Talk:Function (mathematics)/Archive 4

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For reference: Dunford and Schwartz, Part I

For future reference, I thought it might be helpful to see how Dunford & Schwartz (1958, p. 3) introduce functions.

The terms function, mapping, transformation, and correspondence will be used synonymously. The symbols f: A→B will mean that f is a function whose domain is A, and whose range is contained in B; that is, for every a∈A, the function f assigns an element f(a)∈B. If f: A→B and g: B→C, then the mapping gf: A→C is defined by the equation(gf)(c) = g(f(a)) for a∈A. If f: A→B and C ⊆ A, the symbol f(C) is used for the set of all elements of the form f(c) where c∈C. If f: A→B and D ⊆ B then f⁻¹(D) is defined as {x|x∈A, f(x)∈D}. The set f(C) is called the image of C and the set f⁻¹(D) is called the inverse image of D. If f: A→A and C ⊆ A, then C is said to be invariant under f in case that f(C) ⊆ C. The function f is said to map A onto B if f(A) = B and into B if f(A) ⊆ B. The function f is said to be an extension of the function g and g a restriction of f if the domain of f contains the domain of g, and f(x) = g(x) for x in the domain of g. The restriction of a function f to a subset A of its domain is sometimes denoted by f|A. If f: A→B, and for each b∈f(A) there is only one a∈A with f(a) = b, f is said to have an inverse or to be one-to-one. The inverse function has domain f(A) and range A; it is defined by the equation a = f⁻¹(b). Thus the domain and range of f⁻¹ are the range and domain, respectively, of f. The characteristic function χ_E of a set E is the real function defined by the equations χ_E(s) = 1, s∈E, and χ_E(s) = 0, s∉E.

Sometimes, when the range of a transformation f: A→B is to be emphasized at the expense of f itself and its domain, we shall write f(a) as b_a. Then f(A) is said to be an indexed set, and A is said to be a set of indices. If B is a collection of sets, the union ⋃f(A) will sometimes be written as ⋃_a∈A b_a, and ⋂f(A), as ⋂_a∈A b_a.

A relation in (or on) a set A is a collection r of ordered pairs [x,y] of elements of A. It is customary to write xry for [x,y]∈r. Other symbols for relations are =, ≤, ⊂, ⊆, ∼, and ≡.

Dunford, Nelson James; Schwartz, Jacob T., Linear Operators, Part I: General theory, John Wiley & Sons, ISBN 978-0-470-22605-6

Several features are of interest. One, in particular, is that functions are not defined using relations; in fact, relations are introduced after functions. --KSmrq^T 21:57, 17 September 2007 (UTC)

Wonderful text. Thank you very much. Here invertible is synonymous of injective (one-to-one), rather than bijective (this alternative definition should be explained in the article Inverse function). Paolo.dL 08:58, 18 September 2007 (UTC)

Thanks, but wrong conclusion. The book was written half a century ago, and it would no longer be acceptable to call a non-bijective function invertible. --KSmrq^T 05:54, 19 September 2007 (UTC)

Wonderful. That's good news. Robert Norwood seemed to maintain the contrary (see above), but possibly I misunderstood his point. I am waiting for his answer. Paolo.dL 12:50, 19 September 2007 (UTC)

I should qualify my remark. Given f: A→B and g: B→A, we can define inversion as g(f(a)) = a, f(g(b)) = b, or both (for all a∈A and all b∈B). When we say "inverse" without qualification, we usually mean a two-sided inverse unless the context implies something else. Otherwise, for the path

A{\xrightarrow {f}}B{\xrightarrow {g}}A\,\!

we must have (at least) that f is an injection and g a surjection for the composition to be an identity. Examples are commonplace. With matrices, an example is

{\begin{aligned}F&={\begin{bmatrix}0.48&0.36\\0.64&0.48\\0.60&-0.80\end{bmatrix}},\\G&={\begin{bmatrix}0.48&0.64&0.60\\0.36&0.48&-0.80\end{bmatrix}},\\GF&={\begin{bmatrix}1&0\\0&1\end{bmatrix}}.\end{aligned}}

Here we speak of left and right inverses, even though neither F nor G would be called an invertible matrix. (An invertible matrix must be square, although we do have pseudoinverses.) Other routine examples are the split of a short exact sequence, and the section of a fiber bundle. --KSmrq^T 19:46, 19 September 2007 (UTC)

When you define inversion as g(f(a)) = a (as in the article inverse function), I guess you can deduce that f and g are bijective by assuming that they are true functions (i.e. single-valued and total relations, according to the formal strict definition). In other words, you need to assume that the meaning of the word function (in the expression "inverse function"), is not "weakened" by the adjective "inverse" (as well as in "partial function" or "multivalued function"). Am I correct? Paolo.dL 21:40, 19 September 2007 (UTC)

Eh? I just said that knowing only one-sided inversion we cannot deduce bijection, and I gave three examples. But a function with a two-sided inverse must be a bijection. (And, yes, of course, f and g are both honest functions.) --KSmrq^T 01:02, 20 September 2007 (UTC)

I am sorry, but I am confused because some editors of inverse function seem to disagree with you, and I had not enough time to figure out with certainty whether you are saying the same thing with different languages, or you really disagree and one of you is wrong. Jim Belk (a mathematician) recently imposed the adoption of the one-sided definition in the definition section of the article Inverse function. And in the same section, he wrote "...a function is invertible if it is ... a bijection". Moreover, in another section ("Characterization") he wrote that the two-sided definition (i.e. your "both") is equivalent to the previously given one-sided definition. Oleg Alexandrov and Sam Staton wrote that they liked Jim's new definition. But I trust you. So, I was trying to understand whether, under some particular assumptions or conditions, your statement may become compatible with the text by Jim Belk. Paolo.dL 09:44, 20 September 2007 (UTC)

NOTE. We are discussing about main and alternative definitions of inverse function in Talk:Inverse function. Please post there your comments about this topic. Thanks, Paolo.dL 20:36, 21 September 2007 (UTC)

Definition section and article structure

The recent changes to the Definition section by user Wvbailey (talk · contribs) undo all the hard work of trying to make this article helpful for general readers as well as experts. The rewrite was well-intentioned, I'm sure, but a bad idea. (I also have other complaints, but there is no need to go into those.) Therefore I have restored the previous version and saved the excised material at Function (mathematics)/Definitions. Perhaps some of this material would be appropriate for an article devoted to exploring diverse definitions of functions, but not here. --KSmrq^T 09:50, 29 September 2007 (UTC)

I don't have trouble with spinning off a new article -- after I saw my changes the detail seemed to bloat the article -- but I do have problems with this article. It certainly will be of no use to experts; it has no in-line references and has just a small number of references in general. It doesn't seem "glued together" with an "umbrella-concept" or -concepts, but rather a confusing pastiche of little topics pulled from here and there. The little set drawings are wrong/incomplete (cbm and I corrected the caption in one of them) the origins of the terminology is not well documented (with inline references) and it does not reflect modern usage. The notion of "Cartesian product" would be very useful: A collection of ordered pairs, some of which are plucked out to make "the function" and "graphed", make the link between "set theory" and "analysis" clear, *see below. I'll work it from that angle on the linked article. We need some other experts to weigh in -- Travatore and cbm may have opinions...

I was hoping to do that in this section. Also, make a stab at wondering what a "function" really is (Minsky 1967 makes a stab at this when he writes about "criticism, that the interpretation of the rules is left to depend on some person or agent" (p. 106)). Also. Re points in planes and "graphs", "coordinates", etc, only a mention of "infinitesimals" would be needed to pull set theory/computation theory and analysis together cf Enderton p. 176-177 in Chapter 2: First-Order Logic, subchapter Non-standard Analysis: Algebraic Properties. Also William I. Mclaughlin, Nov 1994, Resolving Zeno's Paradoxes, Scientific American.

wvbaileyWvbailey 15:26, 29 September 2007 (UTC)

If you will glance back over the discussion above, you will see that there are two major points of view. One is that, at least in the opening paragraphs, articles should address the general reader. The other is that the article should be aimed at mathematicians. My understanding is that Wikipedia policy is to address the general reader -- the professional will be looking in a graduate text or monograph, not in a general encyclopedia. Rick Norwood 21:35, 30 September 2007 (UTC)

I ~~suscribe~~ (how you spell it? jeezum... subscribe... ) to both opinions simultaneously. (One cavaet: get some Harvard referencing in the article, and do the job right, or I will have a hissy-fit). Secondary or subsidiary pages can be for the experts. But they're okay to have. KSmrq's movement of my rewrite was, in that sense, the right move. (BTW I had to rename this spun-off article because the wiki name-space no longer allows " / " in the so-called "name-space").

Encyclopedia Britannica articles seem to follow the same course you are proposing. They start off slow and have links to much more complex topics. The difference is: they've had a lot of practice and they do a good job of it. Here's their lead:

"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. For most functions the variables range over classes of numbers...[etc]{EB 2006 Ultimate Referenece suite DVD 30 sept. 2007)

(They quickly get into "Geometric representations of functions", i.e. x-y plots, by the way.) I like the above definition. What I don't like is a crass, random, (but set-theoretically quasi-correct) scattering of arrows between little set-bubbles. This is not what most people think of as "a function". What is missing is the notion of "relationship" in the sense of a "predictive" (as in extrapolative) dependence, because an algorithm is at work. The Merriam-Websters that I own has two definitions, the first is the set-theoretic, the second is similar to the Britannica quote, and has to do with "y being a function of x", and by virtue of their example, y having a (clear-cut, non-random) dependence on x. Their example:

"height is a ~ of age>

Here we see a (predictive-) "rule" at work, something like height(age) = H*(1-e^-k*x), or whatever.

Anyway, I won't link the spun-off article to this article until I can sort this out and get my thoughts into some kind of shape ... that is, if the 'bots leave me alone. In the past I've had problems with over-zealous 'bots deleting pages that are not linked because they're works-in-progress. wvbaileyWvbailey 23:01, 30 September 2007 (UTC)

For your convenience, when I removed the considerable material you had written for Definition I placed it in a temporary holding page. It was clearly unsuitable for a standalone article without further work, and also I did not want to force a name upon you. We may use subpages (indicated by the slash) for work-in-progress, but the page was never intended to be a final resting place. My apologies for not explaining that.

One problem that is obvious to me is that you have a final footnote, styled unreadably small, so long and chatty it could be its own article. Others are almost as bad. Either don't say these things, or find a way to incorporate them into the body. If you have a dedicated article to spread out in, structure it well so you don't need the footnotes.

I am not advocating sprawl; I suggest a visit to our editor resources page for guidance as to what we strive to achieve and ways to improve.

To respond to your concerns about this article, "Function (mathematics)": Yes, it can be improved. The first improvement would be to back out some of Paolo's "improvements". We can easily fix clumsiness like the position of the Notation section, which comes long after we have already been using the notation. However, everybody has a stake in "function". Kids are asked to graph functions long before university; every science, hard or soft, uses functions; mathematics both uses and studies functions in many forms, from recursive function theory to calculus to functional analysis. Computer science has its own rich and diverse take on functions, overlapping some with mathematics interests, but best kept separate. (See semantics of programming languages, computability theory (computer science), automata theory, and so on.)

Most editors have seen almost none of this broad range, nor have any reason to. We can give a taste by displaying the Y combinator, λf·(λx·f (x x)) (λx·f (x x)). The definition of "function" required to give meaning to this expression is of no use (and incomprehensible) to the child asked to "graph a function" like y = 3x+2, and also of no use to the functional analyst working with the Hilbert space of square integrable functions.

I believe we should speak first to the child and to the lay adult, then to the scientist and to the engineer, then to the first-year calculus student and to the undergraduate mathematics major, and only then to any specialists.

I don't mean to discourage efforts to improve. Just, when you write, imagine that you are speaking to your precocious ten-year old daughter who asks, "Daddy, what's a function?", or to your beloved 18-year old son taking calculus who asks, "Mom, what's a function really?".

Improvements here will require patience, and wisdom, and patience, and negotiation, and patience, and learning, and patience, and careful writing, and patience, and flexibility, and … did I say patience? --KSmrq^T 05:55, 1 October 2007 (UTC)

Let me second what KSmrq said above, with the exception that I would aim first at the layperson rather than the child. Wikipedia is not a children's encyclopedia, and has content unsuitable for children. Also, I think we want, while avoiding technicalities, to avoid giving the false impression that a function is a "formula". Rick Norwood 12:46, 1 October 2007 (UTC)

With respect to ~~Paolo's edits~~ "inverse function section": too complicated and too much: the article seems way out of balance with too much detail in this particular area and perhaps too little in others -- ~~his~~ stuff could go into the linked article inverse function. (But it is pretty, with the graphs). I frankly don't understand this emphasis on "inverse function" ~~-- it appears prominently in the Britannica article too -- so I guess I have to go with the flow on this one~~.... Given that the two major equation-formation "schemas" of primitive recursion are: "definition by substitution" and "primitive recursion" (cf Godel 1931, Kleene 1952:221) I would think that these should be discussed in detail before "inverse function". (Kleene deals with "inverse" from the point of view of formal number theory's inverse laws for 0 and 1, and for + and * (i.e. multiplication) cf p. 186, which is correct but not much help, or is it?).

With regards to the spun-off page, it will remain linked to this article but not be linked from this article to it, until I feel it is okay, which could be a long time -- I'm in the discovery phase right now. I agree with KSmrq's comments about the footnotes, with this caveat: wikipedia is encouraging "in-line citations". Most articles I am involved with now are using footnotes. I experimented with in-text, in-line cites but have come to footnotes because they don't clutter the text. There is probably some way to make the dinky footnote-text bigger, but I don't know what it is. See Hilbert, for example-- after I inserted the citations I didn't like what I saw; at the outset another editor disagreed with my use of them, and when I saw the light and came to agree with him, he went ahead and converted the in-line cites to footnotes.

