Why does the "Principle Of Permanence" have two different definitions?

Question

This question is a sub-question of previous question on MSE. I feel that on this website I have better chances of knowing more things.

For quite some time now, I have been searching about the "Principle of permanence". Mainly it is because of George Peacock and is found in old literature. It is a very powerful argument. But I couldn't found it in any modern literature!, why is that?

• Why is it not used(perhaps used implicitly but not mentioned) nowadays?

Another quite surprising thing that I found is that wikipedea(link) mentions an altogether different definition of "Principle of permanence".

The usual definition goes like:

"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."--taken from the book "Beginning algebra for college students."

• Why it has two different definitions?

I want to study about the Principle Of Permanence as much as possible so please tell me about all the references/books which mention/explain it. I've already downloaded, Peacock's "A Treatise On Algebra" vol-1 and vol-2, along with this some other sources, like Felix Klein's "Elementary Mathematics From An Advanced Standpoint", etc. The main sources are G.B Fine's "The Number-System of Algebra" and George Peacock's book.

• I want more references/resources for this term, and want to study all the historical perspective of this term as much as possible.

Thank you.

Answer 1

As you correctly noted, the first "principle or law of the permanence of equivalent forms" was formulated by the English algebraist George Peacock in his book A treatise of Algebra (1st ed.1830), page 104 :

"Whatever form is Algebraically equivalent to another, when expressed in general symbols, must be true, whatever those symbols denote."

Peacock was an eminent member of the so-called Analytical Society with Charles Babbage and was contemporary to other eminent English algebraists of that time, like William Rowan Hamilton and George Boole.

The said principle was a powerful heuristic tool in the discovery of new "algebraic structures" like Boolean Algebra but it is clearly not "absolutely sound".

We can think at Quaternions; they are:

a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative.

In this case, contrary to what we may expect on the basis of the principle, we have a number system featuring all the "symbolic patterns" common to previous number systems except for commutativity.

The principle has been forsaken in modern mathematics due to the systematic adoption of the axiomatic method.

A mathematical structure is now defined by its axioms: there are structures endowed with a binary operation that is commutative and structures where the operation is not so; they are simply characterized by differnt axioms.

Answer 2

The principle of permanence was a heuristic principle serving an important role for discovery of new results, historically speaking. In modern mathematics, it has been formulated as a rigorous logical principle called the transfer principle. This exists in a number of contexts but is most famously present in Robinson's framework which involves a hyperreal extension $\mathbb{R}\subseteq{}^{\ast}\mathbb{R}$.

Answer 3

I've come to the conclusion that the Wikipedia article is simply wrong, and I've completely rewritten it (permalink to original). It described the Identity Theorem, not any kind of "Principle of Permanence".

The Wikipedia article was clearly based on this MathWorld article; the Wikipedia article's other citations were either about Peacock's principle of permanence, or were about mathematical facts that were unrelated to whether the described facts should be called "the identity theorem" or "the principle of permanence". The MathWorld article was not written by a regular contributor to MathWorld; "Principle of Permanence" appears to be the only MathWorld article that person has written.

The MathWorld article has only one citation, a footnote from Stephen Wolfram's A New Kind of Science that says nothing like what the MathWorld article says, and instead supports the book you quoted (Beginning Algebra for College Students), the extensively-sourced top answer to this question, and the approachable and well-written top answer to your earlier question. To quote the footnote:

Systems that have evolved from the basic notion of numbers provide a characteristic example of the process of progressive generalization in mathematics. The main such systems and their dates of earliest known reasonably formalized use have been [...]: positive integers (before 10,000 BC), rationals (3000 BC), square roots (2000 BC), other roots (1800 BC), all integers (600 AD, 1600s), [...]. New systems have usually been introduced in connection with extending the domains of particular existing operations. But in almost all cases the systems are set up so as to preserve as many theorems as possible—a notion that was for example made explicit in the Principle of Permanence discussed by George Peacock in 1830 and extended by Hermann Hankel in 1869.

Wolfram is clearly describing the same Principle of Permanence described by Beginning Algebra for College Students and the top answers to both this question and your earlier question. What the Wikipedia and MathWorld articles described is the Identity Theorem, which is unrelated to the Principle of Permanence.

Answer 4

Yes, there are at least four somewhat-different things that have either been called a "principle of permanence" historically, or might be called so currently.

The easiest to explain (and legitimize) is in complex analysis, more often nowadays called the Identity Principle. It had also been called "The Principle of Permanence of Analytic Relations" (though, sadly, I am not able to easily find a source for the latter). The literal assertion is that two holomorphic (=complex-analytic=complex-differentiable) functions on a connected, non-empty set which agree on a subset with an accumulation point in the set must agree everywhere. In practice, this can be used to show that a formula that holds for real values of a parameter holds for complex values... where the formula for real values can be proven by changes of variables, etc., that would be harder to justify with complex values.

There certainly is Robinson's "transfer principle" in non-standard analysis, as mentioned already by @Mikhail Katz. Depending on one's tastes, this might be construed as going in the opposite direction... namely, very roughly, that if something holds in the large world that includes (legitimate) infinitesimals and such, and the assertion does not directly mention them, then the assertion holds for standard real numbers. EDIT: for example, existence... is interesting, in this context.

Also using some model theory and other things, the theory of motivic integration and transfer ideas can move results in the function field case to the number field case (culminating in Ngo's work on the trace formula, etc.) I am not competent to explain why this works. :) See papers of Francois Loeser, for example, for background.

The older "purely algebraic" principle going back to Peacock or earlier has an easy modern interpretation in terms of Zariski closures, or universal mapping properties. The most trivial case is that, if an algebraic identity can be proven "for integers" $x_1,\ldots,x_n$ in a fashion that does not use anything other than associativity, commutativity, and distributivity of the operations, then it is true in any commutative ring, EDIT: because the proof actually gives the result for the polynomial ring $\mathbb Z[x_1,\ldots,x_n]$. By its universal property, it maps forward to any commutative ring generated by $n$ things. This was simply not possible to say directly in Peacock's day. Subtler versions, e.g., that sufficiently many integer points determine a polynomial, or such stuff, verge into Zariski-closure issues. Although I know nothing about it, I would imagine that "universal algebra" (in the modern sense) could formalize broad versions of this.