Talk:Function (mathematics)/Archive 2

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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 Y^X notation is in the wrong register - too advanced. Charles Matthews 20:34, 6 Jul 2004 (UTC)

Even and Odd functions

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)

See Even and odd functions. I added the link. Donar Reiskoffer 06:28, 27 Aug 2004 (UTC)

Great. Thanks a lot, man. --pie4all88 20:53, 27 Aug 2004 (UTC)

Category

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)

Old discussion from the talk-page of "function" (Talk:function)

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)

Computer Science vs. Math

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)

A function which takes multiple parameters is perfectly OK, it's just mapping from some n-dimensional space (each parameter is a dimension). You don't need to get complicated. f(x,y) is simply a function from R^2 or whatever.

we all agree. — MFH: Talk 14:32, 20 May 2005 (UTC)

f(ANYTHING) = 0 is a function. It satisfies the above definition. It is a counter example to every definition on the page. I deleted two examples that contradict the items on the page, e.g. determinism. Lemonade sales CORRELATE with temperature, but, since they are not deterministic, are not a FUNCTION of the weather.

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)

There are lots of different special properties functions can have that are useful to know about, probably the most important (Mathematically) are homomorphisms, isomorphisms, one-to-one, and onto functions. None of these are mentioned. Most of the functions Computer Scientists are interested in are down the boring pathological end of things (i.e. they lose information).

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)

functions that are not functions

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)

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)

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)

ambiguous function

Is this terminology widespread enough for the name of a heading? I think it would be better to name it relations. See [1]. --MarSch 1 July 2005 09:50 (UTC)

seesubarticle

I don't like this "seesubarticle" terminology. It makes it look as if some articles are subordinate to some other articles. That is not the case on Wikipedia. All the articles are independent of each other, and there are only loose connetions between them via links. There is no bigger superstructure, and neither are there often encountered articles considered part of a series.

As such, the subarticle terminology is inappropriate. Of course there are exceptions, sometimes it is convenient to view a bunch of articles as part of a bigger structure, but that is seldom and needs to be discussed on a case-by-case basis. So, one should not use liberarly the subarticle terminology in just about anything. Oleg Alexandrov 1 July 2005 15:38 (UTC)

Please take this to template talk:seesubarticle --MarSch 5 July 2005 12:46 (UTC)

Definition of total?

The definition of a total function given in the formal definition, is different from the article on total relations (http://en.wikipedia.org/wiki/Total_relation).

The definition used here does not require coverage of every element of the range; however, the definition from the article on total relations does.

As far as I know, the latter is correct. I do not know how the requirement to cover every element of the domain only is called.

• (edit) I seem to recall that that requirement is called 'functional'.
• (edit) This also implies that this article should be extended with a note on total functions. (example of the difference: f(x) = x² is not total; f(x) = x is)

Hugo, Tue Jul 5 12:26:23 CEST 2005

gaussian function

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)

ok i figured it has Erf(x) in the primitive. but i don't know why differentials doesn't work on them, like if you replace -x^2 in the exponent of e and then replace it by u, then du=-2xdx and you have to put a (-2x)^-1 to balance the integral, but when you try to express x in terms of u so you can convert the (-2x)^-1 to u it surely is a complex function. why?

-protecter

More on definitions

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)

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)

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)

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)

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)

EPSILON

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)

Also, the "is an element of" symbol is not the Greek letter epsilon. Rick Norwood 23:50, 22 December 2005 (UTC)

Comment

Explanations of range and codomain are weak and vague. The range of a function is the set of all possible output values, not "actual." The codomain is NOT the set of all possible output values. --anon

new beginning

The old introduction did not aim at any consistent level -- some parts were oversimplified, others too advanced. I have taught functions to students with widely differing abilities for many years, and so I wrote about what they seem to need to know. Functions as sets of ordered pairs naturally comes later. Rick Norwood 00:35, 23 December 2005 (UTC)

I read what you wrote and looks good to me. One remark: you need to make the variables italic. Oleg Alexandrov (talk) 02:44, 23 December 2005 (UTC)

Well I think Rick has done some good things here, but I'm worried that now the article is too didactic. It reads more like a highschool text book than an encyclopedia article. Paul August ☎ 20:54, 23 December 2005 (UTC)

You make an interesting point, Paul, and I think I agree with it. I'll try to avoid "textbookese" in the future. Rick Norwood 00:16, 31 December 2005 (UTC)

I actually read more carefully through the article now. Perfectly agree with Paul. The article reads bad, like a silly dialog between teacher and student, not appropriate even for a book. So, Paul, I guess Rick wants to say above that the article is all yours for hacking, if you are inclined so (which you may, as you worked on it before and know it better than others). I may join in too. Oleg Alexandrov (talk) 00:59, 31 December 2005 (UTC)

I agree with Paul. The current intro paragraph made my eyes glaze over; I liked the old intro paragraph much better (however faulty it may have been). The house style for encyclopedia articles is to have the first paragraph of the article summarize, in 3-6 sentences, what the rest of the article will be about. The first major section of the article could very well be written as "functions for simpletons", but the simple language should be limited to such a section. The later sections should be written in a style and sophistication appropriate for the topic covered in that section. linas 06:09, 31 December 2005 (UTC)

Paul says the intro was too didactic and Linas says it makes his eyes glaze over. I've rewritten it, trying to make it more user friendly.

Reading over the article, the thing that stood out was that there were two history sections, two sections on language and notation, and two sections on functions of several variables. I moved the last pair so they are next to each other -- they still need to be combined. I gave a very brief introduction to language and notation, leaving all technical matters fro a later section, and combined and shortened the two history sections -- the older history section has been around a long time and seems to me very good.

Here I pause for comments and corrections. A lot more still needs to be done. Rick Norwood 15:38, 9 January 2006 (UTC)

Missing topic

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)

Why don't you add it here? It is certainly a useful notation. Rick Norwood 17:21, 1 January 2006 (UTC)

I assumed that Linas was going to improve my rewrite, but a recent message from Oleg Alexandrov suggests, if I understand it correctly, that I should do that myself. I'll give it a try. Rick Norwood 14:52, 9 January 2006 (UTC)

Thanks to Oleg

Thank you for improving my rewrite. Rick Norwood 15:30, 10 January 2006 (UTC)

I am happy you were able to read through my grumpy edit summary. :) Oleg Alexandrov (talk) 16:19, 10 January 2006 (UTC)

Jorge Stolfi edit

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)

Sadly, I saw no way to avoid a revert, because the changes made today essentially restructure the article in a form appropriate for a college textbook, removing anything accessable to the layman as "silly". Let me explain the essential point, here. Mathematicians already know the definition of function, and are unlikely to turn to Wikipedia for help in understanding that definition. The people who are likely to turn to Wikipedia are people who are interested in mathematics, or who are learning mathematics, but who do not have a deep understanding of mathematics already. This article and other fundamental math articles in Wiki should be aimed at them, rather than at professional mathematicians. Rick Norwood 15:54, 16 January 2006 (UTC)

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)

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)

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)

I'm happy that you want to contribute -- we need all the help we can get. My only concern was that you wrote an introduction that only a mathematician could read, and then went on to change the order of the subsections in such a way as to make an edit more moderate than reversion difficult.

You ask that you be allowed to go first, so I will wait until tomorrow before making any further changes. (I can't speak for others.)

The E=mc2 function is one of the few functions that most people "know" in the sense that they recognize it as "science". They don't know what it means, but if we explain that this is a function, it gives them an example of the one scientific function they have already heard of. I think a familiar example really does help. There has been some solid research recently that suggests that the concept of a function is where most math students get lost -- it is one level of abstraction too many.

Yes, it is important to distinguish between the use of the word function in math and in computer science. But most people are on a much lower level than that. Most of my students don't understand multiplication! So keep in mind that, to the average reader, math and science are more freightening than Dracula and Frankenstein. Rick Norwood 20:47, 16 January 2006 (UTC)

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)

I've waited a day, as I offered to do, but the article still starts out aimed at the -- what, one percent? five percent? -- the people who already know how to read mathematics, all of whom already know everything in the article. So, I'm going to put back the material aimed at the layperson. The idea is to provide a gradual learning curve.

I recall a professor in graduate school, lo these many years ago, telling me that when I gave a talk, the entire audience should understand the first 10% of the talk and 10% of the audience (at least) should understand the entire talk. I think that should be a minimal standard for wiki articles.

And of course E = mc2 is a function. Input the mass, output the energy. I'm not sure what you mean when you say that it can be many different functions, unless you want to say that c does not have to be the speed of light. Rick Norwood 19:24, 17 January 2006 (UTC)

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)

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)

Restore user friendly introduction

I've made the introduction readable by a non-mathematician, and done a little work on the next section. I'm going to pause now for comments and corrections.

I do think the history section should immediately follow the introduction. Rick Norwood 20:44, 17 January 2006 (UTC)

Rick, sorry, but I cannot agree at all with the current style of the article. First, the most important thing one should teach to *any* audience about mathematics is that mathematics is all about precise definitions and reasoning, and not, not, not fuzzy thinking --- just the opposite of it. If a student who doesn't know that, he/she does not know what mathematics is. In particular, if the reader wants to know what a function is in mathematics, he/she must given the mathematical definition, not something else that is not mathematics. The article as it stands should better be called "function (non-mathematics)".
Second, the definition given in the current head parag is not just fuzzy, but totally wrong: a function in mathematics is absolutely not "a process to compute a value"; that is what an algorithm is (and a formula is a special case of algorithm). Functions and algorithms are very distinct concepts; for instance, two very different algorithms may compute the same function, most functions do not have any algorithm, algorithms can only work on discrete values, and so on.
Imagine an encyclopedia article titled "United States (geography)" that beings saying "The United States is the most massively wonderful kingdom in the World. Briefly put, it is the hotter half of North America. For example, Texas, Disneyland, and Mexico are in the United States". Maybe 90% of the fifth graders may understand this definition, whereas only 10% may understand the correct one. Well, sorry, but that is not a reason to teach them the wrong thing.
All the best, Jorge Stolfi 21:10, 17 January 2006 (UTC)

The flaw in your analogy is that people do not, in general, have difficulty reading geography. They do have a great deal of difficulty reading mathematics.

I did not say that a function was computational. I said that the output was determined by the input. I did not say that functions were "wonderful". I said that the concept was important. A layperson has no way of knowing which mathematical concepts are important and which are unimportant.

I cut you a lot of slack, here, and let you rewrite to your heart's content. But I still maintain that mathematicians already know the definition of a function. A function is a set of ordered pairs with the property that if (a,b) and (a,c) are both in the set then b=c. This is Greek to non-mathematicians. If you think you can do better than I did with an explanation of function that a layperson can understand, please try.

By the way, I always read the talk page before editing an article. Rick Norwood 23:54, 17 January 2006 (UTC)

Let me try again. The first problem with the current head section is that it is not a definition of function; it is a motivation, something that one tells students before giving them the real definition — to get them interested, to help them understand, and so on. Now, starting with the motivation is fine for a textbook, or a tutorial, or the homepage of a course — but not for an encyclopedia article. Here you are supposed to start with the defintion. Motivation and value statements, if needed at all, should take second place.
The concept of function in mathematics depends on the concepts of "set" and "ordered pair". Any "definition" that does not use those concepts is not a a correct definiton; it will mislead the reader, by defining something that is not a mathematical function. Any reader who does not know those concepts is not going to understand what a mathematical function is, no matter how nice was the motivation. It makes no sense to give a wrong definition for the sake of those readers who are not able to understand the correct one.
Sure it is important to make the definition as plain as possible. Perhaps we should break it into three separate sentences instead of one; or say "if" instead of "if and only if"; or write "one, and only one," instead of "exactly one". We may skip some details by saying just "relation" and pushing the details to a separate relation (mathematics) article. But the first paragraph must say somehow that a function is a subset of the Cartesian product of two sets, because that is what "function" means in mathematics.
There are other serious problems with the current version. For instance, the words "input" and "output" are not used in mathematics (they are computer science jargon). Again, they may be great for a motivation talk, but an encyclopedia article should use standard terminology, not invent its own.
Also, a mathematical function cannot "take the temperature and output the density", because those are physical concepts. A physicist may use a function to model the way density changes with temperature; but its inputs and outputs are real numbers, not the physical quantities. How one goes from the latter to the former is a fat chapter in any physics textbook. This may seem a trivial detail to you, but in fact a reader who does not understand this point does not understand what a mathematical function is.
Also, the codomain is definitely not "the set of all output values"; that is the definition of range. And so on.
As for the reader's difficulty: the only way students will learn to understand mathematical language is by reading mathematical language. How can we expect them to learn mathematics, if we avoid using mathematics even when writing about mathematics???
All the best, Jorge Stolfi 09:06, 18 January 2006 (UTC)

"Now, starting with the motivation is fine for a textbook, or a tutorial, or the homepage of a course — but not for an encyclopedia article. Here you are supposed to start with the defintion." I disagree. As far as I know, even traditional encyclopedias provide something in the way of motivation for material that is difficult to understand to most people. The set-theoretic definition is only useful to laypeople once they have an intuitive grasp of the function concept. I prefer section headers: "Motivation" followed by "Formal definition", which I feel provides the best of both worlds and makes it clear that the motivation is something distinct from the formal definition. The motivation should be the launch platform that starts intuitive and slowly increases the abstraction to the point where newcomers can understand why the formal definition is what it is. - Gauge 16:38, 18 January 2006 (UTC)

Four more problems with the current version:

• It fails to say that a function is a special case of relation. This is essential, since many important concepts like composition and inverse are better defined for relations than for functions.
• The example "y = x^2" has the same problem of "E = mc^2". Namely, "y = x^2" is an equation, "x^2" is a formula; neither of them is a function.
• Computer scientists use mathematical functions to model data structures and the effect the algorithms; they also use the word function to mean subroutine. But these are two completely distinct senses of function, even to computer scientists. The current revision of the text makes it seem that computer scientists have their own definition of function, that covers both senses; which is not the case at all.
• The paragraph "functions may be treated abstractly" is almost nonsensical, and replaced a very important statement that most mathematical functions can be neither defined nor computed, and many can be precisely defined but not computed.

Jorge Stolfi 10:13, 18 January 2006 (UTC)

I agree that it is important to distinguish between the computer science definition and the mathematical definition - this is something belonging most appropriately to the "Formal definition" section where the reader will at least have enough background to appreciate the distinction. - Gauge 16:38, 18 January 2006 (UTC)

I've done another rewrite of the intro, always striving for greater clarity, and done the same for the next few sections. Again, I will pause for comments.

Any thoughts about the position of the history section? It seems to me of more general interest than the technical details in the rest of the article. Rick Norwood 01:36, 18 January 2006 (UTC)

Note: it is better to respond at the bottom of the page. That is where people look first.