With regards to the bizarre and arcane. λ-calculus expressions: ick. Why bother? The λ-calculus was eschewed by Gödel at the outset, and he later came to eschew even his own recursion theory (i.e. Hilbrand-Godel recursion, mu-recursion) in favor of Turing's machines (much more about this at the expansion article off Church-Turing thesis).

With regards to computer science -- I don't agree with you here. Our disagreement is probably a chasm. I am not a computer scientist, I'm an engineer educated into advanced calculus and advanced complex variable theory. But that was then (late '60's), and this is now, 40 years later. I used that stuff extensively (especially Fourier transforms, expansions, exponentials), but also programmed in assembly language, throughout my whole career. Basically my sense of it is this: the "industry-consensus" is now and has been since the mid-1960's (since Godel blew off λ-calculus and recursion theory), that the Church-Turing thesis is pretty much the "Turing" thesis and to reduce calculation to computation removes the mystery of tacit knowledge that most of us bring to a calculation. What we are seeing in the more advanced elementary/undergrad texts, e.g. Boolos-Burgess-Jeffrey 2002 (used at UC Riverside) and Enderton 2001 (used by the Dartmouth College undergrads) is that "analysis", at the core, is just an extension of number theory, and number theory is an extension of "logic" (aka Turing machines and/or counter-machines together with "set theory" -- I am confused about that apparent linkage -- I suspect they're two parallel paths [??]). But although all this might provide a good foundation to build upon ... folks still need to know how to graph and present data in tables and work with what they think are 'continuous functions' and calculate fourier coefficients, and to do so they don't need to know how the foundation is built.

With regards to the distinction between function and formula: I agree. This is important. Kurt Gödel did it very succinctly (too brief, but accurate), in his 1931. Immediately before the following quote he defines the "primitive symbols" [e.g. ʄ is the successor primitive symbol], "variables of the first type" (natural numbers), "variables of the second type" (classes of individuals), "variables of the third type" (classes of classes) etc:

"Remark: Variables for functions (relations) of two or more arguments are superfluous as primitive symbols, since one can define relations as classes of ordered pairs and ordered pairs, in turn, as classes of classes [here he defines "ordered pair" ! The guy was thorough!]

"By a term of the first type we mean a combination of symbols of the form:

" ʄa, ʄa, ʄʄa, ʄʄʄa, ..., etc.

"where a is either 0 or a vairiable of the first type. For n>1 we mean by a term of the n-th type just a variable of the n-th type. Combinations of symbols of the form a(b), where b is a term of the n-type and a is a term of the (n+1)st type, will be called elementary formulas." (Gödel 1931 reprinted in Davis The Undecidable 1965:10-11)

The above would need explanation of how, after the "formula" comes into existence, that a "function" makes use of the "formula". My sense of it is this: "the function" "assigns to output Y" (as symbolized by an "equals" symbol, or a <= symbol ) a formula, i.e. "the symbol stuff" -- the string of symbols that includes things called "variables" and "parameters" -- usually written on the right:

Y <=_{assigned to} [place formula here]
More thoroughly Y(x1, x2, ..., xn) <=_{assigned to} [place formula here with "numbers" chosen from "the domain" to be plugged into free variables x1, x2, ..., xn and in the case of the primitive recursion schema the parameters a1, a2, a3 ... ]
Even possibly (Kleene 1952, Godel 1931): There exists a sequence of primitive recursive functions (schemata) that starts with one of the three initial functions, and yields at each "step" [via logical implication ?] either via the substitution function or the primitive recursion function, a primitive recursive function as an "immediate consequence".
The Turing machine equivalent of the above.
The lambda-calculus or other similar equivalent of the above. wvbaileyWvbailey 21:08, 2 October 2007 (UTC)

But is this "function" to be viewed merely as yet another string of symbols, or as/in_context_of a "computer/computor" interpreting the symbols? The word "function" comes from L. functus pp. of fungi to perform [Merriam Websters 9th New Collegiate, 1990]. My sense is (in particular see Minsky 1967:132ff) that an active agent is in here somewhere. And the agent must know beforehand what symbols the "variables" are, where to find numbers to stuff into the variables, and how to do all this good stuff... The same problem exists at algorithm.

With regards to good writing, and the number-theoretic notions of power series, [Cauchy expansions or "Dedekind cuts" and rational numbers -- I found all this in Minsky 1967 with regards to using Turing machines to define numbers between 0 and 1], Fourier expansions, "algebraic function", "transcendental function" etc. I called Britannica for permission to cc here the first page of their CD-article so we can look at it. It's always good to see what the competition is doing. They're supposed to send me an e-mail granting permission, or not. wvbaileyWvbailey 16:25, 1 October 2007 (UTC)

Wvbailey, I did not write the new "Inverse function" section. It was written by KSmrq. I wrote a previous much shorter version. I just edited the first paragraph of KSmrq's new version (see explanation above). Paolo.dL 16:20, 2 October 2007 (UTC)

Sorry. Anther has spoken too, as shown below. Set theory's three-liner:

* DEFINITION 42: f is 1-1 ←→ f and converse(f) are functions [i.e. ordered pairs]

* DEFINITION 43: f is 1-1 → f^-1 = converse(f) (Suppes 1972:88)

*Converse(f) swaps the domain and range, cf DEFINITION 6(Suppes 1972:61):

* DEFINITION 6. converse(A) = { <x, y>: y A x }

wvbaileyWvbailey 21:08, 2 October 2007 (UTC)

Britannica article strikes me as too narrow in scope, and at the same time, rather devoid of actual mathematical content. On the other hand, the section on inverse functions here is monstrously long and unfocussed. The first three paragraphs need to be edited for clarity, and the rest should go. We also need a (sub)section on implicit functions, possibly, before inverse functions. Arcfrk 17:13, 2 October 2007 (UTC)

Unfocused and long but very interesting. I agree that "the rest should go", but I hope it can be used somewhere else. Paolo.dL 18:04, 2 October 2007 (UTC)

Yes, it was I, KSmrq, not Paolo, who wrote the present overlong section on inverse functions. Ironically, I also wrote the short-and-sweet opening for inverse function, which probably says all we need to say for this section. First sentence:

In mathematics, if ƒ is a function from A to B then an inverse function for ƒ, denoted by ƒ⁻¹, is a function in the opposite direction, from B to A, with the property that a round trip (a composition) returns each element to itself.

I believe this sentence concisely states the essence of an inverse, yet carefully avoids wording that would commit to a specific flavor. Is B a codomain or a range? We can choose. Must a round trip start at A or at B? We can choose either or both. Is an inverse unique? That's left open; we see only "an" inverse, not "the" inverse. Not shown here, the next sentence does warn that we are not guaranteed existence. (I realize that competent mathematicians see all these possibilities at a glance, but this page has other readers.)

In fact, I have said privately that here is perhaps not the ideal place for the curve example. So why write it? Because Paolo was repeatedly mangling both articles trying to "fix" language and definitions of the various forms of inverse, and this seemed a way to put a stop to that.

To my sorrow, the attempt has failed; Paolo persists.

As for the exploration, does it lack focus, or is it simply too condensed? I tried to cover half a dozen issues in one extended example.

Uses of functions: direct, parametric, implicit
Partial inverse
Extent of inverse: codomain versus range
Impact of injectivity and surjectivity
Two-sided inverse, left inverse, right inverse
Inverse image

Almost all these details are better left to the dedicated article. I would be content to see the example moved there and explored more leisurely, though I'm not sure it's needed. (I do think it's a pretty example.)

The real issue is Paolo. While he is welcome to find an appropriate venue to learn and practice mathematics and English, this is not the place. I'm sorry to be harsh, but he doesn't know the mathematics and he doesn't know the language. His inept attempts to "explain" in the article, preempting subject experts and native speakers, must stop. --KSmrq^T 11:31, 3 October 2007 (UTC)

KSmrq should stop reverting without explaining why. In my edit summary, I explained that some sentences in the first paragraph of "inverse function" were linguistically incorrect. These sentences, for the second time, were stubbornly restored by KSmrq. Being harsh may be useful, being biased is harmful, and behaving as the owner of a page or section is not allowed in Wikipedia. It is sad that an excellent mathematician and writer such as KSmrq prefers to waste his time to write a generic attack and defend his destructive behaviour, rather than use his undeniable skill to briefly and constructively explain why he doesn't like my edits. Here is the comparison:

	KSmrq's text	Paolo's version
Identity function	[in "inverse function" section:] ... Like multiplication, we have a analogy to 1: for any set X the function id_X maps each element to itself. And like 1, these identity functions are neutral for composition: ƒ o id_X = ƒ and id_Y o ƒ = ƒ, where ƒ is any function from X to Y.	[in a separate section:] ... Like multiplication, composition has an identity: for any set X the identity function id_X is a function which maps each element to itself. ~~And like a multiplication by 1, a composition involving an identity function has no effect:~~ An identity function is neutral for composition: ƒ o id_X = ƒ and id_Y o ƒ = ƒ, where ƒ is any function from X to Y.
Inverse function	For some functions we can go a step further and define an inverse function, denoted by ƒ⁻¹, such that ƒ⁻¹ o ƒ = id_X. Clearly, when ƒ is a function from X to Y, ƒ⁻¹ must be a function from Y to X for the composition to be defined.	The analogy between composition and multiplication can be brought a step further. Like for some elements x we can define a multiplicative inverse denoted by x^-1, for some functions we can define an inverse function, denoted by ƒ⁻¹, such that ƒ⁻¹ o ƒ = id_X, and ƒ o ƒ⁻¹ = id_Y. Clearly, when ƒ is a function from X to Y, ƒ⁻¹ must be a function from Y to X for the composition to be defined. Intuitively, ƒ⁻¹ is a function that undoes and can be undone by ƒ.

Notice that my edit is compatible with the two definitions given in the article "inverse function", and that I created a section for identity functions. About my "preemting ... native speakers", notice, in KSmrq's text:

"...a analogy..."
"Like multiplication, we have a analogy to 1" (implies that multiplication has an analogy to 1)
"And like 1, these identity functions are neutral for composition" (implies that 1 is neutral for composition)

Again, I am very sad for being forced to fight against KSmrq's bias, but reverting a constructive edit without explaining the reason is not allowed, and this is not the first time he behaves this way. Paolo.dL 18:10, 3 October 2007 (UTC)

I found an ambiguous statement in my version (see text with strikethrough format in the table above). I corrected it. Repeating the analogy with respect to "multiplication by 1" is not necessary. But if you want to stress the concept, then you need a longer sentence such as "And like 1 is neutral for multiplication, an identity function is neutral for composition". Paolo.dL 11:54, 4 October 2007 (UTC)

Alternate example <Q = N/D, R> for use in talking about "inverse", etc

It may be that the example is too hard, and there is too much development in that one section, without adequate leadup/development and examples, in the earlier sections. I've run into what may be a simpler example: N - Q*D = R, or N/D = Q with remainder R, (usually presented as equivalence classes of residues). This appears in a couple places e.g. Minksy 1967 Finite and Infinite machines, Saracino 1980 Abstract Algebra: A First Course, Kleene 1952 Introduction to Metamathematics. ~~I was going to explore how to use this on the "definition" page (actually, its talk page), but I might as well do here informally, since this will help me think it out.~~ (We do not want talk about: "generating equivalence classes with modular division" -- abstract -- only about what happens. The example turns out to be quite rich in potential).

re "inverse", as the now-moved example showed, once the "converse set" is created, whether or not you have an invertible function is easy to demonstrate. Also the quote from Saracino:

"The idea of the inverse function is that it is supposed to undo everything f did..." (Saracino 1980:62)

The "R" residue/remainder part of the example is used by Minsky:

"Perhaps the most usual definition [of a function] is something like this:

A function is a rule ...

"For example, suppose the rule that defines a function F is "the remainder when the argument is divided by three". then (if we consider only non-negative integers for arguments) we find that

"F(0) = 0, F(1) = 1, F(2) = 2, F(3) = 0, F(4) = 1, etc.

"Another way mathematicians may define a function is

"A function is a set of ordered pairs <x, y> ...

"If we think of a function in this way, then the function F above is [observe italics] the set of pairs

{ <0,0>, <1,1>, <2,2>, <3,0>, <4,1>, <5,2>, <6,0>, . . . }

Is there any difference between these definitions? Not really, but there are several fine points. [etc]" (Minsky 1967:132-133)

Saracino begins his chapter 7 FUNCTIONS with the following, then by chapter 9 gets to EQUIVALENCE RELATIONS: COSETS:

"DEFINITION If S and T are sets, then a function f from S to T assigns to each s∈S a unique f(s)∈T.

"As a definition this is somewhat strange,in that it tells you what a function does rather than what it is. Sometimes this difficulty is avoided by saying that function is a "rule" ... but this isn't any better because "rule" isn't defined." (Saracino 1980:59)

Kleene 1952 gives an example in mod 2

"EXAMPLE 4. Let X and Y both the re residues modulo 2, i.e. X = Y = { 0, 1 }. The functions x' [successor] and x*y can be defined by the following tables ... [they look just like Karnaugh maps]." (Kleene 1952:32-35).

wvbaileyWvbailey 16:46, 4 October 2007 (UTC)

Comparison with Brittanica

I cc'd the following over from my talk page so I'd have a record here re opinions etc. Paolo can now take solice in the fact that it isn't just he who has had his tail-feathers singed; it's a fact of life on wikipedia. Nevertheless, sometimes, like I'm sure Paolo knows, it hurts. Also this is an illustration that nothing is private on wikipedia, that what you write may come back to haunt you.

Your extended "thinking out loud" at [[Talk:Function (mathematics)#Alternate example for use in talking about "inverse", etc|Function]] may be fascinating to you, but it is an unwanted imposition on editors like me who watch the page.

Also, I must caution you that we are not going to warp the function article in the direction you seem to be headed. We already have articles for, say, recursion theory and primitive recursive functions. These are fine topics, in their place; not here. It is as if we want directions for how to drive from one city to another, and you want to explain the internal combustion engine. --KSmrq^T 11:14, 4 October 2007 (UTC)

I also came here to point out that extremely long posts (like this 10k post) are not a particularly effective way to communicate with other editors. I know that you enjoy source-based research, which is great. If you limit your comments to one or two paragraphs, though, it will make it easier for others to understand the point you are trying to make. It helps me to write short comments because it forces me to find a concise way of saying what I want to say, which improves the clarity of my comments. You can always summarize what the sources say, and be judicious in your quotes.

