Talk:Function (mathematics)

Edits, 4th February 2012

When we define a function as a subset of D × C, the codomain is C. This definition is standard.

Please edit the article to improve it, rather than reverting everything.

Rick Norwood (talk) 01:09, 4 February 2012 (UTC)

You gave me a total of 19 seconds to add my comment here. I submit that is not enough time for me to reasonably respond. I was the person who added the "ordered triple" definition in the first place. But, challenged by the request for a reference, I read a number of standard sources, and none of them used the "ordered triple" definition. Convinced that it is non-standard, I attempted to say what my sources (Holstein, Manin, Halmos, Rudin) say. You've restored it, and claim you will find a reference. I hope you do. I certainly remember it from grad school. But it does not seem to be standard today. Rick Norwood (talk) 01:13, 4 February 2012 (UTC)

You seem to be doing a lot of editing, and I think we are both serious about wanting to improve the article, so I'm going to call it a night, and see what you've accomplished in the morning. Rick Norwood (talk) 01:37, 4 February 2012 (UTC)

I have finished, there is very little there except putting in a citation and moving the things around a bit plus I removed the big triple that I though added nothing.

The comment should have been added before saying go to the talk page to discuss.

I'll try and have a look at the books you've mentioned to see what they do about the domain and codomain. Dmcq (talk) 01:56, 4 February 2012 (UTC)

I've put in a bit about defining a 'function from X to Y' as a set of pairs. The set of pairs isn't a 'function' with this, it is a 'function from X to Y'. Dmcq (talk) 02:39, 4 February 2012 (UTC)

---

So far I like Dmcq's edit. It's nice and succinct. . . excepting this, a sentence that was there before Dmcq's edit:

"The domain X may be void, but if X = ∅ then F = ∅. The codomain Y may be also void, but if Y = ∅ then X = ∅ and F = ∅. Such void functions are not usual, but the theory assures their existence."

It's entirely possible for an algorithm instantiated in a computational mechanism (the whole assemblage a "function box") to have no input at all (i.e. input is void) but have output (cf Knuth: "An algorithm has zero or more inputs" . . . these inputs are taken from specified sets of objects", and "an algorithm has one or more outputs" Knuth 1973:5). An example is the busy beaver function with this function-triple: ({∅}, {|, blank}, F: busy beaver algorithm instantiated a Post-Turing machine).

This "triple" definition helps me think about a two-column table-as-function. There's an input alphabet of symbols, an output alphabet of symbols perhaps the same, perhaps not, and the ordered pairs that define each row in the table, the symbols of which are drawn from the appropriate alphabet, i.e. <input-symbol, output-symbol>. Applying my question to this tabular function, it's entirely possible to have a table that has this row <Ø,☹> perhaps written as < ,☹> i.e. with null input the table outputs a frowny-face. What am I missing here? Thanks. BillWvbailey (talk) 15:48, 4 February 2012 (UTC)

An algorithm may have no inputs and still have outputs, but this is not true of a function, at least not as the word is now understood in mathematics. If the domain or codomain are empty, the cross product is empty, and the only subset of the empty set is the empty set. Rick Norwood (talk) 16:20, 4 February 2012 (UTC)

I think the problem here is an algorithm has a tuple of inputs, for functions this tuple is the input. It is perfectly possible for an algorithm to output a constant every time it is called with a tuple of no inputs. This would correspond to a domain consisting of a single element the empty set rather than the domain itself being empty. Basically the empty set is not the set consisting of just the empty set. Dmcq (talk) 18:49, 4 February 2012 (UTC)

There is a slight difference between the function viewpoint and the mathematical viewpoint. An algorithm that runs with no input, when interpreted as a function from $\mathbb{N}$ to $\mathbb{N}$, should be interpreted as a constant function rather than as the empty function. For example, the algorithm "return 3;" computes the constant function f(x) = 3. The issue is just in picking the right function to correspond to the algorithm. — Carl (CBM · talk) 00:35, 5 February 2012 (UTC)

This is the most convoluted, abstruse definition anybody ever conceived.

The person who recently added this comment to the article put it in the wrong place, but I hope he or she will help us to understand the source of their confusion. Please explain here just what it is about the definition you find hard to understand, and we will try to improve it. Rick Norwood (talk) 13:33, 9 February 2012 (UTC)

Function is a rule

"Function is a rule that" seems more readable than "function associates, etc." but my change was reverted. Tkuvho (talk) 19:38, 9 February 2012 (UTC)

In the 18th Century, most mathematicians would have agreed that a function is a rule. In the 19th century it became apparent that there were serious problems with the idea that a function is a rule, and by the 20th Century, with Cantor and set theory, it was possible to prove that identifying a function with a rule led to contradictions. We don't need in this article to go into the technicalities of countable and uncountable sets and, yes, the idea that a function is a rule is more readable. But we should not say something that is false just to be more readable. Rick Norwood (talk) 19:44, 9 February 2012 (UTC)

Saying that a function is a rule only makes sense if we redefine "rule" to mean "function", at which point it becomes tautologous. The usual meaning of rule, which has to do with a finite expression that defines the function, goes horribly wrong in general, because there are too many functions for us to possibly describe each one with a finite expression. The current language - "a functions associates ... " - is meant precisely to avoid all these issues about how the association is performed. We don't want to say in the first sentence "a function is a set of ordered pairs", either. — Carl (CBM · talk) 21:52, 9 February 2012 (UTC)

This is the usual confusion between pre-mathematical intuitions and their mathematical implementations. The point is that the word "rule" has an intuitive meaning for a general public, whereas "function" does not. Imposing a straitjacket of a rigid definition of "rule" in the context of complexity theory upon the common term "rule" is one of the misconceptions I would have hoped to clarify by now. Tkuvho (talk) 08:24, 10 February 2012 (UTC)

The intuitionists would never have his problem in the first place and could quite happily write rule! :) I had a quick look at if rule could be put in quietly with function or algorithm but I couldn't see a decent way of phrasing it. Dmcq (talk) 09:23, 10 February 2012 (UTC)

The substitution of rules for functions in some intuitionistic settings is one reason those settings use intensional equality. But ordinary mathematics never uses intentional equality for functions (and sets), only extensional equality. — Carl (CBM · talk) 12:11, 10 February 2012 (UTC)

"Rule" certainly does have an intuitive meaning, but that meaning is the one that cannot be used to define the concept of a function, because most functions do not have a rule in that sense. For example, there are uncountably many functions from N to N but there are not uncountably many rules in the intuitive sense. Moreover, although 2x and x+x are different rules, they define the same function on N. The concept of a function is just a completely different concept than the concept of a rule; the relationship is that a rule can be used to define a function, but not that a function is a rule. — Carl (CBM · talk) 11:53, 10 February 2012 (UTC)

That's an important point that's probably worth spelling out in the article I think, that two algorithms or formulae may be completely different and yet correspond to the same function. Dmcq (talk) 14:07, 10 February 2012 (UTC)

This point can be found in Rogers 1987:1-2. The following appears in Algorithm: " Algorithm versus function computable by an algorithm: For a given function multiple algorithms may exist. This will be true, even without expanding the available instruction set available to the programmer. Rogers observes that "It is . . . important to distinguish between the notion of algorithm, i.e. procedure and the notion of function computable by algorithm, i.e. mapping yielded by procedure. The same function may have several different algorithms". " Bill Wvbailey (talk) 15:06, 10 February 2012 (UTC)

Thanks I've stuck in that cite. Dmcq (talk) 01:58, 11 February 2012 (UTC)

At the level of a reader who does not know yet what a function is, the notion of a function is closely related to that of a rule, just as it was historically. The tried and true method of introducing the notion of a function to a novice is to describe it as a "rule that does such and such". I am not sure what high-minded aspects of intensional equality have to do with this. Tkuvho (talk) 19:51, 11 February 2012 (UTC)

I undid another edit to the first sentence which said, correctly, that a function was historically defined as a rule. The first sentence is not the place to go into history, I think; compare all of our other articles. Really we don't want to encourage the reader to think of a function as a rule, we want them to think of it as an arbitrary association between elements of the domain and codomain, which might "not have a rule" in the informal sense of "rule". — Carl (CBM · talk) 12:52, 12 February 2012 (UTC)

I undid one more and I'm done for the day. I believe it is a mistake to use the word "rule" in the intro in that way. A function is not a rule in the informal sense, and clearing up the confusion between the two is vital for understanding what a function is. The relationship is that a function can be defined by a rule, and I suppose I would not mind saying that in the intro. But I object to any sentence which tries to claim, even with some hedging, that a "function" is or should be thought of as a "rule". — Carl (CBM · talk)

I've chopped out rule completely in the first paragraph and also made the initial sentence more nouny rather than verby or whatever the word is for that. Dmcq (talk) 13:25, 12 February 2012 (UTC)

• teh word for that is probably objectificationing, or management speak, or materialism, or grounding, at a guess, though good copy-edit in this instance. NewbyG ( talk) 14:46, 13 February 2012 (UTC)

No problem, although it does remove the "defined by" point. I was typing the following and got an edit conflict. This is a quote from Devlin's book Sets, functions, and logic page 92:

"This definition of function equality means that we should not really speak of a function as being a rule that takes arguments from the domain and produces values in the codomain. Rather a function is determined by such a rule. It is not the rule itself that is the function, even assuming that we are careful to specify the domain and codomain (as we should). It is the argument-to-value association the rule determines that is "the function".

