Math bits 1: Natural numbers 2

We all are familiar with the basic mathematical operations that can be performed with natural numbers. The addition is the most natural one: We have $x$ elements on one hand, $y$ elements on the other, and if put all of them together we get... $x+y$ elements! We all learned at school that this basic operation satisfies a number of properties. Two of them are very important:

Associative Property: Given three natural numbers $x$, $y$, $z$, then it is always the case that $(x+y)+z=x+(y+z)$. To see what this means, let us suppose we have three hands. On the first hand we have $x$ elements, on the second $y$ elements, and on the third $z$ elements. We can put together the elements on our first and second hands, obtaining $x+y$ elements, and afterwards we add the elements on our third hand to obtain $(x+y)+z$ elements. But we can proceed in a different way: We can put together first the elements on our second and third hands to get $y+z$ elements, and then proceed to join the elements on our first hand with the others to obtain $x+(y+z)$ elements. Associativity tells us that with these different procedures we always get the same result.

Commutative Property: Given two natural numbers $x$ and $y$, then always $x+y=y+x$. This amounts to say that the operation of putting together $x$ elements on my left hand and $y$ elements on my right hand, which gives a total of $x+y$ elements, produces the same result (represented by $y+x$) as putting together $y$ elements on my left hand and $x$ elements on my right hand.

Even though most of us have internalized these properties up to the point that we consider them self-evident, you can dare to ask yourself "Why are these properties true?". You better don't do it... :)