What is the "Principle of permanence"?

Question

While reading the book "The Number-System of Algebra (2nd edition)." (Henry Fine, 1890, 1907 2nd ed.) the term "Principle of permanence" occurred to me. I remember I had read this in the book "Beginning algebra for college students." (Lowenstein, 1953). I do not have the copy of the later mentioned book. On page-74 of the book "Beginning algebra for college students" the author writes:

"This principle states that we employ rules under circumstances more general than are warranted by the special cases under which the rules were derived and have validity."

This ${}{}$ statement seems fine to me from the remaining context of the book "Beginning algebra for college students". In Fine's book it seems to have the same meaning but there is no definition of that term given.
I googled a bit and got surprised, because I found a totally different meaning of the term "Principle of permanence", e.g. here and on wikipedea. "Principle of permanence" is defined something related to the complex functions.

• Could you guys explain me what the "Principle of permanence" actually is?
• I want some references related to this term.
• I also want to study the historical perspective of this term and want to know how and why it has two different definitions.

P.S: I've asked another similar question on https://hsm.stackexchange.com/ , because I feel I could get the answer to the last point there. This is the question: https://hsm.stackexchange.com/q/606/141

Answer 1

The principle of permanence is best observed as the (often unmentioned) guiding light when extending the realm of "number".

We start with $\mathbb N$ and addition and multiplication and observe that certain rules hold. Among these are associativity and commutativity of addition and multiplication, the distributive law, cancellation (i.e. $a+b=a+c$ implies $b=c$ and also $ab=ac$ implies $b=c$) etc.

Unfortunately, we cannot always solve $a+x=b$ as long as we work only in $\mathbb N$ and therefore construct an extension $\mathbb Z$ to handle such cases. (I say construct, because elements of $\mathbb N$ occur "naturally" as sizes of sheep herds and the like, but negative numbers don't - to relate negative numbers to everyday life we resort to debts and similar abstract concepts). Also, we define an addition and multiplication on $\mathbb Z$ in a more or less straightforward way (even though pupils often wonder why $(-1)\cdot(-1)$ is $+1$), namely not only so that the restriction to $\mathbb N\subset\mathbb Z$ is the good old addition/multiplication, but also so that (almost) all the well-known laws hold forth: Addtion and multiplication are still associative, commutative, distributive, and cancellation holds for addition. But cancellation law for multiplication gets a dent: $ab=ac$ need not imply $b=c$; it does so only if $a\ne 0$. Using the cancellation law would be an abuse - and is hidden behind many fun pseudo-proofs that $1=2$.

The same happens at further stages up along the number system hierarchy $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$: We construct larger sets of numbers to solve more problems while trying to preserve most basic laws, but always sacrificing something. For example in the last step to the complex numbers we sacrifice linear order. And there are further extensions where we start to sacrifice really important things such as commutativity of multiplication or even associativity ...

In the end, the principle of permanence is the guiding light to answer questions such as "Why don't we define $\frac10$ to be $\infty$ (or $0$ or $42$ or $\mathbb R$ or ...)?" - "Because we would have to sacrifice the permanence of certain beloved rules that are 'more important' than the ability to divide by anything we want".

Answer 2

The "Principle" dates back to XIX century algebra, with Hermann Hankel (see : Vorlesungen (1867), page 10 : "Princip der Permanenz formaler Gesetze") and George Peacock :

The principle here indicated by means of examples was named by Peacock the "principle of the permanence of equivalent forms," and at page 59 of the Symbolical Algebra it is thus enunciated:

"Whatever algebraic forms are equivalent when the symbols are general in form, but specific in value, will be equivalent likewise when the symbols are general in value as well as in form."

For example, let $a,b,c,d$ denote any integer numbers, but subject to the restrictions that $b$ is less than $a$, and $d$ less than $c$; it may then be shown arithmetically that $(a-b)(c-d)=ac+bd-ad-bc$. Peacock's principle says that the form on the left side is equivalent to the form on the right side, not only when the said restrictions of being less are removed, but when $a,b,c,d$ denote the most general algebraic symbol. It means that $a,b,c,d$ may be rational fractions, or surds, or imaginary quantities, or indeed operators such as $d/dx$. The equivalence is not established by means of the nature of the quantity denoted; the equivalence is assumed to be true, and then it is attempted to find the different interpretations which may be put on the symbol.

Compare with Henry Fine, The number-system of algebra treated theoretically and historically (2nd ed 1903), page 12 :

By the assumption of the permanence of form of the numerical equation in which the definition of subtraction resulted, one is thus put immediately in possession of a symbolic definition of subtraction which is general.

Roughly speaking, having proved that a law in "symbolic form" holds for a specific type of values (e.g.integer numbers) we must "generalize" it to other types (e.g.rational numbers), provided that they are subject to the "fundamental laws" (in Fine's book : I-V and VII; in modern terms : the axioms).

Answer 3

The principle of permanence mentioned at wiki is not the same as the heuristic principle mentioned in "beginning algebra for college students", but they are related via the notion of analyticity of complex functions. The principle you found at wiki is related to the fact that a complex analytic function which has a zero of infinite order necessarily vanishes identically.

Meanwhile, the heuristic principle seems to be a reformulation of fruitful (but not always rigorous) techniques often employed in the 18th century by such greats as Euler, sometimes also referred to as the generality of algebra. It is also related to Leibniz's law of continuity.

Answer 4

It is not well known that the permanence principle almost compel us to the usual definition of complex exponentiation. This is proved here

https://www.fsb.unizg.hr/matematika/download/ZS_peacocks_principle.pdf

Answer 5

The principle of permanence is the idea that as we extend the natural numbers to the integers, then to the rationals, etc., the original properties of the natural numbers and its arithmetic are retained.

Tobias Dantzig, in his landmark work Number: The Language of Science (pp. 94-95) formulates the principle as a definition:

A collection of symbols infinite in number shall be called a number field, and each individual element in it a number,

First: If among the elements of the collection we can identify the sequence of natural numbers.

Second: If we can establish criteria of rank which will permit us to tell of any two elements whether they are equal, or if not equal, which is greater; these criteria reducing to the natural criteria when the two elements are natural numbers.

Third: If for any two elements of the collection we can devise a scheme of addition and multiplication which will have the commutative, associative, and distributive properties of the natural operations bearing these names, and which will reduce to these natural operations when the two elements are natural numbers.

Answer 6

The principle of the prolongation of algebraic identities, also known as (Weyl's) principle of the irrelevance of algebraic inequalities, is perfectly explained and exemplified in the following expository paper:

The irrelevance of algebraic inequalities - Berry

The starting idea is that if $f,g$ are multivariate polynomials over a field such that $g\neq 0$ and $f(a)=0$ for every $a$ such that $g(a)\neq 0$ then $f=0$; but this can be used along other smart, related ideas.