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Latest comment: 16 years ago14 comments7 people in discussion
Rfv-sense: (mathematics, of a set) Countable; being either finite or countably infinite. I've never heard of this. MathWorld doesn't seem to support it. The WP article w:discrete says "a discrete set is a countable set" (and that seems to be where we got it from: see our edit history), but I don't believe WP on this point. (That definition was added by w:user:Patrick, who is (currently at least) an admin there; I have asked him for details on this word.)—msh210℠20:28, 28 January 2008 (UTC)Reply
This seems absolutely right and not worded badly. The concept is fundamental and in "clearly widespread use" in mathematics, if that isn't an oxymoron for our purposes. MW3 contains a sense very similar to the one challenged. The sense is even linguistically important, connecting to the notion of countability in pluralization. Something must be considered discrete to be countable. Do we have to cite this? DCDuringTALK11:31, 29 January 2008 (UTC)Reply
But it should be noted that it is restricted to certain branches of mathematics - "discreteness" and "countability" are certainly not synonyms in topology, for example. (It is easy to construct uncountable, topologically discrete sets - just add the discrete topology to *any* uncountable set). Wp mentions that this sense is used in "finite mathematics" aka "discrete mathematics" aka "decision mathematics", so I'll add that. \Mike19:11, 29 January 2008 (UTC)Reply
I know topology better than any other branch of math, and can assure you all that discrete means something totally different from countable in topology; moreover, I've never heard discrete meaning "countable" in any math course that I took before I started in on topology, as far I recall. (Discrete, as far as I know, means having points separated from one another by empty space (that's in lay terms, of course). Note that the set of rational numbers (fractions), a subset of the real line, while countable, is not discrete; on the other hand, the well-ordered set of ordinal type ω₂ is uncountable but discrete.) Moreover, I just checked the two discrete-math textbooks that I own: one, →ISBN, does not have "discrete" in its index, nor any obvious section to look up the word in; the other, →ISBN, writes (page 76):
The set of real numbers is ordinarily pictured as the set of all points on a line. This line is called continuous because it is imagined to have no holes. The set of integers is the pictured as a collection of points located at intervals one unit apart along the line. These points are called discretebecause each is separated from the others.
(Boldfacing mine. Italicization in the original: note that italics are used in math texts to indicate a word being defined.) This conforms to my understanding of the word discrete, not with our entry's sense under discussion here.—msh210℠19:32, 29 January 2008 (UTC)Reply
Well, certainly the countability of the set of all rational numbers is a challenge to the use of the word "discrete" in this sense. Would any normal person use the word "discrete" to characterize the set of all fractions (integer numerator and denominator)? Influenced by early exposure to the mathematics of countability/uncountability (but not to topology) I think of the set of all fractions as discrete. So I would, but the question stands: would a normal person? DCDuringTALK19:58, 29 January 2008 (UTC)Reply
(I noted that the set of rational numbers, as a subset of the real line, is countable but not discrete. I should have said what it looks like as a subset of the real line: Between any two rationals is another. Thus, any small interval of the real line has infinitely many rationals.)—msh210℠20:24, 29 January 2008 (UTC)Reply
Here's another quote from →ISBN (meaning, in addition to the one I quoted above). It's from her Preface, where she's outlining the general subject matter of the book:
Discrete mathematical structures are made of finite or countable infinite collections of objects that satisfy certain properties.
(Boldfacing mine.) This seems to be saying that, in discrete math, discrete means "finite or countably infinite, and also satisfying ". That is, "finite or countably infinite" as a definition of "discrete" is wrong even within discrete math, according to this.—msh210℠20:33, 29 January 2008 (UTC)Reply
I have striken my earlier statement that the term is in widespread use in mathematics. I had that impression but can't find evidence. Few of my math books (mostly pop math, decision math, statistics) include discrete in the index. Those that do don't use the word discrete technically, limiting it to introductory exposition and using "countable" exclusively thereafter. The MW3 equates the terms, declaring "countable" a synonym for "discrete". Their def. is as follows: "capable of assuming, containing, or involving only s finite or countably infinite number of values, items, or objects." Mathematics pushes our intuitive concepts to their paradoxical limits so we shouldn't be too surprised at this kind of awkwardness. But if integers, the paradigm of countability, are not also a paradigm of discreteness, what are they? DCDuringTALK20:55, 29 January 2008 (UTC)Reply
Like you, I had thought initially that it was a good mathematical definition, but also like you I have not found this to be supported in mathematical works. I couldn't find anything useful in my books on functions and analysis. My topology books define (deprecated template usage)discrete quite differently; in that subfield it seems to be applied only to the "power set" of all subsets of a topology. I have yet to look through my statistics books, but my own gut feeling is that (deprecated template usage)discrete is only applied to elements or members of sets, never to the sets themselves. --EncycloPetey05:38, 2 February 2008 (UTC)Reply
I believe the original claim is that a discrete subset of euclidean space is countable, which I believe is true under the standard topology with the definition that a discrete set is a set of isolated points; under other topologies and on other spaces this may not be true (see: Topology by Munkres and Principles of Mathematical Analysis by Rudin). On the other hand Discrete mathematics is the area of study of functions defined on the natural numbers, a totally different area of study than topology or analysis (see Concrete Mathematics by Knuth or any number of books titled Discrete Mathematics, say, Rosen). -- mconlen
