Why do we have to prove $1+1=2$?

Question

I have 0 knowledge on number theory, but I currently have to take a course about proving.

So I wonder why do we have to prove something that is "trivial", in the sense that we are using it, without noticing it in everyday life? What is the significance of proving such statements? Like the well-ordering principle, $1+1=2$, etc.

Answer 1

The Well-ordering principle is not so "trivial".

About the proof of $1+1=2$ it is trivial to prove it from Peano axioms and some definitions :

1) $0$ is a natural number --- Axiom

2) for all natural number $n, \ S(n)$ is a natural number --- Axiom, where $S(n)$ is the successor of $n$

3) for all $n, \ n+0=n$ --- Axiom for sum

4) for all $n, m \ n+S(m)= S(n+m)$ --- Axiom for sum

5) $1=S(0)$ --- definition of $1$

6) $2=S(1)=S(S(0))$ --- definition of $2$

Thus :

$1+1=S(0)+S(0)=S(S(0)+0)=S(S(0))=2$.

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In general, in mathematics we "like" to prove statements, also (if possible) the most "elementary" ones.

Answer 2

Historically, there have been many "trivial" things that turned out to be wrong. You can only be sure about an assertion if you have a proper definition and a proof using that definition.

To the Greeks it was "obvious" that every number was a quotient of two integers; until they bump on the fact that the length of the diagonal of a square of side one cannot be such a quotient.

The Well Ordering Principle is actually not liked by a good number of mathematicians. It seems simple enough, but it turns out to imply many assertions that are not that easy to take (namely, the Axiom of Choice, Zorn's Lemma and all their implications). So accepting well-order implies accepting a whole bunch of not-obvious-at-all things.

People in general and mathematicians have been using numbers for a very long time. But, like I mentioned, not everything is so rosy: if you want to talk about $\sqrt2$, they you need to tell me what kind of object it is (and for a start, you could try to explain what "number" is). This problem took mathematicians a very long time to sort out.

There are also versions of this in analysis and algebra, too.

Finally, the assertion "$1+1=2$" in itself means nothing. You need to define what the symbols, $1$, $+$, $=$, and $2$ mean. There are several ways of doing that. In some (me preferred point of view), "$2$" is defined to be $1+1$ (or, more properly, $S(S(0))$ as Mauro mentions in his answer. If you define the integers in some way and you define what it means to add them, then you need to prove that $1+1=2$.