# 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]:

According to G.Polya [Po1, Po2], analogy seems to have a share in all discoveries... And the oft-mentioned legend ascribes the first ever *mathematical proofs* to Thales of Miletus (c. 600 B.C.) who 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 (actually 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.

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