The italics are in the original. — Carl (CBM · talk) 13:26, 12 February 2012 (UTC)

I was thinking about moving the 'Different algorithms may implement the same function, only the outputs need be the same' at the end of the third paragraph a bit up but didn't see how to do it cleanly before mentioning a function could have many different implementations. Dmcq (talk) 13:32, 12 February 2012 (UTC)

I was looking for a place lower in the article to expand on this issue, but didn't really see a good one there, either. — Carl (CBM · talk) 13:37, 12 February 2012 (UTC)

I've tried putting it in a version at the end of the first paragraph. There's also a citation in the overview section just after the Square definition. Dmcq (talk) 13:53, 12 February 2012 (UTC)

Discussion break

"a function is an association" is much worse than the "a function is a rule". That's not a meaning of the word "association"; not in English, nor in math. For the former just go take a look at wiktionary, where the closest match would be saying the a function is an "act", which it is not; the latter is clear. The fact of the matter is that every function is a rule, i.e. the rule x maps to f(x). While Carl derides this as "tautological", I'd call it "circular"; what's wrong with saying something in the first sentence that is tautologically a synonym anyway. And who cares if we're circular in the first sentence? The point of the first sentence is to be accessible; mathematicians have this thing they "call" a function, but really everyone else would call this a "rule". I think the "anti-rule" people are conflating two issues: (1) that some people think that all functions are given by explicit rules, (2) that a function itself can be thought of as a rule. Worse comes to worst, how about "a function is a way to associate..."? RobHar (talk) 17:47, 12 February 2012 (UTC)

The "association" thing dates from... today and hopefully will not remain until tomorrow. What you are proposing is similar to what the page had yesterday: "a function associates, etc." Your proposed "a way to associate" is better. I still think "a rule that assigns" is even better. The concern that not every function is a "rule" understood in a technical sense can be addressed by writing that "a function is roughly a rule, etc" or something of that sort, which was exactly my edit that was reverted most recently. The current version is WP:PEDANTRY at its best. Tkuvho (talk) 17:51, 12 February 2012 (UTC)

• wp:notpedantry,anyone ?!! NewbyG ( talk) 15:00, 13 February 2012 (UTC)

I found a nifty description in a book with the title "How to prepare for the Virginia SOL":

"A function is a special way of matching the members of one set, called the domain, with the members of a second set, called the range. You can think of a function as a machine that takes the domain elements as inputs, and transforms them into range elements as outputs. But a function must also follow this important rule: Each input must be paired with exactly one output."

Perhaps we could write something similar in the lead of this article? Isheden (talk) 18:15, 12 February 2012 (UTC)

"Special way" is certainly better than "association". Then again, the pedants will say that there are only countably many "special ways". After all, how special can each and every member of a continuum be? Tkuvho (talk) 18:29, 12 February 2012 (UTC)

Just take out the 'special way' and say it matches instead of associates. I can see people having problems with that though as matching sounds like regex to me. Dmcq (talk) 19:35, 12 February 2012 (UTC)

"Matches" sounds both too anthropomorphized and too informal to me. I don't see anything wrong with "association", it seems like a perfectly reasonable way of describing the "functionality" of a function. But I'm not tied to "association" being in the lede as much as I am tied to "rule" not being there. — Carl (CBM · talk) 22:28, 12 February 2012 (UTC)

Re Robhar: I was using "tautological" in the sense of tautology (rhetoric) which basically means "circular". But I think we can do better than give a circular (i.e. meaningless) statement in the first sentence of the article). I also don't like "a function is a way to associate" because a function is not a "way", it is a mathematical object, which is an association between the input set and the output set. — Carl (CBM · talk) 22:26, 12 February 2012 (UTC)

That is not an appropriate use of the word "association", as I stated above. I agree that a function is not a "way", which is why I prefer "rule". As for being a mathematical object: as a mathematical object a function is its graph; so then what is the graph of a function? A function? My point is that while we have a specific standard way of defining formally what a function is, that does not mean that that is what a function is. A natural number is not "an element of every inductive set", it's something we use to count. Similarly, a function is something that models the idea of transforming an input into an output. Like a rule (or a transformation for that matter). And sure it also associates the output to the input, and I'm not against using a word that has as its root "association", but the current use is not an accepted use of the word "association" by anyone. RobHar (talk) 00:15, 13 February 2012 (UTC)

I included a quote above that uses "association" in that way. I see from Google that Serge Lang also used it that way in his Basic Mathematics, first sentence of ch. 14: "We note that a function is an association.". But I'm not really tied to the word "association", which was not in the article until today and is not in the article as I write this sentence.

The main issue I have is that a function is certainly not a rule in the informal sense. It is a "rule" only when we have redefined "rule" to mean "function", which is how authors manage to sometimes say "a function is a rule" with a straight face. We can do better than that circular definition, I believe. Moreover, if someone reads "a function is a rule" without realizing that "rule" has been redefined, they are likely to get exactly the wrong idea, namely that the expression 2x is a function. — Carl (CBM · talk) 00:30, 13 February 2012 (UTC)

I forget which critical originally quipped this sentence should be taken outside and shot, but it could well apply to the current opening sentence In mathematics, a function is a correspondence that associates each input with exactly one output. Using the word "rule" is by far the lesser evil. But whether we eventually settle on "rule" or "association" or something else, surely we can do better than this.

To my mind, the statement "a function is a rule..." is only problematic if you redefine "rule" to mean "finite composition of elementary functions". The usage of the word "rule" in contemporary conversation is not the same as the usage in 19th century mathematics. I think it's reasonable for the lead to describe something informally, noting that rigorous definitions appear further down the page. Jowa fan (talk) 00:52, 13 February 2012 (UTC)

The difficulty is that "a function is a rule" would only be helpful to a reader who doesn't already know what a function is if that person already knows what a "rule" is. But the meaning of "rule" that they will have in mind is something like "a finite expression that defines the function", which is exactly the problematic definition from the 19th century. So by saying "a function is a rule" we will make them think of the 19th century definition. What we want is for them to think that a function is some fixed but arbitrary correspondence or association between elements of the domain and elements of the codomain, which may not be given by a "rule" in the sense they are familiar with. — Carl (CBM · talk) 01:03, 13 February 2012 (UTC)

What you think people think "rule" means and what I think people think "rule" means are quite different. But here is what Michael Spivak thinks (from Calculus, introduction to Chapter 3, "Functions"):

...We will not even begin with a proper definition. For the moment a provisional definition will enable us to discuss functions at length, and will illustrate the intuitive notion of functions, as understood by mathematicians. Later, we will consider and discuss the advantages of the modern mathematical definition. Let us therefore begin with the following: Provisional definition. A function is a rule which assigns, to each of certain real numbers, some other real number.

I think this approach, beginning with ideas that are easily grasped intuitively, and leaving until later the examples that are classically considered "pathological", is a good model for us. Jowa fan (talk) 01:18, 13 February 2012 (UTC)

That is fine for a textbook, but our aim is to be an encyclopedia, which is somewhat different. Unlike a textbook, we can't use a "provisional definition"; our lede should say more directly what's going on with the actual matheamtical definition, because that's the topic of the article. At the very least, when read literally our lede should not be vacuous and should not be wrong. I have a suggestion below. — Carl (CBM · talk) 01:25, 13 February 2012 (UTC)

Bravo Spivak and User:Jowa fan. Carl's concern seems to be mainly that the same function may be defined by different rules. I would suggest modifying a "rule"-based opening to clarify this point. Thus, we could write "a function is given by a rule, etc". We could add a sentence later making the point even clearer, e.g. adding the italicized comment in the previous line. Tkuvho (talk) 08:22, 13 February 2012 (UTC)

break 1

One reason the first sentence is somewhat odd is that, in the past, editors agreed not to put the words "domain", "codomain", "range", or "set" there, to try to keep it simple, but at the same time there is a goal to keep it from being vacuous. If we use these words, we can say:

In mathematics, a function gives a correspondence between two sets, the domain (set of inputs) and the range (set of outputs), so that each element of the domain corresponds to exactly one element of the range. A function can be viewed as a tranformation that takes an input and returns the corresponding output. For example, the function f defined by f(x) = 2x, when given a number x as input, returns the number 2x.

The reason to avoid "codomain" is that "range" is more familiar, and the sentence above is literally correct as written, even though the function might also specify a codomain in addition to giving a correspondence between the domain and range. Note that the sugggestion does not say what a function is, it says what a function does, avoiding the identity issue. — Carl (CBM · talk) 01:16, 13 February 2012 (UTC)

If you think that a distinction between "is" and "does" allows us to avoid certain technical issues, then why not "A function can be viewed as a rule...". I'm looking at Wikipedia:Manual_of_Style/Mathematics#Article_introduction, which says rather clearly that the lead should be informal and not rigorous. I'd prefer not to mention sets in the very first sentence, for the sake of accessibility.

I notice that the MOS gives topology as an exemplar. That article begins with the informal notion of "continuous deformations", which isn't an accurate description of the whole of topology. We don't kick off with "a topology on a space is a collection of subsets satisfying the following axioms..."; the correct definition appears half way down the article, but not in the lead. Likewise, I think the lead here should convey an intuitive (if not completely accurate) idea of what a function usually is, but should not contain a complete and correct definition.

In fact, I wouldn't object too strongly to the article starting "In mathematics, a function can be viewed as a tranformation that takes an input and returns the corresponding output. For example..." I think transformation is actually more misleading than rule (it implies preserving some sort of geometric structure), but at least the prose style isn't so forbidding. Jowa fan (talk) 05:08, 13 February 2012 (UTC)

Reading as far as here Furthermore, functions need not be described by any expression, rule or algorithm: indeed, in some cases it may be impossible to define such a rule. For example... in the fifth para of the Function (mathematics)#Overview does not seem too difficult, I think. Up to there, the lede that is, seems well-worked. Um, this is from the perspective of someone who did first year math many years ago, so I ought to have said IMHO. NewbyG ( talk) 06:41, 13 February 2012 (UTC)

I am not sure I agree with the claim that a function defined, say, by an invocation of the axiom of choice cannot be described as being defined by a rule. In a way it is indeed defined by a rule. The rule is, apply the axiom of choice, then apply Hilbert's epsilon, and you got your function. One only runs into a problem with "rule" if one agrees to wear the straitjacket of a computational-theoretic definition of "rule", which we needn't feel duty-bound to do. Tkuvho (talk) 08:27, 13 February 2012 (UTC)

I think transformation is a bad name for it. A function whose output is 5 when the input is 2 does not transform the 2 in any way. The standard meaning of transform is you keep the original but bend or mould it in some way like producing car bodies out of sheet steel. Dmcq (talk) 08:39, 13 February 2012 (UTC)

Yes, the word transformation has been transformed to the word correspondence, <23:14, 12 February 2012‎ CBM .. (90,015 bytes) (→Lede: Is "Correspondence" better?)> which seems more suitable, IMHO NewbyG ( talk) 09:27, 13 February 2012 (UTC)

Re Tkuvho: if you interpret "rule" in that way, it simply means "function", at which point trying to explain what a function is by referring to a "rule" is circular. But most readers will not think of "rule" as a synonym for "function", so by using "rule" we make most readers think the wrong thing". Both "Association" and "Correspondence" are better in that they do not connote a reason for the association, just that it exists. On the other hand "rule" connotes... a rule. — Carl (CBM · talk) 12:09, 13 February 2012 (UTC)

(ec) And the current footnote 10 has if two variables x and y are so related that whenever a value is assigned to x there is automatically assigned, by some rule or correspondence, a value to y, then we say y is a (single-valued) function of x.... NewbyG ( talk) 12:14, 13 February 2012 (UTC)

proposed first paragraph of lede

I propose the following:

In mathematics, a function is a rule that assigns exactly one output to each input. The output of a function f with input x is denoted f(x) (read "f of x"). For example, f(x) = 2x defines a function f that assigns to any input number, the number twice as large. If x = 5 then f(x) = 10. Two different rules may define the same function if they make the same assignments, for example f(x) = 3x−x defines the same function as f(x) = 2x. Leibniz originally introduced the notion of function in the context of the study of curves. A planar curve can often be viewed as a rule (function) assigning the y-coordinate to the x-coordinate of a point on the curve.

