I am really confused after reading wikipedia...
What I don't get is how can something "bring" information, and in mathematics, how a mathematical object (like a set) can "have" information.
For example If I think about a set (a collection of objects), the only information that has is his cardinality? When we cansider a $n$-tuple (and is a set to in the sense of Kuratowsky) it has the order too. But we can easily see that two different $n$-tuples can have a very different amount of information.
For example if we have a big book of Category theory it has a huge amount of information.
Let say that that book $C$ is $n$-tuple, in other words an element of $A^n$ (where $A$ are letters of the alphabet + "space" + "new line" + "." + "," ... exc..)
We have that evry element of $A$ in the book $C$ appears $\mathcal l_C(x)$ times ($\mathcal l_C:A\rightarrow \Bbb N$)
We can create a new book $N$ where every letter appears in the the book often as in the book $C$ or in other words $\mathcal l_C(x)=\mathcal l_ N(x)$.
In easy words, the book $N$ is made of the same amount of stuff of the book $C$ but I want to write the book $L$ typing $\mathcal l_C(a)$ times the letter $a$ then we type $\mathcal l_C(b)$ times the letter $b$ and so on...
But the book $N$ does not say anything about the categories ... the only thing (information) that give us is the number of letters used in the category theory's book.
That really make a big confusion in my mind.
So my qestions are:
$1-$ Why the books $C$ and $N$ are made of the same components but $N$ lost all the informations of $C$? Where is all the information of $C$?
$2-$ How we define the information (as a quantity) of a mathematical object? How is it relatad to the Information as physical quantity?
PS:I don't know what is the correct tag.