- An obvious symmetry is invariance with respect to space which dictates that if the location at which an experiment is performed is changed the results of the experiment will nonetheless be the same.
- Another symmetry, invariance with respect to time, mandates that the results of an experiment will stay the same even if the time that an experiment is performed changes.
In 1918 symmetry became even more relevant to (the philosophy of) physics when Emmy Noether proved a celebrated theorem that connected symmetry to the conservation laws that permeate physics. The theorem states that for every continuous symmetry of the laws of physics, there must exist a related conservation law. Furthermore, for every conservation law, there must exist a related continuous symmetry.
- For example, the fact that the laws of physics are invariant with respect to space corresponds to conservation of linear momentum.The law says that within a closed system the total linear momentum will not change and the law is “mandated” by the symmetry of space.
- Time invariance corresponds to conservation of energy.
- Orientation invariance corresponds to conservation of angular momentum
Noether’s theorem had a profound effect on the workings of physics. Whereas physics formerly first looked for conservation laws, it now looked for different types of symmetries and derived the conservation laws from them. Increasingly, symmetries became the defining factor in physics.
Physics also respects another symmetry, which as far as we know has not been articulated as such. The symmetry we refer to is similar to the symmetry of time and place that was obvious for millennia but not articulated until the last century. Namely, a law of physics is applicable to a class of physical objects such that one can exchange one physical object of the appropriate type for another of that type with the law remaining the same. Consider classical mechanics. The laws for classical mechanics work for all medium sized objects not moving close to the speed of light. In other words, if a law works for an apple, the law will also work for a moon. Quantities like size and distance must be accounted for, but when a law is stated in its correct form, all the different possibilities for the physical entities are clear, and the law works for all of them. We shall call this invariance for a law of nature, symmetry of applicability, i.e. a law is invariant with respect to exchanging the objects to which the law is applied.
To sum up our main point, philosophically the change in the role of symmetry has been revolutionary. Physicists have realized that symmetry is the defining property of laws of physics. In the past, the “motto” was that a law of physics respects symmetries. In contrast, the view since Einstein is: that which respects symmetries is a law of physics. In other words, when looking at the physical phenomena, the physicists picks out those those that satisfy certain symmetries and declares those classes of phenomena to be operating under a law of physics. Stenger summarizes this view as follows:
“. . . the laws of physics are simply restrictions on the ways physicists may draw the models they use to represent the behavior of matter”. !!!
They are restricted because they must respect symmetries. From this perspective, a physicist observing phenomena is not passively taking in the laws of physics. Rather the observer plays an active role. She looks at all phenomena and picks out those that satisfy the requisite symmetries.
Our main point is that this uniform transformation and the fact that statements remain true under such a transformation is a type of symmetry. Recall, a symmetry allows us to change or transform an object or “law” and still keep some vital property invariant.
As with physics, in the past whereas we used to understand that:
"A mathematical statement satisfies symmetry of semantics".
We now claim that:
"A statement that satisfies symmetry of semantics is a mathematical statement."
In other words, given the many expressible statements a mathematician finds, her job is to choose and organize those that satisfy symmetry of semantics.