Proposed

Proposed

Tkuvho (talk) 09:13, 13 February 2012 (UTC)

Views

• Oppose. The first paragraph cannot be contradictory/circular: A function is defined as a rule, and then two different rules may define a function? Also, "assign" is used in the article in another meaning: Assign a symbol or value to a variable. Isheden (talk) 10:23, 13 February 2012 (UTC)

Here is the text of our current footnote 10 using the word "assign" in the usual sense: Eves asserts that Dirichlet "arrived at the following formulation: "[The notion of] a variable is a symbol that represents any one of a set of numbers; if two variables x and y are so related that whenever a value is assigned to x there is automatically assigned, by some rule or correspondence, a value to y, then we say y is a (single-valued) function of x. The variable x . . . is called the independent variable and the variable y is called the dependent variable. The permissible values that x may assume constitute the domain of definition of the function, and the values taken on by y constitute the range of values of the function . . . it stresses the basic idea of a relationship between two sets of numbers" Eves 1990:235. Your point concerning "rule" is well-taken. I therefore propose the following:

In mathematics, a function is given by a rule that assigns exactly one output to each input. The output of a function f with input x is denoted f(x) (read "f of x"). For example, the rule f(x) = 2x defines a function f that assigns to any input number, the number twice as large. If x = 5 then f(x) = 10. Two different rules may define the same function if they make the same assignments, for example f(x) = 3x−x defines the same function as f(x) = 2x. Leibniz originally introduced the notion of function in the context of the study of curves. A planar curve can often be viewed as a rule (function) assigning the y-coordinate to the x-coordinate of a point on the curve. Tkuvho (talk) 11:49, 13 February 2012 (UTC)

• This is self contradictory: "a function is a rule" ... "two different rules may define the same function". The word "is" is a powerful word. But if you change that to "correspondence" or "association", it would be OK with me. — Carl (CBM · talk) 12:07, 13 February 2012 (UTC)

You seem to have overlooked the fact that my second version says "a function is given by a rule". My impression is that most editors here don't like 11-letter words and prefer a 1-syllable word. Tkuvho (talk) 12:15, 13 February 2012 (UTC)

I see. The issue there is that not every function is given by a rule in the sense of "rule" that naive readers will have in mind. It's much more direct to say "a function is a correspondence" or "a function gives a correspondence" than to say "a function is given by a rule". — Carl (CBM · talk) 12:20, 13 February 2012 (UTC)

I'd say a function is a relation between the domain and the range, which associates every element in the domain with exactly one element in the range. Put simpler, a function is a relation between (a set of) inputs and (a set of) outputs, which associates every input with exactly one output. Isheden (talk) 13:01, 13 February 2012 (UTC)

Yes, me too, this is the mathematical definition. But I think it is too hard for the first paragraph. In the past we made a particular effort not to have the set-theoretic definition up front. — Carl (CBM · talk) 13:22, 13 February 2012 (UTC)

• Oppose rule as well. A function just isn't a rule or algorithm. There's no point saying it is one and then that it isn't. And like Isheden assigns sounds wrong to me too. Dmcq (talk) 12:45, 13 February 2012 (UTC)

A rule is not an algorithm. Applying the axiom of choice and then Hilbert's epsilon so as to produce a function is also a kind of a rule. The advantage of a definition in terms of a "rule" is that it is both immediately accessible to a novice, and also technically correct with a sufficiently broad interpretation of "rule". Tkuvho (talk) 13:07, 13 February 2012 (UTC)

I believe the lede as is is reasonably accessible to a novice reader, presuming that reader is interested enough about functions to want to read a few hundred or what ever words. It would be more confusing by starting out as a rule, and then admitting that rule is imprecise, and not in line with current terminology, with good reason. NewbyG ( talk) 13:13, 13 February 2012 (UTC)

Wouldn't "relation" be as accessible as "rule" or "correspondence", with the important difference that relation is also what a function is to a mathematician (with the additional requirement of one output for each input)? Isheden (talk) 13:22, 13 February 2012 (UTC)

I think the only advantage of "correspondence" (or "association", although other people seem to dislike that) is that it is more in line with the regular English meaning, while "relation" is a more specialized term. — Carl (CBM · talk) 13:28, 13 February 2012 (UTC)

(indenting reply) Well, relation has many specialized usages, otherwise it is rather a common-place word, intuitively understood, however there is the slight verbal clash with the even commoner *relationship* in the marital sense!?! NewbyG ( talk) 13:55, 13 February 2012 (UTC)

OTOH correspondence ‘’could’’ give rise to thoughts of Ye olde letter-writing, whilst ‘’association’’ gives us, far-fetchedly, football, or trade-unionism. Interesting?!? Correspondence works fine, association is fine, relation would be fine, I don't favour it though. NewbyG ( talk) 14:05, 13 February 2012 (UTC)

The function of the lede is not to give a complete and accurate definition, but to introduce the topic in an accessible way (see MOS for mathematics as mentioned above). I suggest this:

In mathematics, a function can be thought of as a rule assigning to each possible input exactly one output <footnote: quotation from Spivak as above> ...

Since Spivak asserts that people do frequently think about functions this way (and the discussion so far suggests that he is not entirely alone in this), we have an accurate and sourced statement, and people can easily scroll down to the section headed "Definition" if they want the formal version. Jowa fan (talk) 13:35, 13 February 2012 (UTC)

I quoted above the following from Keith Devlin:

This definition of function equality means that we should not really speak of a function as being a rule that takes arguments from the domain and produces values in the codomain. Rather a function is determined by such a rule. It is not the rule itself that is the function, even assuming that we are careful to specify the domain and codomain (as we should). It is the argument-to-value association the rule determines that is "the function."

So we also have a sourced and accurate statement that a function should not be spoken of as being a rule...

The lede is meant to be an introduction to the article, but it should also be accurate the to extent we can manage. We may choose not to cover fine details in the lede (such as the two definitions of "domain") but the difference between a "rule" and a "function" is not a minor detail, it's a fundamental thing that a reader has to grasp to understand the concept of a function. This does not mean we should have a formal definition in the lede, but the definition should not be misleading. — Carl (CBM · talk) 13:57, 13 February 2012 (UTC)

• Oppose. I'm with Carl on this issue. The notion of "function" is not trivial and "rule" is simplistic, as my example below shows. However, there does seem to be a "meta"-issue here having to do with the notion/generalization/rule/process/method by which one creates a "function". I noodled on it last night (you folks have been busy!) then cut it so I could sleep on it: This is what I wrote (more or less):

"I like associates. A nice strong active verb. It means: "to bring together or into relationship". [ And what happened to relationship? ] But corresponds works too. “To be” is the most powerful, as in “exists, is”: "The collections C of c-things exists, and the collection D of d-things exists and to every d-thing exactly one c-thing is assigned [placed in correspondence, or associated]". This is the core of a *potential* modus ponens, *potential* because it requires an actual D (aka universe of discourse) from which the d are drawn, before their associated c can be produced.

∃C & ∃D & ∀d ∈ D: d → E! c ∈ C. "

From my dictionary: pp of L. fungi to perform: "a mathematical correspondence that assigns exactly one element of one set to each element of the same or another set." (Webster's 9th Collegiate). Except for the definition itself, nowhere in this is the notion of a "rule". To demonstrate the point: here's a listing of ordered pairs generated with random assignment from two collections:

=CONCATENATE("<", INT(9*RAND()), ",", INT(3*RAND()), ">")

Here's an interesting "object" created by the above:

{ <3,0> <6,1> <0,2> <5,2> <1,2> <0,1> <4,2> } : not a function because of <0,2>, <0,1>, but . . .

{ <3,0> <6,1> <0,2> <5,2> <1,2> <4,2> } : is a function because I manually pulled out <0,1>.

This specific object { <3,0> <6,1> <0,2> <5,2> <1,2> <4,2> } embeds no specific rule for the individual assignments inside the ordered pairs, guaranteed by the rand() functions. But it is an object created by a generalized-to-all-functions rule/process/method for the formation of any function; we can see this in the order of the symbol-assignment inside the CONCATENATE instruction, plus the (random) extraction of symbols from two collections { 0-9 }, {0-3}. The concatenation-process itself failed to be a function; there was still the matter of me checking by hand to see to be sure the assignment not one-many. BillWvbailey (talk) 16:42, 13 February 2012 (UTC)

Certainly if it's not single-valued it's not a function. It is hard to see what you have added to the discussion. Tkuvho (talk) 16:47, 13 February 2012 (UTC)

Devlin quote

The Devlin quote is the best proof that we should use "rule" in the lede. The reason Devlin elaborates on this is because "rule" is the best intuitive way of introducing the notion of function, and he warns against interpreting it in certain ways that are not correct. Clearly describing a function as being given by a rule is enough to respond to Devlin's concern (note that Devlin himself wrote "Rather a function is determined by such a rule", which presumably means that he thinks this statement is correct). Similarly, formalist pedantry was recently overcome at the lede of Dirac delta function where common sense prevailed in the end, and the reader has the advantage of learning the necessary intuitions before proceeding to the more technical part of the page. Tkuvho (talk) 14:02, 13 February 2012 (UTC)

This isn't "pedantry" or formalism, it's entirely about how to help a naive reader. Describing a function as a "rule" is a pedagogical disservice to them. Readers should not build the intuition that there is any sort of "reason" or "rule" behind a function, because that intuition is wrong and has to be overcome to learn what a function actually is: an arbitrary association of inputs to outputs. The word "rule" has a false connotation that makes it harder to learn what a function is. — Carl (CBM · talk) 14:10, 13 February 2012 (UTC)

I understand that this seems to be your position. However, note that the quote from Devlin does not support this at all. Tkuvho (talk) 14:19, 13 February 2012 (UTC)

My argument is exactly that "we should not really speak of a function as being a rule". It's fine if we say that a function can be defined using a rule, that is perfectly correct. In fact I put that in the lede [1] but someone else removed it. I am just saying that any implicit or explicit claim that a function is the same as a rule should be avoided. — Carl (CBM · talk) 14:24, 13 February 2012 (UTC)

There could be this, err, compromise. We should not really speak of a function as being a rule though we often used to  think of a function as being a rule ??? No NewbyG ( talk) 14:34, 13 February 2012 (UTC)

This is close to Isheden's proposal below which I support. Tkuvho (talk) 14:37, 13 February 2012 (UTC)

Regarding the lead: According to MOS:MATH, the lead section should contain an informal introduction to the topic. According to WP:LEADSENTENCE, the first sentence should give a concise definition: where possible, one that puts the article in context for the nonspecialist. I'd suggest mentioning that a function can be thought of as a rule in the lead section, but not in the first sentence. Isheden (talk)

Sounds like a good idea. Regardless of the "rule" issue, I think it is useful to have some mention of history in the lede, as in the suggested paragraph above. Tkuvho (talk) 14:17, 13 February 2012 (UTC)

History, yes. A sub=section on historical usage of terminology, or development of terminology or... NewbyG ( talk) 15:06, 13 February 2012 (UTC)

Discussion, per

The Dirac delta function, or δ function, is (informally) a generalized function on the real number line that is zero everywhere except at zero, with an integral of one over the entire real line.[1][2][3] The delta function is sometimes thought of as an infinitely high, infinitely thin spike at the origin, with total area one under the spike, and physically represents an idealized point mass or point charge.[4] It was introduced by theoretical physicist Paul Dirac.

