What is "Data" involved in a mathematical construction? What exactly do mathematicians mean when they refer to "the data" involved in a construction?
I've encountered this many times and I can usually figure out what's going on, but I am curious about the terminology, how it was introduced, and its connotations.
A concrete example: In pages 35-6 of "Differential Geometry" by R.W. Sharpe, there is a section entitled "Construction of Bundles". The author outlines three conditions that hold for a given $G$ bundle, and then later refers to these three conditions as "data" from which a $G$ bundle can be constructed. This usage confuses me.
Is there a semantic reason why standard words, like "component", "part", or "constituent" are not used?
 A: "Data" is the plural form of the Latin word "datum", which means, among other things, "thing that is given". Viewed this way, it makes perfect sense, doesn't it?
A: The word "data" (singular datum) comes from the Latin and means "thing(s) given". In mathematics, a notion is typically introduced by saying something like, "an operad consists of the following data... subject to the following axioms". It's the stuff that has to be given in the first place before you can begin putting conditions on them in the form of axioms. 
In logic, data is described by a signature (which specifies what sorts of operations and relations one is dealing with); once the signature is given, then you can specify the axioms in terms of logical formulas which are written in terms of the function and relation symbols of the signature. 
A: I don't know.  However, this time I won't let that stop me from answering.
If you talk to a carpenter or a craftsman about a geometric or algebraic
construction, they might look at you in a funny way, since your product
is not material.  Also (unless you are doing topological surgery or concatenation
of words), you usually don't put parts or components together.  Instead you
follow a recipe of low or high level operations applied to things already
constructed, which are as intangible as the result.  The stuff you feed to
the operations is some form of structured data, some of it numeric, some
linguistic, some relational.  Also during the process, you need to check that
the inputs satisfy appropriate requirements, and/or that the recipe can
guarantee the desired outcome.
That's why I am comfortable with the use of the word 'data', and prefer
it to something like 'component' or 'subassembly'.
Gerhard "Except When It Seems Apt" Paseman, 2013.04.14
A: I read it as saying that: if I (the author) were implementing this on a computer I would write a function that took DATA as input and which returns an instance of whatever we are constructing. In this setting DATA is a finite tuple and we specify what type of thing each term is. Sometimes there are conditions on DATA in which case the function would first check that these conditions are satisfied.
If you don't want or like to think in terms of programming it is saying that there is a function from the set of possible DATA (possibly satisfying conditions) to the set of things we are interested in.
A: People have already given several good answers, but I'd like to emphasize the way I usually use "data" as a term. As has been mentioned, "data" means "things given", but I usually use it (and have seen it used) in the following context:
Say we have a theorem that is as follows.

Let $f \colon M \to N$ be an $L$-Lipschitz mapping between two Riemannian manifolds of equal dimension. Then there exists a constant $C$ depending only on the data such that...

In my eyes this would mean that $C$ is a constant depending on $L$ and the dimension $n$ of the manifolds, but independent of the specific choice of the mapping or the manifolds.
