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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?

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It means what Webster's dictionary says it means. – Chris Gerig Apr 15 '13 at 4:06
I guess you're saying that I'm overthinking things? In any case, "data" in the mathematician's sense doesn't seem to match the definition you're referencing: 1: factual information (as measurements or statistics) used as a basis for reasoning, discussion, or calculation <the data is plentiful and easily available — H. A. Gleason, Jr.> <comprehensive data on economic growth have been published — N. H. Jacoby> 2: information output by a sensing device or organ that includes both useful and irrelevant or redundant information and must be processed to be meaningful 3: information in numerical... – pre-kidney Apr 15 '13 at 4:17
I think Chris is referring to the abstract mass noun version of data. Basically the information that you use as input for the construction. – Ryan Budney Apr 15 '13 at 4:30
Data tends to be some kind of n-tuple. For instance a group, $G$ is a triple, $(S,*,e)$ such that conditions hold. The "set" of data is a bit like a "subset" of some product of "sets". A construction then is like a function from this product to some other "set". – Spice the Bird Apr 15 '13 at 4:37
up vote 4 down vote accepted

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

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As an example, suppose you have to produce a "power" of a poset. This will be a partially ordered set where one of the "inputs" is a small poset P, say a chain of 7 elements. A piece of data needed is the degree of power, say n which I will set to 6 for this example. It is reasonable to think of P as input and n as data and that the result will have 117649 elements. However, you use P as a template and instead "manufacture" the product out of thin air, so it is reasonable to think of P and n as data used in the construction. Gerhard "Likes Dealing With Few Primes" Paseman, 2013.04.16 – Gerhard Paseman Apr 16 '13 at 19:48
If you are dealing with classification problems, it is nice to reverse the process, and associate the result with the data P, n, and the recipe for making the result. For sake of clarity, I like such associations to call out the recipe as well as the data when stating such in a theorem. Gerhard "Eats Different Things Different Days" Paseman, 2013.04.16 – Gerhard Paseman Apr 16 '13 at 19:53

"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?

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I like the concept of your answer, but no, it doesn't make perfect sense to me. If I'm understanding you correctly, you're saying that "data" is a synonym for "given". So my objection here is that this doesn't seem to match practical usage of the term "data", outside of mathematics. So are mathematicians using their own vocabulary here? This is where my confusion stems from. For instance, if I know what the word "notebook" means, and then I talk to some mathematician and they use the word "notebook" to mean "French horn", I would like to be notified in advance. That's all. – pre-kidney Apr 16 '13 at 0:22
You would like to be notified in advance? – Angelo Apr 16 '13 at 13:41
Sure, just the same way I am notified in advance what a "vector" is, what a "bundle" is, what a "Topological space" is, etc. – pre-kidney Apr 16 '13 at 18:36

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.

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See my response to Angelo above; I think you guys are essentially saying the same thing. – pre-kidney Apr 16 '13 at 0:23
Actually, I don't think we said essentially the same thing. I was actually trying more to answer your first question, "What exactly do mathematicians mean when they refer to "the data" involved in a construction?", according to how I understand the common usage. I don't think Angelo's answer addressed that at all. – Todd Trimble Apr 16 '13 at 0:48

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.

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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.

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I've also seen this usage, and I don't like it, because it leaves it up to the reader to guess that, in your example, the dimension is part of the data while the mapping $f$ is not. – Andreas Blass Apr 15 '13 at 14:18
Agreed, there is a risk of confusion. On the other hand I somehow find it natural, since the data consists only of "the numbers". And by this I mean that in situations like these we usually have (?) a dependence of the type $C = C(n,L,K,R)$ or something like that. So the constant $C$ depends only on the other constants, not on anything that is freely selectable by the reader. – Rami Luisto Apr 16 '13 at 7:00

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