What Do We Mean by the Term Number of Things?

e.

post-1930) set-theoretical notions.

I think that :group must mean a finite collection of objects (things)and that :number of things in a group is "clearly" (from the discussion) the equivalent of modern cardinality (restricted to finite collections) and it is called a "property" of the collection (group).My interpretation is that things are "individual", concrete or abstract (if any). Of course, it is easy to think to them as concrete objects, like peebles in a pocket or soldier in a platoon.

A platoon is a group of soldiers and the number of things in the platoon is the number of individual soldier forming it.This interpretation makes sense also with regards to the ensuing definition of addition (see CoolHandLouis's answer).Please, note that here group has the "generic" meaning of collection or aggregate; it has nothing to do with the technical term "group" of group theory.

When we "abstract" from the "characters" of the individual things (i.e.form their individual properties, like colour, size, shape for a colelction of balls) and from the order of the objects in the collection (it is the same for the "modern" set concept: A,B,C is "the same" set as C,B,A ) what we obtain is the "number" of the things in the group (the number of the members of the collection).Remember that Cantor's original notation for representing the Cardinal number of the set A was a "double overbar" over A :the symbol for a set annotated with a single overbar over A indicated A stripped of any structure besides order, hence it represented the order type of the set. A double overbar over A then indicated stripping the order from the set and thus indicated the cardinal number of the set

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 definitions of the term number of things in general English?

My understanding about the term number of things is that when we talking of some concrete things then we are interested in knowing how many concrete things(tokens) are there. We do not bother whether the concrete things under consideration have similar properties or not.

When the things under consideration are "abstract objects" then we are only interested in knowing how many different types of "abstract things" are there. For example consider a child learning English alphabets. The student writes the letter "A" 10 times, the letter "B" 3 times and the letter "C" 2 times. the teacher asks the student:

"How many alphabets you have learnt to write?"

The child will reply:

"I have learnt to write three English letters, namely "A","B" and "C"."

The child has actually written 103215 letters but it is understood that the teacher meant to ask "how many types of letters".

Mr.Fines book is quite old. I want to read some latest literature for understanding the term The Number of things. Which field of study deals with this term(Number of things)? Dose Modern Math or Modern Philosophy deals with this term? Which subject I should read for the formal study of this term. Does modern set theory deals with this term? Could you guys tell me about some modern book which formalizes this term. I have downloaded the book "Recursive number theory (1957)" but this seems to be old.