In the philosophy of mathematics, the principle of permanence, or law of the permanence of equivalent forms, is the idea that algebraic operations like addition and multiplication should behave consistently in every number system, especially when developing extensions to established number systems. Before the advent of modern mathematics and its emphasis on the axiomatic method, the principle of permanence was  an important tool in mathematical arguments. In modern mathematics, arguments have instead been supplanted by rigorous proofs built upon axioms, but the principle can still be applied as a heuristic for discovering new algebraic structures. Additionally, the principle has been formalized into a class of theorems called a transfer principle, which states that all statements of some language that are true for some structure are true for another structure.
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. Before the advent of modern mathematics and its emphasis on the axiomatic method, the principle of permanence was considered an important tool in mathematical arguments. In modern mathematics, arguments have instead been supplanted by rigorous proofs built upon axioms, and the principle can only be used as a heuristic for discovering new algebraic structures. Additionally, the principle has been formalized into a class of theorems called a transfer principle, which states that all statements of some language that are true for some structure are true for another structure.
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