Why does the “Principle Of Permanence” have two different definitions?

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.

• While it is interesting to study George Peacock and to see what he originally meant, I do not think there is any generally understood "principle of permanence (of form)" today. – Colin McLarty Dec 8 '14 at 2:02

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.

• As you said that in Quaternions the POP doesn't work so we use axiomatic method. Then when we move from Natural numbers to define Rational numbers why do we axiomize multiplication of rationals in such a way so as it could follow commutativity. I know that rationals must follow commutativity because the concept of multiplication wherever used, e.g. lengthbreadth must be same as breadthlength(area remains same), has to be commutative. But again we don't have a proof for every case, e.g. why must momentum=m*v must follow commutativity, e.g. the momentum of a ball of 2kg moving at 3metre/sec – user31782 Jan 14 '16 at 6:42
• [cont] is same as that of a ball of mass 3kg moving at 2metre.sec. It is just a kind of observable fact. From the point of view of pure maths we don't need to care about applied math. But then why do we define the multiplication of rationals to be commutative from a pure mathematical point of view? The most sound reason that I can reckon is that N must be a subset of Q, but this fact completely kicks out the applied mathematical point of view. We can't just get away by saying the maths is a game with a set of rules. We can't forget that multiplication is not a part of game... – user31782 Jan 14 '16 at 6:48
• It is a principle used in enormous number of situations in real life. – user31782 Jan 14 '16 at 6:50

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 $\mathbb{R}\subseteq{}^{\ast}\mathbb{R}$.