Talk:Function (mathematics)/Archive 2

message: SUMMARY: Every dyadic function can be represented as a dyadic relation. However, not every relation can be represented by a function.
Dyadic relations include: mere associations, functions and series.
1. DYADIC RELATIONS 
Dyadic relations (xRy) or R(x,y) are predicates about relationships of two objects [Pe33] [Mad91].
(x > y), (x loves y), (x includes y), (x friend-of y) and (x son-of y) are examples of dyadic relations [R&W10].
x and y represent individual values. The set of x values is the domain and the set of y values, the co-domain. 
x and y can be tuples of degree n (n-tuples) but they continue being individual values. E.g.: a point of coordinates, a full name or an address. 
Sometimes, x and/or y has specific role. For instance, given (x references y), x is the referencing object and y, the referenced object.
Each R relation can be viewed as a class {(x,y)|xRy}. Then, R definition as (x loves y)is called "intension" (or functor) of R [Dea93]. x and y can be substituted by individual values and they are the arguments of the functor. Each pair of ordered values (Romeo,Juliet) is an instance of (x loves y). The set of such instances is the extension of R. Each instance is a member. The number of members is called cardinality.       
The adjective of dyadic relation is "relative" [Pei33].
2. RELATIONS AND FUNCTIONS 
Functions [y=f(x)] are monadic operations upon zero or more objects giving another object. Therefore, relations (xRy) being propositions are not functions.
A dyadic relation [xRy] is a fact between two existing objects. But in a dyadic function: a new object (y) is calculated using existing object (x).
Propositions of the form xRy are called functors (a word similar to functions) because arguments (a word similar to variables) are substituted by individual objects.
At the moment of the substitution, the functor became a proposition; which, in turns, can be <absurd> or not ; <meaningful> sentences can be <falsable> [Pop59] or not; being, finally, the <falsable>, <true> or <false>. Nothing of that is applicable to functions.
However, every dyadic function can be represented as a dyadic relation: Generating the set of pairs of the class of related values. But not every relation has an implicit transformation (or operation or algorithm) that can be represented by a function. Social convention is the driving force of the persistency of 'CA' ->- 'California', 'NY' ->- 'New York', etc. 
Note. Calling to relations without operation, "mere associations", Russell understands that dyadic relations include: mere associations, functions and series [R&W10].
BIBLIOGRAPHY
- [Dea93] Deaño, A., Introducción a la lógica formal, Alianza ed., 1993. 
- [Ham02] Hammer, E., "Peirce's Logic", The Stanford Encyclopedia of Philosophy (Winter 2002 Edition), //plato.stanford.edu/archives/win2002/entries/peirce-logic/
- [H&L65] Hughes & Londey, The Elements of Formal Logic, Methuen, 1965.
- [Mad91] Maddux, R. D., "The Origin of Relation Algebras in the Development and Axiomatization of the Calculus of Relations" in Studia Logica 50(3/4): 421-455, 1991. //www.math.iastate.edu/maddux/>>origin2.ps
- [Pei33] Peirce, C. S., "Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic." in Memoirs of the American Academy of Sciences 9: 317-78. Reprinted in Peirce, 1933.
- [R&W10] Russell & Whitehead, Principia Mathematica, 3 vol., Cambridge University Press (1910, 1912, 1913). 2nd ed., 1925 (Vol. 1), 1927 (Vol. 2, 3). Abridged as Principia Mathematica, Cambridge University Press, 1962.
[Enrique Villar; mailto:evillarm@capgemini.es]
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I removed this:
:Suppose the domain X is the set of all married men and the codomain Y is the set of all married women. The formula f(x)=the wife of x is clearly not a function. Given a married man a, f(a) may either not unique or not exist. In mathematics, it is not a good idea to write down such a fuzzy notation. One way to get around is to consider the formula as the subset of all (husband, wife) pair in X×Y. Clearly, if an explicit formula for f(x) is really a function, we can still construct the set of pairs f; so nothing is lost by this definition. 
Since it's such a confusing example. The real reason it's not a function is that the relation "married" is obviously too wide - if it means the set of all men who are currently married and the set of all women who are currently married in the usual sense, then we would expect it to be a function - each man is married to exactly one woman. On the other hand, if it means ever married, then it's not. Or we were to take a realworld sample of men who claimed to be married, meaning that they had multiple wives... etc. The final sentence is repeated below under the advantages of the set theoretic approach. Chas zzz brown 10:52 Feb 26, 2003 (UTC)
Thanks for removing this paragraph.  I am just too lazy to do it myself. :-P  The writer of this confusing example
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From the main page, the first graphic has this text: "This is not a function in usual sense because the element 3 in X is associated with two elements a and b in Y (Condition 1 is violated). It is a multivalued function." But element 3 in X is actually associated with elements b and c in the diagram.
:Yes. Don't be shy, change it! - Patrick 00:27 Apr 10, 2003 (UTC)
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The one thing missing from this article was an intro that ahh actually explains what the article is about ;-) - David Gerard 00:08, Mar 23, 2004 (UTC)
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The recent addition on YX notation is in the wrong register - too advanced. Charles Matthews 20:34, 6 Jul 2004 (UTC)
thread_title:  Older topics 

message: Can someone put something in the article that tells the difference between even and odd functions?  I came to Wikipedia to look it up, and I can't seem to find it there. --pie4all88 01:30, 27 Aug 2004 (UTC)
replies: text: Great.  Thanks a lot, man. --pie4all88 20:53, 27 Aug 2004 (UTC)
thread_title: Even and Odd functions

message: This article is in the category :Category:Set theory. While totally correct to be included there and not in a supercat of Set theory, I'd like this article to be included directly in :Category:Mathematics (also). This because so many use functions without having the slightest clue of Set theory, and that many kinds of functions are in Cat:Maths but not in Set theory (the exponential function, for example). ✏ Sverdrup 12:55, 25 Sep 2004 (UTC)
thread_title: Category

message: On the talk page for "function" (Talk:function), there is some material that should belong on this talk page instead. Pehraps it could be moved here or archived in this talk-page's archive. Ae-a 11:25, 9 Feb 2005 (UTC)
thread_title:  Old discussion from the talk-page of "[[function]]" ([[Talk:function]]) 

message: A lot of articles on Wikpedia seem to have problems when they confuse mathematical concepts with Computer Science concepts. This definition is confused in this and other manners, and I don't seem to be able to fix the first part, which starts badly. I've deleted the word "unique" where I could, and pointed out the example which contradicted it anyway.
The way I learned about functions (as a Math student) is this:
A function is a MAPPING from some SET A to some other SET B, such that if you pick an element from set A it takes you to some element in SET B.
That's it. There's no guarantee of uniqueness at the destination (that's a special type of function).
::In this encyclopedia, a multivalued function is not a function (or, it is a set-valued function of codomain equal to the power set of the "original domain"). This is commonly accepted concensus. On the other hand, things like "random" functions are functions, but with somehow "hidden" parameters (i.e. depending on what is called "global variables" in Computer science; including here: random seed, system time and entropy,...). — MFH: Talk 14:32, 20 May 2005 (UTC)
replies: text:  we all agree. — MFH: Talk 14:32, 20 May 2005 (UTC)
replies: text: this is not a counter example to anything. Of course constant functions are functions. "Unique" means not having multiple "y" values associated to one "x", not the converese: many "x" may give the same "y"! — MFH: Talk 14:32, 20 May 2005 (UTC)

text:  yes they are (morphisms mentioned). (Now at least - b.t.w, you forgot to date and sign your post...) — MFH: Talk 14:32, 20 May 2005 (UTC)
thread_title:  Computer Science vs. Math 

