1 plus 1 makes 2. Really? Why is it?

When I was  seven, I was learning arithmetical addition in my elementary school. “Need help?,” asked my father.  “Yes. Why is 1 + 1 = 2?”. So, my father offered me an illustration on his fingers.
- See? Here is one finger, here is yet another finger, all together there are two fingers. Got it?
- No.

After that, I was on my own, as far as mathematics is concerned. Thinking of it, parents did not offer me a help in any homework ever since. Not that I needed a help.

Now with my Master’s in math (my Ph.D. in computer science is not needed here), I  know how to answer the damn question. One needs to know arithmetic. But I do not mean the “arithmetic” we studied in the elementary school.

In real arithmetic, there is a unary operator “+1”, it maps any natural number onto another natural number called “next”. The number “next” for 1 is denoted “2”, it is called “two”. Therefore, 1 + 1 = 2. Quod erat demonstrandum. Thus, we just have proven the Theorem 1: 1 + 1 = 2.

But now here is the next question. Why is that one finger plus one finger equals two fingers? Fingers are not arithmetical objects, there is no an operation “+” on them. However, we may express an operation with close meaning in the language of set theory. Particularly, we will be interested in the binary operation “set union” and the concept of a “cardinal number” ("cardinal"). Roughly speaking, this is a number of elements in the set. When the sets are  finite and have  no common elements, the cardinal of union of two sets is a sum of theirs cardinals. So, we may consider the first set consisting of one finger, the second set consisting of another finger. Cardinal of each of these sets is 1. There are no common fingers between the two sets. Then, union of these two sets contains all the fingers in the first and the second set. Accordingly, cardinal number of this union is 1 + 1, and by the Theorem 1, 1 + 1 = 2. Thus, we have proven the Theorem 2: one finger combined with another finger yields two fingers.

I just explained why 1+1 =2 and one finger plus one finger equals two fingers.  But can we claim  these explanation to be an absolute truth?

Think about it. Mathematics has (1) an abstract language made up by mathematicians and (2) logical reasoning about the objects of this language.   All the mathematical concepts were designed in such a way, so the results of logical reasoning agree with our observations (the same may be said about so called “laws of nature”).  But there may be a different language for arithmetic. In this other arithmetic, 1 +1 =2 too, but the explanation is different.

So, I suspect that not only the question “Why is 1 +1 =2?”, but any other “why” question may have an answer only when we have a particular theory of reality in mind. If we have no theory, we may only describe what happens with given objects, like “one this finger and  that other finger make two fingers”. We need theory to predict what will happen with other fingers.

Conclusions.
1. In arithmetic, there is a proof that 1 + 1 = 2. This proof explains why 1 + 1 = 2.
2. Most likely, there may be an arithmetic with another language yet with the same conclusion 1 + 1 = 2. This other arithmetic will have different explanation for the fact that 1 + 1 = 2.
3. Observing somebody’s fingers does not explain the observation. But the theoretical explanation is good only until another theory is considered.