KSmrq is right that the function article is concerned with the broader mathematical notion of a function, not particularly with notions of effectiveness. Certainly the function article should mention, and link to, an article on computable functions. But computability is a very small field within mathematics. — Carl (CBM · talk) 13:40, 4 October 2007 (UTC)

My point was: Try to find a simpler example -- I proposed <Q,R> = N/D -- used by published authors; develop this example from the start of the article and carry it through to the end. Develop along the way all the notions (function as TABLE, as ordered pairs, as rule), along with the vocabulary/definitions. The example I'm proposing needn't be developed re counter machines or recursion theory -- they are just vehicles, examples, along the lines of number theory, really (i.e. using positive integers -- go from fractions (the rationals) to infinite series (the irrationals) with e.g. Cauchy sequences, as I was exploring on the other page. This is the approach that Britannica takes). But I couldn't describe that in a few paragraphs. It's really a shame that Britannica didn't come back with a permission to cc their whole article here to look at, because they take an approach that is similar to what I would prefer to see here (in particular they don't even mention domain and range, converse, codomain, surjective, injective, bijective, and all the other set theoretic stuff. But they do mention infinite series and graphing.)

This page does not even begin to tell us what a function is, only what it does. And I've come to agree with the other editor who warned me that to define everything in set-theoretic terms is a cheat -- too abstract, too much a plethora of definitions. Move that s**t off this page into set-theory where it belongs. Here's Kleene's take on it:

"We have described a function as a many-one correspondence. One may go further in saying what a many-one correspondence is to be, according to the kind of theory one is working in. In set-theoretic terms, the correspondence can be identified with the set of all the ordered pairs (x, y) of corresponding elements X and Y₁ [Y₁ was previously defined as "the range of the dependent variable y" or f(x) is the subset Y₁ of Y comprising the elements of Y used in the correspondence..."]. One may speak instead of the law or rule establishing the correspondence, at least in dealing with such functions that a law or rule in some understood sense can be given for each function. In the case that X is a finite set, a function can be given as a table.

wvbaileyWvbailey 17:59, 4 October 2007 (UTC)

I just read the Britannica Online article on functions, and it's quite bad compared to what I expected to find. It's written at an extremely elementary level and leaves out many important things:

Domain and range
Bijective, injective, and surjective functions
Any mention of category theory
Any mention of computability theory

Basically, it just gives a brief definition of a function and then lists some examples. I expected it to be more comprehensive, and now I'm sure this is another article that WP can do better than Britannica. By the way, I disagree with the assertion that the first two bullets above are "set theory" - they are part of the basic vocabulary of functions, and arise even in elementary calculus books. — Carl (CBM · talk) 18:13, 4 October 2007 (UTC)

Okay. We're giving and getting bad guidance here. Previous to this was a long dialog about the level to which this should be written. Now, what I throw out as an example (Brittanica) of how "function" is described (not defined), it's too simple.

I'm also a bit surprised that as a recursion theorist you didn't see where I was going with the now-defunct example (that was suggested by Minsky 1967 and Enderton 1972, and Turing 1936, not me, so as this chain of inference is also suggested by them):

Axiom: The Turing thesis: All that can be computed or calculated, by man or machine, can be computed by a Turing machine or by recursion. (If you have a counter-example, for godsakes please show me!)
Turing machines/algorithms and/or recursion theory schema compute only with integers.
Axioms: Set theory. The following reassurance is provided to us by set theory (Suppes 1972), also formal number theory (e.g. Kleene 1952, Hardy and Wright 1979):

Cauchy sequences use only integers.
Any number that can be created (described at all) can be created by a Cauchy sequence (e.g. Turing 1936, Suppes 1972)

Therefore, any number that can be created (described at all) can be created (described) by a Turing machine or by a person working with recursion.
Definition: the notion of function involves an input and an output, or the notion of many-one "correspondence".
Ergo... a function is either a Turing machine/algorithm or a recursion algorithm or an equivalent. Period.

Even a TABLE is just an instance of a recursion-theory CASE function or a "parsing tree" of a Turing machine. It's not me, its Minsky, Enderton, et. al. wvbaileyWvbailey 18:55, 4 October 2007 (UTC)

I think the issue is that you are identifying functions with effective rules for computing them. But the modern conception of a function allows for the rules to be ineffective. For example, the characteristic function of the Halting problem is a function but not computable. — Carl (CBM · talk) 19:11, 4 October 2007 (UTC)

I'm identifying not so much the rule, but machine+rule=function, or man+rule=function. In other words, the uber-function "chr(Halting problem)= ??" never terminates with an answer? This would violate the definition of "function", would it not. What you are saying is that function need not "terminate" with an output?? Also see the next post re division by zero. The characteristic function, even if it is returning an oracle's answer as {1, 0}, must be a function that terminates with a single number { 1, 0 }, correct? It would seem so, by definition. Otherwise it is something else (i.e. maybe a relation, but not a function). Thus every function must terminate, otherwise it is (similar to) "division by zero". Can you point me to something (book, article) I can read up on about this? This would seem to be an important point in the definition of "function". wvbaileyWvbailey 19:47, 4 October 2007 (UTC)

The equivalence is more like "rule = function", but this is a false definition because these mean essentially the same thing in natural language. Yes, a function must always "terminate" in the sense that it must be defined for every input in its domain. But there is no requirement that the rule has to be implementable on a Turing machine - the rule can just be an oracle that tells what output to give for every input. Such an oracle would usually be called a single-valued binary relation. — Carl (CBM · talk) 19:50, 4 October 2007 (UTC)

I'm sticking to my guns on this one. Again, the word "function" comes from L. fungus to perform. This is an active verb, and therefore requires an agent: there is no "performance" without an "performer". A TABLE (aka "list of ordered pairs") is not a performance (L. function). A TABLE is symbols on a paper, an inert object. Therefore, for the TABLE to be a function it needs a "functionary": "one who serves in a certain function" -- "the action for which a person or thing is specially fitted or used or for which a thing exists: purpose" (Merriam-Websters 9th New Collegiate). So I stick with machine+rule=function, or man+rule=function. There's a good discussion of this in Herbert Breger 2000:221ff "Tacit Knowledge and Mathematical Progress" in Groshoz and Breger (eds.). The Growth of Mathematical Knowledge, p. 221-230. Hilbert (1905) "considered it "absolutely necesssary" to have an "axiom of thought" or "an axiom of the existence of an intelligence" before stating the axioms of logic." (p. 227). There's lots more in the article, including quotes and discussion from Polya, Veronese, and Weyl "mathematics is an activity rather than a doctrine" (p. 228). I argue that this is the modern view brought forward by Minsky et al. Set theory is just another one of many possible "theories", as noted by Kleene.

Searle the philosopher has this to say about that (immediately above):

"Nothing is intrinsically computational. Computation exists only relative to some agent or observer who imposes a computational interpretation on some phenomenon" (Searle 2002:17).

wvbaileyWvbailey 00:03, 5 October 2007 (UTC)

Hmm. What's wrong with the f as a function being defined as going from an input x to the unique y such that the ordered pair (x, y) is in the graph of f. That makes a mathematical function (i.e., the graph, together with appropriate domain and co-domain) a function in your sense. The reverse is more complicated, but the concepts are basically equivalent. I think we need go with the graph, or the graph plus domains as the primary mathematical definition, even if the mapping is the primary non-mathematical definition. More to the point, two functions are equal if they have the same graph (and domains); they don't need to have the same algorithm. — Arthur Rubin | (talk) 01:07, 5 October 2007 (UTC)

I agree with your definition. It is where I started from, when I launched into this (see my little pictures at the top of this article, if they are still there ... the "cartesian product of domain u and range v" creating the "graph" i.e. the potentialities for the ordered pairs, the function (aka "ordered pairs" selected as elements from the "graph") "plotted" onto (appearing in) the "graph", their "projections" to the X and Y axis (i.e. notions of the first and second coordinates, domain and range) and "the function" remaining within "the graph". There was the quote from Manin (I added it to the article but KSmrq reverted it: (i) f is a subset of u x v, (ii) projection of f onto u coincides with all of u, (iii) each element of u corresponds to exactly one element of v). This set theory stuff is okay, I suppose, but how to explain to us skeptical folk what is happening with the "cartesian product" (aka graph) when the function is "continuous" (whatever that means ... so the graph is no longer specific points in a plane)?

I agree with your last sentence. Even ole Minsky corroborates it: "For each function there may be many definitions or rules that tell how find the value y, given the argument x. Two definitions or rules are equivalent if they define the same function" (Minsky 1967:134).

Where I am still stuck is this: your "going" is not defined, it's been abstracted out, so now it's just some magical journey that spontaneously produces output from some ethereal object when, somehow, input spontaneously enters from a magically-declared domain. Eeuu. That's my only real problem, a "functor" must be considered to be part of "the function". Perhaps the functor can be "abstracted out", but the fact that that has happened must be noted as one of the fundamental notions/assumptions behind the definition. wvbaileyWvbailey 02:54, 5 October 2007 (UTC)

I cc'd these over from a previous page. Do we need such drawings? Are they helpful? If so, what should they be like? If these are flat-out wrong, and/or my word usage is wrong, please correct them and me. Clearly they are a bit overwhelming and need a progressive development. The trick will be to find a better example (that was what I was after with Q = N/D + R), and captions and drawings that don't overwhelm. wvbaileyWvbailey 03:14, 5 October 2007 (UTC)

Well known mathematics.

While it is fun exploring the implications of elementary mathematics, all of this is well known and well understood by mathematicians, and an explanation for the non-mathematician should be direct and to the point.

(Let me mention in passing that, if you ignore the fact that division by zero is undefined, you can not only prove that 0/0 = 1, you can prove that 0/0 = 7, or any other number. Just note that 7x/x = 7 and then substitute 0 for x, so 7 = 7x/x = 7*0/0 = 0/0. Then you get 1 = 0/0 = 7, so 1 = 7. The "rule" against division by zero exists for a good reason: division by zero leads to wrong answers. I keep running into people who think the "rule" against division by zero is an insult to their freedom of expression, who write countless pages "proving" that they can divide by zero if they want to. But that has nothing to do with the subject at hand.)

Here is how things stand in modern mathematics. There are two different ways to define the inverse of a function. One is to say that f inverse (x) = y iff f(y) = x. In this case f inverse is a function iff f is injective, and the domain of f inverse is the range of f, not the codomain of f. The other way is to say that only bijections have inverses. Both are useful, and it is almost always clear from context which is intended. The little a arcsin, capital A Arcsin convention makes the distinction nicely. The first is an inverse in the first sense but not a function. The second is an inverse in the second sense, but only when the domain and codomain of the sine function are restricted to [-л/2, л /2] and [-1, 1].

The article needs to say this, simply, directly, and briefly. Rick Norwood 14:32, 4 October 2007 (UTC)

division by zero:

Actually, it is defined in the real, practical, operational sense, because we can watch it happen. Of course the problem is, the example never terminates with a single answer, in fact, it never terminates at all. Thus as a (failed or weird) relation it is one-many in nature: 0 in, anything out. But it is not a function. (This seems to imply that a function must terminate with an answer !)

When confronted with N/0 the little counter-machine algorithm is fascinating to watch in action. As I mentioned in the post (now removed) it creates every natural number, one after the other, ad infinitum. If another algorithm were to come along and pluck one of these numbers, it could have its choice of any number it wanted. As to what the invertible example showed, I'm not sure, exactly. You would think that 0*1/0 = 0*any_number = 0.

This is why I like these examples. They are very, very vivid. wvbaileyWvbailey 19:27, 4 October 2007 (UTC)

That's why the examples are appropriate in computation theory, but not in the theory of mathematical functions. They are vivid, and the examples are difficult to abstract into the abstract definition of a mathematical function. — Arthur Rubin | (talk) 01:10, 5 October 2007 (UTC)

The "abstract definition of a function" is just that: "abstract". You probably mean the word to be understood in the sense of "dissociated from any specific instance", but the word also means "difficult to understand, abstruse". In both senses it is not appropriate for all audiences. It (the abstract definition) is not the only definition. Perhaps every mathematical "theory" has its own definition. wvbaileyWvbailey 01:51, 5 October 2007 (UTC)

The "abstract definition of a function" is exactly what the mathematical definition is. It should be primary in the detailed definition, although I have no objection to a "rule" being in the "common" definition. — Arthur Rubin | (talk) 08:32, 5 October 2007 (UTC)

The mathematical definition of a function is a set of ordered pairs such that <x,y> and <x,z> in f implies y=z. Do you really want to try to explain that to a layperson? Rick Norwood 13:06, 5 October 2007 (UTC)

What are the issues here? Please leave a comment.