• The problem with defining a function as a special case of a relation is that anyone who knows what a relation is also knows what a function is. Address the layman first, the mathematician later. Logically, relations precede functions, but on the learning curve, functions come first.

• Your objection that y = x² is not a function is similar to the objection that "2" is not a number but rather is a numeral that represents a number. It is correct, but too technical, and excessive concentration on technicalities renders prose unreadable.

• I'm a mathematician, not a computer scientist. If you have expertise in that area, please fix the paragraph. My changes were aimed at cleaning up the prose. I did not intend to change the meaning.

• I mentioned earlier a study about why so many people find mathematics so difficult. I wish I remembered where I read it -- one of the AMS journals, I think. In any case, most people seem to be able to handle specific functions, such as y = sin(x), which they view as tools for getting from x to y, but are unable to handle the idea of a function as a mathematical object in its own right. Certainly, there is a difference between specific functions and abstract functions. If I say, "Let f be a function from X to Y,..." most people's eyeballs roll up into their head. So, while I agree that this is an important topic, we need to be very careful how we write about it, if we want to be understood (by anyone who doesn't know about it already). I don't think my attempt is perfect. Please feel free to try to improve it. Rick Norwood 13:55, 18 January 2006 (UTC)

Request for a third viewpoint

Jorge Stolfi and I disagree on the level of mathematical sophistication appropriate for the introduction in this and other mathematical articles in Wikipedia. My view is that the introduction, at least, should be comprehensible to the non-mathematican.

My version begins:

The concept of a function is one of the most important in modern mathematics. Functions have inputs and outputs, and the defining property of a function is that a given input always produces the same output.

An example of a function is y = x², the squaring function. Given the input 3, the output will always be 9. The square root is not a function, however, because a positive number has two square roots. The square roots of 9 are 3 and − 3. To make the square root a function we need to choose just one of the two square roots, usually the non-negative square root.

His version begins:

In mathematics, a function is a relation from a set X to a set Y such that every element of X is paired with exactly one element of Y.

An example is the squaring function sqr, from the set Z of all integers to the set N of all natural numbers, that pairs each integer x with its square x². Another example is the function from the real numbers R to R , that associates to each x in the number x³ - 9x.

I have attempted several different compromises, all of which he finds unacceptable, so I think we need an outside opinion. Rick Norwood 14:14, 18 January 2006 (UTC)

Rick, I think Jorge, makes some good points. I'm particularly concerned that a formal definition is now nowhere to be found! I think, if we are creative enough (more than one article?), we should be able to accommodate the needs a broad range of readers, from high school students to mathematicians. I would suggest trying to work things out with actual text here on the talk page — line-by-line, if need be — rather than keep making dramatic changes to the article. I'm going away for a few days, so I won't have time to say much more until I get back on the 22nd, but I will leave a message at Wikipedia talk:WikiProject Mathematics, asking that other editors get involved. I will check in when I get back. Paul August ☎ 15:11, 18 January 2006 (UTC)

Above, Jorge says "The first problem with the current head section is that it is not a definition of function; it is a motivation, something that one tells students before giving them the real definition — to get them interested, to help them understand, and so on. Now, starting with the motivation is fine for a textbook, or a tutorial, or the homepage of a course — but not for an encyclopedia article." This does not seem to be the general opinion as recorded in Wikipedia:Manual of Style (mathematics).

Secondly, the lead is quite long at the moment (about five paragraphs in the versions I looked at). Since the definition of a function is only one line, I think it should be possible to stick it in the lead section. So, I would try to combine the first paragraphs of the two versions listed above and write something like:

"The concept of a function is one of the most important in modern mathematics. Functions have inputs and outputs, and the defining property of a function is that a given input always produces the same output. More precisely, a function defined to be a relation from a set X (of inputs) to a set Y (of outputs) such that every element of X is paired with exactly one element of Y."

Finally, I want to parrot Paul: Go through it line-by-line and do not make dramatic changes. I know, it's a laborious process, but big changes are likely to be reverted as a whole. -- Jitse Niesen (talk) 16:01, 18 January 2006 (UTC)

Agree with Jitse. Start with some motivation "each input gives unique output", then say the rigurous definition ("pairs each element in X with an element in Y"). Many people view Wikipedia not just as a reference, but a place to learn new things. It is very important to give motivation and intuitive explanations. Oleg Alexandrov (talk) 16:20, 18 January 2006 (UTC)

See my suggestion above for separate "Motivation" and "Formal definition" sections. The "Lead" should be understandable to laypersons who want a vague idea of what a function is. Starting the article with the formal definition as Jorge has insisted is absolutely a bad idea. That being said, it is very important that this article does provide a formal definition along with all its nuances, and this is what the "Formal definition" section should be for. The lead suggested by Jitse above is an acceptable compromise that at least suggests that a more precise definition awaits in the article. Finally, while I welcome Jorge's contributions, I feel that they were initially misplaced and much too bold for an article this fundamental to the entire subject. The line-by-line approach is a very good suggestion. - Gauge 16:38, 18 January 2006 (UTC)

WOFWYW (watch out for what you wish)

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)

JA: By way of pre-ramble, let me say where I have my stake in this. I've been working to coordinate the various articles and thus to mediate the sundry POV's on what has historically been called every name from the 'logic of relatives' to 'relational algebra' to the 'theory of relations', just to name a few. Before I could get much of a start on this, I found that I had to rework the 'relation (mathematics)' article in a way that would be both accessible, general, rigorous, and useful. That is still a work in progress, but there's no need to rush. Since functions are but special cases of binary relations, I have to be concerned with how they are treated. In connection with all that, part of the dispute about the 'function (mathematics)' article has to do with negotiating the various levels of treatment in the novice-expert shift, as we used to call it, but part of it is really the more difficult difference in POV's that arises between "Mat" (eponym of the mathematical attitude) and "Phil" (eponym of the philosophical attitude). And that's a big part of what we always wrestle with in trying to craft pedagogical and popular introductions to mathematics proper. To make a long story break off for now, do you think that there should be 1 article on the topic, that gently leads from casual and common sense notions to a rigorous treatment, or do you think that there should be 2, 3, ... articles that take up the subject at different levels and merely refer to each other as appropriate? A prime consideration is that we not misinform or mislead in the process of making the needed approximations and simplifications. Jon Awbrey 16:30, 18 January 2006 (UTC)

what is a function?

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)

Definitions of function, extensional & intensional

• 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)

• 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)

• JA: Randall, I don't use the word 'regretably' with irony. This is an important issue, and I hope that you will save it to the talk page for further discussion, but it does not go in the initial invitation to the topic. You find this article already in hot dispute about the most elementary questions, even of what level to pitch it at, and I'm afraid that some of that dust will need to settle before anything more tricky can be tackled. Jon Awbrey 17:58, 18 January 2006 (UTC)

• 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)

• JA: Been there, read that, taught that. Please read some of the talk page to get a sense of where the present TWIC's are at. Jon Awbrey 18:08, 18 January 2006 (UTC)

Subjectivity is unavoidable?

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)

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)

Dear Jorge, cheer up, it will all work out in time.
But for the Time Being — Not Yet — you can read my
forthcoming article on the Theory of Sighs — maybe
Sighs Matter? — even the title is a Work In Egress.
Exorcise for Readers: Define "subjective function".
Jon Awbrey 20:20, 18 January 2006 (UTC) !?

JA: PS. The piece about codomain and range will be changed, and right up front, where it belongs, just as soon as I can get back to it. I have written up tutorials on this several times before in other settings, and will eventually find those files. Jon Awbrey 21:02, 18 January 2006 (UTC)

Functions & Relations

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)

Thanks, Jon, for fixing my capitalization error, and adding the adjective "binary". I intended my reference to Jorge to be friendly -- evidently it didn't read that way. I never doubted Jorge's expertise, nor his good intentions.

I was taught <a,b> for ordered pairs. You changed one of those to the form (a,b). I've seen both, so I assume it is a matter of preference. But we should settle on one or the other and stick to it, at least throughout this article -- even better if we could establish a wiki style for ordered pairs. Either form is fine with me, as long as we are consistent.

Is there a consensus about where the history of the word "function" should go? I suggest right after the intro. Rick Norwood 21:31, 18 January 2006 (UTC)

• JA: I've spent most of my e-life happy as a lark in Asciiland, so I like <a, b>, too, but I've been told since I got here that that's "deprecated", due to HTML parsers. There's "langle" and "rangle", but I was too lazy for that, and the angle bracken in my browser window look too big -- more like generator brackets. Probably somebody else will know what's undefecated hereabouts. I'm fond of historical approaches, but aside from a liberal sprinkling of etymologies -- since I don't know yet what the community wants this article to be -- I don't know yet if maybe it shouldn't be saved for a more dissertationoid article. "On jugere", as Galois said. Jon Awbrey 22:10, 18 January 2006 (UTC)

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)

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)

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)

• JA: Again, I have barely skimmed the 'binary relation' article so far. I did put two days worth of BST&T into the 'relation (mathematics)' article, if you want a hint as to where I think the next base camp might be, by no means the summit. I think that chunking the scary def into a couple of "intellectual quanta" (IQ's), there called the "frame" and the "graph", may help to tack the slope (against the wind?) a bit. But we'll see. I really wanted to call them the "ground" and the "figure", respectively, but that would have been too radical an innovation, plus transposing the F and the G. As far as math folk go, it will be Cat.Work.Math all the way up. Maybe not today, maybe not tomorrow, ... Jon Awbrey 23:04, 18 January 2006 (UTC)

Perspective from a set theorist

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)

I never heard the word "codomain" until after I got my Ph.D., but it seems to be taking over. You can't fight progress. The same is true of "input" and "output", but that's the computer science influence, and while they grate on my ears, I prefer them to "abcissa" and "ordinate". I think all three words are here to stay. And if I can stand the Bourbaki definition of a ring, I can stand anything. Rick Norwood 00:15, 19 January 2006 (UTC)

I like Randall's most recent edit -- it improves on mine. Except -- do we really need to get into projections? Rick Norwood 00:30, 19 January 2006 (UTC)

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)

Line by line

(Sorry, Jitse and I had an edit conflict here; trying to restore his edits Jorge Stolfi 15:09, 19 January 2006 (UTC))

Jitse's comments

It seems that Jorge had to break of, so let me continue. Current version, more or less line by line, reads:

1. The concept of a function is fundamental to modern mathematics.
2. A function associates an input and an output.
3. A familiar example of a function is [linebreak] $f(x)=x^{2}$, [linebreak] which, given the input 3, produces the output 9.
4. 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".)
5. An essential property of a function is that for each input there must be one unique output.
6. 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).
7. 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.
8. 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)

Stolfi's comments

Well, let me give it one more try. Let's look at the current version (as of 11:50, 19 January 2006 (UTC)), line by line:

The concept of a function is fundamental to modern mathematics.

This is a zero-content sentence. What purpose does it serve? If it weren't there, would the reader miss it?

A function associates an input and an output.

The words input and output sound friendly and "non-mathematical", but in fact they are used here in a very specialized sense. In common parlance, input has two senses: something that you put in ("the input is oil, the output is gasoline") and a channel through which you put things in ("this pipe is the input"). If we use the first meaning, this sentence says that a function is a single pair. If we use the second meaning, it says that a function is something that connects two pipe-like things, like a water heater or a unix filter. By this sense, one could have say, a "squaring" function, wher you put 3 in and get 9 out, then put 5 in and get 25 out; and also a "summing" function where you drop 3 5 6 1 in, type CTRL-D, and get 15 out. Either way, sentence may seem to be readable, but in fact it is more likely to take the reader astray than to help him understand what a mathematical function is.

A familiar example of a function is

Familiar to whom?

f(x) = x²

There are at least 7 (seven) big problems with this single formula.

1. The name f appears from nowhere. To us it is obvious that f is being defined right here, as an arbitrary and temporary name for the function that is about to be specified. But this "silent attribution of a variable name by an implicit universal quantifier" is a stylistic device that is very specific to mathematical jargon. Students take a couple of years before they can understand the nuances of these implicit definitions, their scope, etc. A reader who does not know what a "set" is will probably be lost here; he would not guess the missing sentences "We are going to describe a function. Let's call it f, an arbitrary name, for the time being." Will he perhaps think that this f is the same f that appears in the figure caption?
2. Anyway, there is no point in giving the function a temporary name which is not used anywhere. (Ditto for the figure caption, by the way.)
3. On the other hand, this function must be given a permanent name like sqr, because it is used later in the article's body.
4. The x also appears from nowhere. Again, this is another instance of implicit variable naming. How is the reader supposed to guess the missing sentence "this function takes a number, lets call it x"
5. Nowhere it is said that x is a number, much less what type of number. So the important concept of "domain" (and the fact that one must know the domain in order to make sense of the formula) is completely ignored here.
6. The notation f(x) is explained only two or three screenfuls later. If we assume that the reader knows this notation, then that explanation is unnecessary; if we assume that he does not, then we cannot use it here. Moreover, readers who have seen this notoation only in programming languages or spreadsheets may read it as "call the procedure f, substituting the current value of x for the parameter" --- i.e. a procedure call and not a function definition. To pre-college kids, it may mean "multiply f by x" (since parens are just for grouping things, arent't they?)
7. The "=" symbol here too is a very specialized math jargon. To C programmers, this means "assign x² to f(x)". To Pascal programmers, it means "evaluate f(x), evaluate x², then return TRUE iff the results are equal". To a pre-college kid, it may mean the same thing as "x² − f(x) = 0, i.e. an equation to be solved. Only the reader who has had some exposure to mathematician's jargon will know that, here, the symbol "=" means "returns".

That should be enough for this line, so there is no need to discuss what x² could mean to a reader who does not know what "set" means. But, just in case, we should put a footnote numbered "²" at the bottom of the article, explaining that x² is math motation for "the square of x", and "square" is math jargon for "a number multiplied by itself" (except in geometry classes, of course, where it means "a kind of shape with four corners, that is neither too fat nor too tall".)

which, given the input 3, produces the output 9.

One shouls insert "for instance" somewhere, just in case.

The set of all inputs is called the domain.

Oops, there goes the dreaded word "set"! And, pray, what does "all inputs" mean for the squaring function example above?

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".)

This tells the reader that "range" and "codomain" are usually the same thing. They are not. The word "codomain" was coined by people who define functions as triplets (domain,codomain,graph), precisely in order to distinguish that set from the range. For people who use the older "naked graph" definition, a function has only a domain and a range, both determined implicitly by the graph, and there is no "codomain". Now, the main purpose of an encyclopedia should be to clear up such confusions, not create them.

An essential property of a function is that for each input there must be one unique output.