message: I think the introduction should be a little less categorical in the first phrase (delete "unique") and/or maybe cite the (counter-)example of multivalued functions.
Also, rather than (or: before) citing "operators" right here, I think a link to partial functions could be useful.
Concerning the latter, I doubt that Dirichlet (cited in "History") used the term "function" as here, as in Europe (at least France and Germany),
"function" usually means "partial function", and "map/mapping" (application / Abbildung) means "total function".
In some sense, this is not an objection to WP's use of "function", but of "domain", which imho should mean the set of points where they are defined - i.e. play the role analogue to "range" and not to "codomain" (whose corresponding notion should be something like "origin" or "departure set").
To justify my opinion, I refer to Google search results for "with dense domain" and "its domain is dense", e.g. (put into quotes as it stands to avoid irrelevant results). This usage (i.e. "let f:X->Y be a function from X to Y with (= defined on its) domain D(f)⊂X") is the most common one according to my experience.
PS: and maybe (even more) links should be provided to make it clear that this article is not about function (programming). — MFH: Talk 15:11, 20 May 2005 (UTC)
replies: text: It's a bad idea, in this case, to begin with the history. 'Function' probably is still often used for relations of non-functional kind; but I don't think any professional mathematician would see that as more than abuse of language. Charles Matthews 15:26, 20 May 2005 (UTC)
replies: text: I don't think that function should be used as a synonym for partial function. --MarSch 1 July 2005 09:54 (UTC)
Is sin not a function? because sin isn't a function as defined on this page "such that each element of a set (the domain) is associated with a unique element of another set". Each element in Sin is not associated with a *unique* element of another set. If sin isn't a "function" it would serve as a good example of a non-function. However, I think its important that someone explain why the definition "function" is important in math - why it is a very usful property that each element of a set be associated with a unique element. I have only the vaugest notion of that reason myself. User:Fresheneesz 68.6.101.96 18:53, 10 November 2005 (UTC)
thread_title:  functions that are not functions 

message: i tried to integrate the gaussian function but it seems to be a complex function which is off course not true. i went to the integrator at the wolfram site and it gave me nonsense. can someone tell me what is its integral?
:No, but error function says what can be said. Charles Matthews 16:51, 11 October 2005 (UTC)
thread_title:  gaussian function 

message: I have found your maths pages very helpful.  Can I suggest one change which might be useful for people interested in Category Theory?  When I first met this subject, many years ago when these things were less well explained, I was perplexed by the fact that morphisms seemed to be both similar to normal (set theoretic) functions and different from them.  It took me a while to realise that morphisms were NOT the analogue to functions defined as subsets of a Cartesian product, but that they had a vital extra ingredient: the explicit naming of the codomain.  
The modern view of what a function is seems to take this into account: it is a triple consisting of domain, codomain and subset of Cartesian product.  But the "function" page in Wikipedia does not make this clear enough.  In fact, the function given explicitly under the paragraph headed "Formal definition", is given as being - explicitly - equal to {(1,D), (2,B), (3,A), (4,A)}, with no mention of the object C that is also in the codomain.  
Can I suggest that to avoid confusion, the codomain should always be cited in any explicit definition of a function.  --Tireasyas 17:04, 21 November 2005 (UTC)
replies: text: I agree with Tireasyas. I've edited the formal definition section so as to say explicitly that a function is an ordered triple. As well as modifying the language associated with the third example. I hope this helps. Paul August ☎ 22:17, 28 November 2005 (UTC)
replies: text: I have a similar issue that Tireasyas seems to mention. A function can be described as a set of ordered groups - and i was going to add that in the definition after relation ( "relation (a set of ordered groups)" ) - but I thought ordered groups doesn't sound quite right, but ordered pairs would not cover everything a function can be. Fresheneesz 20:01, 28 November 2005 (UTC)
replies: text:  The notion of an ordered group doesn't have anything to do with the definition of a function. As Tireasyas mentions above (and the article now makes more explicit) formally, a function is an ordered triple (X, Y, G(f)), where X is the domain of f, Y is the codomain of f and G(f) called the graph of f is a set of ordered pairs (x, y) where x is in X and y is in Y, such that for each x in X, there exists a unique y in Y, such that (x, y) is in G(f). So when you are talking about  a function being a set of ordered pairs, what you are really talking about is the graph of the function, which is often informally identified as the function. Does this help? Paul August ☎ 04:55, 29 November 2005 (UTC)
replies: text: Yea it does, thanks. i think i've figured it out, though the specific way it is formally written and talked about seems unneccessarily cumbersome to me. Fresheneesz 07:58, 29 November 2005 (UTC)
thread_title:  More on definitions 

message: Please change all the "in"s to epsilons  example. x in X --anon
: δisagreε. No need to emphasize too much on math notation. I like x in X more than x ∈ X or x ε X. Besides, the ∈ symbol does not show up in many browsers. Oleg Alexandrov (talk) 22:55, 7 December 2005 (UTC)
thread_title:  EPSILON 

message: I have someone over at talk:affine space complaining that they don't understand the notation \mapsto, apparently struggling with f:X\to Y: x\mapsto y and I'm disappointed to see that this article doesn't explain this notation. :( linas 06:09, 31 December 2005 (UTC)
replies: text: Why don't you add it here?  It is certainly a useful notation. Rick Norwood 17:21, 1 January 2006 (UTC)
thread_title: Missing topic

message: Thank you for improving my rewrite. Rick Norwood 15:30, 10 January 2006 (UTC)
replies: text: I am happy you were able to read through my grumpy edit summary. :) Oleg Alexandrov (talk) 16:19, 10 January 2006 (UTC)
thread_title:  Thanks to Oleg 

message: Certainly, there are parts of this article that need a rewrite, especially in the last half.  But, Jorge, you have made drastic changes, apparently without being aware of what has gone before.  Since I may have to revert those changes (I'm commenting here first) I want to explain that we have discussed both here and in other Wikipedia articles whether Wikipedia should aim its math articles at mathematicians or laypersons.  A consensus has been reached that the introduction at least should be readable by a layperson.  Your new introduction, while mathematically correct, does not meet this standard.  Only a mathematician could understand it. Rick Norwood 15:42, 16 January 2006 (UTC)
replies: text: Sorry, I was trying to correct a set o fmajor changes to function (mathematics) , map (mathematics), injective, surjective, bijective, which, as far as I could tell, had been done with no discussion and left the set much worse than in t was before (and with a lot of duplicated material too boot).  Jorge Stolfi 15:59, 16 January 2006 (UTC)

text: I agree that Wikipedia articles should be accessible to the "layman", however an encyclopaedia is *not* a textbook!  In computer terms, it should be more like a "reference manual" than a "tutorial".  So while one must avoid unnecessary use of jargon, one must also try to be succint and precise.  Jorge Stolfi 16:04, 16 January 2006 (UTC)

text: Could you at least let me finish the cleanup before reverting? I don't think I am changing the style that much, and if you think that a gentler intro or more examples need to be added, we can add them.  (But the example E=mc2 is just the sort of "explanation" that is more confusing than illuminating.) Also, someone who comes to an article called "function (mathematics)" presumably already has a foggy notion of what a function is; and this article should make it clear that the mathematical concept is NOT AT ALL the same thing as the computer science concept.  It should also make it clear that, in mathematics, "definition" means "PRECISE definition", not "intuitive concept". All the best, Jorge Stolfi 16:20, 16 January 2006 (UTC)

text: Thanks. I hope to clean up the mess I left in injective function, surjective function, and bijective function before I go home.I also think that we should trim function (mathematics) by moving some of the sections to ancillary pages like inverse function, inverse relation, composition and the like. Putting everything on the same article is bad for readers who looking for a specific topic.  As for lowering the level, targeting exclusively for k12 is not good either; the article should be appealing to more educated readers (college or more), too.  At least, this seems to be the tone of articles in other specialties, like biology, chemistry, etc.  Hopefully we can do both in the same article.  As for E=mc2, the problem is that it is *not* a function, but only a formula that can be many different functions; and this distinction is one of the points that the article must make clear.  All the best, Jorge Stolfi 01:41, 17 January 2006 (UTC)

text: First, "E = mc2" is a formula, not a function. You can use that formula to *specify* a function, but for that you must also specify the function's arguments and their order: is if f(E,m,c), or f(m,c,E), f(m,x,y,z,E,h,t), or what?  Second, "E = mc2" is, strictly speaking, an equation, i.e. a formula that given E,m,c returns either true or false.  Since you say "put in the mass, return the energy", then you must be thinking of "mc2" alone.  However, to evaluate that formula you need m and c, not just m.  (The c may be a constant in physics, but physics is not mathematics! There is all that stuff about measurement units, for one thing; not to mention those physicicsts who speculate that c may change with time.)So, you see, this is a very bad example. It does not help the student to understand the concept of mathematical function. It may give him the feeling that he understood, when in fact he got a completely wrong idea. That wrong idea will not help him understand the correct definition; it will only give him an excuse to skip it.  Or, worse, will give him the idea that the formal definition is just a complicated way to say the same thing.  Worse still, if he is smart enough to realize that the formal defintion does not say the same thing as the informal one, he will learn that "math sucks"...As Einstein once said, "make it as simple as possible, but no simpler".All the best, Jorge Stolfi 21:53, 17 January 2006 (UTC)