Looks like the lede from Dirac delta function. Alles in ordnung. smileyface NewbyG ( talk) 14:15, 13 February 2012 (UTC)

The discussion there was whether to say that the function is zero everywhere except at the origin, or whether not to say that. The difficulty is that there are really two different things being discussed in that paragraph: the δ function and the δ distribution. I argued that the lede was fine because the δ function is defined to have to properties in the first sentence (except that, alas, no such function exists). On the other hand, this article is just about regular mathematical functions, so we don't have quite the same issue, because a function is not defined to be a rule. — Carl (CBM · talk) 14:18, 13 February 2012 (UTC)

Devlin and Spivak say the same thing

English prose can be subtle. When Devlin says "we should not really speak of a function as being a rule..." the word really flags the fact that it often happens even though it's not technically correct, which is the same point made by Spivak. (Notice that "we should not really..." has quite a different meaning from "we really should not..."!) Then Devlin goes on to say "...a function is determined by such a rule." So he's cautioning us regarding the word "rule", but not trying to ban it entirely.

By now it's clear that having the word "rule" in the first sentence is not going to be supported by a consensus any time soon. But there are several of us who think it should be mentioned somewhere near the top of the article. What about this:

In mathematics, a function is a correspondence <footnote 1> that associates each input with exactly one output. The output of a function f with input x is denoted f(x) (read "f of x"). For example, the rule <footnote 2> f(x) = 2x defines a function f that associates any input number with the number twice as large: if x = 5 then f(x) = 10. Two different rules define the same function if they make the same associations; for example f(x) = 3x−x defines the same function as f(x) = 2x.

where footnote 1 is the Halmos quotation that was recently added, and footnote 2 mentions both the Spivak and Devlin quotations given above? Jowa fan (talk) 23:42, 13 February 2012 (UTC)

A function can certainly be determined by a rule as he says, but when he says a function is not really a rule that is also true. What's the point of dragging in another word here though? It doesn't seem to aid in anything that I can see. Dmcq (talk) 23:54, 13 February 2012 (UTC)

The point is to help mathematically unsophisticated readers to gain some intuition as to what functions are and how they're commonly used in practice. Or, to put it another way, the point is to make the lead appear less technical and more accessible. Jowa fan (talk) 05:02, 14 February 2012 (UTC)

I added some comments as you suggested. If you think it is appropriate, I will include the sentence about Leibniz as above. Tkuvho (talk) 15:23, 14 February 2012 (UTC)

I think the discussion about function as a rule would fit much better in the third paragraph, which starts out with "There are many ways to describe or represent a function." Isheden (talk) 08:35, 14 February 2012 (UTC)

I restored CBM's comments about rules. Tkuvho (talk) 15:52, 14 February 2012 (UTC)

Quick Google Books survey

• "a function is a rule" -- about 28,000 books
• "a function is a correspondence" -- about 8,940 books

I would support the wording "a function is a rule" as the most common and the most comprehensible. The phrase "a function is a correspondence" is taken from Bourbaki, who do not give a terribly clear definition of a function (not a good basis for Wikipedia). In any case, the word "correspondence" is a poor translation of what Bourbaki said: they actually define a function to be a particular kind of binary relation. But that is really defining a model of a function, not an explanation of the concept. The phrase "a function is a rule" defines the concept more clearly. -- 202.124.75.226 (talk) 05:17, 16 February 2012 (UTC)

Please register with a username rather than using an IP if you get a chance. Tkuvho (talk) 07:35, 16 February 2012 (UTC)

summary of correspondence vs rule

The opinions seem evenly divided on whether the first sentence should describe a function as a "correspondence" or as a "rule" (I count about 4 users supporting each position). Let me summarize the relevant points.

Users supporting "correspondence" object to "rule" on the following grounds:

• A function cannot be said to be a rule because two different rules may define the same function.
• Describing a function as a rule implies that a function is necessarily given by an algorithm, which would only be true in certain kinds of mathematical constructivism but not in the classical approach.

Users supporting "rule" object to "correspondence" on the following grounds:

• The term "correspondence" is too long, ambiguous, and awkward.
• The term "rule" is immediately recognizable to a general reader, whose intuitions the page should build upon.
• Reliable sources such as Spivak use "rule" to describe a function.
• The term "rule" occurs more frequently (by a factor of 4) than "correspondence" on the web in describing a function.

Furthermore, supporters of "rule" note that that the term does not necessarily imply "algorithm". Thus, $x\mapsto f(x)$ is a rule whenever f is a function. Describing a function as a rule is not tautological because the intuitive meaning of "rule" is familiar to the general reader. The objection that different rules may describe the same function can be addressed by saying that "a function given by a rule", or "a function is often given by a rule", which is sufficient for a lede. Tkuvho (talk) 07:46, 16 February 2012 (UTC)

Add me to the supporters of correspondence and anti-supporters of rule, for basically the reasons you outline. People think they know what it means when you say that it is a rule, and are generally wrong; better to confuse than to mislead. Spivak's "We will not even begin with a proper definition" is fine for a textbook but not very encyclopedic. —David Eppstein (talk) 07:51, 16 February 2012 (UTC)

It is better to build upon a student's intuition than "to confuse". I have not seen any serious evidence that calling a function a rule has misled students. After all, complexity theory is typically taught much letter than pre-calculus. I agree with you that defining a function as an algorithm may be inappropriate. But thinking that it is better to confuse than to build on the student's intuitions is a Bourbakist error. Tkuvho (talk) 08:11, 16 February 2012 (UTC)

Oppose changing to rule. A function is not a rule. A function is a correspondence. If somebody can thnik of a better phrasing or a different word from correspondence fine. Saying rule is not that better phrasing or word. I might as well say a colour is a type of pigment or an mammal is a type of cow. Dmcq (talk) 08:36, 16 February 2012 (UTC)

Allow me to repeat that according to WP:LEADSENTENCE, the first sentence of the lead should give a concise definition: where possible, one that puts the article in context for the nonspecialist. In the first sentence, relation or correspondence is appropriate. The interpretation as a rule fits more naturally in the third paragraph, which starts out with "There are many ways to describe or represent a function." If needed, this paragraph can be moved closer to the top, before the definitions of argument, value, domain and range. Isheden (talk) 08:45, 16 February 2012 (UTC)

I am also opposed to describing a function as a rule in the lead. Building on readers' intuitions is all well and good when those intuitions are generally heading in the right direction. But in this case the intuitive understanding of "rule" by the man in the street (which will, more or less, be the type of finite deterministic algorithm that they have come across in elementary arithmetic and algebra) is exactly the wrong way to describe a function. The intuitive "rule" description gets strained beyond repair once the reader goes beyond simple functions and encounters functions such as the Riemann zeta function, Lambert's W function or the busy beaver function. Gandalf61 (talk) 09:10, 16 February 2012 (UTC)

I'm surprised at the depth of feeling this issue has revealed, and surprised at the variety of meanings attributed to the word "rule". Currently we have a lead paragraph in which the word "rule" is mentioned only by way of example, and not in the first sentence. I suspect this is as close as we can get to representing both points of view in a reasonable way; I don't think we're going to see a clear consensus for either point of view. Perhaps this is a good time for me and the other supporters of "rule" to let go of this particular campaign. Jowa fan (talk) 13:46, 16 February 2012 (UTC)

I'm not convinced the matter has been resolved. I'm seeing two issues being muddled. When a student confronts the notion of "function", he asks himself, but what IS it? Is it a thing? Does it do anything? Or is it just there, just symbols, or what exactly? But even a phone book (Halmos' example (Halmos 1970:30)) has a structure, and somehow the symbols got there (via two rules -- a structural rule of pairing + order, and an assignment rule). So at least there there must be a structure-rule that defines the nomothetic "function" [yes: pairing + ordering]. But does a function require an agent to "make it work"? Is the agent part of the function? Or is the agent separate from the function, something that merely brings effective action to the function to "make it work?"

You see this difficulty in Frege's description of a function as something "unsaturated" (incomplete -- it needs a value assigned to the empty place in order to complete it). This was taken up by Russell in his 1903 where he considered f( ). Halmos observes the same issue:

"The connotations of activity suggested by the synonyms [map, mapping, transformation, correspondence, operator; i.e. " f sends or maps or tansforms x onto y" (Halmos 1970:30)] listed above make some scholars dissatisfied with the definition according to which a function does not do anything but merely is. This dissatisfaction is reflected in a different use of the vocabulary: function is reserved for the undefined object that is somehow active, and the set of ordered pairs that we have called the function is then called the graph of the function." (Halmos 1970:30).