Apart from the oxymoron "one unique", this again ignores the fact that a function has a domain, and it is not required (nor allowed) to produce an output for for any "input" outside that domain. Also, note that this sentence cound be interpreted as saying that all inputs must produce the same output, i.e. the function is constant. Note that we can read the sentence in the "correct" way only because we already know what it is trying (but failing!) to say.

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).

This sentence starts out by saying the square root, then says that there are two square roots then again we are supposed to make the square root into a function, by specifying which square root. Please note that the sentence is mixing up three different senses of the phrase "square root":

1. any number whose square is a given number: "the square roots of 9 are 3 and -3"
2. the mathematical relation from Z to Z (or is it R to R?) that associates each number to all its square roots in the sense (1): "the square root is not a function". Note that this sense is mathematicians jargon, not common usage.
3. a function that we are going to build from sense (2): "to make the square root a function". Note that this usage belong more in computer programming than in math, where one does not normally "make" something "into" something else.

Also, it is not clear how we are supposed "specify which square root". Does it mean that the function returns both roots, and the user picks one? Or that the function flips a coin before outputting the result? And what is the "square root" function supposed to do with negative numbers, or integers that are not squares?

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.

This could be a good example if names and initials were well-defined mathematical concepts. We mathematicians know that those concepts can be made precise (strings on a finite alphabet and all that). But I am afraid that the non-math reader may get the idea that a function is a fuzzy concept that can take fuzzy things like real names, or physical things like people, and output other fuzzy or physical things. In that case, this article will be teaching the wrong thing; it will have negative educational value, it will be worse than no article at all. And we need not even go into the technical difficulties of the example: is f(William Der-Tsai von Zuben III) = "WmD-TZIII" or "WDTvZI" or...

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.

"Determined by the association" is no better than "associates". "Unique element" has the same abiguity noted above (same for all inputs?). Note also that the words "domain" and "codomain" are not used in the head section, so there is no need to define them here.

Well, enough. To restate my point: if we want to explain a mathematical concept, we must give a definition that is mathematically correct, at least within naive set theory. Informal pseudo-definitions may seem easier to understand, but the use of more familiar words does not make the concept clearer. The concept of function involves sets and Cartesian product; a reader who does not know those things will not understand what a function is. On the contrary, when we replace the real thing by some concept that is only vaguely similar to it, we are taking the reader away from the goal.

I was told to look at Wikipedia:Manual of Style (mathematics). Here is their example of a good head paragraph:

In topology and related branches of mathematics, a continuous function is, loosely speaking, a function from one topological space to another which preserves open sets. Originally, the idea of continuity was a generalization of the informal idea of smoothness, or lack of discontinuity. The first statement of the idea of continuity was by Euler in 1784, relating to plane curves. Other mathematicians, including Bolzano and Cauchy, then refined and extended the idea of continuity. Continuous functions are the raison d'être of topology itself.

Here is my head text, before the last revert:

In mathematics, a function is a relation from a set X to a set Y such that every element of X is paired with exactly one element of Y.

An example is the squaring function sqr, from the set of all integers to the set of all natural numbers, that pairs each integer x with its square x². Another example is the function from the real numbers to , that associates to each x in the number x³ - 9x.

Functions are extremely important in all branches of modern mathematics. They are extensively used in every quantitative science, to model relationships between all kinds of physical quantities—especially when one quantity is completely determined by another quantity (or by several other quantities). Thus, for example, one may use a function to describe how the temperature of water affects its density; or how the local gravity acceleration vector depends on latitude and longitude. Functions are also used to model relationships between non-numerical data, e.g. the connection of each country of the world to its capital city.

Note that the Wikipedia model article starts out with a precise mathematical definition, that not only uses the word "set", but is in fact much more technical than my sentence. In my text, the reader who wants to know the precise meaning of "relation" needs only clik on the link, and in a few minutes he will know. The reader who chooses to skip that step will get at least an intuitive understanding of "function" that may be incomplete but is not incorrect. In contrast, a reader who does not know what an "open set" means will be completely lost: it taks many hours of study and exercises to understand that concept, and most examples will probably use other hard concepts like limits, sin(1/x), etc.. The reader who skips that step will remain clueless, or come out thinking that abs(x) is not continuous because it is not smooth.

So, I claim that my header section text for function (mathematics) is well within the Wikipedia guidelines; whereas the current text violates them in a grand way.

Well, sorry to waste your time again. I really should go back to my real work, but simply could not resist... All the best, Jorge Stolfi 14:56, 19 January 2006 (UTC)

actually, the definition of continuous function in the sample article is wrong (an open map is not the same thing as a continuous map)! At the very least, it is quite imprecise. A justification for the informal treatment at the outset of "function" in particular is that this is an entry level concept; a reader who knows little about mathematics is more likely to start here. I agree with you that the actual discussion here is somewhat vague right at the beginning (that's why I added the last sentence, which is a little more precise). But I agree with the authors of the earlier material about the appropriate level of the intro of this particular article. Randall Holmes 15:04, 19 January 2006 (UTC)

I'm right here -- well, most of the time ...

• 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)

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)

• I teach math., too. $y=x^{2}$ is standard rather than $x^{2}=y$ (the reader will expect this and will find the other direction distracting without explanation not appropriate for this intro. Your discussion of alternative notation for the square function introduces notations which are not usual, and some which are just weird (the last one). At the very least, such a discussion of notation is not needed in the introduction. Randall Holmes 15:23, 19 January 2006 (UTC)

• By the way, I quite like the paragraph you added -- without the extra bit. It is a good (the canonical) simple example clearly stated... Randall Holmes 15:25, 19 January 2006 (UTC)

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)

• JA: Then you will understand that teaching, providing information in general, involves violating some expectations, breaking through the mindset that the learner brings to the lesson, but gently, of course. The fact remains that this particular article has so much misinformation in it that no responsible teacher can let it stand as it is. What is not usual to you may be standard to others. I gave the <x, y> in that order in order to emphasise the status of the equation as equation, and to set up the extensional definition in terms of ordered pairs. My intoduction of the "square" language is a standard gambit in computer science texts -- and I'm not the one who dragged in the input/output langauge, but I'll use it to the hilt if it's there. The "f() = square()" bit, maybe it would help to write it ""f(...) = square(...)" the way Quine does, is also standard pedagogy in computer science, and makes for a sly way of seeding the ground for later discussions of "anonymous" functions, lambda abstraction, intensional definitions, and so on. Thus, by the third paragraph, we have prepared the way for a balanced treatment later in the article of both extensional and intensional definitions of functions. Jon Awbrey 15:44, 19 January 2006 (UTC)

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)

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)

• JA: On Pedagogy, on Popularity, on Donner und Blitzen, ... Again, until a little while ago, I had no part in the pith or the pitch of this article, but merely started with what was in place, the hot dispute included, and tried to go with the flow. I know what the eventual target is, the question is where to drive the pitons into the cliff. Jon Awbrey 15:56, 19 January 2006 (UTC)

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)

• JA: Last time I say this. I did not set the level of this article. From what I can see in it, this article has already abandoned all hope of being a concise reference, and is trying to be a work of popular introduction. Not my choice. We can always write another one in a different vein, or vain. All I can do is try to keep this article from being a positive obstruction to correct information about functions. I did not introduce the computational metaphor about input-output, and probably would have started another way, but if it's already in there I will try to make the most of it. Jon Awbrey 16:18, 19 January 2006 (UTC)

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)

Proposal to split article(?)

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)

I think this is an excellent idea. Fxn is too central to maths to be dealt with in a single page, disambiguation pages not withstanding. The introduction/lead paragraph or first major section should address the scope and level issues. Then we can piece this back together. E.g. The (former?) Injection/Bijection/Surjection/Partial/Total map should have a summary mention here but deserves a page of it's own.

There are many sides to this issue, we need to address the breadth; then show where to look. LarryLACa 03:40, 20 January 2006 (UTC)

Strongly disagree with any split. The problem seems to be the appropriate level for the introduction, rather the difference between functions in calculus or in set theory. Oleg Alexandrov (talk) 04:08, 20 January 2006 (UTC)

Consolidating content - fixing redirects

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)

Wikipedia style, again

Many of the bytes on this page are about the issue of whether one should start with the definition, or give some motivation/approximation/analogy first. However, that is not an open issue. The Wikipedia style guidelines (for all subjects, not just mathematics) are very clear on that: the first sentence of every article should be the definition of the head word. Another rock-hard guideline, stamped all over the place, is that "Wikipedia is not a textbook".
I have now looked at the relation (mathematics) article. It is very nicely written, and hopefully it can be used in a Wikibook. But it clearly violates both of those guidelines.
All the best, Jorge Stolfi 16:51, 19 January 2006 (UTC)

There's another subtlety here: it can be argued (forcefully) that the mathematical definition which you gave simply is not the definition of the concept [nor is it the only mathematical definition of the concept; see lambda-calculus]. It is one implementation of a more abstract concept whose description is necessarily informal and should be given in the opening section (with the pedagogical side-effect that the informal concept is closer to what we learn in calculus...). Randall Holmes 17:11, 19 January 2006 (UTC)

• JA: There are definitions and then there are definitions. In loose talk, which is what the general style guide amounts to, "definition" does not mean "definition in the mathematical sense", the sort of thing that we math-minded-minions would rubricize as Definition x.y.z. ..., but a description sufficient to get the article going. Look around, you won't see many articles in WikiWorld starting out with nec-&-suf defs. Jon Awbrey 17:02, 19 January 2006 (UTC)

Jon, here are the first two paragraphs of that article (modulo markup bugs):

A relation is a mathematical object of a very general type, the generality of which is best approached in several stages, as will be carried out below. The basic idea, however, is to generalize the concept of a binary relation, such as the binary relations of equality and order that are denoted by the signs "=" and "<" in statements of the form "5 + 7 = 12" and "5 < 12". The concept of a relation is also the fundamental notion in the relational model for databases.

A finitary relation or a polyadic relation — specifically a k-ary relation, a k-adic relation, or a k-place relation when the parameter k, called the arity, the adicity, or the dimension of the relation, is known to apply — is conceived according to a formal definition to be given shortly. But it serves understanding to introduce a few preliminary ideas in preparation for the formal definition.

That's two heavy paragraphs into the article, with lots of jargonesque words like "binary relations of equality and order", "polyadic", "arity", "adicity", "dimension", "parameter". And still no definition, not even a pseudo-definition that would at least "get the article going". The definition is only given in paragraph #4, and then in such a complicated way that even a mathematician is likely to wonder whether this is the same thing he has been using since his Algebra 101 course.
Sorry, that may be good for a textbook perhaps, but is very bad for Wikipedia. All the best, Jorge Stolfi 17:45, 19 January 2006 (UTC)

• JA: Although it may eventually be pertinent here, I think we have enough turkey on our plates on this table for the time being, so maybe it's better to critique that article on its own talk page. Once again, I did not initiate the article, but merely tried to construct an "on-ramp" for it. The orginal def, of a perfectly standard and correct order of sophistication, was already there when I got there, but I know from experience that it would simply not be accessible to many folks who might otherwise benefit from it. Def's like that are an acquired taste, and I just tried to conduct a well-planned tasting of the available and the best vintages. Jon Awbrey 18:22, 19 January 2006 (UTC)

Jon and Randall (and I see Jorge is back after all); a few comments. First, saying "this article is just plain horrible" are not helpful. Criticism should be impersonal and specific. Second, brevity is the soul of wit. Third, it is not necessary to rewrite entire articles as fast as you can type. I suggest we take it a little at a time. Rick Norwood 18:29, 19 January 2006 (UTC)

• JA: The article is not a person. I make specific criticisms and specific (attempts at) correction. It seemed necessary to emphasize the general seriousness of the situation. Jon Awbrey 18:34, 19 January 2006 (UTC)

Articles are written by people, and nobody likes to have their prose called "horrible". Why not just set out your specific criticisms, and allow others to decide for themselves how serious the problem is? Rick Norwood 18:50, 19 January 2006 (UTC)

Introduction

On reading the discussion above, and reading the current introduction to the article, what seems to me most needed is to take out asides and parenthetical remarks. I don't think anything new needs to be put in, but too many cooks have spoiled the broth. I am going to try to make the introduction shorter and -- for the time being -- not do anything else. Oops -- I am also going to correct the error that I introduced about the use of "range" and "codomain". Rick Norwood 18:36, 19 January 2006 (UTC)

I've read through the introduction now several times, and it seems to flow smoothly. I pause for comments and corrections. Rick Norwood 18:48, 19 January 2006 (UTC)

In fact, the present first sentence, "The concept of a function is fundamental to mathematics", is a waste of the most valuable space in an article (after the title). Kill it or move it, and the article immediately becomes stronger.

On a procedural note, it there are too many edit conflicts on the article, or competing major rewrite visions, the talk page can also be used as a sandbox. Alternatively, use Function (mathematics)/sandbox.

And everyone is advised to be patient. The opening paragraphs of a major service article like this are a tremendous challenge to write well, to the point of consensus satisfaction. --KSmrq^T 19:02, 19 January 2006 (UTC)

• JA: Nobody here has been worrying over the first sentence. There are more substantive issues. If you see a potential improvement there, the normal procedure is just to make it, perhaps blowing a horn first before traipsing into a too trafficky intersection. Jon Awbrey 19:58, 19 January 2006 (UTC)

I reiterate here my suggestion that perhaps it is impossible to write a single article, because at least two quite different interests are involved. How about function (set theory) and function (analysis), with the latter more naturally including the math pedagogy issues (each naturally referring to the other)? I'm not touching this article again; it is too frustrating. I'll draft function (set theory)... Randall Holmes 19:18, 19 January 2006 (UTC)

• JA: Several people, in the blitheness of their own familiarity with notations like f(x), are missing the fact that an article is written to explain that very notation to someone who may not already be familiar with it. Hence, my previous explanation, which I will now proceed to re-enter. Jon Awbrey 18:54, 19 January 2006 (UTC)

KSmrq is right, these conflicting versions just make it very frustrating for everybody. Therefore, I protected the article. Try to come to a consensus, starting with the very first sentence. Do we want to begin with "The concept of a function is fundamental to mathematics"? Or with a formal definition? Or with an informal one? -- Jitse Niesen (talk) 19:22, 19 January 2006 (UTC)

• JA: Here is my current proposal for the introduction. I can reverse the order of x and y if that's really such a big deal, but if we can't present this topic the way that it's standardly presented these days, then I will simply steer every use of it to some other article.

The concept of a function is fundamental to mathematics. In intuitive terms, a function associates an 'input' with an 'output'. Functions can be described in many different ways.

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, and so on. This function is called the 'squaring' function, and it is common to express it any of the following ways:

• y = x²,
• y = square(x),
• y = f(x), where f(…) = square(…).