text: PS. I think that Wikipedia should aim to the 10% who are willing to learn, not to the 90% who just want to be told that they don't have to. Jorge Stolfi 21:57, 17 January 2006 (UTC)
thread_title:  Jorge Stolfi edit 

message: JA: Okay, having argued this issue from both sides of the table, I humbly qualify myself as a veteran.  I will give it a more careful look and get back to you later today.  Jon Awbrey 15:00, 18 January 2006 (UTC)
thread_title: WOFWYW (watch out for what you wish)

message: A function (similarly, a relation, etc.) is an abstraction which has an implementation.
A function is, intrinsically, a way of associating exactly one element of a range (a set or possibly a class) with each element of a domain (a set or possibly a class).  One way of implementing this is to regard a function as a set of ordered pairs (actually, in set theory we sometimes have occasions to consider classes of ordered pairs as well).
This is slightly unsatisfactory, as it does not allow us to specify the range, so there is 
a further technical definition of the function as the ordered triple of its domain, range and graph.
The latter object is obviously an implementation of a function with arbitrary details, and not a good definition of function for the layman.  The definition of a relation (and thus of a function) as a set of ordered pairs is possible to sell to the layman; the addition of the domain and range as components should be given (if at all) as a technical refinement, with an explanation of why it is needed (the domain is redundant in the case of a function!)  But it is also possible to implement the notion of function in set theory without any reference to sets of ordered pairs at all, or to take the function notion as primitive.  The relevance of all of this to the matter of how the article is written is that the article really should say what a function is in the intrinsic sense, and then say that a function is implemented in set theory as a set of ordered pairs, or a triple of domain, range and graph, etc. The layman needs to know what a function does for him, and the layman can see perfectly well that a function is not precisely identical to the set-theoretical concept which implements it (the latter has too much detail); it takes technical training to dull this perception!  Randall Holmes 16:52, 18 January 2006 (UTC)
thread_title:  what is a function? 

message: * JA: More subtleties about the definition of function have arisen, historically associated with the difference between relations and functions defined via their extensions versus relations and functions defined via their intensions.  Both sorts of definition are important and necessary to coordinate with each other, but I will suggest that extensional approaches are better for a start, as they are more easily connected with concrete experience.  To avert one potential misunderstanding, this is a question that is partly independent of, that is, orthogonal to, the need for abstractions, as both extensions (sets) and intensions (properties) are abstract objects.  Jon Awbrey 17:44, 18 January 2006 (UTC)
replies: text: * one needs to be aware, though, that the layman is more familiar with an intensional understanding of both relation and function.  There is method in my madness; I'm not really thinking about "abstract data types" but about pedagogy. Randall Holmes 17:49, 18 January 2006 (UTC)

text: * not, it belongs precisely in the initial intro to the topic (though the level of abstraction needs to be addressed, perhaps).  Read the definition of "function" in a calculus book.  Randall Holmes 18:01, 18 January 2006 (UTC)
thread_title: Definitions of function, extensional & intensional

message: Sigh....When I started working on this article, I thought that it was a real mess. After some hours of tweaking and twiddling, I honesly thought that I had managed to get at least the head paragraph into a fairly good shape --- a reasonable compromise between correctness, completeness, terseness, and clarity. Then Rick came along and "improved" it to something that I found simply awful. After couple of drastic reverts, and several kilobytes of barely urbane discussion, the head section is again in a state that, to my eyes, is obviously much worse than my last version --- clumsy, needlessly verbose, jumping right away into hair-splitting details that are only relevant within axiomatic set theory. (And wrong too --- "codomain" is not "range"!)It is obvious now that the problem is entirely subjective: my taste is just completely different from that of the other editors.  What seems a polished jewel to me, looks like doggie-doo to them, and vice-versa.Well, I just don't know what to do about it.The amazing thing is that in the numerous radical rewrites that I did before, in articles like Abrahamic religion or psychic surgery, or even mass lexical comparison, I never ran into such radical differences of taste as I found here in this case.  The few disputes I faced there were always punctual and easily resolved. (The closest experience I had was in fact over another math article, aliasing).Is there something wrong with me, or is it something to do with mathematics? Perhaps "good style" in mathemtics is 100% subjective because mathematics is 100% about mental constructs, without the slightest connection to reality?Sadly, my vacations are already over, and my colleagues are already mad at me for working at Wikipedia instead of doing what I am paid to do.  So I must give up on these articles for now. Hopefully by next Christmas Wikipedia will have a well-defined standard for mathematical articles --- and I hope that it will be something that I can learn to like and work with.So long, all the best, and thanks for all the fish... Jorge Stolfi 19:54, 18 January 2006 (UTC)
replies: text: I do know that range and codomain are not the same thing.  But I didn't want to add another parenthetical remark; the distinction is clearly explained below in the article.  (Besides, I belong to the minority school which says that a function or relation is best implemented as its graph -- just a set of ordered pairs -- note that I didn't try to impose this view...). Have a good semester. Randall Holmes 20:33, 18 January 2006 (UTC)
thread_title: Subjectivity is unavoidable?

message: I've given the first few paragraphs my best shot.  And here I pause for comments and criticisms. No more changes by me until tomorrow. Rick Norwood 21:11, 18 January 2006 (UTC)
replies: text:  I think that this is now too low-level and needs further attention.  There must be a happy medium which avoids my purple prose and Jorge's diving in directly with the mathematical definition, but which is nonetheless reasonably precise.  But I must go to a department meeting. Randall Holmes 22:36, 18 January 2006 (UTC)
replies: text:  Also, the definition of binary relation claimed in the "mathematical definition" section is not the definition given in the binary relation article (it is the less favored of two alternatives).  I am not saying that I object to this definition:  I actually belong to the minority school that prefers it.  But I believe it is a minority.  The mathematical definition section needs to acknowledge the nasty ordered triple version up front... Randall Holmes 22:36, 18 January 2006 (UTC)
replies: text: I disagree.  I think the nasty ordered triple version is a minority view, only being used in category theory and related topics.  But category theory is being used as a substitute for set theory these days, so it may be a majority view now.  But I suppose we should give both views some standing. Arthur Rubin | (talk) 23:45, 18 January 2006 (UTC)
thread_title: Functions & Relations

message: I have a strong dislike for "input" and "output" in the introduction of the article -- and a mild dislike for confusing the (formal) domain of a (partial) function (used only in category theory and related concepts) and the set of points on which it is defined (also, and more often,  called domain).  Since the mathematical concept here seems to be that of a total function, there is no difference. 
I admit my perspective is mostly from the 80s and earlier.  See my late mother's books on set theory and the axiom of choice.  In those books and articles, the domain and range of a relation are defined from the graph of the relation, and the concept of "codomain" doesn't exist.  It's possible that "modern" mathematics takes a different approach -- however, one then needs to redefine an ordered tuple of not-set proper classes, which wouldn't otherwise exist.
Arthur Rubin | (talk) 23:45, 18 January 2006 (UTC)
replies: text: How do we say first component and second component of a pair?  I think "projection" is standard for this purpose (I do not mean projection of a set!) Randall Holmes 00:54, 19 January 2006 (UTC)
thread_title: Perspective from a set theorist

message: It seems that Jorge had to break of, so let me continue. Current version, more or less line by line, reads:
# The concept of a function is fundamental to modern mathematics.  
# A function associates an input and an output.  
# A familiar example of a function is [linebreak]  f(x) = x^2 , [linebreak] which, given the input 3, produces the output 9.
# The set of all inputs is called the domain.  The set of all outputs is called the codomain.  (Some authors call the codomain the range, others assign a different meaning to the word "range".)
# An essential property of a function is that for each input there must be one unique output.  
# Thus, for example, the square root is not a function, because there are two square roots for any positive number.  The square roots of 9 are 3 and − 3.  To make the square root a function, we must specify which square root (usually the non-negative one).
# A function need not involve numbers.  An example of a function that does not use numbers is the function that assigns to each person's name their initials, so that, for example, f(Andrew Wiles) = AW.
# A more precise, but still informal definition follows.  Let A and B be sets.  A function from A to B is determined by any association of a unique element of B with each element of A.  The set A is called the domain of the function; the set B is called the codomain of the function.
My comments: Line 1 and 2 are fine. Line 3 is open to the criticism that it confuses functions with formulas or something else; how about "A familiar example of a function is given by …". I'm not sure line 4 is necessary (by the way, I think I also learnt "range" for what is here called the codomain). Line 5 is okay. Line 6 might be confusing: I seem to remember that I learnt in high school that square roots are per definition positive. Line 7 is good. I'm not very happy with line 8, without being able to say why. Overall, I think it is pretty good. -- Jitse Niesen (talk) 14:00, 19 January 2006 (UTC)
thread_title: Jitse's comments