So what exactly IS a function? My take is that most mathematicians consider the nomothetic "function" to be a passive [object? entity? structure?]: just a pairing + ordering relation presented in an effective format suitable for a target agent, and it need not be symbolic (discrete): e.g. an analog device like a slide rule or an X-Y plotting device (e.g. two pins in a gridded board with a string around them + a pencil, to draw ellipses.) Bill Wvbailey (talk) 17:16, 16 February 2012 (UTC)

This is a very interesting and thought-provoking comment, but we are not really discussing the "true nature", if any be found, of mathematical entities such as functions. Rather, we are discussing the best way of presenting them on this page. It seems clear that saying what a function "does" is more accessible than a static definition. The term "rule" suggests a dynamism that is simply not there in "correspondence". Tkuvho (talk) 17:50, 16 February 2012 (UTC)

I agree with you just enough to keep trying to clarify what I see is the issue (formation rule F versus assignment rule R). I don't have an answer yet. Since I'd prefer the word "assigns" (action) or "assignment" (outcome) that's what I'll use: Our choice here is whether the symbol string " function " indicates an active process, a passive (static) object, both in various contexts, or neither. For example: "a function assigns exactly one output to every input" is much different than: "a function is the assignment [correspondence, association] of exactly one output to every input". In the first case the "function" is an agent always following a generalized formation rule F that is usually, but not always, under guidance of a specific assignment rule R. In the second case the function is the outcome of the process, a static "object" (table, algorithm, formula, Halmos' telephone book). In this latter case we need to the "function's" (aka graph's) formation rule F so as agents we can use the "object". So just what is a "function"? An agent or an outcome? This is almost identical to the debate around Gurevich's stance that an algorithm is the programmed Turing machine as opposed to a symbol-string to be instantiated in a Turing machine. No wonder students get confused. Bill Wvbailey (talk) 20:01, 16 February 2012 (UTC)

The "function is a rule" language works better in the classroom, especially when it comes to talk of "applying" a function. -- 202.124.74.119 (talk) 05:07, 17 February 2012 (UTC)

@WvBailey: It seems to me that from a Brouwerian point of view a function is F rather than R. @IP: Can you source this claim in educational material? As far as "correspondence" is concerned, the term is inappropriate because it does not reflect the basic asymmetry between domain A and range B of a function $f:A\rightarrow B$. According to most of the meanings given at the "correspondence" page, there is on the contrary a symmetry between A and B, as for example in a "relation". The term "rule" clearly indicates that we are going from input to output, and this direction breaks the symmetry as it should. As far as wiki policy on definitions is concerned, obviously it is not referring to a mathematical definition. CBM stated above that it would be inapropriate to include a set-theoretic definition in the lede, and I don't see how "correspondence" is more of a definition than "rule" (if anything it is the opposite, because of the asymmetry issue I just mentioned). I also don't see why "correspondence" is more or less encyclopedic than "rule". It is certainly more confusing. Tkuvho (talk) 08:36, 17 February 2012 (UTC)

Guess I'm old fashioned, specification as domain (inputs) plus rule (processor, f(x)=) (each input -> unique output) seems fine to me (correspondence is an exchange of letters, isn't it?....)Selfstudier (talk) 10:39, 17 February 2012 (UTC)

You are characterizing the problem as rule versus correspondence. There is no problem there. Rule is wrong and correspondence is exact. If you don't like correspondence try something else like relation or map or something else that doesn't cause too much of a problem. Or try phrasing the lead differently and you might be able to avoid any problems that way. There is no need to make a false dichotomy here. Dmcq (talk) 10:49, 17 February 2012 (UTC)

Perhaps I wasn't clear, correspondence seems to me inexact and it is not just rule but rule plus domain (ie a function is something you specify).

A function is not a rule. The link is the other way round. As it says in the lead at the moment two different rules can define the same function. Your argument about correspondence applies to anything here if you take the wrong definition, and even with your definition correspondence is clearer. One person corresponds with another, they are linked by a relation to each other. If everyone on the input side sends just one letter then that is a function between the senders and receivers. With rule it matters if they use postcodes or the mail is sent via the North Pole. Dmcq (talk) 11:34, 17 February 2012 (UTC)

Umm...that is precisely why you specify a function.Selfstudier (talk) 11:50, 17 February 2012 (UTC)

If I type function into Wikipedia search I end up on a disambiguation page that says "Function (mathematics), an abstract entity that associates an input to a corresponding output according to some rule".Selfstudier (talk) 12:13, 17 February 2012 (UTC)

That's slightly better than saying it is a rule. I think it would be even better if it just said an abstract entity that associates an output with each input. Dmcq (talk) 12:31, 17 February 2012 (UTC)

OK I think I see where you are coming from now, you appear to want to cater for cases where there is no "rule" other than association eg 2 lists (array). As you can see from my use of the word array, I tend not to regard such a thing as a function; since there is nothing to compute with, why would you need to call this a function?Selfstudier (talk) 13:26, 17 February 2012 (UTC)

See Gandalf61's comment above for a few other examples of mathematical functions that cannot be described by any simple rule. Isheden (talk) 13:47, 17 February 2012 (UTC)

I never said the rule had to be simple.....Selfstudier (talk) 14:15, 17 February 2012 (UTC)

@Tkuvho: Correspondence in this case is an alternative term for relation. You are right that relation is more general than function, but the characteristic that is specific to a function follows immediately afterwards: "that associates each input with exactly one output." This is analogous to the first sentence of linear map: "a linear map (...) is a function (...) that preserves the operations of vector addition and scalar multiplication." A reliable source for this can be found here: [2] Isheden (talk) 13:08, 17 February 2012 (UTC)

See my comment above, you also have the idea that the key or only part is merely the association between input and output which is fine if everyone would standardize on that, however I can see why this gives rise to cognitive dissonance on the part of many (myself included)Selfstudier (talk) 13:40, 17 February 2012 (UTC)

Computation is irrelevant to functions, they just are. If you produce a sorted list using binary sort or quicksort you still end up with a sorted list. The function is the same. In computing one would have different procedures binary_sort(list) or quicksort(list) but as far as functions go all that happens is for each list you have a sorted list. Dmcq (talk) 13:45, 17 February 2012 (UTC)

In fact function and procedure epitomise the difference. A person's function may be to ensure the doorknowbs are shiny. The procedure might be for him to get metal polich and cloth and shine them. As far as the functionary is concerned the output s shiny doorknobs. Dmcq (talk) 13:54, 17 February 2012 (UTC)

I am talking about the difference between lists with a fixed number of entries and lists where new entries are capable of being added (computed with a rule). As I said, you are defining functions in a particular kind of way that is not universally accepted. In practice, one will always specify (well define) a function and it will be perfectly obvious what the intention is. I would cover the general situation that you appear to be describing with some other word, relation probably.Selfstudier (talk) 14:03, 17 February 2012 (UTC)

Could you point to somewhere that defines them differently in the way you're thinking of thanks. Dmcq (talk) 14:18, 17 February 2012 (UTC)

This is as good an approach to the question as any...do you have a favourite?Selfstudier (talk) 14:26, 17 February 2012 (UTC)

I don't see how any of that is particularly relevant to this discussion. That's an introductory text where they spend all day and still don't end up with a proper definition. This is an encyclopaedia. Dmcq (talk) 15:57, 17 February 2012 (UTC)

I thought that it answered the question of why we are having this discussion rather well, actually...Selfstudier (talk) 16:03, 17 February 2012 (UTC)

I saw a sort of working definition where rule, relation or correspondence could be used according to the author. I did not see a definition. The text went on and on and is far too long as a basis of a summary for a lead sentence. It did not talk about computation being needed or that a function only applied when large of infinite numbers of inputs were involved. Dmcq (talk) 16:12, 17 February 2012 (UTC)

I must have misunderstood you, is it that you simply want a source using "rule" rather than "relation" or "correspondence", that is quite straightforward since the vast majority of educational establishments use such a definition. The other part I am not too clear about is the level, the link I gave was at Calc 1, are you looking for something more technical?(I had thought the lead in Wiki articles was supposed to be aimed low, so to speak).Selfstudier (talk) 16:24, 17 February 2012 (UTC)

Look at [3] pp. 153-155 for an alternate basic-level explanation. It defines function in terms of a relation as well. Isheden (talk) 16:36, 17 February 2012 (UTC)

Yes, the one headed "3.1 What is a function?" is fairly typical of introductory level explanations, the idea of a machine or processor between values (inputs and outputs) is also very common.Selfstudier (talk) 16:49, 17 February 2012 (UTC)

A quite recent discussion is in The Princeton Companion to Mathematics, I will quote some bits from the pages there (2.2 on page 10 and 11 followed by 2.3 Relations on 11 and 12):

"One of the most basic activities of mathematics is to take a mathematical object and transform it into another one,sometimes of the same kind and sometimes not."....."A function is, roughly speaking, a mathematical transformation of this kind.It is not easy to make this definition more precise. To ask, “What is a function?” is to suggest that the answer should be a thing of some sort, but functions seem to be more like processes. Moreover, when they appear in mathematical sentences they do not behave like nouns."......"If f is a function, then the notation f(x)= y means that f turns the object x into the object y. Once one starts to speak formally about functions, it becomes important to specify exactly which objects are to be subjected to the transformation in question, and what sort of objects they can be transformed into. One of the main reasons for this is that it makes it possible to discuss another notion that is central to mathematics,that of inverting a function."

Selfstudier (talk) 18:10, 17 February 2012 (UTC)

I think the 'special way of matching' in the one Isheden got is more straightforward. I could live with that if people really are dead set against correspondence. The Princeton one just doesn't work except as hand waving. One doesn't turn or transform objects, the original value stays around and is unchanged. Dmcq (talk) 18:20, 17 February 2012 (UTC)

The Princeton one (Gowers) (and Lamar) is simply a recognition of the difficulty that exists in arriving at a hard and fast definition without rendering the definition so general as to be worthless as a description of what mathematicians actually say and do.

"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. Formerly, when a new function was invented, it was in view of some practical end. Today they are invented on purpose to show that our ancestor's reasoning was at fault, and we shall never get anything more than that out of them. If logic were the teacher's only guide, he would have to begin with the most general, that is to say, the most weird functions." (Poincare, 1899) Selfstudier (talk) 18:32, 17 February 2012 (UTC)

"Rule" seems fine to me, although I don't really have a horse in this race. Sławomir Biały (talk) 23:05, 17 February 2012 (UTC)

I vote for correspondence. It can later be characterized in terms of rules, but it ought to be introduced in terms of correspondence.—PaulTanenbaum (talk) 23:20, 17 February 2012 (UTC)

I vote for simply deleting the first paragraph so that the lead is "In mathematics, there are many ways to describe or represent a function" etc....Selfstudier (talk) 23:39, 17 February 2012 (UTC)

Another vote for correspondence.Rick Norwood (talk) 13:05, 18 February 2012 (UTC)

summary of rule AND correspondence

The previous discussion helped clarify the issues. What emerges is that many editors, including "correspondence" supporters, feel that "rule" is the more intuitive term immediately accessible to a novice. On the other hand, there is a persistent concern that "rule" may be misleading as it is too close to "algorithm". I would therefore propose the following version that attempts to kill two birds with one stone:

In mathematics, a function is a rule that associates each input with exactly one output. Here "rule" is used in a broad sense encompassing correspondences that are not necessarily defined by any specific algorithm. The output of a function f with input x is denoted f(x) (read "f of x"). For example, etc.