Functions are applied in particular settings. Such a setting is specified by giving a set of allowable inputs and a set of contemplated outputs. The first set is called the domain or the source of the function. The second set is called the codomain or the target of the function. The type of a function f, as it is contemplated in a given setting, is indicated by writing f : X ? Y, where X is the domain or source of f and Y is the codomain or target of f.

The set of outputs that results from applying the function to all the elements in its domain is called the range of the function. The range is necessarily a subset of the codomain, by may not be the whole codomain.

An essential property of a function is that for each input there must be one unique output. Thus, for example, the relation ' y is a square root of x ' is not a function, because every positive number has two square roots. The square roots of 9 are 3 and - 3. To make the square root a function, we must specify which square root to choose for each input value (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 nation its capital. In this case f(France) = Paris.

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.

Jon Awbrey 19:30, 19 January 2006 (UTC)

Jon, you are not getting things off to a very good start when you begin by saying "if we can't present this topic the way that it's standardly presented these days, then I will simply steer every use of it to some other article". There are some good points and some bad points in the intro you propose, but unless you are willing to compromise, the article will just stay locked.

• JA: I am simply saying what I will do. That course will be forced on me by the practical necessity of providing useful information that is needed in other articles that refer to the concept of a function. It's a compromise just starting with what's here and trying to make it better.

So, taking it one paragraph at a time:

Paragraph One -- we agree.

Paragraph Two -- I agree that we need to explain what "f" is all about. I do not think we need to give three examples of this simple function, however. The intended reader may not know much math, but she is not stupid. One example should suffice.

• JA: I am writing this off the cuff. Assuming good faith means considering the possibility that there's a reason coming down the pike. I have already stated my rationale for this 3-fold rep -- what they call the 'Märchen rule' in German -- it sets up a concrete illustration of the extensional-intensional distinction in the def of functions, all without invoking any fancy words.

Wiki style manual says to avoid bullet points.

• I did not see that. Certainly few people follow that.

And calling the function "square" as in square(3) = 9 is not standard notation, and does not make the article clearer. Here is my proposed Paragraph Two, taking yours as a starting point:

• Assuming good faith mean that I should not have to go into the dustier parts of my library just to prove that this is standard in all sorts of contexts, especially intro and advanced comp sci books, which I-O language I have compromised in using. But I will. Attaching the concept to the vernacluar is good in the beginning. Starting out with an abstract meta-variable like f begs the question that is here to be explained. I have much experience with how big a step this is for some folks.

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 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

$f(x)=x^{2}$

• JA: The purpose of writing what I did was to help make the leap of abstraction from a concrete function like "square" or "squaring" to a generic metavariable like f. Stating the relationship between the function and the equation is actually tricky and has to be done with care if people are not to confuse them. I know quite a bit about the types of misconceptions that actually arise, and what I wrote was calculated to avoid them.

Jon Awbrey 20:42, 19 January 2006 (UTC)

What do you think? Rick Norwood 20:06, 19 January 2006 (UTC)

Paragraph two, second attempt

On reading the discussion above, and reading the current introduction to the article, what seems to me most needed is to take out asides and parenthetical remarks. I don't think anything new needs to be put in, but too many cooks have spoiled the broth. I am going to try to make the introduction shorter and -- for the time being -- not do anything else. Oops -- I am also going to correct the error that I introduced about the use of "range" and "codomain". Rick Norwood 18:36, 19 January 2006 (UTC)

I've read through the introduction now several times, and it seems to flow smoothly. I pause for comments and corrections. Rick Norwood 18:48, 19 January 2006 (UTC)

In fact, the present first sentence, "The concept of a function is fundamental to mathematics", is a waste of the most valuable space in an article (after the title). Kill it or move it, and the article immediately becomes stronger.

On a procedural note, it there are too many edit conflicts on the article, or competing major rewrite visions, the talk page can also be used as a sandbox. Alternatively, use Function (mathematics)/sandbox.

And everyone is advised to be patient. The opening paragraphs of a major service article like this are a tremendous challenge to write well, to the point of consensus satisfaction. --KSmrq^T 19:02, 19 January 2006 (UTC)

• JA: Nobody here has been worrying over the first sentence. There are more substantive issues. If you see a potential improvement there, the normal procedure is just to make it, perhaps blowing a horn first before traipsing into a too trafficky intersection. Jon Awbrey 19:58, 19 January 2006 (UTC)

I reiterate here my suggestion that perhaps it is impossible to write a single article, because at least two quite different interests are involved. How about function (set theory) and function (analysis), with the latter more naturally including the math pedagogy issues (each naturally referring to the other)? I'm not touching this article again; it is too frustrating. I'll draft function (set theory)... Randall Holmes 19:18, 19 January 2006 (UTC)

• JA: Several people, in the blitheness of their own familiarity with notations like f(x), are missing the fact that an article is written to explain that very notation to someone who may not already be familiar with it. Hence, my previous explanation, which I will now proceed to re-enter. Jon Awbrey 18:54, 19 January 2006 (UTC)

KSmrq is right, these conflicting versions just make it very frustrating for everybody. Therefore, I protected the article. Try to come to a consensus, starting with the very first sentence. Do we want to begin with "The concept of a function is fundamental to mathematics"? Or with a formal definition? Or with an informal one? -- Jitse Niesen (talk) 19:22, 19 January 2006 (UTC)

• JA: Here is my current proposal for the introduction. I can reverse the order of x and y if that's really such a big deal, but if we can't present this topic the way that it's standardly presented these days, then I will simply steer every use of it to some other article.

The concept of a function is fundamental to mathematics. In intuitive terms, a function associates an 'input' with an 'output'. Functions can be described in many different ways.

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, and so on. This function is called the 'squaring' function, and it is common to express it any of the following ways:

• y = x²,
• y = square(x),
• y = f(x), where f(…) = square(…).

Functions are applied in particular settings. Such a setting is specified by giving a set of allowable inputs and a set of contemplated outputs. The first set is called the domain or the source of the function. The second set is called the codomain or the target of the function. The type of a function f, as it is contemplated in a given setting, is indicated by writing f : X → Y, where X is the domain or source of f and Y is the codomain or target of f.

The set of outputs that results from applying the function to all the elements in its domain is called the range of the function. The range is necessarily a subset of the codomain, by may not be the whole codomain.

An essential property of a function is that for each input there must be one unique output. Thus, for example, the relation ' y is a square root of x ' is not a function, because every positive number has two square roots. The square roots of 9 are 3 and − 3. To make the square root a function, we must specify which square root to choose for each input value (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 nation its capital. In this case f(France) = Paris.

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.

Jon Awbrey 19:30, 19 January 2006 (UTC)

Jon, you are not getting things off to a very good start when you begin by saying "if we can't present this topic the way that it's standardly presented these days, then I will simply steer every use of it to some other article". There are some good points and some bad points in the intro you propose, but unless you are willing to compromise, the article will just stay locked.

• JA: I am simply saying what I will do. That course will be forced on me by the practical necessity of providing useful information that is needed in other articles that refer to the concept of a function. It's a compromise just starting with what's here and trying to make it better.

So, taking it one paragraph at a time:

Paragraph One -- we agree.

Paragraph Two -- I agree that we need to explain what "f" is all about. I do not think we need to give three examples of this simple function, however. The intended reader may not know much math, but she is not stupid. One example should suffice.

• JA: I am writing this off the cuff. Assuming good faith means considering the possibility that there's a reason coming down the pike. I have already stated my rationale for this 3-fold rep -- what they call the 'Märchen rule' in German -- it sets up a concrete illustration of the extensional-intensional distinction in the def of functions, all without invoking any fancy words.

Wiki style manual says to avoid bullet points.

• I did not see that. Certainly few people follow that.

And calling the function "square" as in square(3) = 9 is not standard notation, and does not make the article clearer. Here is my proposed Paragraph Two, taking yours as a starting point:

• Assuming good faith mean that I should not have to go into the dustier parts of my library just to prove that this is standard in all sorts of contexts, especially intro and advanced comp sci books, which I-O language I have compromised in using. But I will. Attaching the concept to the vernacluar is good in the beginning. Starting out with an abstract meta-variable like f begs the question that is here to be explained. I have much experience with how big a step this is for some folks.

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 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

$f(x)=x^{2}$

• JA: The purpose of writing what I did was to help make the leap of abstraction from a concrete function like "square" or "squaring" to a generic metavariable like f. Stating the relationship between the function and the equation is actually tricky and has to be done with care if people are not to confuse them. I know quite a bit about the types of misconceptions that actually arise, and what I wrote was calculated to avoid them.

Jon Awbrey 20:42, 19 January 2006 (UTC)

What do you think? Rick Norwood 20:06, 19 January 2006 (UTC)

Paragraph two, third attempt

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)

Function (set theory)

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)

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)

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)

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)

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)

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

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)

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)

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)

Jorge Stolfi's proposal

Here is my proposal:

Partial plot of a function that to each real number x associates the number x3 − 9x. 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 integer squaring function, from the set Z of all integers to the set N of all natural numbers, that pairs each integer x with its square x2 — namely, 0 with 0, 1 (and also −1) with 1, 2 (and −2) with 4, and so on. 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 relationships between quantities—especially when some quantity (or several quantities) completely determine another quantity. Thus, for example, a physicist may use a 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. Domain, codomain, argument, image If f is a function from a set X to a set Y, then X is called the domain of f, and Y is called its codomain. Each element of the domain is called an argument of the function. For each argument x, the corresponding unique y in the codomain is called the function value at x, or the image of x by (or under) the function. The value of a function f at an argument x is traditionally written f(x). The parentheses may be dropped in some cases, e.g. when using the logarithm function, as in log x, or the trigonometric functions, as in cos x and tan x. In other contexts, such as automata theory, the postfix notation x f is used instead. The notation f$\;:\;$x${}\mapsto {}$y is sometimes used to mean f(x) = y.

The rest is as in my last edit (see page history), modulo "things to do".

Comments

I immodestly claim that this proposal is

1. conforming to the Wikipedia guidelines,
2. correct and "honest" to the reader (modulo involuntary mistakes)
3. fairly exhaustive
4. fairly readable (modulo the author's broken English)
5. appropriate and useful for technically-educated users (including mathematicians 8-)
6. fairly helpful even to pre-college kids (at least as much as or better than any of the other proposals)
7. succint
8. not paternalistic
9. educational (teaches the non-math reader how to think mathematically)
10. well organized, with concepts introduced at proper time and in the proper order

(Well, if I don't praise my work, who will? 8-) All the best, Jorge Stolfi 21:47, 19 January 2006 (UTC)

We seem to be moving toward two articles, one introductory and one technical. Rick Norwood 21:53, 19 January 2006 (UTC)

(I have edited Jorge's text trivially to use x³ instead of x³, and minus instead of hyphen.) Where you see a bifurcation, I see two streams trying to flow together into a greater whole. :-D

To proceed further, perhaps rather than merely displaying text we could continue to clarify goals. Especially, what facts must be stated, in the opinion of the participating editors? Jorge has made a good start at stating the intangible goals he values. What audience is targeted by each segment of the intro? And so on. --KSmrq^T 22:08, 19 January 2006 (UTC)

To respond to KSmrq's question, the audience for the introductory paragraph is any intelligent reader who wants to know more about mathematics, but is not by training a mathematician. Since I'm told the Mathematics article is the fifth most viewed article in Wikipedia, I think there are probably a lot of people in that category, some students, some simply interested readers. The first paragraph must give a layperson some idea of what a function is and what a function is used for. And I think the first paragraph should also mention that functions are a fundamental concept (as it does now). Beyond the first paragraph we need a correct but not fussy mathematical definition. I would put the history of the concept next, since that is likely to have more readers than the more technical bits. Then the basic ideas of domain, range, codomain, one-to-one, onto, and finally the idea of multi-valued functions and functions of several variables. I would put everything else in articles this article links to. Rick Norwood 22:30, 19 January 2006 (UTC)

My take on the first sentence may be illustrated by analogy. Suppose someone asked me who Jane Fonda is, and I replied: "Jane Fonda is physically fit and many people find her attractive." While this true, it is not the right place to begin.

Much as I enjoy including history in articles, your proposal to place history before a formal definition is a disservice to most readers. They would prefer to know what a function is before they care about how the concept arose. --KSmrq^T 00:53, 20 January 2006 (UTC)

It has been claimed that my introduction can be read only by mathematicians. I disagree. The words "relation" and "set" are plain English words, and their ordinary sense is a good approximation to the mathematical sense. A naive person who reads that paragraph, using the ordinary sense of the words, will still get a fairly correct picture. Moreover, until he learns the mathematical meaning of those words, no amount of explaining and examples can bring his understanding closer to the truth.

The only "jargonesque" thing in my first paragraph is the implicit naming convention: "a set X", instead of "a set, which we will call X. However, those two instances of the convention are fairly mild, and I bet that even the most naive reader will have encountered enough examples of that convention to understand its use here.

As for the examples: writing "f(x) =x²" is both unnecessary (we don't need a name for the function), distracting (here we must illustrate the concept of function, not its notation), and confusing (it is two uses of the implicit naming device, much more complicated that the X and Y of the first parag). The notation "y = x²" is not good either: the convention that the argument is called x and the result is called y is not even a mathematical tradition; it is at best a custom that is seen in many, but not all, pre-college classrooms.

Jorge Stolfi 22:38, 19 January 2006 (UTC)

Clearly, you work in a very different mathematical tradition that the one I work in, and it may be (if you are a teacher) that you have brighter students. I have been teaching functions for mumblety-mumble years, and I can only assure you that none of my Freshman college students could understand your first sentence.

As for your second point, I was taught (at MIT) and find in the textbooks I teach from (Thomas' Calculus, for example) y = x². In fact, there are sound mathematical reason to distinguish between y = x², a function, and x², a monomial. In particular, when mathematics majors reach the level of abstract algebra, polynomaials are usually viewed as a ring, a set of objects, while functions, though they also have a (pointwise) ring structure (if the codomain is a fixed ring) are usually viewed as mappings, a set of morphisms.

In any case, these are technical questions far beyond the scope of this article. The main point where we differ is pedagogical -- you think laypeople can read sentences that my experience tells me they can't. Rick Norwood 22:53, 19 January 2006 (UTC)

Next topic

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)

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)

Another proposal

I am finally beginning to understand what I have been trying to say. 8-) Please consider this version:

Diagram of a function from the set {1,2,3,4} to the set {A,B,C,D} 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. Partial plot of a function that to each real number x associates 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)

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)

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)

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)

Let me comment on the first sentence. I now agree with Jorge and KSmrq that we should give a definition in the first sentence, which should be as precise as possible. I can see three difficulties for people trying to understand Jorge's first sentence: the word relation, the word set, and the variables X and Y. However, I think most people's concept of a set is good enough for getting the gist. It is a bit more difficult for relations between sets: I think in common parlance, a relation between X and Y would be something like X is a subset of Y, while in mathematics we actually want relations between elements of X and Y. Using letters to denote the set is probably okay for most, but it is perhaps not necessary.

