Hi Alan,
Brief response as time is limited:
The things you say my definition of number has overlooked are important concepts, but they are not part of the definition of number.
Much of what you describe, e.g. shapes, sounds, and names, refer to numerals and not to numbers. There is an important difference. Numerals come in many flavors, such as '5', 'V', '1010', 'five', etc. The purpose of a rigorous definition is to find some one "thing" which is unambiguous, understandable by everyone, and unique, to which we assign the name of the number, for example 'five'. One problem is that we can't be sure what "things" are. We might have doubts about anything anyone might propose to be the "thing" we are going to call 'five'.
As Kyle pointed out, it was Frege who came up with the "thing" that was acceptable to mathematicians. The number five is defined to be the set of all pentuples (I hope that is a word; if not, you know what I mean). Now we could wonder if there really are any real sets that contain five things. Bishop Berkeley would say, no, there are no such things as material objects so we can't have a set of five material objects. But we could mollify him by explaining that the "things" need not be material. They could just as well be ideas which should certainly satisfy the good Bishop.
The mathematical idea of a set is primitive and is accepted without definition. Zermelo showed how to develop an axiomatic theory of sets which is used as the basis for the definition of numbers.
Beyond the numerals, which are not part of the definitio but simply an agreement on symbols we choose to use to communicate with, you mention operations such as addition. You imply that such operations are implicit in the definition of number. They are not. Actually, I guess they are, in a way.
What goes on is that natural numbers are defined in the way I described and then once we have them defined, we go on to define operations such as addition. This is not a natural or implicit outgrowth of our definition, however, as it seems you think it must.
For example consider the numbers two and three. By definition, two is the set of all pairs and three is the set of all triples. Now, what do you get when you combine them? Well, not much. If you combined them using the set-theoretic union, you get a bigger set containing some pairs and some triples but not a single pentuple. So the combination has little or no relationship to the number five. So how do we combine two and three to get five?
It is not straightforward, but it is rigorous. The operation of addition is defined as a function. A function is defined to be a set of ordered pairs, i.e. each element of the function is a pair of things where the sequence of which thing is first and which is second is important and maintained. When we define a function, we say "it is a function on A to B", where the first things in the ordered pairs are members of A and the second things in the ordered pairs are members of B and each member of A appears exactly once as one of those first things.
So to define an addition operation on a set A, we say that addition is a function on A [cross] A to A, (I don't trust my ability to get a big gothic "X" to appear on your screen to indicate the cross product so I wrote [cross]), where A [cross] A is defined to be a set of pairs of elements of A.
This is not sufficient, as there are many ways of forming such a function. So, we make some conditions, or axioms, which dictate the rules of how those "sums" are to be defined. From those axioms (typically they are Peano's axioms) and from the definitions, you can proceed to prove the theorems necessary to establish the familiar properties of sums of numbers, such as the one you cited: 1 + 1 = 2.
That equation, or number fact, is anything but obvious if you intend to be rigorous. Furthermore, it is strictly a consequence of our agreement to make certain definitions, accept certain axioms, and use certain logical rules to make inferences from them. There is no "coercion" or "compulsory association" involved at all, cars and buses notwithstanding.
I didn't mean to get carried away like that, but it is fun for me. Please forgive me.
It's good talking to you Alan; I wish I had more time.
Warm regards,
Paul |