Tkuvho (talk) 12:51, 19 February 2012 (UTC)

• Oppose Rule is in the lead paragraph in two places and used properly there. How about assuming that people reading this article will at least read the whole first paragraph? What you have is hand waving and not right. If we're going to stick in something different can't we at least try and get it better than what's there already? Dmcq (talk) 13:10, 19 February 2012 (UTC)

• I don't think this would be an improvement over the current article, either. Why use a word like that just to turn around and redefine it? In the end mathematicians don't think of functions as rules; my calculus students often misunderstand functions to be rules, but that's because they don't know what a function is yet. — Carl (CBM · talk) 13:16, 19 February 2012 (UTC)

• Oppose Rule should not be mentioned in the first paragraph at all, unless it is properly established in the article (with sources) that any function can be thought of as a rule. Otherwise, it fits better after "Some functions may be described by ..." further down in the lead. Isheden (talk) 14:39, 19 February 2012 (UTC)

Just out of interest, did you have some function in mind where there is no process or rule for getting the output from an input?Selfstudier (talk) 19:27, 19 February 2012 (UTC)

• Can't say I like that very much, one useful test is substitution into another phrase in common usage such as "limit of a function" or "distance function" and see whether it sounds sensible or not...Selfstudier (talk) 15:03, 19 February 2012 (UTC)

• Oppose for reasons already discussed. Rick Norwood (talk) 15:15, 19 February 2012 (UTC)

• I think I am becoming confused as to whether,in the lead, we are seeking to define function or whether we are seeking to explain function. We already have a (long) section titled Definition so I am thinking we must be trying to explain and the fact is there are probably as many ways to do this as texts or courses that you wish to examine....Selfstudier (talk) 16:11, 19 February 2012 (UTC)]

WP:LEAD explains what the lead is for. Its main functions are to introduce the topic and summarize the contents. So in fact most of the lead should appear later in the article in expanded form. Dmcq (talk) 16:37, 19 February 2012 (UTC)

Mathematics articles in Wikipedia should be accessable to laypersons, but not at the expense of mathematical accuracy. I think "correspondence" is a word most people can understand. Rick Norwood (talk) 16:31, 19 February 2012 (UTC)

My main objection to correspondence is its static nature, it seems at variance with input and output which imply some sort of process (rule)....Selfstudier (talk) 16:59, 19 February 2012 (UTC)

I think what you're talking about is the application of a function. A function is pretty much static,, a graph of a sine function just is, it doesn't do anything. A car doesn't go anywhere unless the engine is switched on and iit is driven (well not normally anyway!) Dmcq (talk) 17:44, 19 February 2012 (UTC)

For instance we can talk about x "squared", the input is x, the process is squaring....Selfstudier (talk) 18:09, 19 February 2012 (UTC)

I think that you specify a function by giving a domain (allowed inputs) and a process (rule) for converting inputs to unique outputs and maybe even specify the output set as well for more complex cases. (Notice that I am not saying that a function IS a rule, that is more of a convenient usage, I am saying a function is something you specify.....Selfstudier (talk) 20:24, 19 February 2012 (UTC)

Any straightforward meaning of rule doesn't really work however you put it. Even the real numbers can't all be given by rules and yet we think of them all as existing. Dmcq (talk) 20:56, 19 February 2012 (UTC)

Hmmm, not sure what this is really a reply to, can you elaborate?....Selfstudier (talk) 00:00, 20 February 2012 (UTC)

This article must reflect mathematics today, when countless (literally!) functions cannot be given by any rule. A function is NOT something you specify, the choice function being a prime example. Rick Norwood (talk) 21:54, 19 February 2012 (UTC)

"The Axiom of Choice is obviously true; the Well Ordering Principle is obviously false; and who can tell about Zorn's Lemma?" Is that all you have been able to come up with? In any case it still falls within my definition for all practical purposes even if not for all set theoretic purposes (who cares?)...Selfstudier (talk) 23:57, 19 February 2012 (UTC)

This whole debate seems to result from a confusion of "rule" with "algorithm." But a rule need not be given by an algorithm -- consider the characteristic function of the rationals, where f(x) = 1 if x is rational, and 0 otherwise. That's obviously a rule, but there is no algorithm to do that. Since the vast majority of books say "a function is a rule" and "rule" in that sense is considered broadly, Tkuvho's suggestion is a very reasonable one. I really dislike "a function is a correspondence" because:

1. "correspondence" is even less clear to the reader than "rule"
2. it's easy to misread "correspondence" as implying all functions are injective
3. if you formalise "correspondence," you wind up defining a function to be its graph (i.e. {(x, f(x))|x in D}), which makes the phrase "graph of a function" rather strange. -- 202.124.75.33 (talk) 23:52, 19 February 2012 (UTC)

I voted for eliminating the first paragraph altogether and I still think it is pretty hopeless but in the absence of support for that, I will join the "rule" supporters who by my count now have the edge....Selfstudier (talk) 00:11, 20 February 2012 (UTC)

Well I just did a google search of 'function is a' and the first few entries I got were

a function is a special type of relation where

A function is a "well-behaved" relation

A function is a relation

function is a useful tool

function is a relationship

A function is a preset formula

a function is a rule

A function is a group of statements that is execute

function is a generator;

A function is a rule

a function is a type of procedure or routine

A function is a subroutine

function is any algorithm or subroutine

A FUNCTION IS A PROGRAMMING UNIT

A function is a block of organized, reusable code

A function is a mathematical process that uniquely relates

a function is a mapping of some domain onto some range

a function is a method

A function is a specific relation

If you're going to go on google it looks like relation is the one that is most definitive and it is used in the top entries. I could try this on your 'vast majority' of books too but I'm pretty certain rule is not the winner. I think your counting of people is also a bit awry. Dmcq (talk) 00:31, 20 February 2012 (UTC)

Google searches (in quotes) 1)"A function is a rule" 19.4 million 2)"A function is a relation" 2.5 million and 3) "A function is a correspondence" 1.4 million Seems about right to me, sure you can argue that it isn't correct but that's not a decision for us to makeSelfstudier (talk) 00:51, 20 February 2012 (UTC)

I emphatically disagree with your position that a majority of internet hits should overrule mathematical correctness. —David Eppstein (talk) 01:09, 20 February 2012 (UTC)

My position? Don't blame me, I didn't start it....:-) Anyway,we disagree about what is "correct" regardless of Google, right?Selfstudier (talk) 01:14, 20 February 2012 (UTC)

The more relevant thing is not Internet hits (which nobody actually suggested), but Google Books hits, since a book is more likely to be a WP:RS. In fact we cannot apply our own definition of "mathematical correctness" as some would suggest, but we need to follow the books. In general, the books say:

• "A function is a relation" 36,000
• "A function is a rule" 28,100
• "A function is a mapping" 16,400
• "A function is a correspondence" 8,900

So "relation" in fact the most popular, and "correspondence" the least. If we restrict attention to books published by Springer, Addison-Wesley, OUP, CUP, Wiley, and McGraw-Hill (which are even more likely to be a WP:RS), we get "rule" on top by a very slim margin:

• "A function is a rule" 266
• "A function is a relation" 252
• "A function is a mapping" 87
• "A function is a correspondence" 54

Basically, the books support "rule" or "relation" but not correspondence. -- 202.124.75.245 (talk) 07:50, 20 February 2012 (UTC)

User:Selfstudier wrote above: I will join the "rule" supporters who by my count now have the edge. I similarly support the "rule" wording which is the most helpful to novices and used by Spivak, therefore arguably mathematically correct. However, the assertion that "rule" supporters "now have the edge" is somewhat optimistic. In the current discussion, only Selfsdudier, myself, and an IP (again, I suggest registering with a more stable user name) support "rule". Unless there is further input by interested editors, pursuing this further would not be consistent with wiki guidelines for resolving content disputes. Tkuvho (talk) 07:55, 20 February 2012 (UTC)

Since Wikipedia is not a democracy (WP:DEM), voting is pointless. And instead of endless debates about the article lead, I think the way forward is to rework the Definition section in the article. The lead should merely reflect what is being expanded later in the article. Isheden (talk) 08:41, 20 February 2012 (UTC)

I'm more happy with the content than with the lead at present, I agree with you provided that you don't simply intend a transfer of the whole of this argument to a different place in the article...Selfstudier (talk) 10:41, 20 February 2012 (UTC)

A relation and a correspondence are practically the same thing, it is like the difference between 'a function from X to Y' and 'a function'. Correspondence is slightly more accurate but I'm happy with relation. As to google you really need to check the actual references, their counts can be quite widely out and as shown by the ones I listed they can refer to lots of things besides mathematics. Also Wikipedia says we should use the most reliable sources in WP:RS not all sources. Also note that 'function is a relation' even if it worked would not get three of the six relation entries in my list ' a function is a special type of relation ', 'A function is a "well-behaved" relation', and 'A function is a specific relation'. As to counts on checking the references I have found google counts can very easily be out by a factor of ten even when there are only a few thousand responses, and the figures can go down if you say OR or up if you say AND. This is why I checked the references the way I did.

What you say here is correct,(binary)relation will always need to be qualified in order to arrive at function since it is a generalization and for purposes of this article and the lead in particular, an unnecessary one as reference to the existence of a generalized version can be made at the foot of the article for those interested in pursuing the matter further.Selfstudier (talk) 10:35, 20 February 2012 (UTC)

If you want to check the books properly do something like 'function is a' and actually check the first few pages of returns. Chuck away any that are irrelevant and then actually look at what is left and the standard of the books they were in. Dmcq (talk) 08:51, 20 February 2012 (UTC)

The last Google Books search above was restricted to a particular set of reliable publishers: Springer, Addison-Wesley, Oxford University Press, Cambridge University Press, Wiley, and McGraw-Hill. The phrase "a function is a specific ..." occurs only 9 times with those publishers (and only 2 of those occurrences relate to mathematics), so it isn't really a contender.