So, my proposal for the first sentence is:

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)

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)

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). --KSmrq^T 05:19, 20 January 2006 (UTC)

Relations

• 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)

• JA: In this setting, X and Y are not 'variables', they are sets that are called the domains of the binary relation. Jon Awbrey 05:07, 20 January 2006 (UTC)

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)

• JA: In discussing relations in general, "domain" is a generic term that covers all of the domains of a relation. The domains are distinguished by their place in a sequence, and are referred to as the "first", "second", "j^th" domain, and so on. Even in the case of binary relations, the term "codomain" is less often useful, partly because binary relations tend to be applied and composed both "on the left" and "on the right", and partly because there are so many different conventions about the syntax that is used to denote applications and compositions. When it comes to functions, "domains" in the plural is still generic for both domain and codomain, and the terms "source" and "target" have lately come into favor, by way of paving the way for category-theoretic language that becomes practically essential after some point, both for advanced mathematics and computer science. Jon Awbrey 13:28, 20 January 2006 (UTC)

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)

The competition did this ...

Here are the two opening sentences of the Encyclopedia Britannica article:

"function, in mathematics, an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). In its most general usage in mathematics, the word function refers to any correspondence between two classes."

The rest reads rather like what I had in mind for function (analysis). I would certainly not have written the second sentence quite that way, but this is a fairly informative and accessible article about functions of a real and complex variable which does not on the whole talk about sets... Randall Holmes 13:54, 20 January 2006 (UTC)

while I certainly don't entirely endorse this article, it is worth looking at. Randall Holmes 13:56, 20 January 2006 (UTC)

• JA: The first sentence betrays a confusion between sign and object, thing denoting and thing denoted. The second sentence, in trying to be general, has simply become vague. The default meaning of "function" in mathematics, even before 1911, is given as a certain type of binary relation on two sets that is total on the first domain and single-valued. If folks want to speak of "partial functions" or "multi-valued functions", then they are required to add the adjective if they want to be understood. Jon Awbrey 14:16, 20 January 2006 (UTC)

• I agree that the second sentence is vague: I don't think that the author was intending to talk about partial or multi-valued functions, but simply making a mistake (which could be corrected without too much verbiage). In the first sentence, the notion of function as it is actually understood by many perfectly competent users of the concept is described: moreover, this notion of function can be formalized (in typed lambda-calculus, for example) [I am not suggesting introducing such considerations, merely refuting your specific objection to this kind of definition]. It is not necessarily the case that an essential confusion of use and mention is involved. (I add that you might be right about the mention of the dependent and independent variables as such; I was defending the reference to expressions)Randall Holmes 16:49, 20 January 2006 (UTC)

• A major point about this article is that it has reasonable coverage of what a typical reader might want to remind himself about functions of one or more real or complex variables, and says almost nothing about sets ("classes" in the article) at all. It comes from the 1992 edition, so it is current thinking of editors of a major encyclopedia. There is a considerable gap here between the layman's understanding and the understanding of a committee of mathematicians. Randall Holmes 17:01, 20 January 2006 (UTC)

• JA: We can discuss whether "all learning is anamnesis" in another setting, perhaps under the shade of an old plane tree, but I suggest that our aim should be set somewhat higher than merely reinforcing popular misconceptions, no matter how common or familiar, and no matter how comforting that may be to teacher or student either one. When it comes to our relation with what should be "no competition" here, I think an old bit of motherwit just about covers it: "Two wrongs do not make a right". Jon Awbrey 17:16, 20 January 2006 (UTC)

Two points: First, Stolfi reiterates (and I think that he is right) that it is not the purpose of an encyclopedia to teach. It is a reference, and articles on a topic may presuppose some familiarity with the actual prerequisites for understanding that topic. Pedagogy really is somewhat beside the point here (not entirely beside the point, though). Second, I simply deny that knowledge of set theory is appropriate prerequisite knowledge for understanding what a function is. The actual definition of function appropriate to calculus for most consumers of calculus involves very little set theory, if any at all (more set theory sneaks in as the level goes up...) The best students in my calculus classes do understand what a function of a single real variable is, and their understanding is not mediated by any knowledge of set theory. The foundations of mathematics in set theory are an encoding of knowledge that existed already before sets were heard of, and the insistence on explaining concepts in terms of set theory which are not understood in terms of set theory by most consumers is somewhat wrong-headed. I am a set theorist, so I'm not saying that set theory is a bad thing, nor that it cannot be used as a foundation for mathematics: but, de facto, at least where I am, it is not the foundation of school mathematics (the New Math failed), and it is in terms of their understanding of school mathematics that most readers will start reading the article. Addressing the immediately preceding post, the conception expressed in the first line of the Encyclopedia Britannica article is not a misconception at all; it is probably the dominant concept of function, and the article should probably start there (and ideally point the reader to the official set theoretic conception if he or she needs it, which the Encyclopedia Britannica article does not do). Randall Holmes 17:53, 20 January 2006 (UTC)

Another competitor is the American Heritage Dictionary, 4/e. Their online definition is:

Mathematics a. A variable so related to another that for each value assumed by one there is a value determined for the other. b. A rule of correspondence between two sets such that there is a unique element in the second set assigned to each element in the first set.

Presumably the first version would cover y as a function of x. The second version does not say how the rule is given, so it needn't be, say, a formula or primitive recursive function. Notice they don't shy away from mentioning sets. Nor does Merriam-Webster Online:

a mathematical correspondence that assigns exactly one element of one set to each element of the same or another set

With three tiny words this latter definition explicitly allows the codomain to be the same as the domain. Both dictionaries choose the word "correspondence" rather than "relation". --KSmrq^T 00:27, 21 January 2006 (UTC)

Pause, take a deep breath...

KSmrqT wrote: "Much as I enjoy including history in articles, your proposal to place history before a formal definition is a disservice to most readers. They would prefer to know what a function is before they care about how the concept arose."

I agree. What I said was that the history should come after both an explanation and a definition but before one-to-one, onto, etc. Now, if functions looked like Jane Fonda, we wouldn't have a problem. Rick Norwood 14:06, 20 January 2006 (UTC)

We seem to be back on the first paragraph, and there seems to be a consensus that we should begin with a definition. I know from experience that most people do not have any idea what the word "set" means. What is elementary to mathematicians is very difficult for laypeople. Most people learn from examples, and I am sorry to say that many never move to understanding abstractions. One class of people I teach are people who are going to be teachers, and about half of every class will swear on a stack of bibles that the sentences "If I were rich I would buy a car." and "I am rich and will buy a car." say exactly the same thing. They process words differently from the way I process words. Teaching them elementary set theory -- even to the point of telling the difference between an element and a subset, is like pulling teeth. I would estimate only about half of them ever grasp the concept. So, with that in mind, here is a definition of function that is practical rather than mathematical. The mathematical definition can (and must) come later.

A function in mathematics associates an input and an output. For example, the squaring function y = x² associates 1 with 1, 2 with 4, 3 with 9, and so on.

Short and to the point. No mention of relations, sets, domain, range -- just the naive idea of what a function means. Rick Norwood 14:17, 20 January 2006 (UTC)

[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)."

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?

It reminds me of this definition:

"The Internet: Learn what you know. Share what you don't."

Perhaps we should replace "Internet" with "Wikipedia" and paste that slogan at the top of this article... 8-)

Or perhaps we could put a note at the top

"If you do not know what a "set" is, and are not willing to learn, then perhaps you will find this other article a lot more interesting than this one. Enjoy, and thanks for cliking us."

8-)

Sorry, but I still think that mathematics articles should be written for people who are at least willing to learn mathematics.
All the best, Jorge Stolfi 17:09, 20 January 2006 (UTC)

I object specifically to the sentence "A function in mathematics associates an input and an output". It is necessary to speak precisely. Even if the input/output metaphor is granted (and while I don't especially like it, I could grant it in an introductory sentence), one must say something more precise. It is permissible to be formal; it is indefensible to be vague to the point of being wrong. Your opening sentence covers the case of a slot machine: one puts in an input (pull the lever) and gets an output -- not uniquely determined by the input! Stolfi's opening sentence is better: it is at least true! Randall Holmes 17:35, 20 January 2006 (UTC)

One could say "a function associates input with output, with each output being uniquely determined by the input from which it is derived. Each function takes input of a particular kind and gives output of a particular kind (which may be the same as or different from the kind of input it takes)." Randall Holmes 17:35, 20 January 2006 (UTC)

The converse danger associated with the concern I express above is that one ends up with "purple prose"; but the fact that it is difficult to write with both precision and pithiness does not relieve one of the obligation to attempt to do both. And in mathematical text, precision is more important, particularly when one is dealing (as one is here) with a reference rather than a textbook. Randall Holmes 17:35, 20 January 2006 (UTC)

Good luck!

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)

Paideia

• 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)

Another deep breath

[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)

The words input and output jar on my ear, also. Would you have me say a function associates with every abscisa an ordinate? Like it or not, input and output are here to stay. I hate Big O, little o, but I accept the fact that they are here to stay.

Still, while I, too, have other important work to do, I am not ready to give up. Rick Norwood 20:25, 20 January 2006 (UTC)

Another proposed Paragraph One

We need to move forward. Here is what I propose. Everybody put their Paragraph One here and we vote. I know a vote with so few people is not the best solution, but our differences seem irreconsilable and at least a vote will allow us to move forward. Rick Norwood 20:33, 20 January 2006 (UTC)

Candidate A (the paragraph now in the locked article):

The concept of a function is fundamental to mathematics. In intuitive terms, a function associates an 'input' with an 'output'. Functions can be described in many different ways.

Candidate B (Rick Norwood):

A function, in mathematics, associates an 'input' with a unique 'output'. For example, the squaring function, usually written

$y=x^{2}$

or

$f(x)=x^{2}$

or

$SQUARE(x)=x^{2}$

takes input 1 and produces output 1, input 2 produces output 4, input 3 produces output 9.

Candidate C (Jitse Niesen):

In mathematics, a function associates an 'input' with a single 'output'. An example is provided by the squaring function, which associates each number with its square, so that the input 2 is associated with the output 4, and 3 is associated with 9, and so on.

Comments

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)

I do like Jitse's the best, with candidate B a close second. I have a pet peeve about calling equations like "y = x²" and even "f(x) = x²" as "functions". I also like the use of the word "single" rather than "unique", as it is arguably more understandable and still correct. The notation should probably be the next thing to address. Something like:

"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)

I like the first part of Gauge's suggested second paragraph, but think it goes on too long. Let's wait until tomorrow to see if anyone else wants to comment. If not, then Jitse's paragraph is Paragraph One and we can (hooray!) move on to Paragraph Two. Rick Norwood 00:18, 21 January 2006 (UTC)

Voting may force a choice, but that is not a consensus. Why is Jorge's sentence not represented among the candidates? And why not some (paraphrase of) one of the dictionary definitions I quoted? Of the three options shown, I have already objected to A, and explained why. I also strongly object to B, in part because it immediately and crudely introduces three notations, injecting topics which need to be handled later and more delicately. As for C, it's the best of the bunch, but I, too, would prefer to avoid saying "input" and "output". So I vote for D, none of the above. I prefer any of the omitted choices. --KSmrq^T 16:32, 21 January 2006 (UTC)

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)

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)

Suggestion

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)

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)

I think even better is to write

$f\colon x\mapsto y$ 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)

• JA: It's a hard choice. I recently had to reformat an entire article just to avoid the "ransom note" effect that I got from TeX on a couple of different browsers. Normally TeX handles spacing for you, but a little extra white (or grey) space and set-off display text is often useful on the first few intros of a novel notation. Jon Awbrey

Common types of functions by domain and codomain

$\mathbb {N} \to \mathbb {R}$

list

$\mathbb {R} \to \mathbb {R}$

2-dimensional curve

$\mathbb {R} ^{2}\to \mathbb {R}$

3-dimensional surface

$\mathbb {R} \to \mathbb {R} ^{2}$

parametrized curve in 2D

$\mathbb {R} ^{2}\to \mathbb {R} ^{3}$

parametrized surface in 3D

$\mathbb {R} \to \mathbb {R} ^{3}$

parametrized curve in 3D

could this be included in the article? --Joris Gillis 08:52, 21 January 2006 (UTC)

• JA: Between sets one uses \to, between elements or vars, \mapsto. Jon Awbrey 09:00, 21 January 2006 (UTC)

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. --KSmrq^T 16:47, 21 January 2006 (UTC)

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)

Dysfunctional misconceptions to be avoided

• JA: As a rule, it's better to give positive examples of what ought to be included under a concept, as opposed to inditing negative examples of what ought to be excluded from it, but ...

• JA: The current direction that the article is taking is quickly rounding up all the usual suspects in the way of misconceptions about mathematical functions and placing them on the platform for emulation rather than immolation, so ...

• JA: I will shift to the other foot, and simply make an account of the misconceits that must be avoided, if this article is to qualify as a reference, a survey, a tutorial, or any mixture thereof, on the subject of functions, as she is spoke in mathematics.

• JA: But I've had to put this pot/kettle on the back burner now, so ...

• JA: Jon Awbrey 16:50, 21 January 2006 (UTC)

Confoundation No. 1

• Thou shalt not confuse a function with one of its values.

Confoundation No. 2

• Thou shalt not confuse a function with one of its tuples.

Confoundation No. 3

• Thou shalt not confuse a function with one of its formulas.

• JA: If the aim of the article is to communicate some inkling of how functions are thought of in mathematics, then from the beginning it must be conveyed that a function is a mathematical object, that is, an object of mathematical conception and discussion. The only rule that limits what can be regarded as an object of mathematical conception and discussion is that it can be conceived and discussed without a logical inconsistency. It is true that some people have problems with what they would call "platonism" here, but mathematicians as rule have other sorts of problems than that, and they normally assume that if they are talking consistently then they are talking about some object that is the "moral equivalent" of a reality. Jon Awbrey 03:04, 22 January 2006 (UTC)

• JA: In mathematics as anywhere else it is indispensable to distinguish the role of an object from the role of a sign, using the short word "sign" to cover the whole panoply of characters, codes, equations, expressions, formulas, indices, notations, numerals, sentences, symbols, terms, variables, and so on that are used to denote objects. The distinction is trickier to make than it seems at first precisely because it is a distinction of roles in practical context and not a distinction of ontological essences, but it is nonetheless "essential" to make it, if utter confusion is to be avoided. Jon Awbrey 03:04, 22 January 2006 (UTC)

• JA: The distinction between objects and signs is sometimes treated in terms of the related distinction between use and mention, but there is a subtle distinction between these two distinctions. This comes about because there are different perspectives on the use-mention distinction. In particular, people of particular philosophical persuasions say that the normal use and even the normative use of signs does not of necessity invoke either the existence or the reality of any such denoted objects. This is something to to think about, but it does not reflect the 'naive' ('native' or 'natural') standpoint of normal mathematical practice, which tends to go on about its normal business as if mathematical objects like numbers and functions really exist and are merely denoted by a host of different systems of numerals and notations. Jon Awbrey 16:44, 22 January 2006 (UTC)

JS: Once you accept that the notations f(x)=x^2, lambda.t:t*t, and {(z,z^2): z \in R} denote the same function, you must agree that the concept of function is quite independent of any signs. You use signs to "describe" a function, but they *are not* the function.