"For each input, there is a single (but not unique) output". —Preceding unsigned comment added by 201.80.248.51 (talk) 23:27, 6 November 2007 (UTC)

Conversation on this talk page has drifted through many topics, including both inverse functions and the definition of a function on its own. I am somewhat confused what is being discussed. In order to make the discussion more productive, I have a suggestion. I'm asking everyone who is interested to leave a brief comment here expressing the concerns they have with the current article text, or saying the current article text is OK. That would let us see whether there is actually a disagreement that we need to discuss, or whether we are just tilting at windmills. Please don't go into great detail; the goal is just to see which areas actually require discussion. I will read over the article carefully and leave my own comments later today. — Carl (CBM · talk) 14:03, 5 October 2007 (UTC)

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I think the B+ rating is fair. The big problem is so many people doing major rewrites that the style shifts abruptly from sentence to sentence, but that problem is not likely to go away. If someone took the time and trouble to do a careful copy edit, rewrites would wipe out any progress in a day or two at most. This is a topic where every math major wants to put in his two cents worth. Rick Norwood 15:13, 5 October 2007 (UTC)

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This is exerpted from a week ago:

I do have problems with this article. B rating at best. It has all the worst characteristics of wiki-writing. It certainly will be of no use to experts; it has no in-line references and has just a small number of references in general. The "formal definition" is inadequate (e.g. Manin's is better, and could be given in full). It does not integrate well with other articles. Nor is it of use to beginners: It doesn't seem "glued together" with an "umbrella-concept" or -concepts. It needs an umbrella-example, one that starts at the beginning and goes to the end, used throughout to illustrate the "modern" vocabulary. The article is bloated, an unstructured pastiche of little topics pulled from here and there leaving the reader with no feeling of unity. There is no sense of what is important and what is optional. The little set drawings are wrong/incomplete (cbm and I corrected the caption in one of them) the origins of the terminology is not well documented (with inline references) and it does not reflect modern usage. The notion of "Cartesian product" and "graph" would be very useful: A collection of ordered pairs, some of which are plucked out to make "the function" and "graphed", make the link between "set theory" and "analysis" clear, is visual and worth development. (originally posted wvbaileyWvbailey 15:26, 29 September 2007 (UTC) but trimmed and edited.) wvbaileyWvbailey 15:26, 5 October 2007 (UTC)

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The primary issue here is that some editors do not like the formal, mathematical, definition of "function", and want to introduce the question of the definition of "rule" in the informal definition. The reason it's an "informal definition" is that it cannot easily be made formal, and hence is not used in actual mathematics. I don't think the question of what a rule might be and types of rules, algorithms, primitive recursive functions , recursive functions, etc. belong in this article or any closly allied article. Perhaps function (mathematics) definitions could be renamed to map (mathematics). Actually, I think I prefer the version a couple months ago which mapped (LOL) to both definitions of relation (mathematics), as to whether the domain and codomain are specifed as part of the definition of relation/function.

The confusion as to whether an inverse function is a left- or right- inverse function should be left to that article, with at most a sentence here to indicate that some functions may have left- or right- inverse functions, but not a two-sided inverse function.

— Arthur Rubin | (talk) 17:23, 5 October 2007 (UTC)

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I know of nobody here who does not 'like' the formal defintion of a function, and I don't know anyone who wants to define a function as a rule, which is mathematically incorrect -- for one thing, the number of real valued functions is uncountable, while the number of rules is countable.

A simple and correct informal definition is that a function has an input and an output, and has one and only one output for a given input.

Rick Norwood 17:55, 5 October 2007 (UTC)

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I've haven't done a recent sweep through the article for a check against my usual criteria (correct, clear, compelling, …), in part because the thrashing has been too distracting. Rick Norwood is too generous when he posits math majors; that would be a step up in expertise. He also may be too optimistic about agreement on a formal definition. I've chosen to mostly limit my role to major damage control, but there's been a lot of that lately.

Most of the thrashing has, indeed, been confined to questions about inverses and about how to define "function".

Paolo still struggles to understand the variations available for inverses, yet mangles the English language to explain them to the world. The results are not pretty, and definitely not helpful.
Wvbailey has been taking a self-taught crash course in function definitions. He relates his discoveries at enthusiastic length both on this talk page and in this material which I excised and reverted from the Definition section.

While I sympathize with poet Robert Browning's sentiment that "a man's reach should exceed his grasp, or what's a heaven for?", the consequences here have been hell. --KSmrq^T 01:16, 6 October 2007 (UTC)

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Based on the comments above, I think it should be possible, if everyone focuses on moving the article forward, to come to some agreement about the content. I remember editing the article in December 2006 [1]. When I wrote that version of the definition section I was interested in providing concrete references, pointing out that the definition of a function as a rule is important, and also pointing out the definition of a function as a binary relation.

After reading the article as it stands, I would put it at a B to B+ rating, but not A. It does have a good selection of information, and uses summary style appropriately, but could use significant copyediting. — Carl (CBM · talk) 03:05, 6 October 2007 (UTC)

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Whole article suggestions

I would group areas for improvement in two categories: strategic and "quick fixes".

Overall, the article tends to over-emphasize formal aspects, such as definitions and notation, over the substance: how are functions used and what place do they occupy within mathematics. Geometry Guy had interesting comments a while back on the general aspects that are still worth looking at.
At Paolo's insistence, we have a lot of set-theoretical diagrams, but not enough illustrations involving graphs (including the multivariable case), tables of values, etc which are considerably more common ways to visualize functions, and would appeal to a much wider audience.
A few sections bring down the quality of the article, in my opinion. "Inverse function" is disproportionately long (first few sentences ending with the Farenheit – Centigrade would suffice) and unfocused. "Notation" can be reduced to at most two sentences, and merged into "The vocabulary of functions". "Other properties" in "Is a function more than its graph?" is a really bad idea. Some of these topics qualify for "See also" section, and the others are just a random list and should better be removed. "Restrictions and extensions" should be merged into "The vocabulary of functions". "Lambda calculus" and "Functions in category theory" do not contain anything substantial and can be merged into some general discussion of various types of functions across mathematical disciplines. "Computable and non-computable functions" would benefit from better explanations (some of which were given on this very talk page). "Pointwise operations" is very crude. Maybe, it should be expanded to a section on various function spaces (i.e. concentrating on a collection of functions as opposed to a single function). A lack of such a section is in fact another strategic weakness. Also, a section on implicit functions is urgently needed.

Given the shortcomings listed above, I wouldn't presently support the B+ rating. On the other hand, I disagree with Wvbailey on just about every item that he thinks needs to be "improved". I propose to restore some of the earlier discussion that was banished to the archive due to his prolific contributions to this talk page and join Carl and KSmrq in pleading to keep the discussion focused on specific areas of improvement, not wandering over whatever anyone happened to read about last. Also, I'd like to ask KSmrq to stop insulting Paolo, whose contribution is not such a disaster as he makes it appear, and to patiently explain his position. For example, if the real issue is that certain sections are so near perfect that any edits (and especially, expansion) actually make them worse – say so! I agree with Carl that careful copyediting will go a long way towards improving the quality; but also with Rick, who pointed out that the balance would be hard to sustain given the past history of the article. Arcfrk 03:47, 6 October 2007 (UTC)

I stand back and laugh a little, because many of us know that this is all a tempest in a teapot. Frankly, given the exposure of this article and the low entry barrier to editing, we're lucky it's not in worse shape and that the disputes are civil and limited.

We seem to have a handful of knowledgeable and experienced editors (Carl, Rick, Arthur, Arcfrk) commenting, and I'm sure others are lurking, so I'm willing to see what we can do in a short time with more intense effort. Here's a list of some things that strike me as I skim:

The set illustration accompanying the lead is too technical too soon. Remove it.
The opening sentence speaks of "primary" and "secondary", an alien concept to my view; and "deterministic" is unhelpful. The second sentence says "of a specified type", a clumsy way to hint at domains and codomains. The third sentence says "[t]his definition covers most …", but we haven't seen a formal definition, and what's "most" about?
The second and third paragraphs should be swapped.
The first section should be called "Introduction" (we have a separate section on "Notation", far too late in the article). Other than that, it's surprisingly decent. It threatens to confound functions with morphisms (mapping operators as well as sets), but stays clean. Yet the square root example makes me uncomfortable, for several reasons: (1) a function can return a set and still be single valued; (2) it's too early to introduce an inverse function; (3) it makes me think of Riemann surfaces and branch cuts, an unwelcome intrusion. But maybe this is the best we can do with an elementary real-valued function.
"Definition" is a known trouble spot. Many a calculus student will be shocked to hear their textbook is "informal"! What about Dunford & Schwartz: "the function ƒ assigns an element"? (The word "rule" is not used, and relations do not appear until a little later — and even then more as a generalization of functions.) Certainly it is a distortion to speak of "the" formal definition. And please, now is the time to meet terms — domain, codomain, range/image, inverse image — and notation. Let's get the ubiquitous colon-arrow (ƒ: A→B) and mapsto notation available early, along with the equally popular ƒ(C) = ⁹⁄₅C+32 and C = ⁵⁄₉(F−32).
Shorten "History of the concept" to "History". Some editors favor history early; I would move it to the end. The inline references such as da Ponte (1992) should take advantage of the {{harvtxt}} template to automatically incorporate an HTML link to that item in "References".
"Functions in other fields" is a waste of space. Use a disambig notice at the top of the page, or nothing.
"The vocabulary of functions" should be "Vocabulary" or "Terminology". I do not understand the criteria for a term to be included, nor can I discern the organization.
"Function composition" is fine. However, it jumps at us out of nowhere.
"Inverse function" is verbal diarrhea in this context, but otherwise exemplary prose. (I wrote it!) Copy the intro paragraph of inverse function, keep the temperature example, stop.
"Specifying a function" is the beginning of a good idea, but not well done. And why here? Now is also the time to remember the inverse function theorem.
"Notation" I have already said comes far too late, and needs to include variants currently mentioned separately. I like commutative diagrams; not here.
"Functions with multiple inputs and outputs" is poorly placed and poorly named. Three paragraphs, maybe; three subsections, no. Why not put this under "Specifying"?
"Set of all functions" is OK (but "all"?); I'd rather see "Function space".
"Is a function more than its graph?" has no business as an independent section. The title: yuck.
Why is "Other properties" a subsection? I'm not thrilled with a bare list.
"Restrictions and extensions" belongs under vocabulary.
"Pointwise operations" belongs under "Function space" (see above).
"Computable and non-computable functions" should be "Computability", and expanded slightly.
"Lambda calculus" belongs under "Specifying", and really should encompass other recursive function formalisms as well.
"Functions in category theory" is silly; morphisms are hardly exclusive to category theory. We could talk about categories as a foundational replacement for sets; let's not.
"See also" is a very strange collection.
"References" is thin. As regular readers know, I like references that instruct and inspire the reader (and can be found online), I like Harvard style inline (sparsely), and I detest footnotes. We could stand to offer our readers a little more here.
"External links" seems somewhat haphazard.
I haven't thought about what important topics might be missing, but we should.

My biggest overall complaint is poor organization. --KSmrq^T 10:21, 6 October 2007 (UTC)

I think the contrast between "rule" and "relation" is worth pointing out here. It integrates with the historical development of the function concept. I dug out a calc book that explicitly says "rule" so that we'd even have a source for it. Could you comment on the text here? — Carl (CBM · talk) 14:37, 6 October 2007 (UTC)

You ask specifically about definition, but I have a broader comment. The entire article looks better in that version! We've had a few minor improvements like the function plot; other than that, it's better written and better organized. Perhaps we should roll back to that version as a better base for further improvements.

I'll open a new section to comment on definitions. --KSmrq^T 22:53, 6 October 2007 (UTC)

That's interesting, because I think the present table of contents is better than the TOC in the old version. Duplication of information has been an ongoing problem in this article, and I was never happy with the length of the old version. If there will be a rollback, I'd rather if it happens before I try copyediting the present article... — Carl (CBM · talk) 23:08, 6 October 2007 (UTC)

Critique → Edit

Having scanned the article for issues, I decided many of my concerns were probably noncontroversial enough to attack quickly. I have made a beginning, but clearly have left rough edges that need polishing. Although I think this version is strictly better, I won't be at all offended if it is seen as too precipitous and reverted. I've replaced the "Inverse function" section (since I know the originating editor won't complain), and made a few changes in wording scattered throughout the page; but mostly I've renamed and shuffled sections. Some sections could do with a more substantial rewrite. And again, don't be shy in reverting and speaking up if it seems that more discussion should occur before such an edit. --KSmrq^T 13:33, 6 October 2007 (UTC)

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Arthur Rubin suggested this at the discussion page of the "fork"-article Function (mathematics) definitions. I am suggesting that as it stands, without further qualification, "Function (mathematics)" a very broad subject, one topic of which is "the definition", and another of which could be how the notion of function is developed in the various theories of mathematics (set theory, number theory, computation theory, abstract algebra, whatever...). Perhaps one could get rid of the "definitions page" altogether with something like this brief list. I thought that his summary was very useful so I brought it over from there to here.Emendations added by wvbaileyWvbailey 20:47, 6 October 2007 (UTC)

The term function, in mathematics may mean one of the following:

(1) A relation (mathematics) such that for every element of the domain there is exactly one associated element in the codomain.

(2) A primitive recursive function.

(3) A recursive function.

(4) A function (1) computable by an algorithm.

(5) A function (1) computable by a Turing machine.

(6) A function (1) which is also definable re a definable set re definable.

(7) A function (1) which is also definable re mathematical logic re definable.

We could add commentary on partial functions and multivalued functions, and possibly "provable functions" ...

