I am reading the book "The Number-System of Algebra (2nd edition)." I have some problems with the the first article: "Number".
The author has confined the concept of number of things to the groups which have all distinct elements, that is the number of letters in a group having elements A,B,C is $3$ iff $A,B,C$ all are distinct.
- What is the definition and explanation, that is what is the meaning of the term number of things in general English.
- Why Professor Fine has confined this term only for those groups which have all the distinct things.
- How the modern Mathematics defines the term Number of things$^{2}$.
My personal understanding is as:
"Thing" refers to both "abstract object" and "concrete object". The "group" of Fine's book is not the "Set" of our modern language. When I say I have 3 coins in my pocket I mean that I have three different "concrete objects" in my pocket. I understand that in real life we do count multiple "abstract objects" of same type as a single one. I am still doubtful on my understanding.
It appears to me that the modern mathematics does not explain the term Number of things when the things under consideration are concrete objects. Read my comment here for my confused mind. I do not have enough knowledge of modern math(especially set theory). There might be some explanation of the term Number of things given by the modern set theory.
${}^2$ Things may refer to both "abstract objects" and "concrete objects".
