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Occasionally I see the use of the word "data" in definitions. For instance, one definition of an exact sequence starts off by saying, "An exact sequence of abelian groups (or modules or vector spaces) is given by the data of two homomorphisms [...]" (Perrin, Algebraic Geometry).

I've heard this term used in class as well once. In these instances data of course does not refer to data as used in statistics, as in data from an experiment.

What is the purpose of using such a strange word in abstract mathematics? Has anyone noticed this word?

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    $\begingroup$ This is not a real question. $\endgroup$
    – Harry Gindi
    Commented Dec 26, 2009 at 8:56
  • $\begingroup$ I suppose it's not mathematical; nevertheless I do think it's important to be careful in mathematical exposition to make the text as clear and beautiful as possible. I'm not saying that the use of the word "data" is overly confusing, but I think the way definitions are written can subtly influence the understanding of the reader, and that we ought to pay attention in formulating them. $\endgroup$
    – user1437
    Commented Dec 26, 2009 at 15:42
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    $\begingroup$ I do not find this "strange" at all. $\endgroup$
    – Gerald Edgar
    Commented Dec 26, 2009 at 18:31
  • $\begingroup$ To the extent that I think this is a good question, it relates to the Baez et al. notion of "stuff, structure, and property". (Lots of links; google for it.) The "data" is the stuff and structure. $\endgroup$
    – Theo Johnson-Freyd
    Commented Dec 26, 2009 at 19:35
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    $\begingroup$ I'm closing this as "not a real question," but it could also be closed as "subjective and argumentative." Latching on to some technical definition of a common word and objecting to any other use is just silly, like objecting to somebody saying "mosquitoes are vectors for disease" or insisting that "my answer is technically more informative then yours because it uses more characters, so therefore carries more information." $\endgroup$
    – Anton Geraschenko
    Commented Dec 26, 2009 at 23:29

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In mathematics, the word "data" is often used for the most general mathematical noun. One may consider a set, a function, a category, a group, or seven groups "data."

But for data to be interesting, it must have some sort of structure. Thus the data must satisfy certain properties.

A set-with-binary-operation consists only of data: a set S and a function f: S x S ---> S. It becomes a semi-group when we add a requirement to this data: the function must be associative.

In general, people speak of "data" and "structure." These are the raw materials and craftsmanship that create the mathematical universe.

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  • $\begingroup$ Is this in line with the shift in recent mathematics from set-theoretic thinking to a more categorical thinking where the objects we study are not necessarily formulated in terms of some set with relations? It would be interesting to trace the appearance of this word. $\endgroup$
    – user1437
    Commented Dec 26, 2009 at 15:46
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This use seems in line -- although perhaps not identical -- with the following dictionary definition:

Data:

  1. Factual information, especially information organized for analysis or used to reason or make decisions.

  2. Computer Science Numerical or other information represented in a form suitable for processing by computer.

  3. Values derived from scientific experiments.

It is often used in mathematics in the way you have identified above. Namely, when defining a mathematical structure, it gives the reader a heads up as to the fact that that the structure is "multi-sorted" and involves more than one object. In more formal language, one might say "tuple", e.g.,

"A topological group is a triple $(G,m,\tau)$, where G is a set, $m: G \times G \rightarrow G$ is a binary operation, and $\tau$ is a family of subsets of $G$, such that...."

One could also have said "A topological group is given by the data G,m,$\tau$..."

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  • $\begingroup$ This is sort of implicit in your answer, but perhaps it is worth pointing out that data is the plural of the latin word datum (something given). The word datum is often used for a single unit of information and so using the word data really does mean that several units of information need to be given in order to specify e.g. a short exact sequence. $\endgroup$
    – Dan Petersen
    Commented Dec 26, 2009 at 8:36
  • $\begingroup$ Thank you for your answer. Usually when I read definitions the words are either fairly primitive, or else mathematical. Like "together with", "tuple", etc. it seems to me that data is slightly less primitive in a way and thus it was a bit unsettling to read. It seems unnecessary :) $\endgroup$
    – user1437
    Commented Dec 26, 2009 at 15:36