As another semiotician put it some 2000 years ago,

"When you use your finger to point at the moon, no one would mistake the finger for the moon. Why is it that when you use words to point at an idea, people always confuse the words with the idea?"

--Lao Tzu

If you liked that (or even if you didn't 8-), here are a few other apt quotes I fetched from my scrapbook:

"Young man, in mathematics you don't understand things, you just get used to them."

--Von Neumann

"Just go on... and faith will soon return."

-- J. D'Alembert, to a friend, about infinitesimals.

"Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore."

-- Albert Einstein

"Mathematicians are like Frenchmen: whatever you say to them they translate into their own language, and forthwith it is something entirely different."

-- Johann Wolfgang von Goethe

"Medicine makes people ill, mathematics make them sad, and theology makes them sinful."

-- Martin Luther

"Math is tough."

-- Barbie

All the best, Jorge Stolfi 22:13, 22 January 2006 (UTC)

Great quotes. "The way that can be traveled is not the true way." Rick Norwood 22:41, 22 January 2006 (UTC)

Ah, quotation time. Here are a relevant pithy few.

"Put it before them briefly so they will read it, clearly so they will appreciate it, picturesquely so they will remember it and, above all, accurately so they will be guided by its light.

— Joseph Pulitzer

"So easy it seemed once found, which yet unfound most would have thought impossible."

— John Milton

"Even if you're on the right track you'll get run over if you just sit there."

— Will Rogers

"Math was always my bad subject. I couldn't convince my teachers that many of my answers were meant ironically."

— Calvin Trillin

Enjoy. :-D --KSmrq^T 04:50, 23 January 2006 (UTC)

Confoundation No. 4

• 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)

But some of us are old-fashioned enough to believe that the geometry of a graph is somehow more than its set of ordered pairs. Rick Norwood 22:43, 22 January 2006 (UTC)

Exegesis, hermeneutics, and assorted homilies

• JA: None of this stuff is new. I know from my own practice that it was being taught as standard material to freshperson and sophomore non-majors at moderately well-respected universities in the U.S. all throughout the 1980's. So there is nothing non-standard or impossible to communicate about these distinctions in particular or this material in general. Jon Awbrey 17:15, 21 January 2006 (UTC)

I look forward to your comments -- and agree completely that most dictionary definitions of mathematical terms should be burned. I'm particularly annoyed by: "topology -- a synomym for topography". Rick Norwood 17:05, 21 January 2006 (UTC)

I sympathize, but strongly disagree.

It is difficult to say where precision becomes pedantry, but I think you've crossed the line. On the other hand, I have just emerged, bloody but unbowed, from a big fight with long time wikipedians who insisted that a polynomial is a function, so I realize that it is possible to go too far in the other direction.

To me, your insistence that f(x) = 3x is not a function is like insisting that one and one is not two (in fact, the number represented by the word one plus the number represented by the word one is the number represented by the word two). I could quote a dozen well respected mathematics books that unabashedly call f(x) = 3x a function (instead of saying: the function whose ordered pairs are of the form (x, 3x) or, better yet, the ordered triple (R,R,(x,y))| y = 3x.

There is no hard and fast rule about how much to put in and how much to leave out, but all of us leave out a lot every time we write, and authors who put in too much are as hard to read as authors who leave out too much. Rick Norwood 20:16, 22 January 2006 (UTC)

• JA: I did not list the Confoundations in their eternal logical order — which would have been more like: "Do not confuse a function with one of its (1) values, (2) tuples, (3) graphs, (4) formulas (al gore rhythms, programs, etc.)", and in that order showing a bias toward extensional definitions over intensional definitions — but I elected to chastize and remedy them in order of their prevalence and seriousness in secular (mal-)practice. I hesitated even to mention the confusion of functions with their graphs — it is not only a far more venial fall from graphs than than the other brands of mathematical turpitude, but one the geometry of whose doctrinal punctilios is better put off until the coils of mortal syntax are suitably shriven first. Jon Awbrey

Both (3) and (4) summarize possible implementations of the function concept. (4) leads to the lambda-calculus (expressions can with (extreme) care be used to represent functions, in certain contexts) and (3) (in its set-theoretical rather than geometric form) is one of the official definitions of what a function is (and was the dominant definition for most of this century). Randall Holmes 00:57, 23 January 2006 (UTC)

[JS] (Back to serious technical mode.) Consider the following theorem:

"Theorem 1: if f is a real function of a real variable that is continuous in an interval [a,b], then for any y between f(a) and f(b) there is a x in [a,b] such that f(x) = y."

The beauty of this theorem (I hope I got it right) is that it holds for any function, not just for any function that can be defined by a formula or any other finite description. If we define "function" as being "finitely describable function", then we are limiting the span of this theorem to an unconceivably small subset of all functions.

Furthermore, consider

"Theorem 2: every linear vector space has a basis".

This theorem holds even for, say, the space F of all continuous functions from R to R. A basis is a function from some index set X to F; as far as I know, to build such a basis for F we need the axiom of choice, and then we get a basis that cannot be described. I do not know what happens if instead of F we use G, the subset of all "finitely specifiable" functions in F. (In fact I am not even sure that G is meaningful, but let it be.) Does G have a finitely describable basis? I suspect it doesn't; but in any case proving this is much harder than proving Theorem 2. So, if we choose to define "function" as "something that has a formula", then we may lose Theorem2.

Actually, to put it more succintly: if we insist that "function" is "formula", we lose the axiom of choice. Are we willing to do that?

We may also lose the notion of "countable set", since to show that a set X is countable we would have to give a formula for the mapping from X to N; and there are sets that can be specified, but not enumerated, by a finite formula.

All the best, Jorge Stolfi 03:57, 23 January 2006 (UTC)

Paragraph Two

We have agreed on Paragraph One:

In mathematics, a function associates an 'input' with a single 'output'. An example is provided by the squaring function, which associates each number with its square, so that the input 2 is associated with the output 4, and 3 is associated with 9, and so on.

Well, *I* haven't agreed at all on that. But since the vote is 2 against 1, I can only give up...Jorge Stolfi 22:01, 22 January 2006 (UTC)

Nor have I agreed. If that makes it 2 against 2, then we have more to discuss. I heartily sympathize with Rick's impatience to say something, but I still think we'd have more success by exposing our concerns and goals than by voting.

Consider the use of a formula. Compare to the description of a group by generators and relations, as in the group with generators a and b and the relation a²b⁻¹ = 1; the group itself exists independently of this particular description. Or compare a polynomial, which is an algebraic object, to a polynomial function, which is something completely different (though obviously closely related). We can use a polynomial like x² to define a polynomial function from integers to integers, or reals to reals, or quaternions to quaternions — all different.

Often we define a particular function using a formula or a rule and often we give the function a name; thus f: N→N, n↦n². Or we say that "y is a function of x", indicating a dependency of one variable on another. As a practical matter, almost every function that we use is defined by a succinct rule of some sort; on the other hand, most functions from the reals to the reals cannot be so described, and are not computable. And sometimes the rules are tricky beasts without a "formula", like mapping each real number to 0 if it is rational and 1 if it is irrational.

I want our readers to see a variety of examples of functions and ways of defining them and uses for them. I do not want them to get the mistaken impression that a function is a formula, or other method of specification. After all, we want to be able to talk about, say, function spaces such as ℓ² (described here). And we need to be able to discuss the subtle difference between a function and a distribution.

Jitse does not begin with a formal definition, and I do not join with Jorge in thinking we must. On the other hand, Jitse's first sentence feels uncomfortably vague. I've got other demands on my time at present, but I'll try to find an opportunity to write some introductory paragraphs myself, rather than just object to others. --KSmrq^T 04:07, 23 January 2006 (UTC)

My version õf the article had a whole section on the topic of "How to specify a function", which listed all those ways of specifying a function (and a few others). The section may still be there. However, that section has nothing to do with the definition of the "concept" of function --- which, I must insist, is independent of how one specifies the pairing, of how one evaluates the function, or even of whether such specification/algorithm exists or not.
Jorge Stolfi 19:08, 23 January 2006 (UTC)

The current Paragraph Two is out of the running, because it repeats the second sentence of paragraph one. I'll offer my paragraph two, others can offer theirs, and we can discuss. Rick Norwood 21:11, 22 January 2006 (UTC)

Proposed Paragraphs Two: Choice A (Rick Norwood):

A function is often named by means of a formula. For example, common names for the squaring function are $y=x^{2}$ and $f(x)=x^{2}$. Once a function has been defined, it can be applied to a specific input. For example, if f names the squaring function, we can write f(3) = 9.

Choice B:

Sentence one, again

I agree that we should spend some more time discussing the first sentence, especially since I hope that it will pay back later if we can come to an agreement for the first sentence. I believe we all agree that we should start with some kind of definition, which should be exact and understandable (though we differ in how much weight to give to the conflicting goals of understandability and exactness). Here are some possibilities that were mentioned:

1. The current beginning

The concept of a function is fundamental to mathematics. In intuitive terms, a function associates an 'input' with an 'output'.

2. Proposed by Rick Norwood

A function, in mathematics, associates an 'input' with a unique 'output'.

3. Adapted from 2.

In mathematics, a function associates an 'input' with a single 'output'.

4. Merriam-Webster online

A mathematical correspondence that assigns exactly one element of one set to each element of the same or another set

5. American Heritage Dictionary

A rule of correspondence between two sets such that there is a unique element in the second set assigned to each element in the first set.

6. Encyclopaedia Brittanica

An expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). In its most general usage in mathematics, the word function refers to any correspondence between two classes.

7. Proposed by Jorge Stolfi

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.

8. Adapted from 7.

In mathematics, a function is a pairing which associates each element from one set with a single element from a second set.

9. Another compromise proposal

Functions have inputs and outputs, and the defining property of a function is that a given input always produces the same output. More precisely, a function defined to be a relation from a set X (of inputs) to a set Y (of outputs) such that every element of X is paired with exactly one element of Y.

Comments

Please add your comments here, refering to one of the sentences above. Of course, you may also add another one. -- Jitse Niesen (talk) 13:40, 23 January 2006 (UTC)

I am certainly willing to keep working until we get it right, so, back to paragraph one. I agree with kSmrg that we must never say or imply that every function can be given by a formula, or that the formula is the function. On the other hand, in giving an example of a function, I think the use of a formula follows current usage. Let me check that, by pulling down three math books at random from the shelf behind me.

Topology by Munkres: "In calculus, a function is often given by a simple formula such as $f(x)=3x^{2}+2...$. Note that this is an example, not a definition.

Advanced Calculus by Buck: "The rule of correspondence may be described by a formula such as $f(p)=x^{2}-3xy$." Again, an example, not a definition.

Braids, Links, and Mapping Class Groups by Birman. Function isn't in the index. But, as they say, two out of three ain't bad.

I could pull down a dozen more books, and I think the outcome would be the same. As an example, not as a definition, $f(x)=x^{2}$ is ok.

Now, to turn to the nine choices we are offered for Paragraph One. 1, 2, and 3 say almost the same thing, using "input" and "output". 4 and 8 say the same thing, but avoid the words "input" and "output" and includes the word "set". 5 and 6 are simply wrong, for the reason kSmrg gives, though 6 goes on to admit that mathematicians don't actually require a rule. 7 starts the article with the mathematical definition. 9 combines features from all of the above. I like it, and it gets my vote, if we are going to vote. I could live with any of 1, 2, 3, 4, 8, or 9. Rick Norwood 20:59, 23 January 2006 (UTC)

[Stolfi:] Well, the words "input" and "ouptut" are almost never used in mathematics, and this is not the place to invent new nomenclature; so, for me, that excludes 1,2,3, and 9. Options 5 says "rule" which implies "finitely describable", which is wrong (it may be the result of a lexicographer "clarifying" what a mathematician told him); so 5 is out. Option 6 is actually two definitions, both wrong (depending on how one reads "relationship" and "correspondence", they define binary relation or bijection, not function) Also the definition in terms of classes is not widespread among mathematicians (who generally use "set" and consider naive set theory sufficient for their needs). Finally 4 and 8 use "corresponcence" and "pairing", which are no more intuitive than "relation", and are not standard nomenclature, so they would be actually confusing to readers who know a bit of math and are using wikipedia as a reference.
In conclusion, the only alternative I consider acceptable is (surprise!) 7. Sorry...
All the best, Jorge Stolfi 23:00, 23 January 2006 (UTC)

PS. However, I would be happy with 4 if it used "relation" instead of "mathematical correspondence":

10. In mathematics, a function is a relation that assigns exactly one element of one set to each element of the same or another set.

That is 100% correct (modulo the set/class issue) and succint. The only nit is the clumsy language "of the same or another"; but I have seen much worse in other model Wikipedia articles.
Jorge Stolfi 23:12, 23 January 2006 (UTC)

Perhaps we can split the difference. Try this one:

11. In mathematics, a function is a correspondence (specifically, a relation) that assigns exactly one element of one set to each element of the same or another set.

Perhaps this adequately conveys naive information (agreeing with the dictionary definition) and technical precision (agreeing with Jorge's original formal statement). I wish it didn't use the word "one" twice, but I'm uncomfortable changing "exactly one element" to "a single element". --KSmrq^T 02:23, 24 January 2006 (UTC)

I don't like the wording "of one set... of the same or another set"; it just comes off as too wordy to me. I think my proposed sentence (below) is clearer. It also attempts to meet Jorge's suggestions halfway. - Gauge 17:59, 24 January 2006 (UTC)

It is correct... and completely incomprehensible to non-mathematicians. I strongly object to suggestion 10 above for this reason alone. I thought 9 above was a good compromise between intuition and rigor. I think it gives the sense to newcomers "ah! i'm at the right article" while at the same time suggesting that a precise definition awaits in the article. I disagree with Jorge's opinion that certain cases (including 9) should be excluded for mentioning "input" and "output". Mathematically meaningless as these terms may be, I think they are very useful to people who are trying to gain some understanding of what a function is. We are not giving rigorous definitions for "input" and "output", so in my opinion this is not really introducing new nomenclature. - Gauge 02:50, 24 January 2006 (UTC)

Gauge, you accidentally removed my comments; there should have been a warning to that effect. I have reinserted them in place. --KSmrq^T 05:26, 24 January 2006 (UTC)

Sorry about that. I don't know how that happened; I didn't get an edit conflict so I presumed everything was okay. - Gauge 17:59, 24 January 2006 (UTC)

Another proposal, say 9′, because I realized I didn't like the wording of 9: In mathematics, a function is defined to be a relation from a set X (of "inputs") to a set Y (of "outputs") such that every element of X is paired with exactly one element of Y.