This seems to be a good summary and may be of use to the article. wvbaileyWvbailey 14:15, 6 October 2007 (UTC)

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The following extremely-formal definition does have a place, perhaps near the end? Arthur Rubin also suggested the same. Because it is supported by references, I suggest that it be included:

This formal definition of function comes from set theory: A function is a restricted kind of binary relation: ^[1] ^[2] ^[3] The following is Manin's contingencies (constraints) on the definition^[4]. Here f is a function (mapping), u is a set made of elements u_i, v is a set made of elements v_i, u x v is the Cartesian product, the set of all possible ordered pairs <u_i, v_i>:

f is a subset of u x v
the projection of f onto u concides with all of u
each element of u corresponds to exactly one element of v

∀z(z ∈ f ⇒ (∃u₁ ∃v₁ (u₁∈u ⋀ v₁∈v ⋀ "z = <u₁, v₁>")))

⋀ ∀u₁ (u₁∈u ⇒ ∃z(v1∈v ⋀ "z=<u₁,v₁" ⋀ z∈f))

⋀ ∀y₁ ∀v₁ ∀v₂ (∃z₁ ∃z₂ (z₁ ∈f ⋀ z₂ ∈f ⋀ "z₁ = <u₁, v₁>" ⋀ "z₂ ⋀ <u₁, v₂>" ) ⇒ v₁ ⇒ v₂ (Manin 1972:12)

woops, forgot to sign. wvbaileyWvbailey 14:31, 6 October 2007 (UTC)

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I had that definition of a function here in December. The formula in the language of set theory isn't needed; we can just use English, and we can paraphrase and reword things. As to the list of 7 things above, I don't know the context in which Arthur Rubin was speaking, but this article is not primarily about computability; it's about the general definition of a function in mathematics. — Carl (CBM · talk) 14:41, 6 October 2007 (UTC)

How about the formal stuff in a footnote? (Or not -- as long as there's a footnote or inline citation. (What is it with the resistance to this??) I strongly encourage that at least the informal words should be used, because of the notion of "graph". I have to think about the other. wvbaileyWvbailey 15:04, 6 October 2007 (UTC)

I don't think the formula in the language of set theory is needed here at all. Anyone who knows enough to want it knows enough to reconstruct it from the English description. Note that I did have an inline citation in the version I linked to above. We do not need to directly quote from the source we cite. — Carl (CBM · talk) 15:10, 6 October 2007 (UTC)

Okay. I urge you to add your version or mine without the formula. Mine would be the three bullet-points with the lead-in sentence. Or yours, either would do. I have no desire to add anything to this article, not after the lashing I've had. I'll leave this decision up to you. About the other, the bullet-points were "examples of how functions are formed in the various branches of mathematics, with links." My understanding of his list was not meant to include only computability theory, but to be more general in nature. wvbaileyWvbailey 18:21, 6 October 2007 (UTC)

My comment was intended to comment on the Definitions fork, which including definitions 1 and 5, with references to 2 and 3. (If I posted it in this page, I apologize.) I just added a few more which I've seen here and there; that nothing "exists" unless it's "definable" in the appropriate sense. — Arthur Rubin | (talk) 18:41, 6 October 2007 (UTC)

Testing this link -- I got ASCII gibberish

Could someone else please test this link to the following article? I got to the "main page" but when I clicked on "pdf" for the article I got an ASCII mess. I have the latest and greatest Adobe Acrobat Standard 8 and the reader 8 + updates so that shouldn't be the issue. Lemme know what you get, Thanks, wvbaileyWvbailey 00:21, 7 October 2007 (UTC)

da Ponte, João Pedro (1992), "The history of the concept of function and some educational implications", The Mathematics Educator, 3 (2): 3–8, ISSN 1062-9017

The link to the Word version still works; it was lost in this edit. But who could predict the journal would have a broken file like that. The Word version is here. — Carl (CBM · talk) 02:50, 7 October 2007 (UTC)

Got it: http://www.educ.fc.ul.pt/docentes/jponte/docs-uk/92%20Ponte%20(Functions).doc

Thanks, BillWvbailey 03:16, 7 October 2007 (UTC)

About KSmrq's behaviour

I did not receive a specific answer to my 8 October comment, in which I asked for explanations about two consecutive reverts by KSmrq. I only received a generic offensive answer. After repeatedly trying other strategies, both personally and publicly, I see no other option except being harsh, to stress the concept that nobody, not even a brilliant and well deserving editor such as KSmrq, can behave as the owner of a page on Wikipedia. I will describe the situation as I see it.

I see that KSmrq eventually created a separate section about identity functions, as I already did in both my previous edits, which he summarily reverted. And he corrected the two incorrect sentences (see my 3 October comment above) in the first paragraph of inverse function, after restoring them twice by undoing my edits, writing that his text was an example of excellent prose, and accusing me of mangling english language. He also removed his pretty "verbal diarrea" (as he called it), which significantly contributed to reduce the readability of the whole article, after complaining about being distracted by others' "trashing". I am glad that KSmrq accepts criticism at least when it comes from himself. Is it so difficult to politely interact with others? Is an editing war more effective than a short but civil explanation or discussion? Regarding KSmrq's writing standards, "I don't like it" or "it's not pretty" is correct and clear, but not compelling and not complete. Paolo.dL 21:41, 10 October 2007 (UTC)

Since this section is "about KSmrq's behaviour", I also feel necessary to recognize again, as I did several times before, that his contributions in this and other talk pages and articles are typically outstanding and extremely useful. I have always held him in high esteem and I hate being forced to fight against him to defend my right to edit and discuss reverts.

NOTE: I received a message about this topic by Arcfrk on my talk page, and answered on his talk page. Paolo.dL 12:02, 11 October 2007 (UTC)

Please, if you're going to quote me, get it right: the spelling is "diarrhea" (with an "h"), from the Greek meaning to flow through; and I said "exemplary" prose.

I am happy to respond to questions or criticisms from you or anyone else here on this talk page, so long as they are productive for the article. I am not happy to see you edit on topics you do not already understand; sensible editors respect their limits. --KSmrq^T 13:15, 11 October 2007 (UTC)

Unless you can prove that I do not already understand, you should write: "on topics I assume you do not already understand". You see, the explanation that you refuse to give (which might even prove that I do not understand!) would also show a way to make the article on inverse function clearer. In my opinion, the definition section given on inverse function is still far below your writing standards. Please also consider that an explanation about a revert is needed, even when you are "not happy". Of course, you can also write that you are not happy, that I am disturbing you, even that I am ignorant and stupid and stubborn. But first you must provide a reason. Paolo.dL 13:39, 11 October 2007 (UTC)

I'm afraid I agree with KSmrq; your (Paolo.dL) early edits in this article clearly show you do not understand the concept of "function". Assuming good faith, I assume that, if your more recent edits were accurate, that KSmrq reverted them because none of your previous edits had been both understandable and accurate. (This was written without checking recent edits; it's possible you're both wrong.) — Arthur Rubin | (talk) 14:04, 11 October 2007 (UTC)

I am totally appalled. Your attack really does not make sense to me. You are absolutely wrong. Please say exactly which of my 50 edits "in this article", most of which were accepted even by KSmrq, showed that I did not understand the concept of function. [By the way, my first edit in this article was on the 5th of September]. It is true that I did not understand the concept when I read the article for the first time, and that's why I discussed it with CBM, Trovatore, you and others, in this talk page and on Talk:Partial function. That's also the reason why I later edited the article, with your and CBM's and Arkfrk's help. Before my edits, this article was much longer and more redundant, and its TOC was even worse than it is now (see Carl's comment above, dated 6 October).

Also, are you really maintaining that, after a mistake or a few mistakes, an editor loses his right to edit and to be explained why his edits are reverted, in the rare occasions when they are? As far as I know, even vandals are not treated that bad in Wikipedia. Paolo.dL 15:00, 11 October 2007 (UTC)

Taking your last note, first. There's a certain editor (User:WAREL, and sock puppets, and IP addresses), who frequently makes edits in mathematics articles which are not correct. There is a general agreement that his edits can be reverted as vandalism, even if they are occasionally correct, until he provides justification. — Arthur Rubin | (talk) 21:00, 11 October 2007 (UTC)

Your analogy with people who frequently makes incorrect edits is totally arbitrary. Nobody ever, except for KSmrq, treated me as a vandal. Everybody can see in the history what I did to improve this article. You should either substantiate your statement about my edits in this article, or apologize for your unfair attack. I believe that your comments are based on a misunderstanding (see your talk page). Do you realize that you are publicly offending me without providing a valid explanation? Paolo.dL 22:15, 11 October 2007 (UTC)

Sorry, your edits were just confusing, rather than being wrong. I agree that KSmrq probably shouldn't have reverted you without further comment, but it appears to me that his first comment was "revert confusing" rather than "revert vandalism"; only when you reinserted the information did he imply it was vandalism. I tend to agree it was more confusing than the section on inverse function and composition that you replaced, but I think I would have tried to improve it, rather than simply reverting. — Arthur Rubin | (talk) 23:51, 11 October 2007 (UTC)

Arthur, thanks for checking. Of course, I do not agree about the fact that my edits were confusing. Neither the "early edits" that you referred to, nor the latest ones, those that KSmrq reverted. They improved this article a lot. The article was very confusing when I first read it (possibly at a negative peak in its readability history), much less after my edits, and still better now, after KSmrq's revision. I am not responsible about the readibility of the article in general! I am responsible about beginning this discussion, which called the attention of many excellent editors (including you) to the bad state of the article. Most of my work consisted of just removing redundancies, moving/creating sections/subsections/figures, and clarifying a few sentences, mainly in the definition section. Next time you express a judgement on my work (even a judgement such as "it's confusing"), I beg you to substantiate, otherwise I will interpret it as bias, rather than a contribution to our common goal. I mean, please bring at least one example. And if you refer to just one or two edits out of 50 (which one?), please do not generalize. I appreciated your help in removing redundancies from this article. I hope I will be able to collaborate with you and KSmrq in the future without being forced to fight against unsubstantiated claims or biased behaviour. That was very hard to accept for me and it will always be. Paolo.dL 09:46, 12 October 2007 (UTC)

Specific comments about latest edits

As for the content of this article,

I was interested in the definition, which was not clear at all when I first read it. Now it is. Carl's original definitions were also quite interesting. However, the current main definition of function is very effective and much easier to remember than Carl's main definition. The reason is that the two words "total" and "single-valued" may act (together with the figures) on the intuition and memory of a layman as two pins on a pinboard. If you decide to restore Carl's definition(s), please do not delete the current definition, and keep it in a separate subsection of the definition section (do not move it back to the end of the article, where it was before I moved it). And please do not forget to say explicitly that it is equivalent to the "main" formal definition.
I was also interested in the definition of inverse function, and when I first read the section in this article it was not consistent with the definition given in the main article. Now it is. But it could be made clearer. In my opinion, if you want to summarize the definition of inverse functions, either you decide to give the most widely used definition (as in the main article), in short:

g(f(x))=x and f(g(y))=y

or you write explicitly:

g(f(x))=x, or f(g(y))=y, or both.

By the way, this can be written using other notations, and I agree with KSmrq: we don't need to explain that Y may be either the range or the codomain of f.

This was pointed out by Jitse Nitsen on a recent editing summary on inverse function: KSmrq's "round trip" does not necessarily start from X. Writing just "g(f(x))=x" (as KSmrq did in his original edit), or just a vague definition together with a two-sided example (as he did in his last edit) is questionable, in my opinion. On that, I perfectly know KSmrq's opinion (see his 3 october comment above), but I would like to know the opinion of others as well.

Paolo.dL 10:25, 11 October 2007 (UTC)

As for the current definition section, it's better than Carl's but I question whether it's better than KSmrq's. I'm willing to accept this as a compromise. — Arthur Rubin | (talk) 14:15, 11 October 2007 (UTC)

RE definition. Maybe keep the first one as it is now, maybe bulletize the second definition (although KSmrq's prose version is pretty darn good) but whatever ... , I too support the use of two definitions here, the first one "easy", the second with its additional notions of "cartesian product" and "graph". Bill Wvbailey 14:37, 11 October 2007 (UTC)

Put the definitions here so we can look at them: Wvbailey 03:06, 12 October 2007 (UTC)

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KSmrq version pulled from history:

First definition

The simplest definition of binary relation is "A binary relation is a set of ordered pairs". Under this definition, the binary relation denoted by "less than" contains the ordered pair (2, 5) because 2 is less than 5.

A function is then a set of ordered pairs with the property that if (a,b) and (a,c) are in the set, then b must equal c. Thus the squaring function contains the pair (3, 9). The square root relation is not a function because it contains both the pair (9, 3) and the pair (9, −3), and 3 is not equal to −3.

The domain of a function is the set of elements x occurring as first coordinate in a pair of the relation. If x is not in the domain of f, then f(x) is not defined.

The range of a function is the set of elements y occurring as second coordinate in a pair of the relation.

A codomain of a function is any superset of the range.

Second definition

Some authors require that the definition of a binary relation specify not only the ordered pairs but also the domain and codomain. These authors define a binary relation as an ordered triple $(X,Y,G)$ , where X and Y are sets (called the domain and codomain of the relation) and G is a subset of the cartesian product of X and Y (G is called the graph of the relation). A function is then a binary relation with the additional property that each element of X occurs exactly once as the first coordinate of an element of G (which is called the graph of the function). Under this second definition, a function has a uniquely determined codomain; this is not the case under the first definition.

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As it appears now: Basic definition #1:

The formal definition of function is as special kind of binary relation. The condition for a binary relation ƒ from X to Y to be a function can be split into two conditions:

ƒ is total, or entire: for each x in X, there exists some y in Y such that x is related to y.
ƒ is single-valued: for each x in X, there is at most one y in Y such that x is related to y.

Advanced definition #2, Carl's version from history:

The consensus of modern mathematicians is that the word "rule" should be interpreted in the most general sense possible: as an arbitrary binary relation.

Thus it is common in advanced mathematics (see Bartle (2001) for an example) to formally define a function f from a set D to a set C to be a set $G_{f}$ of ordered pairs (x,y) in the Cartesian product $D\times C$ . It is required that for each x in D there is at most one pair (x,y) in the set $G_{f}$ . The "rule" of the function is: given x in D, if there is a pair (x,y) in $G_{f}$ then f(x) = y, and otherwise f(x) is not defined. The set of x for which f is defined is called the domain of f; if the domain of f is all of D then f is called total and the notation $f\colon D\rightarrow C$ is used. The set C is called the codomain of the function; this must be specified because it is not determined by $G_{f}$ . The set $G_{f}$ is called the graph of the function.

Advanced definition #2, Bill's Manin-version::

This formal definition of function comes from set theory: Function are restricted binary relations "identified with their graphs; otherwise, we would not be able to consider them as elements of the universe [of discourse]"(Manin 1972:12). The following are Manin's three conditions on the definition. In following, f is a function (mapping), u is a set of elements u_i called the domain, v is a set of elements v_i called the codomain, u x v is the Cartesian product -- the set of all possible ordered pairs <u_i, v_i>:

f is a subset of u x v
the projection of f onto u coincides with all of u
each element of u corresponds to exactly one element of v

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Definition ruminations

Carl says, "I think the contrast between "rule" and "relation" is worth pointing out here." And he asks for comments on this version.