message: * JA: To anybody who's taught math, this article is just plain horrible, and something "will" be done about it.  I have sound pedagogical reasons for my edits, so please discuss them before assuming that I don't know what "usual", etc.  Jon Awbrey 15:18, 19 January 2006 (UTC)
replies: text: Jon, before you waste your time, please note (again) that Wikipedia is not supposed to be a textbook! It is supposed to be a reference work, like a dictionary,  thesaurus, farmacopoeia, herbal, chemical index, etc..  A wikipedia article, like a dictionary entry, should be, first, precise and exhaustive; second, well-organized (so that the reader does not have to read two pages to find the sentence he needs); and third (only third), as clear as possible within the previous constraints. "Pedagogical", in the textbook/classroom sense, is defintely not a goal. All the best, Jorge Stolfi 15:38, 19 January 2006 (UTC)

text: to clarify motives here.  There may be excellent reasons to write x^2=y instead of y=x^2. However, everyone has been taught to write functions in the latter way since the beginning of time, and most readers of the article know this and will be brought up short.  One could write later in the article about notational improvements which would be useful pedagogically (I could write a paragraph about why we should write (x)f so that the notation for composition could go in the right direction) but this does not belong in the introduction where it is distracting.  The notations for the square function that you give are not common (though you might like them):  an intro is not the place for novelties.  You can't pull rank on me:  I'm a professor of mathematics, teaching mostly calculus and sometimes precalculus, and I'm also a set theorist; this article is in my territory in both respects. Randall Holmes 15:34, 19 January 2006 (UTC)

text: It may be standard in CS pedagogy, but it is not standard notation in mathematics, and this is a mathematics article.  One should not violate expectations in the introduction to an encyclopedia article [one can do so later].  The f(...) = square(...) is unintelligible in this context; there is a function(computer science) article.  But take a look at my latest edit:  I added a paragraph on function notation (which even uses 'square'). Randall Holmes 16:01, 19 January 2006 (UTC)
replies: text: Further, on inspection essentially everything in the article is correct at this point (even in the intro, with allowances for poetic vagueness).  Please be aware that "function" is a concept which belongs to a mathematical discipline -- and that discipline is in the final analysis set theory, not calculus pedagogy... (and the computer science use of "function" is not the same at all) Randall Holmes 16:09, 19 January 2006 (UTC)

text: Since several people are actively editing this page at this time, I will wait until later today.  But please note that several people (myself included) are confusing "range" and "codomain".  This needs to be fixed. Rick Norwood 15:59, 19 January 2006 (UTC)

text: I didn't set the level either, nor do I like the input/output metaphor (though I teach about functions using this metaphor often enough, it is not really appropriate as a definition of function).  I'm sympathetic to describing the function notion informally in the opening paragraph, so that a relatively naive reader will understand that the function notion he learned about in school is effectively the same as the set-theoretical notion, and subsequently being appropriately technical.  If I wrote the entire article, it would look different; but it would be quite impossible to attempt this at the moment... What should appear in a "concise reference", in your opinion? Randall Holmes 16:33, 19 January 2006 (UTC)
thread_title: I'm right here -- well, most of the time ...

message: Perhaps the problem is that the range of this article is too broad.  Perhaps we should have a 
function (calculus) article and a function (set theory) article [function (math pedagogy); function (analysis)?].  There are essentially different interests at work here. Randall Holmes 16:50, 19 January 2006 (UTC)
thread_title:  Proposal to split article(?) 

message: When this settles down we need to look at the redirects, to make sure they are still consistent with (any) relocated content. 
LarryLACa 03:40, 20 January 2006 (UTC)
thread_title:  Consolidating content - fixing redirects 

message: Jorge Stolfi's introduction returns to the idea of an introduction that cannot be read by anyone but a mathematician.  I think Jon and I agree that the introduction should be something a layperson can read.
Here is a suggested compromise.  I include your three examples but leave out your SQUARE function.
::For example, the equation y = x2 describes a function that associates each value of the input x with just one value of the output y, in such a way that the equation is true.  Thus, an x of 1 gives a y of 1, an x of 2 gives a y of 4, an x of 3 gives a y of 9.  This function is called the 'squaring function', and it is common to express it as 
::: y = x ^ 2 
::or
::: f(x) = x ^ 2 
If this is acceptable, why don't you try a compromise version of paragraph three. Rick Norwood 21:30, 19 January 2006 (UTC)
thread_title: Paragraph two, third attempt

message: I have drafted this article (function (set theory)).  It is unapologetically more abstract (perhaps Jorge will like it, and perhaps not).  I suggest that an article along these lines should be paired with a more numerically and graphically (and naturally pedagogically) oriented article function (analysis).  Randall Holmes 21:26, 19 January 2006 (UTC)
replies: text: How about this.  We leave the title "Function" on this introductory article, and put a link "For a more theoretical discussion of the concept, see Function (set theory).  That way you can take all of the pedagogy out of Function (set theory), and the article Function can defer some technicalities for the more advanced article.  Jorge, if he chooses, can contribute to the Function (set theory) article and make it as pure as he desires. Rick Norwood 21:52, 19 January 2006 (UTC)
replies: text: Well, I still can't see how one can write a useful article called function (mathematics) without using any set theory, not even the word "set". However, a separate article on function (category theory) or function (axiomatic set theory) would cerainly be quite appropriate. Jorge Stolfi 22:54, 19 January 2006 (UTC)
replies: text: I never proposed that one should write the second article without referring to sets.  The idea is that the function (analysis) article could be less formal (though certainly mentioning sets) and refer to function (set theory) when things seemed too convoluted.  Mind, I actually believe that an integrated function (mathematics) article is perfectly possible -- given cooperation.  Randall Holmes 00:35, 20 January 2006 (UTC)

text: PS. There is already a function page that covers other meanings of function as well, and, for the maathematical sense, says in one sentence all that a naive reader may want to know.  That's another reason to assume that the reader who gets here expects to get more than a "brain pacifier". Peace and love, Jorge Stolfi 23:04, 19 January 2006 (UTC)

text: PPS. Oops, would someone please lock that page, quick!? 8-)  Jorge Stolfi 23:04, 19 January 2006 (UTC)

text: Keep in mind that mathematicians used functions for generations before set theory was even invented.  Modern taste in matematics puts everything in terms of set theory (except, of course, those things that can't be put in terms of set theory).  But that is mathematicians writing for other mathematicians.  "There are five and twenty ways of writing tribal lays, and every single one of them is right." Rick Norwood 22:59, 19 January 2006 (UTC)
replies: text: I know this very well.  This is part of the reason that I maintain that function (set theory) is not precisely the same thing as function (analysis), though the latter has now been officially defined as the former for 100 (150?) years or so.  The difficulty is that many users of function (analysis) do not actually understand (or maybe understand and simply do not like) function (set theory). Randall Holmes 00:35, 20 January 2006 (UTC)

text: About a century ago, chemists already knew about elements, but still wrote things like "sulphate of magnesia" instead of "sulphate of magnesium", because the closest that they could get to the pure element was its oxide (magnesia). Yet, today they do not begin their classes by teaching the old naming scheme; they teach the new one from the start --- because it is much more logical, much more general, and no more difficult to learn.Same thing with functions: for a long time people used only real-valued functions, and there were many definitions, each good for a special purpose but bad for others.  Then came the set-theoretical definition, that covers all the old senses and is not only simpler, but vastly more useful.  The set model made it possible to expand the scope of our many old intuitions about functions to things like integer and non-numeric functions, functionals, differential operators, linear maps, etc.; and at the same time put them in much more solid foundation.  Now, trying to explain the old concept of function to a naive reader is a disservice also for this reason: in the end, he will only gain a very limited grasp of the concept, and is likely to miss the chance to understand its real importance to modern mathematics.  It is like telling a geography student that the Universe is, well, "basically the Earth, plus a bit of space around it".All the best, Jorge Stolfi 23:35, 19 January 2006 (UTC)
thread_title:  Function (set theory) 