The advantage of "relation" over "correspondence" is that "relation" has no hint of suggesting injectivity. However, my vote is still for "rule." -- 202.124.73.189 (talk) 10:52, 20 February 2012 (UTC)

When all is said and done, we have one group seeking to define function as ordered pair and another as rule. In some sense both are correct and this is reflected in sources although I question whether novices would have the mathematical maturity to cope with the higher abstraction of ordered pair. Also many sources simply fudge the issue with something along the lines of "a rule.....assigns......set A to setB blah". To me this suggests that we ought to go with something simple in the lead for the casual reader and cover the actual situation as reflected in sources for the main article.Selfstudier (talk) 10:59, 20 February 2012 (UTC)

I think it is prudent to reflect, in the article section Definition, how function is defined in various sources before attempting to formulate a concise definition for the first sentence of the lead. Isheden (talk) 11:14, 20 February 2012 (UTC)

The definition of a function is certainly as a relation - a set of ordered pairs, possibly with a stated domain and codomain. The reason that we don't say "ordered pair" or "relation" in the lede is exactly because we don't want to go too fast for those who don't know what a function is. Otherwise, that is exactly what we would say. When more advanced math books say "rule", this is because they have implicitly redefined "rule" to mean "relation". But for those who don't know, they will think that "rule" has its normal English meaning, not a jargon meaning. So for them it's better to stick with words where the usual English matches the mathematical English, and "correspondence" does that. — Carl (CBM · talk) 13:09, 20 February 2012 (UTC)

I think saying they are both 'right' is mixed up with how Wikipedia decides things. Wikipedia says 'verifiability not truth' and 'rule' is verifiable as being used in a number of places. In such cases we should use the best sources. Mathematics is even less of a democracy than Wikipedia. What's 'right' is not decided by vote. About the one doing google counts above, you really do need to check the returns properly and I have explained why. Checking the results also can give an idea about how things are actually done. The initial few pages tends to give the most reliable sources about a subject. Doing a google books search on 'a function is a' and looking at the mathematical results in the first ten pages gives

a rule, a relation, a rule, a special type of relation, a formula, a certain kind of correspondence, a single valued dyadic relation, a special kind of relation, a rule, a rule, a relation, a certain kind of correspondence, a set of ordered pairs, a relation, a set, a rule of correspondence, a relation, a special case of a binary relation, a set of ordered pairs, a particular kind of relation, a certain sort of relation, a set of ordered pairs, a correspondence, a set of ordered pairs, a relation, a relation, a set of ordered pairs, a relation, a quantity or mathematical expression, a rule, a relation, a rule, a set of ordered pairs, a rule

There seemed to be a few duplicate books there, for instance James Stewart books with a rule occur a couple of times but we'll ignore that. As far as I can make out that gives 7 for a rule and 14 for a relation. Dmcq (talk) 13:17, 20 February 2012 (UTC)

Personally I prefer correspondence because it doesn't give people the wrong idea from a familiar word like rule, they know they have to read on a bit. Also it is right mathematically. It is terribly hard to get rid of a wrong idea, unlearning is much harder than learning something new, that's why one should try to be reasonably accurate to start with. Dmcq (talk) 13:29, 20 February 2012 (UTC)

We appear to be in danger of just going over the whole argument yet again and it will be just as pointless yet again, let's just agree on a bunch of reputable (reasonably recent) sources and go with what they say for the main article and then summarize it somehow per Isheden suggestion for moving foward...Selfstudier (talk) 13:37, 20 February 2012 (UTC)

I have just carefully reread the lead as it currently stands, and it seems pretty good to me. Selfstudier's view of set theory ("who cares") is original research. The google search is meaningless. Mathematics articles should be based on standard textbooks. Rick Norwood (talk) 13:38, 20 February 2012 (UTC)

I agree about textbooks. The more difficult issue here is that "function" is used at all levels of math, from elementary school to postgraduate. If we look at sufficiently low-level books, we will find all sorts of strange things - the same would be true in chemistry, or quantum physics, or any other technical area, where low-level books routinely stretch the truth, sacrificing accuracy for the appearance of accessibility. If we look at books at the upper undergraduate level and above, there is great uniformity in the definition of a function: it's a relation R such that if xRy and xRz then y=z (there is another minor issue whether a relation includes info on its domain and codomain). So in the main body of this article we also use this definition. But in the very first sentence, it may be too much to say that a function is a relation, so we try to make it a little more friendly. In doing so, we should not give up on being accurate, or give a description that a naive reader would be likely to misunderstand. — Carl (CBM · talk) 13:46, 20 February 2012 (UTC)

This is all very interesting stuff, and it seems we are certainly clarifying our thoughts here. (1) The Google booksearch is interesting, but maybe not the whole story. If some of the texts are computer science, rather than math, the terminology may differ. (2) Google searches do not determine our policies though. (3) Agree with above (WP:DEM), voting is pointless. (4) about the article lead and the Definition section The lead should merely reflect what is being expanded later in the article. (per (User:Isheden| 08:41, 20 February 2012) (5) relation is good, rule must take its place, correspondence is a simple English word, may be the best candidate, I am thinking (6) rule, relation, etc will also appear numerous times, it is not winner-take-all (7) good work, though maybe we've reached the point of re-hashing, good work anyways. NewbyG ( talk) 14:12, 20 February 2012 (UTC)

As Dmcq points out, when students have first learned something wrong, it is very hard for them to unlearn it. Any professional mathematician will tell you that "a function is a rule" is wrong, and that students who have been taught that a function is a rule have a lot of trouble if they major in mathematics. It's ok to say that "some" functions are given by rules, with particular examples. But the first sentence should not make a statement that is mathematically false, and I don't understand the problem with "correspondence". It's not that hard a word.

Most college textbooks are reasonably accurate. A study by the National Council of Teachers of Mathematics found that no American grade school or high school textbook is at an acceptable level of mathematical accuracy. Their choice of the best textbook for schools to use was an English translation of a textbook series used in Singapore. Needless to say, nobody paid them any mind, but we should be careful to rely on college textbooks. Rick Norwood (talk) 15:37, 20 February 2012 (UTC)

Just so we're clear, this debate is merely a reflection of the same debate taking place, particularly over the last decade or so, among researchers and educators, for example, Thorpe 1989 "“We should teach the most intuitive and practical definition and not confuse our students with unnecessary abstractions” and "“A function should be defined as a rule, or perhaps as a certain kind of machine, but certainly not as a set of ordered pairs.” and Malik's views as to the static nature of the "modern" (ie 1939) definition. Of course, just like here, there are pro and con; the only thing that's sure is that things will change and probably faster than they have in the past.....Selfstudier (talk) 18:25, 20 February 2012 (UTC)

You know, there is another approach that I took with the Algorithm article. Rather than clutter the main article with alternatives and debates, I created an Algorithm characterizations sub-article that presented all the different viewpoints around what an algorithm "is". There might be room for something like that here, since there is clearly sourced information that presents alternative characterizations of "function". Then what happens at the top level is to give a tight definition that doesn't weasel around; the debate and alternatives, if an educator, student, etc is curious about the debate, can be entered on the "characterizations" page. I think the references/cites you folks have come up with are really good and should not be consigned to the oblivion of a talk page. BillWvbailey (talk) 19:20, 20 February 2012 (UTC)

Create subsection with formal symbolism?

[[cc'd down from above to start a new thread]

Carl, I looked for your definition in the article: "a relation R such that if xRy and xRz then y=z" but I'm not seeing it. If anyone's interested here's a source. From Suppes 1972:57:

"DEFINITION 1. A is a relation ↔ (∀x)(x ∈ A → (∃y)(∃y)(x = <y, z>)).

The <y, z> is the standard Kuratowski definition of the ordered pair cf page 32. From Suppes 1972:86-87:

"DEFINITION 39. f is a function ↔ f is a relation & (∀x)(∀y)(∀z) (x f y & x f z → y = z).

"DEFINITION 40. f(x) = y ↔ [(E!z) (x f z) & x f y] V [-(E!z) (x f z) & y = 0]

(The symbol 0 indicates the empty set cf p. 14). The formal definition of domain D is on p. 59 and for range ℛ on page 60:

"DEFINITION 3. DA = {x: (∃y)(x A y)}

"DEFINITION 4. ℛA = {y: (∃x)(x A y)}

I find it useful to have all of these in one place, and to have them as simple and as formal as possible, so there's no questions about their interpretation. BillWvbailey (talk) 16:10, 20 February 2012 (UTC)

The article says it in English: "a necessary condition is that every element in the domain is the first element in exactly one ordered pair." In particular, if we have both (x,y) and (x,z) then y=z. — Carl (CBM · talk) 16:42, 20 February 2012 (UTC)

You may find a lot with searches such as function + "Bourbaki Dirichlet"....Selfstudier (talk) 17:55, 20 February 2012 (UTC)

Jeez I looked right at it [the quasi-formal word definition]. Does anyone think it's worth adding a section with formal, symbolic definitions? If not, I prefer the precision and brevity of Carl's "a relation R such that if xRy and xRz then y=z", i.e. a function is a relation "&" with the following restrictions . . .. There's no debate around whether or not a function is a relation, is there? And there's no debate about the restrictions, is there? This is all in the spirit of what Rick Norwood wrote below above: making the definitions "rock-hard and ironclad", so they serve as a reference/resource for educators that is, in the best judgment of the community, as good as it can possibly be. Bill Wvbailey (talk) 19:01, 20 February 2012 (UTC)

There is some differences. The definition as a relation is essentially the version used in set theory as described in the definition section in the fourth paragraph. Probably there should be more about it, there is a related bit in the third paragraph of the definition which links the version used in most other areas and the set theory version. It doesn't really make an awful amount of difference but logically there is a difference. The relation version doesn't work with function spaces directly but 'you know what I mean' about covers the differences. Dmcq (talk) 22:15, 20 February 2012 (UTC)

Very odd that the History does not really mention Bourbaki (I am thinking particularly of "the Dirichlet-Bourbaki definition")Selfstudier (talk) 13:31, 21 February 2012 (UTC)

New "Function Characterization" sub-article?

