WHY KANT SHOULD HAVE STUDIED SNOWFLAKES
SUPPOSE WE WANT TO SAY that a certain theory of the world is objectively true. What might this mean? Well, among other things, it means that the theory must be true for all observers, regardless of their point of view. That is, its validity should not depend on where you happen to be standing, or which way you happen to be looking, or what time it is.
A theory that is independent of perspective is said to possess symmetry. In normal parlance, "symmetric" is used to describe objects, not theories. Human faces, snowflakes, and crystals are symmetric in certain ways. A sphere is symmetric since no matter how you rotate it, it retains the same form. Therein lies a clue as to how symmetry might be defined more abstractly. A thing possesses a symmetry if there is something we can do to it so that afterward, the thing looks the same as it did before. This was the definition that the physicist Hermann Weyl (18851955) came up with. A snowflake, for example, is symmetric with respect to the operation "rotation by 60 degrees," rotation does not affect its appearance. It is also mirror-symmetric, since reflection in a mirror leaves its form unchanged. A theory is said to possess a certain symmetry if there is something you can do to itdisplace its coordinates in space or time, for examplewithout affecting its form or its equations. The more symmetries a theory has, the more universally valid it is.
I have just set the stage for one of the most underappreciated discoveries of the twentieth century: For each symmetry that a theory possesses, there is a corresponding law of conservation that must hold in the world it describes. A conservation law is one that states that some quantity can neither be created nor destroyed in an interaction. If a particular theory is symmetric under displacement in space, the law of conservation of momentum must hold. Similarly, symmetry under displacement in time implies the law of conservation of energy. Symmetry under rotation implies the law of conservation of angular momentum. Other, more subtle symmetries imply still more subtle conservation laws.
The intimate connection between symmetry and conservation is "a most profound and beautiful thing," in the words of the late Richard Feynman, one that "most physicists still find somewhat staggering." Laws that were once thought to be brute facts about the natural worldlike the first law of thermodynamics, which says that energy can neither be created nor destroyedturn out to be preconditions of the possibility of objective knowledge. This revelation has a decidedly Kantian flavor. But Kants transcendental reasoning was fuzzy and often fallacious. The symmetry-conservation link, by contrast, was proved with airtight logical rigorby a woman named Emmy Noether.
Emmy Noether was among the greatest pure mathematicians of the century. Born in Bavaria in 1882, she obtained a Ph.D. at Erlangen in 1907. Though the equal of such illustrious colleagues as David Hilbert, Felix Klein, and Hermann Minkowski, Noether was, as a woman, barred from holding a full professorship. Indeed, classicists and historians on the faculty at Göttingen tried to block her from giving unpaid lectures as a Privatdozent, moving Hilbert to comment, "I see no reason why her sex should be an impediment to her appointment. After all, we are a university, not a bathhouse."
When the Nazis came to power in 1933, Noether, a Jew, was stripped of her semiofficial position at Göttingen. She fled to the United States, where she taught at Bryn Mawr and gave lectures at the Institute for Advanced Study in Princeton. In 1935 she died suddenly from an infection after an operation.
Loud of voice and stout of figure, Noether struck her friend Hermann Weyl as looking like "an energetic, nearsighted washerwoman." In addition to being one of the pioneers of abstract algebra, she had considerable literary gifts, writing poetry, a novel, an autobiography, and, with a collaborator, a play. Her discovery that symmetries in a theory imply conservation laws was published in 1918. It is sometimes called Noethers thoerem.
Does Noethers theorem mean that laws of conservation are not out there in the world, but are merely artifacts of our epistemic practice? Such an idealistic interpretation should be resisted. The world does exert some control over just how symmetricthat is, universalthe true theory of it can be. Some symmetries have failed experimentally. In 1957, for example, Tsung-Dao Lee and Chen Ning Yang won the Nobel Prize for showing that our laws of physics would not be exactly valid for people living in a universe that was the mirror image of our own.
If the law of conservation of energy were ever shown to fail, the consequences would be more serious. The true theory of the world would then, we know from Noethers theorem, depend on what time it waswhich would be a great blow to its objectivity.
The curious thing is that, at many points in the history of science, it looked as though the energy-conservation principle had failed. Each time, however, it was salvaged by making the concept of energy more general and abstract. What began as a purely mechanical notion eventually came to encompass thermal, electric, magnetic, acoustic, and optical varieties of energyall, fortunately, interconvertible. With Einsteins theory of relativity, even matter came to be viewed as "frozen" energy.
Sooner than give up energy conservation, Henri Poincaré once observed, we would invent new forms of energy to save it. Thanks to Emmy Noethers great discovery, we know why: The timelessness of physical truth hangs on it.