message: Jorge -- it would be a big help if you put your new comments at the bottom of the page, instead of making people go look for them somewhere in the middle. "Say goodnight, Gracie."  "Goodnight Gracie." Rick Norwood 23:35, 19 January 2006 (UTC)
replies: text: I will try. But there are also arguments for keeping the reply close to the text it refers to... Jorge Stolfi 23:39, 19 January 2006 (UTC)
thread_title: Next topic

message: I am finally beginning to understand what I have been trying to say. 8-) Please consider this version:
In mathematics, a function is a relation between a set X and a set Y, that pairs each element of X with a single corresponding element in Y.
An example is the function from the set {1,2,3,4} to the set {A,B,C,D} that pairs 1 with D, 2 with B, 3 with A, and 4 with A (see the diagram at right). Another example is the function from the set R of all real numbers to R itself that associates each number x to the number x3 − 9x.
Functions are extremely important in all branches of modern mathematics. They are extensively used in every scientific and technical field to model all kinds of numerical relationships—typically, when some quantity (or a collection of quantities) completely determines another quantity.  Thus, for example, a physicist may use a mathematical function to describe how the temperature of water affects its density; or how the gravity acceleration vector on the Earth's surface changes depending on the position (latitude and longitude). Functions may also represent relationships between non-numerical data, e.g. the pairing of each country of the world with its capital city.
Functions are also important in computer science, where they are used to model all sorts of data structures, and the effect of algorithms.  In that field, however, the word function is also used in the very different sense of procedure or sub-routine; see function (computer science).
In most mathematical fields, the terms map, mapping, and transformation are usually synonymous with function. However, in some contexts they may be defined with a more specialized sense: for example, in topology a map may be defined to be a continuous function.
The rest is as before. 
===Comments===
The problem with my previous version was that it gave the modern set-theoretical defintion, but then tried to illustrate it with examples inspired on the old-fashioned view, namely "a process to compute an output number from a given input number".  Those examples were confusing not because of the set-theoretic wrapper, but because "numbers" and "formulas" are very complicated concepts, much more complicated than "function". 
All the best, Jorge Stolfi 00:17, 20 January 2006 (UTC)
replies: text: Do note that considerations having to do with "numbers" and plugging things into "formulas" will certainly come up in practical applications of "function":  they really are part of the function concept as it is encountered in practice... Randall Holmes 00:53, 20 January 2006 (UTC)
replies: text: Yes, but the point is: the complexity of the old concept of function (the reason why it is so hard to explain to students) lies all in the concepts of "number" and "formula", not in the concept of "function". Specifically, the difficulty lies in the facts that numbers are infinite, that real numbers are dense, that formulas are complicated to parse and evaluate, etc..For instance, when the domain is infinite you cannot just tabulate the function; you must specify it indirectly by some complicated method like a formula, algorithm, implicit equation, etc.. The "input and output" metaphor is only relevant when you specify the function by formula and algorithm; you don't need to use those words when dealing with small finite sets. And so on.If you start with simple finite examples, you will find that the concept of function, by itself, is in fact quite banal, much easier to explain than the notion of "real number" or even "integer number".  Once they got the idea of function, then you can move on to more complicated examples, numeric-valued etc..Makes sense?Jorge Stolfi 05:23, 20 January 2006 (UTC)
replies: text:  Very banal, very simple -- but at an extremely high level of abstraction.  The difficulty is that the reader needs to recognize that he is reading the right article.  I hasten to add that your latest text actually does quite rapidly suggest the connection with the usual definition of functions using formulas. Randall Holmes 16:54, 20 January 2006 (UTC)

text: In mathematics, a function pairs each element from one set with a single element from a second set.
One problem is that this may not be recognized as a definition. It would perhaps be better to write "A function is a specification which pairs …", but specification does not quite seem the correct word here, and I could not think of a better word. -- Jitse Niesen (talk) 04:42, 20 January 2006 (UTC)
replies: text: Could say "... an entity which pairs ...".  However this would be a strictly informal pseudo-definition, whereas if you say "relation" the first line will be  a correct and complete mathematical definition. Jorge Stolfi 04:54, 20 January 2006 (UTC)

text: Great, looks like we're making progress. For the record, I did not state that the first sentence should be a definition, only that it should be put to more focused and productive use. However, if a thesaurus can lead us to a definitional sentence that satisfies all, so much the better. To read like a definition, the wording probably needs to begin "In mathematics, a function is … ." And the next word is … what? Jorge points out the correctness of "relation", but I agree with Jitse that a general audience will be unfamiliar with that as a technical term. Often relation and function are terms learned at almost the same time. Other options are "a pairing" or "an association that pairs". I confess that function is such a simple but deep and familiar concept that it is not easy for me to see it with a "beginner's mind". That's one of the things that makes basic articles challenging. Part of the problem may be trying to make the first sentence carry too great a burden, to be complete and correct for a technical audience yet easily understood by a general audience. Besides simply using more sentences, another way to relieve the burden is to display a variety of both examples and counterexamples. Some effort has already been made in this direction, and I would encourage more (especially counterexamples). --KSmrqT 05:19, 20 January 2006 (UTC)
thread_title: Another proposal

message: The problem with my previous version was that it gave the modern set-theoretical defintion, but then tried to illustrate it with examples inspired on the old-fashioned view, namely "a process to compute an output number from a given input number".  Those examples were confusing not because of the set-theoretic wrapper, but because "numbers" and "formulas" are very complicated concepts, much more complicated than "function". 
All the best, Jorge Stolfi 00:17, 20 January 2006 (UTC)
replies: text: Do note that considerations having to do with "numbers" and plugging things into "formulas" will certainly come up in practical applications of "function":  they really are part of the function concept as it is encountered in practice... Randall Holmes 00:53, 20 January 2006 (UTC)
replies: text: Yes, but the point is: the complexity of the old concept of function (the reason why it is so hard to explain to students) lies all in the concepts of "number" and "formula", not in the concept of "function". Specifically, the difficulty lies in the facts that numbers are infinite, that real numbers are dense, that formulas are complicated to parse and evaluate, etc..For instance, when the domain is infinite you cannot just tabulate the function; you must specify it indirectly by some complicated method like a formula, algorithm, implicit equation, etc.. The "input and output" metaphor is only relevant when you specify the function by formula and algorithm; you don't need to use those words when dealing with small finite sets. And so on.If you start with simple finite examples, you will find that the concept of function, by itself, is in fact quite banal, much easier to explain than the notion of "real number" or even "integer number".  Once they got the idea of function, then you can move on to more complicated examples, numeric-valued etc..Makes sense?Jorge Stolfi 05:23, 20 January 2006 (UTC)
replies: text:  Very banal, very simple -- but at an extremely high level of abstraction.  The difficulty is that the reader needs to recognize that he is reading the right article.  I hasten to add that your latest text actually does quite rapidly suggest the connection with the usual definition of functions using formulas. Randall Holmes 16:54, 20 January 2006 (UTC)

text: In mathematics, a function pairs each element from one set with a single element from a second set.
One problem is that this may not be recognized as a definition. It would perhaps be better to write "A function is a specification which pairs …", but specification does not quite seem the correct word here, and I could not think of a better word. -- Jitse Niesen (talk) 04:42, 20 January 2006 (UTC)
replies: text: Could say "... an entity which pairs ...".  However this would be a strictly informal pseudo-definition, whereas if you say "relation" the first line will be  a correct and complete mathematical definition. Jorge Stolfi 04:54, 20 January 2006 (UTC)

text: Great, looks like we're making progress. For the record, I did not state that the first sentence should be a definition, only that it should be put to more focused and productive use. However, if a thesaurus can lead us to a definitional sentence that satisfies all, so much the better. To read like a definition, the wording probably needs to begin "In mathematics, a function is … ." And the next word is … what? Jorge points out the correctness of "relation", but I agree with Jitse that a general audience will be unfamiliar with that as a technical term. Often relation and function are terms learned at almost the same time. Other options are "a pairing" or "an association that pairs". I confess that function is such a simple but deep and familiar concept that it is not easy for me to see it with a "beginner's mind". That's one of the things that makes basic articles challenging. Part of the problem may be trying to make the first sentence carry too great a burden, to be complete and correct for a technical audience yet easily understood by a general audience. Besides simply using more sentences, another way to relieve the burden is to display a variety of both examples and counterexamples. Some effort has already been made in this direction, and I would encourage more (especially counterexamples). --KSmrqT 05:19, 20 January 2006 (UTC)
thread_title: Comments