The contrast is not about informal versus formal; rules can be formal. The problem with rules is that we limit ourselves. Defining functions by rules based on arithmetic and on series was once acceptable, but important examples burst those bounds (Kleiner 1989 harvnb error: multiple targets (2×): CITEREFKleiner1989 (help); O'Connor & Robertson 2005).

Kleiner, Israel (1989), "Evolution of the Function Concept: A Brief Survey", The College Mathematics Journal, 20 (4): 282–300 (This paper won a George Pólya Award.)
O'Connor, J. J.; Robertson, E. F. (2005), The function concept, retrieved 2007-10-12

Some history

The concept of function was forced to evolve. Although numerical dependency had been observed and tabulated in antiquity, the abstraction we call "function" only became necessary with calculus (later, analysis). Euler first tried:

"A function of a variable quantity is an analytic expression composed in any way whatsoever of the variable quantity and numbers or constant quantities."

His "any way" included infinite sums. But he later found the need for more:

"If some quantities so depend on other quantities that if the latter are changed the former undergoes change, then the former quantities are called functions of the latter. This definition applies rather widely and includes all ways in which one quantity could be determined by another."

He said it, but he didn't really believe it. Everyone still thought of functions as continuous, differentiable (almost everywhere), and (privately, if not publicly) given by some kind of formula. When that, too, proved overly restrictive, Lacroix tried:

"Every quantity whose value depends on one or more other quantities is called a function of these latter, whether one knows or is ignorant of what operation it is necessary to use to arrive from the latter to the first."

Dirichlet shook things up with his famous everywhere-discontinuous monstrosity, defined over the unit interval to be 0 at rationals and 1 elsewhere. The need for something flexible yet precise was becoming more urgent; as Hankel said:

"One person defines functions essentially in Euler's sense, the other requires that y must change with x according to a law, without giving an explanation of this obscure concept, the third defines it in Dirichlet's manner, the fourth does not define it at all. However, everybody deduces from his concept conclusions that are not contained in it."

Producing pathological functions became a cottage industry, and not everyone was happy with the trend. Poincaré protested:

"For half a century we have seen a mass of bizarre functions which appear to be forced to resemble as little as possible honest functions which serve some purpose."

One suspects he would have hated Bourbaki!

Some history

If we are going to give a definition like Bourbaki, we'd better first motivate it. Even among modern authors, few need to be so fussy. It's just as well; Bourbaki's style crippled a generation of students. Here's a reminder of what we're talking about:

Since Bourbaki’s Theory of Sets builds on a foundation of logic and sets, functions are introduced early. A graph G is defined as a set of couples (ordered pairs). It has two projections: pr₁ gives the set of first objects, and pr₂ gives the set of second objects. If (x,y)∈G, the object y corresponds to x. A correspondence between sets A (the source) and B (the target) is a triple (G,A,B), with pr₁G⊂A and pr₂G⊂B. A functional graph is a graph for which at most one object corresponds to any x. A correspondence ƒ = (F,A,B) is a function if F is a functional graph and if pr₁F=A.

Bourbaki verbatim

Here is Bourbaki in English translation, from Google Book Search.

4. FUNCTIONS

DEFINITION 9. A graph F is said to be a functional graph if for each x there is at most one object which corresponds to x under F (Chapter I, §5, no. 3). A correspondence ƒ = (F,A,B) is said to be a function if its graph F is a functional graph and if its source A is equal to its domain pr₁F. In other words, a correspondence ƒ = (F,A,B) is a function if for every x belonging to the source A of ƒ the relation (x,y)∈F is functional in y (Chapter I, §5, no. 3); the unique object which corresponds to x under ƒ is called the value of ƒ at the element x of A, and is denoted by ƒ(x) (or ƒ_x, or F(x), or F_x).

Here is the original French (ISBN 2-903684-03-0).

4. Fonctions

DÉFINITION 9. — On dit qu’un graphe F est un graphe fonctionnel si, pour tout x, il existe au plus un objet correspondant à x par F (I, p. 40). On dit qu’une correspondance ƒ = (F,A,B) est une fonction si son graphe F est un graphe fonctionnel, et si son ensemble de départ A est égal à son ensemble de définition pr₁ F. Autrement dit, une correspondance ƒ = (F,A,B) est une fonction si, pour tout x appartenant à l'ensemble de départ A de ƒ la relation (x,y)∈F est fonctionnelle en y (I, p. 41); l'objet unique correspondant à x par ƒ s’appelle la valeur de ƒ pour l’élément x de A, et se désigne par ƒ(x) ou ƒ_x (ou F(x), ou F_x).

Bourbaki verbatim

Bourbaki gives us many useful terms to work with, but cannot stand alone. We must acknowledge common, rigorous (or not) alternatives and different language. Novices assume mathematics is uniformly consistent in its notation and definitions; it is not! Bourbaki's "function" is not that in "primitive recursive function", nor that in "Dirac delta function", nor that in "fractional linear function".

Incidentally, I don't believe I have ever contributed a definition to the article. The one being attributed to me was there long before my edit reverting Bo Jacoby. The edit history contains a number of interesting attempts stretching back over several years, a cautionary message for present enthusiasts. --KSmrq^T 11:03, 12 October 2007 (UTC)

Comment about Carl's definition

This business of "rule" is tricky and potentially ambiguous. Yes there is a "rule" that does this: "Do until hat empty: pluck an x from hat (domain D containing white balls with 1 thru 10 written on them, for example) and randomly assign a number to the ball (say randomly selected from 1 thru 10 using the =10*INT(RAND()) "function" ), pair each white ball with its randomly-selected number (e.g. written on a sticky-tab stuck to the ball) and throw the pair into the place called 'graph' ". My take on it is: this example is not what most folks think of as a rule in the mathematical sense: most folks think of a rule [fr. L. regula straightedge, regere to lead straight. First definition: (a) a prescribed guide for conduct or action], as a deterministic process as described by a "math formula". Plug in "x", turn the crank, out comes a single "y". And every time you plug in a particular x you get the same y out, not as the hat example shows, if you repeat the experiment you get different outcomes.

Emendation of Advanced definition #2, Carl's version from history:

The consensus of modern mathematicians is that the word "rule" should be interpreted in the most general sense possible: as an arbitrary binary relation. Thus it is common in advanced mathematics (see Bartle (2001) for an example) to formally define a function f from a set D to a set C to be a set $G_{f}$ of ordered pairs (x,y) in the Cartesian product $D\times C$ . It is required that for each x in D there is at most one pair (x,y) in the set $G_{f}$ . Thus in this most general sense, the "rule" of ~~the~~ any function is: given x in D, if there is a pair (x,y) in $G_{f}$ then f(x) = y, and otherwise f(x) is not defined.

However, given that it exists, a "rule" for a specific function must eventually produce a single y (output) given a particular instance of x (as input).

[are there other criteria to be added? Or qualifications Such as "even rules using random elements can be considered functions if they meet the basic criteria"]

In either case, the set of x for which f is defined is called the domain of f; if the domain of f is all of D then f is called total and the notation $f\colon D\rightarrow C$ is used; otherwise if some x do not produce output then f is called partial. The set C is called the codomain of the function; this must be specified because it is not determined by $G_{f}$ . The set $G_{f}$ is called the graph of the function.

In addition to those already linked, I added boldface to the words that might be "imprecisely known" for a newbie. Bill Wvbailey 15:21, 12 October 2007 (UTC)

The definition of a function is not limited to "set theorists" in any way - it is perfectly common in all of mathematics. For example, Bartle's book is an advanced calculus book, and the Nicolas Bourbaki group were not set theorists. The sentence about a terminating process/method just confuses the issue: for most relations, the only "method" that can be ascribed is to look at the relation and choose the appropriate value using it. This is in no way an algorithmic process (for example, an arbitrary binary relation on the reals is not computable in any sense of the word, nor even definable). Algorithmic considerations are not part of the definition of functions in ordinary mathematics. — Carl (CBM · talk) 19:10, 12 October 2007 (UTC)

Okay, try the above. Bill Wvbailey 19:18, 12 October 2007 (UTC)

Definition → Definitions

In light of the discussions we've been having, I felt it was time that we showed readers more variety in definitions, rather than pretend that one definition dominated and subsumed all the others in terms of popularity, importance, and rigor. Despite knowing that everyone will now be obliged to hate the change, I have rewritten the section and retitled it in the plural.

As bullet points I give four rather different forms of definition. As I hope the text makes clear, these should not be considered merely different ways to say the same thing. I inject commentary (too much? too little?) on the definitions to suggest motivation, strengths, and weaknesses for each.

I could probably step away for a day and see a dozen things to improve when I look again. More explicitly, I hold no fantasy about this being polished perfection; I do hope it is a step in the right direction.

Here's what I would like to see preserved:

multiple definition flavors, including rules, graphs, and relations
no false claim that rules are necessarily informal or inferior
no elevated status for computability

The Introduction immediately preceding the Definitions should probably be amended to better prepare for the new variety.

Just for fun, the formula in my first function example, y = 5x−20x³+16x⁵, is the fifth Chebyshev polynomial, T₅, satisfying T₅(cos(θ)) = cos(5θ).

Enjoy (I hope!). --KSmrq^T 10:38, 13 October 2007 (UTC)

small comments

(1) First definition: just an interesting note, that in Kleene's formalization of a number-theoretic "system" along the lines of Godel he defines only two "function symbols" -- + for addition and * for multiplication.

(2) Second definition:Perhaps this is where the notion of a "table" to express an otherwise expressible function comes in?

(3) Third definition: In the definition where "graph" appears, use G instead of F. Also, pr₁ and pr₂ are "projection" I assume. You might want to expand that a bit (altho believe the notion of "image" appears later, perhaps expand this notion of projections and "coordinates" here?). Also, you might want to index x i.e. x_i and y, y_i, or make it clear that the x ∈ X and y ∈ Y. Somehow the notion of Cartesian product has disappeared here but it reappears later as Z x Z, etc. In the spirit of defining words and symbols as they are used, the ⊂ and ∈ require explanation, in fact shouldn't ⊂ be ⊆? You could write "Perhaps every element y in Y (usually symbolized by y∈Y) is associated with some x∈X, perhaps not." Perhaps here the words and notions of "restricted domain aka domain of discourse" ⊆ domain aka universe" and "range ⊆ codomain" could come in.

(4) Fourth definition: Again, make sure that the notionof Z x Z etc. gets defined somewhere.

Other than that, this is more in the spirit of what I would like to see in this article. I agree that the first section could get pruned. Perhaps the later parts (after the definition section) could get pulled into these definitions (thus expanding the definition section), and perhaps even eliminated. Last night I went through and found the following words used in the version just previous to this one (words from the two sections just following the definition section):

argument
value
image under
graph
ordered pairs
subsets

Cartesian coordinates
range
preimage, inverse image
singleton set
set theoretic symbolism { }, the "element" sign ∈, f^-1

injections, one-to-one function
surjections, onto function
bijections, one-to-one and onto
restriction
extension

Other words not used, or used in previous "definition" versions such as Carl's:

binary relation
Cartesian product
partial (function), I think this appeared in the "definitions"
codomain
projection(s)
first and second coordinate

Bill Wvbailey 14:49, 13 October 2007 (UTC)

Thanks for your comments. Looking backing, using the projection notation was a silly blunder; not only was it never defined, but formal notation was intended for later in the section. So I have fixed that and made a few other improvements.

(1) It is quite common in foundation theory to build natural numbers from sets, then to define addition and multiplication inductively/recursively. My notation for the polynomial used exponents, but these can be written out as multiplications. (Or, we can define powers inductively.)

(2) No, you really need to break that mental block! An explicit table is still a way to write down a rule, and we can take it as "syntactic sugar" for a definition by cases. Honestly, we can prove the existence of functions that can never be written down with a finite string of symbols, no matter how clever our notation. Typically these involve the Axiom of choice. We can also use an argument from cardinality. Our notation uses strings of symbols, and we can easily prove that the set of all possible strings ("things we can write down") is countable. But the set of all possible real-valued functions over the unit interval is uncountable. So there is an enormous gap.

(3,4) In Bourbaki we find G for a general graph and F for a "functional graph"; I prefer to stick with F. As noted, I have tried to clean up some problems with notation and nomenclature. Also see my further comments below.

We should combine and rewrite the Vocabulary and Notation sections, and place that immediately following the Definitions. The new material in Definitions should allow us to more easily introduce everything we like.

Concerning the Introduction section, my thought is to better motivate and prepare for the definitions, to give some feel for all the different things we do with functions. Some current material might be removed, but new material might also be added.

By the way, the subsection "Functions versus maps" is far too myopic. Many authors use "map" as a synonym for "function", and explicitly say so. In topology a map is usually understood to be a continuous function, whereas a function might be more arbitrary. The present wording is yet another example of someone too tightly focused on their own tiny copse of trees to see the entire forest of usage. I meant to fix this; maybe later.

Finally, I hate to sound like a broken record, but I'd really appreciate it if Paolo left it alone for others to consider. Lack of sensitivity to nuances of the writing — its meaning, flow, structure — is painful to see in edit after edit. --KSmrq^T 10:40, 14 October 2007 (UTC)

PS: I would especially appreciate comments from the numerous experienced mathematicians — Carl, Rick, Arthur, Arcfrk, … — who had criticisms of earlier content. Looking at the big picture, is this Definitions rewrite helpful or a distracting waste of time? Is it too fussy? Too sloppy? Does it adequately address your stated concerns? Raise others? Does it tell any lies? Omit important truths? --KSmrq^T 10:59, 14 October 2007 (UTC)

(I have to go away and think about the notion of knowing (and constructively proving, or do such proofs use the LoEM extended to the infinite?) the existence of something that cannot be defined; this seems to fly in the face of Turing's thesis. But anyway...) A final thought because I think you're on the right track here: I may not be an experienced mathematician but I am a very experienced writer and drafter (wiki example: see discussion page at Busy beaver); to do a bang-up job here you need more than mathematicians on your team -- you need talent from all walks of life. You are on the right track because the article will be more inclusive of a wide range of readers. RE a heavy emphasis on set-theoretic language (I am agnostic, but the following is a published viewpoint), Ponte's paper "The History of the Concept of Function and some Educational Implications" has this to say about that: "It is certainly possible to define functions in a very general form, for example as sets of ordered pairs, emphasizing an algebraic perspective in elementary mathematics. This is not, however, a suitable basis to produce an accessible elementary mathematical theory, rich in intersting results and in significiant applications." (p. 13). Ponte is aiming his article at secondary-school (high-school) educators: "Most students arrive at secondary school with many difficulties in abstract thinking. For many, dealing with Cartesian graphs and algebraic expressions, is not an easy task. The teaching of functions needs to articulate in a balanced way the three most important forms of representation, namely numberical, graphical and algebraic forms ... tables and computations." (p. 11) Bill Wvbailey 15:55, 14 October 2007 (UTC)

As a very simple example of a function that must exist but can never be defined, consider the function f(x) = k, where k is a constant integer so large that it can ever be written out, writing one digit per second, in the lifetime of the universe. We know such a function must exist, but we can never give a rule for writing down even one ordered pair...it would take too long.

