I recall that I read---in a book by Constance Reid---of a named principle that guided the arithmetic conventions that applied to operations on newly discovered mathematical objects.

For example, when negative numbers were finally admitted into the fold of arithmetic, the product of two negative numbers was defined to be positive (even if it appeared on face value to be counter intuitive) to maintain logical consistency with all that went before it.

I thought this idea was called the "Principle of Arithmetic Consistency" or something like that, but I have not been able to Google successfully for any such candidate expression.

I do not believe my recollection to be false.

Could someone please tell is if such a principle does indeed exist and is recognized, and if so, what its formal name and definition are.


  • 2
    $\begingroup$ Generality of algebra? "The algebraic rules that hold for a certain class of expressions can be extended to hold more generally on a larger class of objects, even if the rules are no longer obviously valid". The idea is due to Leibniz and the term to Cauchy. Although it is more of a claim that suitable conventions can be found than a guide to making them. $\endgroup$
    – Conifold
    Dec 3, 2017 at 21:23
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    $\begingroup$ Thanks, @Conifold. Your answer prompted me toward what I was looking for. In a Long Way From Euclid, Constance Reid mentions the Principle of Permanence of Form. MathSE has a question on this too. I think this expression has gone out of use, though, because it is not found on the Web. $\endgroup$
    – chandra
    Dec 4, 2017 at 10:28
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    $\begingroup$ See also this post. $\endgroup$ Dec 4, 2017 at 10:45
  • $\begingroup$ It seems that "generality of algebra" and Hankel's "principle of the permanence of formal laws" from "Vorlesungen über die complexen Zahlen und ihre Functionen" are not the same thing. "Generality of algebra" was a principle that people used in a proof, effectively some variation on "If an algebraic law holds for numbers/polynomials, it must hold for power series/analytic functions". The latter is a principle for choosing new definitions or axioms, and is more like what the poster is looking for. Hankel uses the definition of negative and fractional powers of numbers as an example. $\endgroup$ Dec 17, 2017 at 19:22
  • $\begingroup$ Additionally, Birkhoff and von Neumann use "Hankel's principle of 'The Perseverance of Formal Laws'" as a justification for type $II_1$ von Neumann algebras, instead of type $I_\infty$, in their original paper on quantum logic. $\endgroup$ Dec 17, 2017 at 19:25


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