message: * JA: The proper language is either "relation on X and Y", or "relation over X, Y", or, if X = Y, then "binary relation on X" or "binary relation over X".  Jon Awbrey 05:01, 20 January 2006 (UTC)
replies: text:  Actually I have heard more often "relation between X and Y". (In fact I don't recall seeing the other two wordings.  I must be too old...) Anyway, the problem with those wordings is that "and" and "between" are sort of symmetric, i.e. "X and Y" is usually the same thing as "Y and X". So it may seem that any relation on X and Y is also a relation on Y and X.  The "from"/"to" wording may not be very common, but it is clearer, and --- important --- carries over to the traditional wording for functions (which are special cases of relations).For the same token, one should call X the domain and Y the codomain.Jorge Stolfi 05:36, 20 January 2006 (UTC)
replies: text: Adopting a certain nomenclature for a general concept does not exclude a more specific nomenclature for special cases of that concept.  So, when speaking of k-adic relations (even for k=2) we can use "the ith domain"; but that does not conflict with also useing "domain" (without index) and "codomain" for the first and second domains of a binary relation.In any case, the domain/codomain nomenclature is well-espablished for functions, and we must respect that — even though they too are special cases of k-adic relations.Jorge Stolfi 16:30, 20 January 2006 (UTC)
thread_title: Relations

message: I'm signing off, unwatching this page, and returning to my narrow haunts in set theory and logic.  Any of you are welcome to visit function (set theory) (which I will keep on my watch list) and (really!) take a look at the Britannica article.  Consider the difference between these two articles and the markets they address.  My professional survival requires me to leave this very interesting discussion; I must do other things :-) Randall Holmes 18:05, 20 January 2006 (UTC)
thread_title:  Good luck! 

message: * JA: There seems to be some misunderstanding about the meaning and the connotations of the word "pedagogy".  There is no hint or intent of condescension or paternalism in the word itself — it is simply a reference to the "gentler way" of learning and teaching that was one of the shining innovations of the Golden Age in Classical Greece.  It may serve understanding to look up the word "paideia" in a moderately good encyclopedia, say this one:  Paideia.  Jon Awbrey 20:08, 20 January 2006 (UTC)
thread_title: Paideia

message: :[Stolfi:] Rick, it seems that we still have totally incompatible ideas about what Wikipedia should be.  Let me paraphrase your position, as it sounds to me: "95% of the readers of this article will never understand what a mathematical function really is ayway. So let's write some sentence that they will understand, even if it is wrong. Then those readers may stop reading right there and go back to their video games, happy and gratified, convinced that they have learned some math; and thinking that Wikipedia is so much more wonderful than their Maths teacher at school, who keeps pestering them with those "set" things. As for those 5% of readers who have some idea of what a "set", and came to this article hoping to refresh their understanding of "function" — well, screw 'em; who cares about their needs (and their opinion)." 
No.  I never said anything of the kind.  I said the wind should be tailored to the shorn lamb.  At least 95% of the readers of the article can understand what a function is.  I would say 100%.  But not if a lot of mathematical jargon is thrown at them in the first sentence.  The opening paragraph I proposed is not wrong.  I would never suggest that Wikipedia should print anything that is wrong.  It just uses everyday language in place of mathematical jargon.  As for people who want more detailed mathematical information, they get what they want, too -- just a few paragraphs down.
"Has it occurred to you that the problems you have with your students were in fact caused by all their previous "Mathematics" teachers who fed them that sort of "intuitive" baby-food, insted of teaching them real mathematics?"
Of course it has occurred to me.  It is a problem I face every day.  But I also know there are two ways to fail to communicate.  One way to fail is to start at zero and never go any faster.  Another way to fail is to go from zero to sixty on the first day and never slow down.  The best way to communicate is to start where people actually are, and take them where they want to go.  It's called the learning curve.  Rick Norwood 20:19, 20 January 2006 (UTC)
thread_title:  Another deep breath 

message: I like B better than A. The first sentence of A is not so important that it has to be mentioned first, and the third sentence of A is also not that important. -- Jitse Niesen (talk) 22:24, 20 January 2006 (UTC)
replies: text: "Functions are often assigned names. In this case, the function could be named 'SQUARE' to indicate what it does to inputs, or just 'f' if one wants to keep the notation concise. This function takes a single number as input, but we do not know what value this input will be ahead of time, so we give it a variable name 'x' as a placeholder. Then we can write the function as SQUARE(x) or f(x), where the symbol "(x)" means that the function takes a single number 'x' as input. In the example above, we can then write f(1) = 1, f(2) = 4, f(3) = 9, and so on. This function f(x) can be more rigorously defined by specifying that the equation f(x) = x2 must be true for all inputs 'x'."
What do you think? - Gauge 23:10, 20 January 2006 (UTC)

text: Everyone was (and is) free to put anything on the list they want.  I, too, prefer consensus to a vote.  But we cannot spend an infinite amount of time wrangling over this.  Rick Norwood 17:00, 21 January 2006 (UTC)
replies: text: What I like about Rick's proposal is that it restricts us to the first paragraph and forces us to concentrate on the text instead of discussing abstract concepts. I don't like the voting part either, but if everybody puts up a proposal and we discuss pros and contras we might get somewhere. -- Jitse Niesen (talk) 17:29, 21 January 2006 (UTC)
thread_title: Comments

message: Instead of:
f\;:\;x{}\mapsto{}y means f(x) = y.
Try:
f\ :\ x\ \mapsto\ y means f (x)\ =\ y.
Jon Awbrey 07:20, 21 January 2006 (UTC)
replies: text: Yes, much better.  I'm still learning how to create wiki math text.  I'll use your example here as a model. Rick Norwood 17:01, 21 January 2006 (UTC)

text: f \colon x \mapsto y means f(x) = y.
(edit this section to see the difference in LaTeX).  LaTeX does all the spacing for you and it tends to look ugly if you force spaces with explicit commands.  Though sadly, Wikipedia renders the first formula for me as a PNG and the second in HTML; yours forces two PNGs but I think the spacing is clunky.  Not that every article isn't littered with inconsistently typeset math....Your mileage may vary. Ryan Reich 17:30, 21 January 2006 (UTC)
thread_title:  Suggestion 

message: ; \N \to \R
: list
; \R \to \R
: 2-dimensional curve
; \R^2 \to \R
: 3-dimensional surface
; \R \to \R^2
: parametrized curve in 2D
; \R^2 \to \R^3
: parametrized surface in 3D
; \R \to \R^3
: parametrized curve in 3D
could this be included in the article? --Joris Gillis 08:52, 21 January 2006 (UTC)
replies: text: As presented, this is full of subtle technical problems. For example, in applications most lists are finite; it is the graph of a function R→R which provides an example of one extremely limited kind of 2D curve; parametric surfaces in algebraic geometry commonly require the domain to include points at infinity; and so on. To be clear, I support the idea of accessible examples; these are just not careful enough. --KSmrqT 16:47, 21 January 2006 (UTC)
replies: text: They could be cleaned up to some extent, but list needs to be changed to sequence".  I don't think that they would be better than specific examples, though. Arthur Rubin | (talk) 04:54, 23 January 2006 (UTC)
thread_title:  Common types of functions by domain and codomain 

message: * Thou shalt not confuse a function with one of its graphs.
::However, many mathematicians define a function as being just a graph, and all the others define it as being a graph with a disclaimer/guarantee sticker pasted on it. And both agree that a function has one, and only one, graph.Jorge Stolfi 22:27, 22 January 2006 (UTC)
thread_title: Confoundation No. 4

message: Are "input" and "output" used by mathematicians?  I just checked Thomas's Calculus, the most widely used calculus book in the United States.  In a sidebar we see: Input (domain), Output (range).  
Just for fun, I looked in Calculus and Analytic Geometry by Thomas (2nd edition) 1958, and found several surprises.   But I also found this on page 15, "We may think of the function which tells what number y is to correspond to a given x as a machine into which we may put any number x (in its range) and the machine gives y as the output."
Now, as an aside, the surprises.  You may have noted above that Thomas calls the set of inputs the range of inputs (he notes that other authors use the word domain).  He also allows y = ±SQRT(x) to be a (multiple valued) function.  Clearly, both of these usages have changed in the intervening years.
In any case, input and output have been around for almost fifty years, at least, and are the words commonly used today.  They are not apt to go away just because some people don't like them. Rick Norwood 15:00, 24 January 2006 (UTC)
thread_title:  Input and output 

