain the same. It is remarkable that enormous variety and complexity arise under meticulous quantitative constraints.
There are intrinsic connections between invariance, symmetry, and conservation. For instance, energy is conserved because the laws of physics don’t change with time.
We now know that for every continuous symmetry there is a law of conservation, and vice versa. [In technical jargon, corresponding to every infinitesimal transformation of the Lorentz group there is a conservation theorem.]
This is one of the most fundamental insights of 20th century theoretical physics. It was proved in 1905 by Emmy Amalie Noether (born: 23 March 1882).
Many people have heard of Madame Curie who was an eminent experimental physicist/chemist. But not as many know about Emmy Noether, the prolific pure mathematician: much of her work is quite esoteric and little related to everyday experiences, which is why she is not as famous. When a person writes a doctoral dissertation entitled On Complete Systems of Invariants for Ternary Biquadratic Forms, one cannot expect its author to become a household word. But among mathematicians Emmy Noether shines like a star of the first magnitude. In the appraisal of the mathematician-historian Eric Temple Bell, “she was the most creative abstract algebraist in the World.”
In 1900 – 1902 Noether was studying French and English because her goal was to become a teacher of these languages. In 1903 she began to audit courses in mathematics at the university. She could not register herself for a degree, because, after all, she was a woman. Constance Reid recalls in her biography of David Hilbert that the reasoning of the mathematics professors was: “How can it be allowed that a woman become a Privatdozent (one with a license to teach at the university)? Having become a Privatdozent, she can then become a professor and a member of the University Senate. Is it permitted that a woman enter the Senate? … What will our soldiers think when they return to the University and find that they are expected to learn at the feet of a woman?” [This was during the First World War.] Few could argue with this impeccable logic. To think that training in mathematics automatically enables one to think rationally and clearly on all issues is a gross error. Rational thinking often melts away in political, religious, and social issues without the victims knowing it, because in these contexts deeply experienced emotions, ingrained prejudices, and unconscious craving for power blind the mind’s eye.
One of the prices societies pay for keeping records is that later generations get to know the absurdities and atrocities of their ancestors. People in societies which have no such records are not only blissfully aware of past horrible deeds, they are all too eager to fantacize how great and flawless their ancestors were, and how they have always treated women and members of the lower strata of society with great respect. They can’t understand that shabby treatment of women was not unique to Germany or the Western world.
Be that as it may, the foremost German mathematician of the time David Hilbert had the wisdom to say, “Gentlemen, I do not see that the sex of a candidate is an argument against her admission as a Privatdozent. After all, the Senate is not a bath-house.” So it was that Noether was allowed into a doctoral program which she completed in three years with the highest honors. From 1919 they let her teach in Göttingen at the insistence of Hilbert, but only as a second class professor, and with hardly any remuneration.
In the 1930s matters turned from the ridiculous to the ominous in Nazi Germany when the absurdity of not allowing a woman to be professor was replaced by the outrage of not permitting Jews to hold any position at all. In 1933, Emmy Noether accepted a position in Bryn Mawr College in Pennsylvania. She went back to Germany, only to come back for good to the United States. A good many future mathematicians derived the benefit of her guidance and inspiration in Bryn Mawr.
When Emmy Noether died in 1935, Albert Einstein wrote a letter to the New York Times in which he described her as “the most significant creative woman mathematical genius thus far produced.”
Fortunately, we have come a long way from the unconscionable prejudices that cloud people’s minds to the conviction that competence in mathematics, music, or whatever is a function of race and religion, gender and nationality. It was the likes of Emmy Noether who gave the lie to that mindless nonsense which fails to see that ultimately we are all human beings endowed with capacities that are invariant under racial, religious, national, and gender differences.
March 23, 2016