Around the time when mathematics was becoming formal, the notion of detaching from attaching "contextual interpretation" to symbols in algebra, up to the point of avoiding inconsistency (contradiction) was becoming recognized by people like Peacock and Hankel.

I know this principle was adopted by Peano, but I'm aware that the generally accepted construction of the number systems we have today are not due to Peano, but rather Dedekind and Cauchy.

I realize the principle of permanence while may not exactly recognized is used throughout mathematics since the 19th century even if the exact logic of it may have changed slightly.

So I'm looking for an explanation of how this principle progressed from the 19th century and its influence in the foundations of mathematics.

Thank you.

  • $\begingroup$ Wikipedia article confuses Peacock's "principle of permanence" and Cauchy's "generality of algebra" with later formalism and transfer principles when it says that Peano "adopted it". The early heuristic idea that one can simply carry over algebraic rules from one set of objects to another was explicitly rejected by Cauchy and was antithetical to Dedekind's and Peano's rigor. It is only in formalizations, where symbols are reduced to rules codified in the axioms, that one can freely shift interpretations and transfer algebraic formulas. $\endgroup$
    – Conifold
    Commented Feb 22 at 2:27
  • $\begingroup$ See this post for 19th Century. No longer used today. $\endgroup$ Commented Feb 22 at 7:02
  • 4
    $\begingroup$ You may want to clarify what you mean by "the principle of permanence". $\endgroup$ Commented Feb 22 at 9:48


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