message: Well, it is obvious that my idea of what "mathematics" means is no longer current. I see that Jon went ahead and rewrote the article into something that presumably everybody finds great, but I simply cannot recognize as mathematics, much less good mathematics.  Sadly, I am unable to contribute within that framework.  It is not at all gratifying to spend several hours cleaning up an article --- making sure that all the information is correct, that the notation and terminology is defined in proper order, that the prose is as simple as possible, that secondary topics are consistently presented in the proper articles etc. etc. -- and then someone comes along, throws all your work away, and replaces it by something that cannot possibly be of any help to anybody -- whether they be laymen, good or bad students of any level, teachers, engineers, mathematicians, or advanced category theorists.Anyway, the good news is that 99.999% of wikipedians do not give a damn about functions (just see how many have joined this discussion), so this article won't do much harm either.I think I'd better spend my time on more fundamental topics like pyramid power and opossum poetry --- which are much more useful to mankind, and where Wikipedia's article quality standards seem to match mine.Fare well and good bye (for good this time)... Jorge Stolfi 16:13, 24 January 2006 (UTC)
replies: text: Sorry to hear that you are leaving. Consensus can be hard to achieve, especially on something so fundamental to the entire subject. I hope that whatever consensus we achieve will adequately address your concerns as well as ours. - Gauge 17:59, 24 January 2006 (UTC)
thread_title: Giving up

message: I can certainly understand the impatience of both Jorge and Jon.  We've spent a week, and a thousand words, on a single sentence.  I think Jon's reaction is probably more constructive than Jorge's.  We will see whether editing what Jon has written leads to a consensus and a better article, or yet another edit war. Rick Norwood 17:30, 24 January 2006 (UTC)
thread_title:  "edit boldly" 

message: I've made a few minor changes in the introduction, mainly to make it consistent.
I notice that sometimes when I use the "math" symbol set, I get larger type -- other times not.  Can someone tell me how to make f(x)=x^2 the same font as f(x) = \sqrt x.
Rick Norwood 21:08, 24 January 2006 (UTC)
replies: text: There are two answers to that.  First, you can't.  Second, you could typeset the first one as f(x)\,=\,x^2, which forces it to be rendered as a PNG.  Wikipedia renders TeX as a graphic when it contains some typesetting data that can't be done in HTML; usually picky spacing will force it.  Unfortunately, you can't unforce it if it's due to a particular gylph not being present in HTML. Ryan Reich 22:16, 24 January 2006 (UTC)
thread_title:  working on the new version 

message: Before the recent brouhaha, my suggestion that we move the history section to a position following the introduction had some support.  Is there an objection to that now?
The article seems to me to be chopped up into too many tiny pieces.  Any objection to grouping the sections under fewer subheads? Rick Norwood 21:08, 24 January 2006 (UTC)
replies: text: I usually prefer to have the definition immediately after the introduction. However, the consequence is often that I don't find a good place to put the history then but near the very end. I'm quite happy to have the history as Section One. -- Jitse Niesen (talk) 23:48, 24 January 2006 (UTC)

text: If you are referring to Woodstone's edit replacing "a function associates an 'input' with an 'output'" with "a function associates a unique 'output' with each of its 'input's", then I agree with you. If you are referring to using "x^2 = y" instead of "y = x^2", then I disagree. -- Jitse Niesen (talk) 23:48, 24 January 2006 (UTC)
thread_title:  Three suggestions for the further improvement of the article. 

message: First consider the domain {1} and the codomain {1} and the function f defined by f(1) = 1.
Next consider the domain {1} and the codomain {1,2} and the function g defined by g(1) = 1.
Is there consensus whether f = g ?
f is surjective while g is not surjective. Is 'surjective' a property of a function, or is 'surjective' a property of a function together with a codomain ?
I consider the following statement to be true.
:The function f, defined by f(x) = x2 where x is a real number, is surjective on the set of nonnegative real numbers, but not on the set of real numbers. 
(The answer to the above question, When are two functions equal ? , is of cause never, because if they are equal, then there are not two functions but only one. When f = g, then the cardinal number of the set {f,g} is =1. Oh dear, math is very different from plain language).
Bo Jacoby 10:26, 25 January 2006 (UTC)
replies: text: Dear Ben:  There is a disagreement between the two (different!) current formal definitions of "function" on this question.  Under the most common (and older) definition, a function is a set of ordered pairs no two of which have the same first coordinate, and y=f(x) means (x,y)=f.  Under this definition, two functions which have the same values are the same function, period.  Under this definition, the domain of f is uniquely determined by f, but the codomain is not (any superset of the range of f is a possible codomain).  Some mathematicians are uncomfortable about this, so they refine the definition of a function to be a triple (X,Y,G), where G is a subset of the cartesian product of X and Y such that each element of X occurs as a first coordinate of G exactly once.  Under this definition, each function has a uniquely determined codomain.  Under this definition, the function from the reals to the non-negative reals defined by y=x2 is not the same function as the function from the reals to the reals defined by y=x2.  The reasons for the latter definition are highly technical; I would suggest that if this issue confuses you, the best approach is to stick with the simpler definition (and I offer you as evidence that the new-fangled definition (which I don't especially like) really is more confusing than the old one!) So far as conflicting statements between articles go, if you read any of the articles relation (mathematics), binary relation, or function (mathematics), the editors of the articles are  mostly aware that there are two competing definitions, and try to qualify their statements when the differences between the two definitions make a substantive difference. Randall Holmes 13:43, 27 January 2006 (UTC)

text: I think that you mean to say that they fall under the old definition (under which functions with the same values are equal).  The new definition is the one adorned with domains and codomains.  But actually the theorems on unique definition of differential equations (and essentially all mathematical theorems) can be phrased in terms of either definition of function.  One simply has to be careful how one phrases them (in the usual theorems about differential equations, one makes sure that all the functions one is talking about have the same codomain, and everything works just fine). Randall Holmes 15:05, 27 January 2006 (UTC)

text: Something which may not be clear to naive observers is that the choice of what definition of function to use is essentially a matter of style; any matter of mathematical fact can be phrased in terms of either definition.  Please note that the precise meaning of the English word "function" is not a matter of mathematical fact:  it has some fuzziness (and any mathematical concept not completely formalized admits some fuzziness) which a formal definition must make precise.  Different mathematicians may choose to resolve the fuzziness in different ways.  This does not mean that the differences in formal definitions make no difference at all:  their technical details make it easier or harder to say particular things.  I claim that the original definition (which is still commonest) wins hands down in terms of simplicity and ease of use.  This seems not to be a fashionable position  nowadays, at least in some quarters. Randall Holmes 15:11, 27 January 2006 (UTC)
replies: text: On the talk page of relation (mathematics) you can see far more discussion of this exact issue (which applies to relations as well as functions) than you could possibly want to see, carried on with some of the fervor of a religious war  :-) Randall Holmes 15:14, 27 January 2006 (UTC)

text: re your practical suggestion, we can't do that in Wikipedia; we can only report actual mathematical usage, and both concepts are called "function":  they are competing formal implementations of the same semi-formal mathematical concept.  So both definitions must be reported. Randall Holmes 15:16, 27 January 2006 (UTC)
thread_title:  When are two functions equal ? 

message: I've moved the "history" section to a position just following the definitions.  There seemed to be no objection, in talk, but since this is a major change, I'm going to pause for a while before making any other changes. Rick Norwood 22:17, 25 January 2006 (UTC)
thread_title:  moving history 

message: The controversy over this page seems to have quieted down a bit.  I think the first few sections (through history) are now in very good shape.  We have a definition that can be understood by non-mathematicians.  We have a precise mathematical definition.  And we have a little bit about how the meaning of the word has changed over the years.
I think the next thing that needs to be done is to combine some of the very short sections that make up the rest of the article, with a heading such as "The vocabulary of functions" or "Properties of functions". Rick Norwood 15:48, 26 January 2006 (UTC)
thread_title:  How things stand. 