Note the use of the quotes around "inputs" and "outputs" which is an attempt to convey that these are to be used as intuition. - Gauge 03:07, 24 January 2006 (UTC)

Input and output

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)

Giving up

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)

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)

"edit boldly"

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)

working on the new version

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)

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)

Thanks! Rick Norwood 18:23, 25 January 2006 (UTC)

Three suggestions for the further improvement of the article.

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)

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)

It seems to me there are valid pedagogical reasons to name the input before the output. Comments? Rick Norwood 22:05, 24 January 2006 (UTC)

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)

I was talking about the former, and several other cases where it has been changed around so the output is mentioned before the input. I'll wait a little while for other opinions, then change that back if there are no objections. y = f(x) is standard -- though one textbook I am currently teaching out of writes xf = y! Rick Norwood 18:29, 25 January 2006 (UTC)

When are two functions equal ?

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)

I can't tell whether you're trolling here. I'll respond in order.

• Two functions are equal if and only if their domains, codomains, and graphs are equal (as sets). So your f and g are not equal even though they are defined by the same formula. It is precisely to cover differences in domain and codomain that these are included in the formal definition of a function. So yes, "surjectivity" is a property of a function, and the statement "a function together with its codomain" is redundant. Your indented statement is also true.

• That said, you've hit on the exact reason the formal definition of a function is confusing. The intuitive notion of a function is that it is defined by a definite process and that the domain and codomain are simply "references" or "context". If you want that, go to Turing machine. Your indented statement, though true, is misleading in its wording since it implies that you think the same squaring function acts on both reals and nonnegative reals: it doesn't; those are two different functions, one the restriction of the other. This is in the same order of confusion as the idea of a "partial function", which is contradictory but means what we think it should.

• The parenthesized comment is why I thought you were trolling. Of course math is different than plain language: it's more precise. "Two functions f, g" means I have just written two letters denoting functions. It's the same confusion, actually, that you get if you introduce a pair of identical twins as "one person": genetically true, but something higher-level makes it false. See also forgetful functor.

Ryan Reich 13:41, 25 January 2006 (UTC)

What he said. Rick Norwood 18:31, 25 January 2006 (UTC)

Only adding my inevitable comment that not all (perhaps even not most) mathematicians incorporate the codomain as a component of the function; if the simpler definition of a function is used, then the functions in question are equal. But in any context, follow the definitions that have been agreed upon (or imposed by the textbook, as in your math class :-) Randall Holmes 20:23, 25 January 2006 (UTC)

Thank you very much, gentlemen.

1. The square functions in my indented statement are not restrictions of one another, as they have identical domains: the set of real numbers. They merely have different codomains.
2. You seem to disagree on the formulation from the Function_(mathematics)#Specifying_a_function subsection: If the domain X is finite, a function f may be defined by simply tabulating all the arguments x and their corresponding function values f(x).

Bo Jacoby 12:23, 26 January 2006 (UTC)

Sorry; I misread your statement to mean that you restricted both the domain and codomain to the nonnegative reals. One way to see that these functions should still be different is that if you restrict each function to the nonnegative reals, then the one whose codomain is the nonnegative reals now has an inverse function (rather than "partial function") but the one whose codomain is the full reals does not. This is part of the reason the domain and codomain are included in a function's definition. Ryan Reich 13:25, 26 January 2006 (UTC)

No, I was not restricting either square function, and neither of them have an inverse. Ok, then consider the two square functions restricted to nonnegative reals. Then we can define an inverse, a square root, on the nonnegative reals, but not on the negative reals. Does this mean that it is practical or necessary or useful to distingush between two square functions, both being defined on the nonnegative reals and both having the same range, the nonnegative reals, but one of them having a codomain extending its range ? Any mathematician will testify under oath that the square function on the real numbers is well defined, but you insist that there exists an infinity of square functions on the real numbers, all having the same domain, and all having the same graph, but having different codomains? Honestly, I think it makes no sense. Still the concept of codomain does make sense. Any superset to the range of a function is a codomain. So a function has one range and an infinity of codomains. So you can define the set of all functions, A^B, having codomain A and domain B. Bo Jacoby 14:40, 26 January 2006 (UTC)

Here are a few points in response:

• Don't put words in the mouths of "most mathematicians". It is possibly true that they would agree that "the square function on the real numbers" is well defined, but by that they would mean the square function on the entire set $\mathbb {R}$ and not some subset, or with a different codomain than $\mathbb {R} .$ The reason is that "squaring" derives from the multiplication of real numbers, which is sensible only in the context of the real number field and not some arbitrary subset, even if the operations still happen to be defined there. Another way to respond to your suggestion is that indeed, the square function on the real numbers is well defined, as are the square functions on any subset of the real numbers, and these are just restrictions of the former. Nor do they have "the same graph", for it is truncated to include only tuples whose first elements are in the restricted domains. It is not even nonsensical to insist that these are different functions: would you object if I insisted that the subsets of the real numbers are distinct from the reals themselves, although they consist of the same elements? Your problem is one of language: just because the functions sound the same you insist that they should be the same, though there are good reasons not to.

• And those good reasons do exist, by the way. There is the matter of inverses, which I mentioned last time: depending on the domain and codomain, "the same" function may or may not have an inverse, but surely it is an intrinsic property of a function whether or not it is invertible. Also, depending on context it might be natural to consider a function as being different in an essential way from some extension to a larger domain. For example, you might be thinking about functions on the integers, say polynomials like $f(x)=x^{2}-x+1$, and you might be curious whether (and where) they achieve a minimum value. Well, on the real numbers this is a calculus question, but the calculus answer doesn't make sense for integers since it gives a fraction: not only is the location of the minimum different, there are actually two of them for integers. The extended function is quite different from the restricted one. You can turn things around, too: take a number-theoretic function like the prime counting function $\pi (x)$ whose value is the number of primes between 1 and x. The same "definition" works for real x but the reason the function is interesting for number theorists completely vanishes when you think of it as a real function: for the real numbers, primes are not interesting at all. And last, of course there are good category-theoretic reasons to remember the domain and codomain, as you observed.

• Finally, you are barking up the wrong tree here. This encyclopedia does not judge whether or not a fact is "reasonable", simply whether it is correct of its subject. Mathematics does define a function as a triple with domain, codomain, and graph, and if some mathematicians like to forget this it is for ease of use when the full information is unimportant, or when it is useful to pass back and forth between restrictions and extensions, or even just because it makes more intuitive sense (like you think it does), it is not because the full definition is somehow unreasonable in general. The place to make recommendations about mathematics itself is in a journal, or at least to a mathematician in an academic context. But even there you are unlikely to receive much sympathy since there is no point in weakening a perfectly good definition whose fine points actually have their uses, even if they don't have them everywhere.

Ryan Reich 01:10, 27 January 2006 (UTC)

"Mathematics does define a function as a triple with domain, codomain, and graph," is NOT entirely accurate. It's true in some contexts (category theory), false in others (set theory), and may be confused as you state in still other contexts. As for my take: Two functions which have the same graph are equal in some senses, and not equal in others. (The example phrasing of a set being equal to a class in NBG set theory comes to mind.) If one function is a restriction of another, they are clearly not equal -- but may still be considered equivalent in other contexts. Arthur Rubin | (talk) 01:48, 27 January 2006 (UTC)

Dear Ryan Reich.

• When you talk about restrictions I think you missed my point. I totally agree that functions, f,g, having different graphs, must be considered different, even if the equation f(x) = g(x) is true whereever it makes sense. The graph of a (proper) restriction of a function is a (proper) subset of the graph of the function.
• My point is about two functions, f,g, having the same graph, {(x,f(x))} = {(x,g(x))}, and consequently the same domain = { x | there exist an y such that (x,y) is in the graph}, and the same range = { y | there exist an x such that (x,y) is in the graph} .
• Any superset to the range of f is a codomain of f. The codomain cannot be uniquely derived from the graph. I fail to understand why f and g, having different codomains only, must be considered different. Usually they are not considered different. One talks about the sine function rather than about one of the infinitely many equivalent sine functions having different codomains.
• If the graph {(f(x),x)} defines a function, then f has an inverse, f⁻¹, having that graph, {(y,f⁻¹(y))} = {(f(x),x)}. The domain of f⁻¹ is the range of f. Bo Jacoby 09:21, 27 January 2006 (UTC)

Dear Arthur Rubin. Thank you for the reference to NBG set theory which I will study. Bo Jacoby 09:21, 27 January 2006 (UTC). PS: Now I read it, but to me it did not answer the question: When are two functions, having the same graph, not equal ?

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)

Dear Randall. Thank you very much. You answered my original question: Is there consensus whether f = g ?, and the answer is: No, there is no consensus. The new definition defines a different concept, which should be given a different name in order to avoid confusion. (The important theorems on unique solutions to differential equations fails under the new definition of 'function'). Bo Jacoby 14:46, 27 January 2006 (UTC)

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)

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)

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)

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)

moving history

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)

How things stand.

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)

shorter but not yet more careful

Norwood was quite right to shorten the definition section (I think the style is better) but his definition was wrong. I have repaired it. I think the total effect is better than my original paragraph, thanks! The main problem was that those who adorn functions with domain and codomain invariably also adorn relations with domain and codomain, so it is not correct to refer to the third component as a binary relation in this definition. In the opening, the reference of the phrases "domain of f" and "codomain of f" was confusing; I replaced them throughout with just X and Y, and defined domain and codomain subsequently (and I do mean "the" domain and "a" codomain). Overall, it's an improvement. Randall Holmes 17:27, 26 January 2006 (UTC)

none of us are perfectly careful: I had to change my fix to the references of domain and codomain; of course, no definition of domain and codomain is needed, since we already know what a binary relation is... Randall Holmes 19:08, 26 January 2006 (UTC)

I also changed the first sentence. More precise definitions are required in mathematics; precise mathematical definitions are not only needed by "professional mathematicians" (the professional mathematicians tend to be the producers of formal definitions, but there are other consumers).

The opening paragraph and naming conventions

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)

The point being that whether you like any of these forms or not (I happen to like all of them for different purposes) all of them occur in high school algebra class; all of them are familiar to our consumers. It is not the business of an encyclopedia article to endorse the particular usage that we favor (the later text of the article, not the intro, could spend time bringing out the advantages and disadvantages of each usage; we can of course freely express our personal opinions on the talk page!). Randall Holmes 20:58, 26 January 2006 (UTC)

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)

OK, I'm glad we're not having an edit war :-) Randall Holmes 22:04, 26 January 2006 (UTC)

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)

Multidimensional functions?

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)

OK, on closer inspection I can see where it ends. Never mind! Randall Holmes 22:09, 26 January 2006 (UTC)

Capital function

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)

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)

Topmost picture is a faulty example

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)

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)

I agree with Randall Holmes. Rick Norwood 19:59, 27 January 2006 (UTC)

Not satisfied with the first sentence

The latest edits look mostly good to me, but the first sentence "In mathematics, a function relates each input to exactly one output" is irksome; it doesn't even attempt to say what a function is, and doesn't explain what these "input"s and "output" are or where they come from. We should at least make an attempt at providing some sort of definition in the first sentence. My recommendation of course would be 9′ or similar, but I am open to other suggestions. - Gauge 08:42, 28 January 2006 (UTC)

This has been discussed at great length. The naive definition of a function is that a function is a "thing" that relates inputs and outputs. What kind of "thing" a function is (a set of ordered pairs) is covered just a little further down the page.

That said, I could live with 9, but not with 9', provided the change could be made without starting up a new round of editing wars. (My objection to 9' is that it defines a conceptually simpler word, "function", in terms of a conceptually more difficult word, "relation".) Rick Norwood 14:05, 28 January 2006 (UTC)

What I don't like about 9 is that it begins with "Functions have inputs and outputs, and the defining property of a function is that a given input always produces the same output" which sounds more like a second or third sentence than a first sentence; in particular, it doesn't say what a function is. The word "relation" in 9′ is intended in the naive sense rather than in the rigorous mathematical sense. We can try to find a different word for that if you insist, but insofar as it is a "thing" which relates inputs and outputs, I figure such a thing can be called a "relation" without confusion. - Gauge 20:47, 28 January 2006 (UTC)

Here we go again.

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)

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)

the mathematical definition is not so easy to rewrite correctly

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)

further, there is actually only one definition of function given, which is read in two different ways (unpacked into two different more explicit definitions) depending on how "binary relation" is defined. The person who changed it did so without realizing the structure of the original single paragraph, and isolated the single definition under his "basic" (now "first") definition. What is present now is correct (I think) [deleted my crabby remarks -- RH] Randall Holmes 14:54, 28 January 2006 (UTC)

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)

Is a function more than it's graph revisited.

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)

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)

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)

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)

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)

Does the Wikipedia model really work for mathematics?

I am developing a more fundamental doubt (see what Norwood says about Oleg's reverts under the previous heading. I don't see how we can possibly have sensible articles on core concepts on whose definition everything else depends unless someone competent writes them and they are then frozen and edited (by their manager or by a limited class) after consultation only. This doesn't apply to all topics, but this article and function (mathematics) for example are about ideas about which many people have ill-informed, strongly held ideas and about which other people, perhaps not so ill-informed, have ideas based on philosophical or pedagogical ideas which deviate too far from the norm for easy accommodation. It was interesting to be able to write an article on New Foundations for people to read -- this is unlikely to attract the attention of too many people of the categories mentioned; articles about obviously technical subjects are not usually subject to this kind of problem, and seem to look pretty good. But central ideas of mathematics (especially ones about which silly statements are prevalent in low-level textbooks or in the popular literature) must require a constant painstaking watch which in the end may not be a sensible use of the time of competent people. (JA should not necessarily assume that I am referring to him). Randall Holmes 02:23, 29 January 2006 (UTC)

I agree that it is probably a good idea to revert major changes to this article that haven't been discussed on the talk page. If we make this a kind of informal policy, it may help the edit sprees. Speaking of which, I just did one myself, mostly cleaning up typos, spelling, notation, and wording. This is the first time I've read through the entire article, and it seems pretty good. The first sentence needs replaced, though— it really doesn't agree with me for reasons I stated earlier. - Gauge 03:08, 29 January 2006 (UTC)

Thanks to Gauge and Randall for improving details in the article. I still need an explanation for the use of the second definition of function and relation. We also need a link to multiplication, and consensus within the article whether the product is written xy or x·y or x×y. The multiplication function should be written f((x,y)) rather than mul(x,y), which is not standard anyway. Regarding Randell's doubts on the wikipedia way, it may happen that the competent mathematician is not a perfect author. For writing an article that nobody can edit, wikipedia is not the right medium; teamwork may be frustrating. Considering the risk of incompetent editing, I find the wikipedia articles surprisingly good. Bo Jacoby 12:00, 29 January 2006 (UTC)

I would say, yes, wiki works for math. For example, all of today's changes have corrected mistakes and improved the article. I'm told that the half life of an error on a wiki page is fifteen minutes.