Now, the above is an explanation for someone who is not a pure mathematician. Pure mathematicians have much more subtle and interesting examples -- but you need a lot of background to understand them. For example, let S be the set of all pairs of group presentations and define f(p,q) to equal 1 iff p and q represent the same group, 0 otherwise. We can prove that f exists but can never be given by any method whatsoever. Counterintuitive, but it follows logically from ZF+choice. Rick Norwood 20:44, 14 October 2007 (UTC)

(Response to Ksmrq) I've been tied up a few days, but finally got a chance to answer your request for comments about the new text. I don't think it has any significant omissions or oversights. The idea of contrasting the various definitions is sound. There are a few things that can be improved by minor copyediting, which I will do next. I think the next to last paragraph is too terse. The last paragraph is a holdout from previous versions, and I think it can be removed. — Carl (CBM · talk) 00:40, 17 October 2007 (UTC)

KSmrq's edit

Good work, KSmrq! Rick Norwood 20:37, 14 October 2007 (UTC)

Peace please

KSmrq: peace, please. I am not fighting against you. We have the same goal. I hold you in very high esteem, I really like your text, you did a precious job, your writing style is a model for me, but my latest edit was useful to make clearer what a layman cannot easily see in your text. You say that I am not sensitive for the nuances and flow of your writing, but in my opinion the explanation about pairs, triples and Cartesian products that you inserted in your latest edit makes your text less fluent that it was after my edit. Your nuances are absolutely fascinating for me when I understand them. In most cases, you seem to be one of the most skilled editors I know, because you can explain math with simple words, without assuming (too much) previous knowledge. And you know how to give tremendous weight to single words and avoid unneded redundancies. But nobody is perfect. I represent non-mathematicians and I can see better than mathematicians when further cleaning is needed to make some text more easily understandable to a broader public. So, I am only asking you not to assume I am not sensitive, but to just humbly step down your pulpit once in a while, and accept peer to peer dialogue without assuming lack of sensitivity or respect or skill. Of course, as a writer I am much less close to an ideal of perfection than you are. So, if you accept a serein discussion you will certainly be able to show that you are right whenever you are, and in some cases even when you are not! As for mathematics, of course, you will be right even more frequently, but I never discredit your words, even when you are wrong (e.g., on the 19th of September, in this talk page, I contributed to remind you that, for some authors, invertible functions do not need to be bijective; but even then, I initially assumed you were right).

May I suggest a new strategy, not too stress you too much? I will create a new comparison table showing the difference between your text and mine, and number the list to make it easy for you and others to refer to each edit. If you don't want or don't have the time to answer, that's ok: others will be kind enough to answer, I hope. And since I am not here to fight a war, I also hope it will not be difficult to reach an agreement. Paolo.dL 12:55, 15 October 2007 (UTC)

Specific comments

Here are a few comments about the latest version of the "Definitions" section by KSmrq:

1) As for partial functions and multivalued functions, the text suggests that for some authors they are functions (with only one restriction rather than both). We discussed this topic in Talk:partial function with Trovatore and Carl, and they maintained that non-total and non-single-valued functions are not regarded as functions, even by those authors who call them partial and multivalued functions. For instance, Carl (CBM) wrote: "If you pick up any common calculus text, undergraduate analysis text, or undergraduate algebra text, you will get the definition of "function" as a total single-valued relation. It is completely standard." Note that, if this were true, the "common" restrictions would become "standard" restrictions.

2) Also, KSmrq wrote:

A function ƒ from a set X to a set Y associates to each element x in X an element y = ƒ(x) in Y.

This definition implies two restrictions. From the word "each" we can deduce that a function is total, and from y = ƒ(x) that it is single-valued. And the "tremendous flexibility" described later can be applied to this kind of definition as well. I mean, for instance X can be defined as Z×Z, and a less restrictive definition of function can be easily obtained:

A function ƒ from a set X to a set Y associates to each element x in X at most an element y in Y.

[By the way, in this case I guess the expression "partial function" would not be necessary anymore.]

However, the article seems to suggest that you have tremendous flexibility only when you define a function as an ordered triple of sets. Paolo.dL 12:55, 15 October 2007 (UTC)

3) KSmrq wrote: "The domain of F, and commonly of ƒ, is the set of all first elements of F". I suggest to add: "for some authors, the domain of ƒ coincides with its source, and not with the domain of F (also called domain of definition of ƒ)." Notice that the common terminology is greatly confusing:

	X	pr₁ F	pr₂ F	Y
1	source	domain	range [why not codomain?]	target or codomain [why not cosource?]
2	source or domain	domain of definition or domain of discourse	range	target or codomain

Can you see the lack of symmetry in the first row of the table? Without the additional sentence I suggested [summarized in row 2], the reader cannot understand the reason why the "codomain" is called "codomain", rhater than "cosource", or why the range is not called codomain! As you possibly know, I like to dream about hypothetical more intuitive terminology, and this is my dream:

	X	pr₁ F	pr₂ F	Y
3	source or domain	input range	output range	target or codomain

We are not here to change the world, but this hypothetical terminology is an example of intuitively appealing terminology, and I hope that, with time, mathematicians will more and more frequently choose to adopt the most intuitive available terminology. That's why intuitive terminologies should be reported in Wikipedia, even when they are not the most commonly used ones. Paolo.dL 14:32, 15 October 2007 (UTC)

I'm afraid the confusing terminology between domain of a function (or relation) and the domain of discourse on which a (partial) function or relation is defined is in the literature. I suppose we couldhoose a consistent terminology in our articles, except there really isn't one in the literature which has names for all 4 sets. We may not create new notation. — Arthur Rubin | (talk) 18:09, 15 October 2007 (UTC)

Yes, I only wrote that mathematicians should use "the most intuitive available terminology" (I meant available in the literature). And I suggested to add to the article a sentence such as "for some authors, the domain of ƒ is X, while the domain of F is called domain of definition of ƒ." This is the "most intuitive available terminolgy" I was referring to. It's option 2 in my table.
And no, I am not maintaining that option 2 is "confusing", and I am not suggesting to use here my hypothetical terminology (option 3). Option 2 is less symmetric than my hypothetical terminology, but it's not confusing at all, in my opinion. The only confusing terminology is, unfortunately, the most commonly used one (option 1), which is the only one described by KSmrq in the "Definitions" section. Paolo.dL 12:01, 16 October 2007 (UTC)

Who provided the modern definition of "function"?

The article says:

Dirichlet and Lobachevsky independently and almost simultaneously gave the modern "formal" definition of function.

However, no citation to the work of either Dirichlet or Lobachevsky is given, and Lakatos strongly disputes the claim:

There is no such definition in Dirichlet's works at all. But there is ample evidence that he had no idea of this concept. In his [1837], for instance, when he discusses piecewise continuous functions, he says that at points of discontinuity the function has two values: ...

(Proofs and Refutations, 151, Cambridge University Press 1976.)

I have tagged this sentence in the article with a {{fact}}. I hope that in the future the article can credit the correct person, whether that is Dirichlet or Lobachevsky or someone else. -- Dominus (talk) 18:39, 12 February 2008 (UTC)

Merge proposal

Someone (I think User:CBM proposed a merge of this article with function (set theory). What is proposed, exactly? Does anything need to be done, maybe function (set theory) can just be deleted? Sam Staton (talk) 08:18, 31 March 2008 (UTC)

We don't delete an article so easily, the content of function (set theory) is merged in function (mathematics) then function (set theory) is redirected to function (mathematics). Cenarium^Talk 12:26, 31 March 2008 (UTC)

I'll wait a few days for comments. If there aren't any objections, I (or someone else) will go through the merged article and add to this one the content that needs to be merged in. Rather than copying the entire other article here, the idea is to carefully merge individual facts to make sure this article covers them. Then the other article will be redirected to this location. It doesn't require any admin tools, just a few edits to the pages. — Carl (CBM · talk) 13:05, 31 March 2008 (UTC)

OK, thanks. As far as I can tell, the only things that are missing from this page, as compared to function (set theory), are the bits about currying, and the mention of category theory. Sam Staton (talk) 13:29, 31 March 2008 (UTC)

Definitions

The article has problems related to references, it lacks in-text citations. I think that the introduction is very computer science-oriented and neglects the set-theoretical aspect. The article states that there are no universally accepted definition of a function, but I think that it should be backed by several references if correct. I've always been thinking that the set-theoretical one was the more general and the more accepted one. I'd like to have more opinions, they are some old discussions on the talk page about this. Cenarium^Talk 20:35, 30 March 2008 (UTC)

The sentences in question,

"Because functions are used in so many areas of mathematics, and in so many different ways, no single definition of function has been universally adopted. Some definitions are elementary, while others use technical language that may obscure the intuitive notion."

seem reasonable enough in the context of the rest of the section that explains what those differences are. — Carl (CBM · talk) 20:41, 30 March 2008 (UTC)

But it's not referenced. I am quite sure that the set-theoretical definition can cover the others. All the sources I use when making maths use this definition per default. Of course it may be other trends unknown to me. In fact, the set theoretical definition: a functional and/or applicative (i.e. left-total) binary relation is the only (mathematical) definition I know. Cenarium^Talk 21:05, 30 March 2008 (UTC)

Which thing are you saying you would like a reference for? Remember that WP:V only requires that things be verifiable in theory, and in this case we are probably all aware of numerous textbooks with numerous definitions of functions. That isn't to say that the article can't be improved; I'm sure it can.

As for other definitions, look at a calculus or pre-calculus textbook. It will be quite unlikely to define a function as a special type of binary relation. Even the set-theoretic definition varies widely from one text to the next - must the domain be specified? The codomain? — Carl (CBM · talk) 21:19, 30 March 2008 (UTC)

Don't you think that the sentence you cited above should be referenced ? There is no way for a reader to verify this. Until mid 2007, the article said almost the contrary and proposed only the set theoretical definition. Textbooks in calculus generally don't define the concept of function. And it's precisely the aim of this section. For example, the first "definition" is not a formal one, it's a way to express a function using other functions. It could be misleading for the reader if we don't make the difference explicit. Cenarium^Talk 21:31, 30 March 2008 (UTC)

I can cite WP:V if you wish: ""Verifiable" in this context means that readers should be able to check that material added to Wikipedia has already been published by a reliable source" and "All quotations and any material challenged or likely to be challenged should be attributed to a reliable, published source using an inline citation.". I think that it applies here. Cenarium^Talk 21:36, 30 March 2008 (UTC)

Taking a quick look at the introduction of this article, it mentions functions in lambda calculus which are certainly not those of set theory (but personally I don't think lambda calculus is about functions at all, but about manipulation of expressions that in certain cases describe functions). I'm also convinced that any statement about lack of formal discipline in the mathematical community can be easily underpinned with many examples from literature. For instance I'm sure there are many instances where a correspondence that fails to be defined for all inputs like

x\mapsto 1/x

or even

x\mapsto {\sqrt {x}}

are called functions without explicitly specifying a suitably restricted domain. In fact I think the term "meromorphic function" describes something that in general fails the totality criterion. As an aside, not really relevant to the English Wikipedia, I mention that here in France the notion of "fonction" seems to be exempt of the obligation of being everywhere defined, while "application" (more or less equivalent to "map" in English) is used when this obligation is imposed. (Apart from this I think that Function (mathematics) can use a lot of improvement, and Function (set theory) even more.) And in general, I think that the notion of "Verifiable" should not be applied to mathematical statements in the same way as for instance statements about living people. Lots of simple arguments are clearly justified in themselves, while on the other hand there are doubtlessly many utterly silly statements/conventions that can easily be corroborated by the published literature. If I see an argument in some Wikipedia math article that has an obvious hole or can in some other way be clearly improved, I feel free to do that without worrying whether the result corresponds to anything published by a reliable source. If one starts throwing around WP:V, I'm sure one could scrap 90% of Wikipedia math content. Marc van Leeuwen (talk) 07:06, 31 March 2008 (UTC)

Here's are a few sources for comparative definitions of functions that you might use. (I haven't actually read them, but their titles look apropos.)

Youschkevitch, A. P. (1976), "The concept of function up to the middle of the 19th century", Archive for History of Exact Sciences, 16 (1): 37–85, doi:10.1007/BF00348305.
Monna, A. F. (1972), "The concept of function in the 19th and 20th centuries, in particular with regard to the discussions between Baire, Borel and Lebesgue", Archive for History of Exact Sciences, 9 (1): 57–84, doi:10.1007/BF00348540.
Kleiner, Israel (1989), "Evolution of the Function Concept: A Brief Survey", The College Mathematics Journal, 20 (4): 282–300, doi:10.2307/2686848.
Ruthing, D. (1984), "Some definitions of the concept of function from Bernoulli, Joh. to Bourbaki, N.", Mathematical Intelligencer, 6 (4): 72–77.
Dubinsky, Ed; Harel, Guershon (1992), The Concept of Function: Aspects of Epistemology and Pedagogy, Mathematical Association of America, ISBN 0883850818.
Malik, M. A. (1980), "Historical and pedagogical aspects of the definition of function", International Journal of Mathematical Education in Science and Technology, 11 (4): 489–492, doi:10.1080/0020739800110404.