message: The opening now has all naming conventions for functions in simple forms which could be seen in algebra class:  y=x^2 (one variable depends on another); f(x) = x^2 (temporary name); Square(x) = x^2 (permanent name).  Randall Holmes 20:56, 26 January 2006 (UTC)
replies: text: A few edits ago we had an edit conflict and I overlooked a few of your tweaks, that got reversed as a consequence. Sorry. I think it's taking good shape now. −Woodstone 21:23, 26 January 2006 (UTC)

text: OK, I'm glad we're not having an edit war :-) Randall Holmes 22:04, 26 January 2006 (UTC)
replies: text: When I saw thirteen new edits this afternoon, I thought, "Oh, no, here we go again."  I am happy to say that is not the case.  Both of your edits look fine to me. Rick Norwood 22:32, 26 January 2006 (UTC)
thread_title:  The opening paragraph and naming conventions 

message: There's a problem with this header; I see that the hierarchy of subsections is right, but the end of this high-level section is not obvious to the reader (at least, not to me).  I don't know that this really adds much to the article.  Randall Holmes 22:08, 26 January 2006 (UTC)
replies: text: OK, on closer inspection I can see where it ends.  Never mind! Randall Holmes 22:09, 26 January 2006 (UTC)
thread_title:  Multidimensional functions? 

message: It occurs to me that I recall reading that Bolivia has two capitals.  From the sidebar:
::::::::La Paz is the seat of government; Sucre, the legal capital.
Perhaps we could come up with another example of a non-numeric function? Arthur Rubin | (talk) 01:55, 27 January 2006 (UTC)
replies: text: The Netherlands has likewise Amsterdam as capital and The Hague as seat of government. Nevertheless, there is still just one capital. No confusion. Let's not worry about those fine details. −Woodstone 08:11, 27 January 2006 (UTC)
thread_title:  Capital function 

message: The topmost picture, showing part of a graph of a function, is a quite bad example. Because the graph runs out of scale in both domain and range, it creates a lot of unclarity. It should be easy to modify to a function on a bounded domain (say [0,1]), which determines the limits of the x-scale and whose graphs stays in the y-scale. Who did the picture? −Woodstone 15:22, 27 January 2006 (UTC)
replies: text: Technically you are right, and one could for example use y = \sqrt{4-x^2}, but the first examples every student sees have infinite domain; I don't think this will make any difference to the article, and in fact, from a real (not ideal) pedagogical standpoint would probably create more confusion on the part of a reader naive enough to be confused. Randall Holmes 15:30, 27 January 2006 (UTC)
replies: text: I agree with Randall Holmes. Rick Norwood 19:59, 27 January 2006 (UTC)
thread_title:  Topmost picture is a faulty example 

message: More than a dozen changes already this morning, many of them restoring things (such as an unsatisfactory definition of codomain) that we have already discussed at length.  Please, please, discuss changes here before making them on the page, lest the page need to be locked again.  Rick Norwood 14:20, 28 January 2006 (UTC)
replies: text: I actually rather like the new intro (but I am noted for sprinkling my text with nested parenthetical remarks).  However, the math definition was trashed. Randall Holmes 14:58, 28 January 2006 (UTC)
thread_title:  Here we go again. 

message: on the same note, several of the changes to the mathematical definition were simply wrong.  {(x,y)} is not an abbreviation for a relation, nor is {(x,f(x))} an abbreviation for a function.  The second definition is not motivated by the paradoxes.  There is nothing "advanced" about the second definition, in the opinion of those who use the first (it is more complicated, though). Randall Holmes 14:45, 28 January 2006 (UTC)
replies: text: I must qualify the remarks about the math definition with the comment that I don't mind the changes to the intro; they appear to be reasonable, though one should be careful about adding too many qualifications to the text. Randall Holmes 15:01, 28 January 2006 (UTC)
thread_title:  the mathematical definition is not so easy to rewrite correctly 

message: With the two definitions in place, is this section really necessary.  I hesitate to delete it myself, but.... Arthur Rubin | (talk) 20:38, 28 January 2006 (UTC)
replies: text: Before I can offer an opinion on the section, I need to reread the entire article again, from beginning to end.  Maybe tomorrow I'll have some ideas. Rick Norwood 23:05, 28 January 2006 (UTC)

text: I'm beginning to understand why Oleg Alexanderov just reverts almost any undiscussed changes to the article mathematics.  Function (mathematics) seems to be an article which most mathematicans and not a few undergraduates who visit wiki feel the need to rewrite.  If there were one or two changes a day, there would be time to judiciously consider each, but when there are a dozen changes a day, every day, it becomes a problem.  I, for one, am going to get more hard nosed about reverting any change that introduces an error.  Randall and Arthur seem to have cleared up most of the outright errors introduced recently, but I'm still not happy with the "two definitions of an ordered pair" business. Rick Norwood 22:38, 28 January 2006 (UTC)
replies: text: where are there "two definitions of an ordered pair"?  We must talk about the two definitions of a binary relation, unfortunately; as we have been reminded, this makes a (more apparent than real) difference re certain statements of mathematical fact... Randall Holmes 02:11, 29 January 2006 (UTC)

text: I've made a few tiny tweeks to the introduction, each explained in the subject line.  Here I pause, so as not to go against my own advice: when an article has been around for a long time, and is in pretty good shape, make only a few changes at a time. Rick Norwood 23:05, 28 January 2006 (UTC)
thread_title:  '''Is a function more than it's graph''' revisited. 

message: There are such things. Randall Holmes 21:45, 29 January 2006 (UTC)
replies: text: Yes certainly, but it is confusing when all the examples of n-ary operations, as opposed to binary operations, are for n=2. Bo Jacoby 08:00, 30 January 2006 (UTC)
replies: text: I agree with Bo.  Binary operations makes a better heading.  You might mention n-ary operations in the last sentence, as an extension of the idea of binary operations. Rick Norwood 13:49, 30 January 2006 (UTC)
thread_title:  n-ary operations 

message: With the busy edits of December and January, the article has picked up quite a bit of repetition.  I'm going to try to cut out some of the repetition, and also combine the stacato sections to improve the flow. Rick Norwood 21:53, 2 February 2006 (UTC)
replies: text: Thanks for the edits Rick; looks good. Is there a reason why you use single occurrences of nbsp instead of simply using a space in your formulas? I don't think it makes things render differently, but I could be wrong. I would prefer to have spaces because the nbsp's tend to clutter things up unnecessarily. - Gauge 23:48, 2 February 2006 (UTC)

text: The entity named nbsp is a non-breaking space, so a formula or equation will not have an awkward line break appear in its midst. An alternative is to paste in a UTF-8 unicode character like thinsp, which should appear as whitespace in the edit window, and (since it is not the "space" character) also prevent line breaking: a  =  b. Here's a list of sample spacing options: ensp (" "), emsp (" "), emsp13 (" "), emsp14 (" "), numsp (" "), puncsp (" "), thinsp (" "), VeryThinSpace (" "). --KSmrqT 06:35, 3 February 2006 (UTC)

text: If you are seeing little boxes instead of white space, then the characters are not displaying properly because of font or browser or configuration problems on your end. This potential problem is one reason non-breaking space is more common, because it displays properly in all known browsers. --KSmrqT 15:12, 3 February 2006 (UTC)
thread_title:  Eliminating repetition 

message: "A function need not involve numbers. An example of a function that does not use numbers is the function that assigns to each nation its capital. In this case Capital(France) = Paris."
I'm not a mathematician, but I question the validity of this. In the example given, the fucntion, Capital(x) = y does not yield exactly one output per input. Capital(France) = Vichy and Capital(France) = Aachen are arguably valid outputs for the input "France." Generally a nation has only one capital (at a time). However it's misleading to equate the mathematical certainty of a fuction (exactly one output per input) with a historical rule-of-thumb (unless a war has disrupted the normal political process, a nation, during modern times, shall usually have only one capital). Perhaps someone could come up with a better example of a Non-Numeric Function? Thanks! 66.17.118.207 15:11, 26 June 2006 (UTC)
replies: text: Absolute non-ambiguity is impossible.  Even f(x) = x + 5, which seems non-ambiguous, would admit a question whether the addition took place in the natural number system, in the real number system, or in the integers modulo 7.  I suspect that any example of a non-numerical function would suffer from similar problems.  One has to use common sense. Rick Norwood 15:37, 26 June 2006 (UTC)

text: Are you serious? First, this is a non-mathematical example. Second, France has only one capital now. Third, if some country happens to have simultaneous multiple capitals we can simply allow the function to return a set. Fourth, if we really need to accommodate different facts at different times we have tools such as modal logic. Finally, are you serious?! --KSmrqT 23:49, 26 June 2006 (UTC)
thread_title: Non-Numeric Functions