In the history of mathematics, the principle of permanence, or law of the permanence of equivalent forms, was the idea that algebraic operations like addition and multiplication should behave consistently in every number system, especially when developing extensions to established number systems.
In modern mathematics, arguments have instead been supplanted by rigorous proofs built upon axioms, and the principle is instead used as a heuristic for discovering new algebraic structures.
The principle was described by George Peacock in his book A Treatise of Algebra (emphasis in original):[4] 132.
[7] Around the same time period as A Treatise of Algebra, Augustin-Louis Cauchy published Cours d'Analyse, which used the term "generality of algebra"[8][page needed] to describe (and criticize) a method of argument used by 18th century mathematicians like Euler and Lagrange that was similar to the Principle of Permanence.
However, when following Georg Cantor's extensions of the natural numbers beyond infinity, neither satisfies both properties simultaneously.