Tribute to Invariance

Importance and Abstraction of Invariance

Various conservation or invariance principles form a foundation of physical and chemical sciences. Writes I.Stewart [Stw]:

In short, nature is symmetric because we live in a massproduced universe --- analogous to the surface of a pond. Every electron is exactly the same as every other electron, every proton is exactly the same as every other proton, every region of empty space is exactly the same as every other region of empty space, every instant of time is exactly the same as every other instant of time. And not only are the structure of space, time, and matter the same everywhere: so are the laws that govern them. Albert Einstein made such "invariance principles" the cornerstone of his approach to physics; he based his thinking on the idea that no particular point in spacetime is special. Among other things, this led him to the principle of relativity, one of the greatest physical discoveries ever made.

According to G.Polya [Po1, Po2], analogy seems to have a share in all discoveries... A legend ascribes the first ever mathematical proofs to Thales of Miletus (c. 600 B.C.) who is reputed to have appealed to the symmetry properties of the isosceles triangle and the circle. Mathematics is often [Dev] defined as the science of patterns while the desktop paradigm revolutionized (in fact, created) home computing. Humans could not have formed the abstraction of number without first developing the concept of sameness which is correspondence and, when strict, is equivalence while, when relaxed, relates to similarity.

All these terms symbolize several related ideas that, of necessity, permeate all branches of mathematics. Continuity is invariance of nearness. Fixed points are invariant under transformations. Commutative and associative laws originate with invariance of the result of counting objects under modification of enumeration order. Patterns are recurrent, regular attributes of world phenomena or abstract examples. Physical laws imply regularity without which there would not be much to talk about. Nor there would be anybody to do the talking.