TITLE: The "constants" of a structure (model theory)
QUESTION [7 upvotes]: A structure is composed of the domain, constants, relation symbols, and function symbols.  I understand all of the ingredients of a structure immediately except for the constants.  The definition in my book (A Shorter Model Theory) is

A set of elements of $A$ called constant elements, each of which is named by one or more constants.  If $c$ is a constant, we write $c^A$ for the constant element named by $c$.

An example of a structure with constants is 
$$\langle \mathbb{R}, +, -, \cdot, 0,1, \le \rangle$$
I understand that $0$ and $1$ are special because they are the identity elements for addition and multiplication, respectively, but why are they the constants, and in what sense do we need to specify them?  I understand that I get a totally different structure if I where to use $<$ instead of $\le$, but I don't see what is so special about the constants.  Why not have a structure of the following type:
$$\langle \mathbb{R}, +, -, \cdot, \frac{1}{2},\pi, \le \rangle$$
Is this different and if so, in what sense?  Viewed as a group $\mathbb{R}$ has many choices of elements but only one identity element so I can make sense of specifying the identity element when talking about a group, but in what sense are the symbols related when we write out the structure (or signature)?

REPLY [10 votes]: I've always thought the treatment of constants as a separate class of symbols is rather silly.  A constant is just a special kind of function symbol: a $0$-ary function symbol.  There's only one way to give no inputs to a function, so such a function must always give the same output.
So, if you understand the role of function symbols, you also understand the role of constant symbols!  The roles are exactly the same; constants are just functions that require no inputs in order to give an output.  Changing the interpretation of a constant symbol from $0$ to $\frac{1}{2}$ as you suggest very much gives you a different structure, just as much as changing the interpretation of the function symbol $+$ to some other binary operation would.  Both $\langle \mathbb{R}, +, -, \cdot, 0,1, \le \rangle$ and $\langle \mathbb{R}, +, -, \cdot, \frac{1}{2},\pi, \le \rangle$ are perfectly good structures, but they are different structures, just as much as $\langle \mathbb{R},+\rangle$ and $\langle \mathbb{R},\cdot\rangle$ are different structures.
As for your final question:

in what sense are the symbols related when we write out the structure (or signature)?

The answer is "not at all"!  The relations and functions that are part of a structure need not have any relationship at all, other than that they are relations and functions on the same underlying set.