—David Eppstein (talk) 21:38, 30 March 2008 (UTC)

I don't question the fact that there are many ways to define/express functions, I just say that the set-theoretical definition is nowadays the most in use as a formal definition, that we should make a difference between the formal ones, and the non-formal ones. The formal ones I know are set-theoretical (with minor differences). In the majority of mathematical articles I read, for example Ring (mathematics), there is a clear formal definition, then the variants are listed. What would make this article different ? Cenarium^Talk 21:51, 30 March 2008 (UTC)

Thanks for your research David, I think that these references can be included in the article. Cenarium^Talk 21:59, 30 March 2008 (UTC)

Re "what would make this article different": the notion of a function is a basic one that needs to be made understandable at the high school level. Ring theory is rather more advanced. Formalisms involving sets, binary relations (and how one goes about defining the ordered pairs necessary to define binary relations set-theoretically, and whether one uses a set theory with atoms or defines everything in terms of sets all the way down, or whether one replaces sets with topoi or whatever) aren't the way to do that. So, while I agree that set-theoretic definitions of a function common in research mathematics needs to be treated here, I think it would be a mistake to pretend that it's the only style of definition in use pedagogically today, or to make it the main content of the article. —David Eppstein (talk) 22:06, 30 March 2008 (UTC)

I agree, the set-theoretical functions shouldn't be highlighted in the introduction of the section definitions. But I think that instead of starting with an example to express a function using others, we should say something like "Intuitively, a function is a way to assign to each element of a given set exactly one element of another given set.". It's not too technical and it's not misleading. We can give examples after that then talk about the formal definition. What do you think ? Cenarium^Talk 22:13, 30 March 2008 (UTC)

(ec) To divert this discussion a bit, I think someone ought to mention that there are (at least) two versions of the set-theoretic definition of functions. A function can be defined (as in our article) as an ordered triple (graph, domain, codomain), but is also often defined to be just the graph. I think I've even seen the definition (graph, codomain) since (if we assume our functions are total) the domain is redudant. Then there's the question of whether functions have to be total or not; are article isn't very clear on this. Algebraist 22:17, 30 March 2008 (UTC)

Not to mention that which set theory you are using is also relevant. —David Eppstein (talk) 22:18, 30 March 2008 (UTC)

I think that we should create subheadings, for example "intuitive definitions", "set-theoretical definitions" and "other definitions". (There is probably better.) Cenarium^Talk 22:24, 30 March 2008 (UTC)

(ec) Then there's the question of which definition of ordered pair... Anyway, I appreciate I'll need to source this for it to go in the article, but to my mind the concept of 'function', like that of 'ordered pair' or 'ordinal number' is independent of the implementation of the concept in a specific set theory. We shouldn't think of the Kuratowski ordered-pair or the (graph, domain, codomain) function as definitions of these concepts, but as implementations that allow as to talk about these objects within our set theory. The important thing is that the implementation should allow us to prove (in the set theory) the key properties of the concept (ordered pairs are determined by their first and second elements, there's an ordinal corresponding to each well-ordering, etc. etc.), other properties of the implementation being profoundly unimportant (my lecturer used this as an argument for the axiom scheme of Replacement!). </rant> Algebraist 22:28, 30 March 2008 (UTC)

In this case, there is no ~~cabal~~ definition. Cenarium^Talk 22:34, 30 March 2008 (UTC)

In my perspective, all that matters is that all these "implementations" of the concept are equivalent in some sense, and then the definition is the (intuitive) equivalence class of these implementations. Cenarium^Talk 22:41, 30 March 2008 (UTC)

And so we can talk of the definition (the equivalence class) and a definition (a representant of the equivalence class). It's my personal view, I precise. I think that my opposition to the sentences above were due to the fact that I thought it was about formal definitions. It's a bit abusive to me but I understand that it's common to call definition things that are only intuitive representations of a concept. I'll try to clear this up in the article. Cenarium^Talk 22:57, 30 March 2008 (UTC)

The discussion of the actual content of the function article ought to move to talk:function. It looks like it will be another few weeks of discussion there. — Carl (CBM · talk) 01:27, 31 March 2008 (UTC)

Function (set theory)

For those interested in expanding the set-theoretical definition of a function, we have already this article: Function (set theory). I just found it out. Cenarium^Talk 23:45, 30 March 2008 (UTC)

Ick. This really needs to be merged into the main function article. — Carl (CBM · talk) 01:27, 31 March 2008 (UTC)

Indeed, we should create a section later in the article where we can put all the set-theoretical aspects. Since it's announced, we can add more advanced content there. Cenarium^Talk 02:05, 31 March 2008 (UTC)

I think almost all the content of function (set theory) is already in the main function article. That's one reason a merge is in order. I don't see that there is really any "set theory" in the definition of a function, any more than there is abstract algebra in the quadratic formula. — Carl (CBM · talk) 02:14, 31 March 2008 (UTC)

I agree with the merge. But can you give a formal definition of a function without using the concept of set, so set theory? To me, the concept of function is fundamental to set theory also because they are the morphisms between sets. The behavior of a function depends heavily on the underlying set theory, for example AC and the surjective functions. Cenarium^Talk 02:31, 31 March 2008 (UTC)

Calling every definition that uses sets "set theory" is overkill; set theorists don't investigate the definition of function in any serious way, they simply use it as part of their work. It's like calling pre-calculus "differential geometry" because it mentions derivatives, or calling the quadratic formula "complex analysis" because it involves i. Functions are part of shared, common core of mathematics, along with the natural numbers, the real line, and many other fundamental concepts. Put another way: many mathematicians who don't know any set theory know what a function is. — Carl (CBM · talk) 02:43, 31 March 2008 (UTC)

I suppose that there are different views on this. It depends if you consider a function as an intuitive object or as a formal object, defined in the setting of set theory. Anyway, we still can add relevant information in function (mathematics) about the properties and behavior of functions in set theory. Cenarium^Talk 03:11, 31 March 2008 (UTC)

The section treating the set theoretic definition of a function is bloated and, in my opinion, redundant. The main concepts are covered below in the section "vocabulary" (of functions). The rest of it reads like a bad textbook. I think that, instead of merging from "Function (set theory)", we should summarize the content of this section in one paragraph and restore the "main" link. A few points deserve to be mentioned:

Set-theoretic approach to functions is the hallmark of the failed "New Math" movement, which clearly demonstrated that excessive formalism should be avoided like plague.
As Geometry Guy explained in detail on the talk page almost exactly one year ago, most of productive uses of functions in mathematics are not based on set theory, nor do they conform to the rigid formal definitions given using sets. Intuition related to functions may be algebraic, analytic, or geometric, but very rarely is it set-theoretic.
The key to improving this article is in illustrating the multiple uses of the concept of the function, rather than regurgigating generalities.

Arcfrk (talk) 04:23, 31 March 2008 (UTC)

Certainly, the use of functions goes beyond any set-theoretical view on functions and was already in massive use long before set theory emerged. But if we want to talk about the definition of a function, the definition using sets deserves to be mentioned. I think too that the use of function in analysis is generally based on R, and the construction of R requires set theory (N, equivalence classes...), it would be a mistakes to think that functions, even in elementary analysis don't use set theory at all. What about the set theory of the real line, and the use of AC in analysis, especially functional analysis ? I don't think that the definition should be overly emphasized, but there are definitely things to say in a set-theoretical point of view. Cenarium^Talk 12:40, 31 March 2008 (UTC)

The construction of the real numbers from the natural is just one construction, not the only one. The article on functions is not the place to worry about the use of the axiom of choice in functional analysis.

Re Arcfrk: I agree it will take a while to find a new consensus about the 'definition' section. But I can explain why the function (set theory) article needs to be merged independent of that section; I was just unaware of the need to merge until I realized the article existed. (1) Function (set theory) is not about the use of functions in set theory, it is just about the ordinary definition of functions. (2) It is, almost completely, a duplicate of the main function article - the merge won't add very much to the main article. This is because (3) There isn't a bunch of stuff about "functions" in set theory to talk about, apart from the definition. The material that Cenarium mentions (set theory of the real line, the axiom of choice) is covered in other articles, and doesn't belong in an article about functions. — Carl (CBM · talk) 13:18, 31 March 2008 (UTC)

All the constructions mentioned in Construction of real numbers requires set theory, more or less elementary. I think that there are things to mention, but not to develop too much of course. Do you think the we should move all the thread to talk:function (mathematics) ? 13:41, 31 March 2008 (UTC)Cenarium^Talk

You could define the real numbers to be a real closed ordered field, for example, without starting with the natural numbers. Yes, it would be better to discuss the function article on its talk page. — Carl (CBM · talk) 14:46, 31 March 2008 (UTC)

Yeah but it doesn't prove the existence of R, the construction of R requires constructions and theorems of a set theory, for example ZF or ZFC. Even if we only want to work axiomatically, for example using real closed fields, the theory uses really a lot of set theory. The point is that even the "elementary" functions used in analysis are based on set theory, one way or another. And we shouldn't forget that. Cenarium^Talk 17:48, 31 March 2008 (UTC)

You're advocating a certain viewpoint - that all of mathematics is based on set theory. The actual state of affairs in research in the foundations of mathematics is much less clear about such things. In one direction, systems much weaker than ZF can interpret the real numbers (for example, weak systems of second-order arithmetic). In another direction, those who pursue categorical foundations would be unhappy with the claim that their work is "based on set theory". It's a more accurate summary of things to look at ZFC as just one of many possible foundational systems, one in which most of mathematics could hypothetically be formalized. — Carl (CBM · talk) 18:23, 31 March 2008 (UTC)

Actually, I'm not a fan of set theory, and I'm more like a category theorist (though I'm still undergraduate). I never said that all mathematics are based on set theory. But the concept of function as I know it is based on set theory. I used R as an example because it's the main object of study in elementary analysis. It's certainly possible to define the real numbers in weaker systems, but I'm not sure that we can have a rich theory about the reals and functions over reals like this. ZF is an illusion for sure. Cenarium^Talk 17:01, 1 April 2008 (UTC)

By the way, I was talking of set theory in the most general sense, not only ZF, there are many systems that can define their own concept of set, and function, for example a system with predicates, or a class theory with a notion of smallness. The second order arithmetic cannot interpret the majority of functions between reals nor the majority of subsets of reals. But, maybe, a third-order arithmetic ? Cenarium^Talk 12:43, 2 April 2008 (UTC)

graphing & evaluation tool

Comprehensive web-based function graphing & evaluation tool

does not seem to work... I tried to plot sin(x)/x and Re(exp(I*x)); both failed. I would exclude this reference. dima (talk) 03:25, 13 April 2008 (UTC)

Works for me - just enter sin[x]/x in the text box, specify min & max values for x, and you're in business. Complex values aren't supported, however, so you're second case won't work... --Cheese Sandwich (talk) 03:40, 13 April 2008 (UTC)

root(x) is a function.

$\pm {\sqrt {x}}$ perfectly is a function. It is just not a function from $\mathbb {R} _{+}$ to $\mathbb {R} _{+}$ . It is a function from $\mathbb {R} _{+}$ to ${\mathcal {P}}(\mathbb {R} )$ (or from $\mathbb {R}$ to ${\mathcal {P}}(\mathbb {C} )$ , depending how one looks at it). Namely, it returns a set $\{r_{1}\in \mathbb {R} ,r_{2}\in \mathbb {R} \}$ . Correct me if I'm wrong though.Niarch (talk) 02:23, 17 April 2008 (UTC)

Mathematically, Niarch is correct: the function root can be defined in such a way. However, in the physical slang, the sign

\pm

may have different meanings. For example, if

x

is coordinate, its value is supposed to be real number, not a set mentioned; then, statement that

x=\pm {\sqrt {y}}

means that

x={\sqrt {y}}

or

x=-{\sqrt {y}}

. Therefore, the answer to the question (wether root is function) depends on the definiiton of symbol

\pm

. dima (talk) 15:56, 17 April 2008 (UTC)

Dima is right here, and his edits clarify things. But I'd like to point out that Niarch's idea comes from something more general: to give a binary relation between sets X and Y is to give a function X→PY. (This observation is sometimes used as a definition of powerset, eg in topos theory.) I'm not sure that this observation belongs to this article though. Sam Staton (talk) 20:00, 17 April 2008 (UTC)

As a function of

The article needs to explain point blank in the intro something like this.

"The initial velocity (v) of an enzyme catalysed reaction is measured as a function of the substrate concentration, (s)."

Then:

"In this case the function would be ________"

It needs real world examples, not blather and silly diagrams from overzealous math graduates.--148.197.5.19 (talk) 15:51, 15 May 2008 (UTC)

Nonsense. This article is about functions in mathematics. If you want to discuss "realistic" functions, that should be in another article entirely. — Arthur Rubin (talk) 23:59, 15 May 2008 (UTC)

I agree with Arthur, and disagree with the anon's assessment of the article as "blather and silly diagrams," but there's certainly value in discussing applications of functions, and in particular in noting that a function isn't always defined by a simple rule but can be any set of pairs of points, even if they're empirically determined, or computed by a complex algorithm. Dcoetzee 01:46, 16 May 2008 (UTC)

Notation-- incomplete

The article uses some notation that it doesn't describe in the Notation section. I'd expand the Notation section myself, but I don't know what the trailing comma and period mean below:

{\begin{aligned}f\circ \mathrm {id} _{X}&=f,\\\mathrm {id} _{Y}\circ f&=f.\end{aligned}}

I've also seen other function notations, including use of the trailing comma followed by one or more subscripts.

--98.31.54.35 (talk) 20:42, 1 June 2008 (UTC)

The trailing comma and period are unnecessary punctuation, and should be removed. Rick Norwood (talk) 13:28, 2 June 2008 (UTC)