The use of mul(x,y) illustrated a point. There is no need for a link to multiplication, or for a wiki standard of how multiplication is to be written. The appropriate symbolism varies with the context. Rick Norwood 14:45, 29 January 2006 (UTC)

I was in an extremely bad mood when I wrote the above; a longer axis of observation will give more information :-) Randall Holmes 21:44, 29 January 2006 (UTC)

Reading The Wisdom of Crowds will enlighten your spirit. Don't worry, my friend, we are doing very well. Bo Jacoby 23:21, 29 January 2006 (UTC)

n-ary operations

There are such things. Randall Holmes 21:45, 29 January 2006 (UTC)

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)

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)

Eliminating repetition

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)

Sorry to make so many changes in one day, but that was the only way I could keep clear in my own mind what had already been written and the necessary logical order of presentation. The material I was working through had many concepts defined twice, some concepts defined in terms of concepts only mentioned later in the article, and a number of references to sections that no longer exist. If anyone feels that the new section is now too long, it can easily be broken up into two or more parts. Rick Norwood 23:01, 2 February 2006 (UTC)

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)

I'm glad you liked the edits. I blush to admit that I don't know the difference between a space and an nbsp. I've been typing the formulas ''f''(''x'') because that's how one of the Wiki gods told me to do it. Rick Norwood 23:52, 2 February 2006 (UTC)

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 (" "). --KSmrq^T 06:35, 3 February 2006 (UTC)

Thanks for the explanation, KSmrq. I've wondered what those little boxes were (made of ticky tacky, I assumed).

I have, with some trepidation, deleted a couple of sections, because I think all of the material in them is covered above. If anyone wants them back, please put them back. Rick Norwood 13:39, 3 February 2006 (UTC)

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. --KSmrq^T 15:12, 3 February 2006 (UTC)

Reverted text

The following text was reverted. It has some useful thoughts, but lacks important features of the text replaced. Also, it mistakenly changed "solutions of integral equations" to "solutions of integrals", something completely different. --KSmrq^T 03:02, 8 April 2006 (UTC)

Right. Next time I'll think again before writing three paragraphs.

Please understand that I'm not striking up an attitude. But put yourself in my place for an instant. --VKokielov 05:43, 8 April 2006 (UTC)

I also don't think I'm an it. Humbly yours, --VKokielov 05:46, 8 April 2006 (UTC)

The pronoun "it" is meant to refer to the text, not the author. I copied the text here so it would be easy for others to see, not lost in a reverted history. I've been in your place, so I know what it can feel like to be reverted for honest efforts. The idea of the talk page is for different editors to work out what's best for the article. See below. --KSmrq^T 10:58, 8 April 2006 (UTC)

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Specifying a function

To define a function precisely, all that is needed is a rule which for every element in the domain specifies a (unique) element in the codomain. All the properties of a function, whether they reflect individual elements in the domain or codomain, or again subsets of elements, may be derived from this kind of rule.

There are two basic ways to define a function.

An algorithmic definition is a way to go from some explicitly specified form of the input to another, again explicitly specified, form of the output. For example, if the domain X is finite, a function f may be defined by simply tabulating all the arguments x and their corresponding function values f(x), because every element of the domain is made to correspond to (one) element of the range. Or a function may be defined recursively, in terms of previous values of the function. It is key to remember that algorithms, while they always define functions, are not in a one-to-one correspondence with them. That is, a given function may be computed in a variety of ways, and some functions (or values of functions) we may not be able to compute at all. For some of these functions, we define algorithms which compute related functions, often functions whose value can be made arbitrarily close to the value of a function we are trying to compute, but cannot (since our representations for, e.g., $\pi$ are non-terminating) be made equal to that value. Taylor approximations work in this way.

More fundamentally, and in practice far more often, a function is defined declaratively. A declarative definition describes the relationship between the input in the domain and the single output in the codomain; but carries no information on how to compute the function. The most common example of a declarative specification is an explicit or implicit formula. A formula combines simpler functions in an unambiguous way to define a more complex function; it is called implicit whenever there is anything other than the name of one variable on either side of the equals sign. For example, we know that $x^{2}$ is a funciton mapping one real number onto one nother, and we know that $x+y$ maps two real numbers to (again) one other. Now we can write $x^{2}+y^{2}$, which takes two simpler functions, the squares, and links them with the third, addition, to get an addition of two squares. Meanwhile, $ax^{2}+bx+c=0$ defines implicitly a function which solves the equation; explicity this is the quadratic formula.

Declarative definitions of functions are the most common and the most useful in mathematics. Common declarative specifications are: closures, limits (in particular, infinite series, which are limits of sequences of real numbers and thus cannot be computed to a finite decimal in general), analytic continuation, and as values of integrals and of solutions to differential equations.

There is a technical sense in which some mathematical functions cannot be defined at all, in any effective way, explicit or implicit. A fundamental result of computability theory says that there are functions that can be defined precisely, but not computed.

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Discussion on the above text

This effort gets off on the wrong foot. We must be very careful not to state or imply that functions are defined by rules. As the previous text says, most (not just "some") functions cannot be defined by rules. So that must be fixed.

Most? --VKokielov 01:56, 11 April 2006 (UTC)

Yes. Most real numbers (uncountably many) are irrational; most functions cannot be defined by rules. --KSmrq^T 02:56, 11 April 2006 (UTC)

What's a rule? If you have the time to explain. I'm really asking, and not arguing. --VKokielov 18:40, 11 April 2006 (UTC)

For purposes of this "most" idea we can be generous and say that a rule is anything we can write down in any language in a finite string of symbols. That makes the set of rule-definable functions countable, hence of "measure zero" in the set of all functions. --KSmrq^T 07:38, 12 April 2006 (UTC)

But why is that what a rule is? Why isn't a rule, say, any description of how the function is constructed? Why isn't a rule a series of implications A-->B-->C-->... which lead to a constraint that narrows the space? For instance, by a simple analysis we can deduce that some functions must be continuous, or differentiable, or...Is there any particular reason for calling a rule a finite string of symbols? --VKokielov 01:27, 13 April 2006 (UTC)

I am, in fact, asking for the criterion. There's no philosophy here -- I'm not htat much of a hack :) -- but the matter of fact remains: these functions which can't be described with rules -- are they important? are they relevant? If we can't describe them, how can we conclude anything about them? --VKokielov 01:29, 13 April 2006 (UTC)

Free your mind. Write down any description you like. Use natural language (English). Use fancy notation. Use sophisticated theoretical constructs. The fact that you can write a description of any kind has limited you to countably many functions, a negligible sample of all possible functions. But useful, nevertheless!

Are the others "useful"? For many some pure mathematicians those are "fighting words". Yes, their existence is vital for various proofs to go through, just like the existence of all the uncountable real numbers we can never calculate. :-D --KSmrq^T 09:15, 13 April 2006 (UTC)

I still don't see the relationship. Who says that what what 'm trying to describe is in any proportion with the language I'm using to describe it? In other words, if I write out the words "if there exists a number N, there exists a number N + 1", haven't I just postulated that there exists no biggest number? And, if that's the case, can't I draw a big infinity symbol next to those words and henceforth call the idea "infinity"? --VKokielov 19:12, 13 April 2006 (UTC)

Better yet, the idea "uncountably many" itself is a derivation. isn't it? For instance, we say that 2^N (where N are the natural numbers) is not countable. We also say 2^N exists. Now when we write down 2^N and say it's the set of all subsets of N, isn't that a good enough definition? --VKokielov 19:14, 13 April 2006 (UTC)

I'm beginning to get lost in this discussion. If I say there are functions f(x)=ax for a is any real number. I have just described more than countably many functions with only a few dozen symbols. So how can you maintain that all functions described by any kind of rule are in a countable set? −Woodstone 19:32, 13 April 2006 (UTC)

Don't confuse the forest with the trees. We can certainly define the set (or field) of real numbers. That is quite different from defining each specific real number. Likewise, we can define the set of continuous real-valued functions over the unit interval, but we cannot write down a description for each function. Does that help? --KSmrq^T 23:56, 13 April 2006 (UTC)

It doesn't help me, unless I understand what you mean by "description". --VKokielov 05:18, 14 April 2006 (UTC)

Then you're going to have to do some work on your own. I suggest you read about computability theory for a related (but different) example. Do you understand the difference between countable and uncountable? If we can write down a description, using English and mathematical notation and so on, we have just written a finite string of letters and relation symbols ("+", "≤", whatever) and numbers and so on. I don't care how you write it; it's a finite string. Therefore, the set of all such descriptions is countable. If this kind of thinking is new to you, read about Cantor's diagonal argument. It's fun stuff, not very hard, but surprising. It's also a basic part of a mathematician's education. --KSmrq^T 10:07, 14 April 2006 (UTC)

Thank you for leading the discussion. I'll see what I can find out. --VKokielov 13:21, 14 April 2006 (UTC)

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Calling a tabulation an algorithm is slightly perverse, although a finite tabulation can always be made into an algorithm. It is a mistake to claim that an algorithm involving π cannot be exact; mathematical algorithms need not use computer floating point.

The term "declarative" may be unfamiliar to most mathematicians. This paragraph has numerous typos. And the examples make no sense; for example, the roots of a quadratic equation would not generally be considered a function (see the square root example at the beginning of the article).

Two functions.  :) --VKokielov 01:58, 11 April 2006 (UTC)

Note that recursive functions are already discussed later in the article. Also, there is some delicacy involved in assigning meaning to recursive definitions, with different fixed-points giving different functions.

The original grab-bag of ways to define a function includes possibilities not covered in the rewrite. (The problem with integrals, as opposed to integral equations, is again noted.)

For all these reasons, the text really had to be reverted. This is not to say that the previous text is ideal; for example, it is awkward in places. But the replacement needs work before it can succeed. --KSmrq^T 12:04, 8 April 2006 (UTC)

function notation

In my fairly short university career I have be taught that if you define a function as f(x)=x^2, for example, then you can also say f=x^2. Similarly if you say y=x^2 then you can also say y(x)=x^2. i.e. There is no difference between f(x) and f they mean the same thing but with slightly different stresses on what the function is used for.

What I am suggesting is that this should be be flagged as different notation "f(x)=y" saying that instead "f(x)=f" or "y=y(x)".

The classic notation is y = f(x). In the past twenty years, I see just f and also y(x) more and more frequently, and see no problem as long as these notations are used unambiguously. If they are mentioned in the article, it should be mentioned that these are modernisms. Rick Norwood 14:13, 27 May 2006 (UTC)

Study of foundations and use of computers has forced more careful use of notation, though convenient brief forms persist. Contrast

$f\colon \mathbb {C} \rightarrow \mathbb {R} \,\!$

$x\mapsto x{\bar {x}}\,\!$

with

$f(x)=x{\bar {x}}\,\!$

or

$y=x{\bar {x}}.\,\!$

More provocatively, contrast with Standard ML's

fun f x = x * conj x

or the C programming language

float f(struct complex x) {return ctimes(x,cconj(x));}

or the Common Lisp

(lambda (x) (* x (conjugate x)))

In the SML case type polymorphism broadens the domain and range possibilities, while in the C case the result is restricted to a float rather than, say, a double. The Lisp case does not name the function, nor does it restrict the domain and range. --KSmrq^T 18:45, 27 May 2006 (UTC)

Non-Numeric Functions

"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)

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)

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?! --KSmrq^T 23:49, 26 June 2006 (UTC)

notation for functions.

See Talk:Fourier_transform#Where_to_put_it. The conventional notation for functions, where you write f(x) meaning f, is insufficient when it is not obvious what the independent variable is. Please comment on the suggestion for a solution. Bo Jacoby 11:44, 7 August 2006 (UTC)

I think what this article does, begin with the simple case and go on to more advanced cases, is the only way to go. Another advanced problem is free variables vs. bound variables. Probably worth discussing, but only near the end of the article.

A discussion of independent and dependent variables should be discussed in the context of implicit functions. Rick Norwood 13:27, 7 August 2006 (UTC)

Bo, a simple request: provide a mainstream citation showing that your proposed notation (e.g. f = (x^y←x) instead of f(x) = x^y) is an established notation in mathematics. When you were asked for citations for this notation in Talk:Fourier transform, the only thing you provided was a loosely similar syntax in the J programming language. (I've seen ← used to denote functions, and I've seen "=", but I don't recall seeing this kind of combination of "=" and ←; it seems obscure at best.)

Wikipedia should stick with the conventional notations for a given topic; more obscure/uncommon ones can be mentioned but should be presented as such. Notations not used in mainstream mathematics publications should not be included in Wikipedia at all, according to policy.

—Steven G. Johnson 05:36, 11 August 2006 (UTC)

Steven, I appreciate your entering a discussion rather than to delete my contribution. You are slowly growing civilized.

The use of the colon in the notation, f$\;:\;$x$\mapsto {}$y is unconventional and pointless and confusing. Equality should always be written with the equality sign: f=x→y. Parentheses should be used to fix the order of operations: f=(x→y).

The arrow points from the independent to the dependent variable. Usually an asymmetric symbol signifying an asymmetric connection is reversed to mean the reverse connection. For example, (a<b) is the same thing as (b>a). Both notations are free to be used. So (x→y) has the same meaning as (y←x), that x is mapped to y and that y is a function of x.

The lambda notation, λ x.y, uses the symmetrical symbol (.) to signify an asymmetrical connection between independent and dependent variable. This makes it hard to read. Using the arrow instead of the dot is an improvement: λ x→y . Now the lambda is no longer needed and we are left with x→y.

The convention that letter x implicitely signifies the independent variable, is useful for simple cases, but insufficient for complicated cases. An explicite indication of the independent variable is needed to distinguish the power function x^y from the exponential function x^y.

You know the equality sign, the arrow and the parentheses. The combination f=(y←x) means what it is supposed to mean. Even if you don't recall seeing this combination of digits: 1329685, you will not request me to provide a mainstream citation showing that it is an established notation in mathematic.

Bo Jacoby 10:20, 11 August 2006 (UTC)