Mathematics: A Discrete Introduction , Edward Scheinerman, 2000 “Intuitively, a function is a ‘rule’ or ’mechanism’ that transforms one quantity into another.”....“develop a more abstract and rigorous view of functions”....“A relation f is called a function provided (a,b) in f and (a,c) in f imply b=c” (Pipe and smoke it,lol)Selfstudier (talk) 19:48, 20 February 2012 (UTC)

Thing is, all these higher level definitions are all geared (even though they don't say so) to making you accept the existence of function as some kind of mathematical object or structure but this point of view is already under attack from a number of directions, category theory for example. My simplistic way of looking at things says to look at it as many different ways you possibly can and don't get too hung up on any of 'em....Selfstudier (talk) 19:54, 20 February 2012 (UTC)

It's clear you know your literature, and you could contribute a lot to such a sub-article. As I wrote a couple sections above, we can find a debate similar to this started by Gurevich at Microsoft re "algorithm" -- algorithm as an actual machine versus a structured symbol-string (or equivalent static object) -- and I can trace the characterization-of-computation debate back to Goedel, Turing, Post and Church. My personal opinions? today a function is a "static object", tomorrow a function is a student with a pencil, a piece of graph paper tacked to with two nails and a string around it as they're drawing an ellipse. I'm so old that I used analog computers and started out doing trig and logs on a slide rule, so I've been tainted. This is why I think a sub-article would be a good thing. Bill Wvbailey (talk) 20:19, 20 February 2012 (UTC)

Slide rule helping my dad out when I was young(er) so perhaps not quite on as you but headed rapidly in that direction; recall doing a Math Logic course where it was decided (heh) that algorithms didn't exist or if they did couldn't be defined,lol.

Hopefully people will dig up a few references and we will see what we got for a sub article..Selfstudier (talk) 20:30, 20 February 2012 (UTC)

In category theory, they generally use the word "morphism" instead of "function" to emphasize that the things they work with may not be functions. But they don't actually change or "attack" the usual definition of a function. — Carl (CBM · talk) 22:09, 20 February 2012 (UTC)

I never said they did?......Selfstudier (talk) 23:38, 20 February 2012 (UTC)

I must have misread your comment from 19:54, 20 February 2012 (UTC) above; apologies. — Carl (CBM · talk) 00:28, 21 February 2012 (UTC)

A more accurate definition

I think the recently added 'a more accurate definition' before the 'precise' definition should just be removed. All the stuff about relations there doesn't add anything, it shouldn't have even been where it was originally. The bit further down about partial functions and multivalued functions is all that's needed about different relations in respect to functions. The definition section already explains about he differences between functions as relations and as a triple. Defining just as a relation is not correct. You have to say 'a function from X to Y' can be defined as a 'relation between X and Y' to make any sense and then you'd have to work backwards to a function. Have a look at the article about relations. It says ' More generally, a binary relation between two sets A and B is a subset of A × B'. It does not say a relation is a set of ordered pairs. Dmcq (talk) 11:09, 21 February 2012 (UTC)

I've moved the text here so it can be discussed and removed it from the article until fixed:

A more accurate definition is as follows. A function is a special case of a more general mathematical concept, the relation (or correspondence), with the additional restriction that each element of the domain appear as the first element in one and only one ordered pair (X, Y). In a relation, an element of the domain may not be the first element of any ordered pair, or may be the first element of two or more ordered pairs. A relation is "single-valued" when if an element of the domain is the first element of one ordered pair, it is not the first element of any other ordered pair. A relation is "left-total" or simply "total" if every element of the domain is the first element of some ordered pair. Thus a function is a total, single-valued relation.

Relation is used in books so there may be some way of fixing this. I think this reference from just below saying about how functions can be treated as ordered pairs and that's how they would do it in the book explains the problem well:

Ethan D Bloch (2011). Proofs and Fundamentals: A First Course in Abstract Mathematics. Springer. p. 131. ISBN 978-1-4419-7126-5. http://books.google.co.uk/books?id=QJ_537n8zKYC&lpg=PP1&pg=PA131#v=onepage&q&f=false. 

Dmcq (talk) 11:21, 21 February 2012 (UTC)

In view of the intense discussion, I believe there is a need for various levels of abstraction in this article. I have attempted to write a more concise definition without introducing to much terminology based on two other sources. Please let me know if you have further concerns. Unfortunately I cannot access the book page you mentioned. Isheden (talk) 14:13, 21 February 2012 (UTC)

A definition should not start of by saying what something isn't. That should be left till later.

The paragraph

Instead of talking about a 'function' and specifying the domain and codomain in its definition, it is normally more convenient to talk about functions with a specific domain and codomain, functions of a given type can then be defined by the set of ordered pairs F. For example one 'function from the reals to the reals' would be given by the set of pairs (x, 2x) where x is a real.

was removed in the midst of those changes. This is at the heart of what that citation is about. The statement ' A function from X to Y is a particular case of a relation' is not really correct and this is supposed to be the more accurate section. A function from X to Y is not a particular type of relation,, it is a particular type of relation from X to Y. The 'from X to Y' matters. If you think of adding 'from X to Y' as turning a triple, the domain, codomain and graph into just the graph then things work out okay. Correspondences were defined as triples as well so they map directly to functions. If we think of relation without the 'from X to Y' as being the same as a correspondence then functions are a typoe of relation. However relations are only defined in the context of from X to Y. If relation is to be mentioned without the from X to Y then either we are talking about the older set theoretic definition of a function as just a set of ordered pairs.or we are assuming something that isn't in the normal definition of a relation. Dmcq (talk) 18:11, 21 February 2012 (UTC)

I have made an attempt to address your concerns. Feel free to revise as you see fit. The "rule" discusssion was already part of the introduction, so I moved it there. If relation and correspondence are not the same (which I assumed), then the distinction needs to be explicitly stated in the article. I guess the first sentence of the lead is also unsatisfactory then. Perhaps it should be revised to something like "A function is a correspondence from a set of inputs to a set of outputs, such that..."? Isheden (talk) 09:42, 22 February 2012 (UTC)

Thanks. I hope I've made it more explicit and clearer. The two terms lead to resolving a bit of ambiguity and making things clearer in logic. I'm happy with either relation or correspondence in the lead and I'm sure somebody in the future will have other ideas for rephrasing it better. Dmcq (talk) 11:11, 22 February 2012 (UTC)

Rename overview to introduction

I think the lead of the article acts as the overview and the current 'Overview' is better renamed as 'Introduction'. It is currently not very marvellous as an introduction but I think the basics are there. This is where the black box with inputs going in and outputs coming out can go. Dmcq (talk) 11:53, 21 February 2012 (UTC)

Cannot be given by any rule?

An editor claims to have found a function not given by any rule. The claim is unsourced. Tkuvho (talk) 13:41, 21 February 2012 (UTC)

Furthermore the claim is contradicted by Spivak and Stewart (leading calculus textbook today), who define a function as a rule. Tkuvho (talk) 13:42, 21 February 2012 (UTC)

I hope we will be able to have a clearer view of things when we have more refs (DMCQ has added a couple already and I have some others somewhere)Selfstudier (talk) 13:52, 21 February 2012 (UTC)

Certainly I do not claim to have "found" a function not given by any rule. The existance of such functions is a commonplace of set theory. The essential point is this. Rules are sentences of finite length in some finite alphabet. The number of such sentences is countably infinite. But the number of functions from, for example, the real numbers to the real numbers is uncountably infinite. Therefore, there are infinitely more functions than there are rules. This is a well-known fact and I trust that unlike some editor above who dismisses all of modern set theory ("Who cares."), you accept the modern view of the subject, modern in this case meaning the last hundred years. I'll provide a reference shortly, but off the top of my head, see Munkres Topology or Halmos Naive Set Theory.

A freshman calculus book is hardly the final authority on the subject, but certainly Spivak, a major mathematician, knows everything I meantion above. I don't know if he qualified his definition or if he deliberately oversimplified (perhaps because so many people said his Calculus on Manifolds was unreadable). Or maybe the definition of function is written by Stewart. I'm sure Thomas is rolling over in his grave because of some of the things later authors put into Thomas' Calculus. I would appreciate it if you provided the complete quote from Spivak and Stewart. It would save me a trip to the library. Rick Norwood (talk) 13:53, 21 February 2012 (UTC)

Note that the term "rule" preceded your narrow definition thereof. Tkuvho (talk) 14:00, 21 February 2012 (UTC)

("Some editor" would be me and I haven't actually attempted to edit this article as yet)If one accepts set theory (ZFC) as being a suitable basis for all of mathematics then one needs to accept the consequences for one's definitions which will of necessity need to somehow be shoehorned into compliance with the ontology. There are quite a few mathematicians (not a majority admittedly) who don't accept this premise (including some who do not even accept the idea of "uncountable"). Why is this? Simple, because when you marry AC to axioms for infinity you can produce peculiar results like Banach Tarski and some people are uncomfortable with a failure to produce exhibits of "objects" claimed to "exist" and there are other reasons.So one should not suppose that set theory is the be all and end all because it just isn't....Selfstudier (talk) 14:17, 21 February 2012 (UTC)

Selfstudier: From Fraleigh, Abstract Algebra, "Zorn's lemma ... equivalent to the well-ordering theorem (itself equivalent to of the Axiom of Choice)...quickly became an essential part of the mathematician's toolbox." Rick Norwood (talk) 15:06, 21 February 2012 (UTC)

Tkuvho: Certainly it is possible to define a function as a rule, provided you define a rule as a set of ordered pairs. But this article is using the word "rule" in its common meaning, "a guide or principle for governing action" (New Merriam-Webster Dictionary). I don't think it is unduly narrow to assume such guides or principles are of finite length, written in a common language with a finite alphabet. Rick Norwood (talk) 15:10, 21 February 2012 (UTC)

I understand that this is your opinion. However, several editors on this page expressed the sentiment that $x\mapsto f(x)$ is a rule that defines the function f. This may seem tautological, but isn't really, since the reader already has an intuitive understanding of the term "rule", upon which its mathematical formalisation is based. Tkuvho (talk) 15:13, 21 February 2012 (UTC)

We are reviving a discussion that took place at the beginning of the 20th century, but which most modern mathematicians consider settled (see Fraleigh, above), and which goes far beyond the scope of this article. Selfstudier is correct in noting that there are a few holdouts. He is also correct in noting that they are a minority. Note that I do not object to the use of the word "rule" in the article. Rather, I want the common opinion of most modern mathematicians to be included as well, and, since it is the majority opinion, not labled dubious. Rick Norwood (talk) 15:20, 21 February 2012 (UTC)

Please explain how the common meaning of "rule" allows for a rule that is not explicit. Certainly, if an ordinary reader reads the word "rule" he or she will understand an explicit rule.

Let me ask you a meta-question, Tkuvho: Do you understand the history behind this discussion, and never-the-less place yourself in the camp of those who reject the Axiom of Choice? Or is this all new to you? Rick Norwood (talk) 15:25, 21 February 2012 (UTC)

Why, I use the axiom of choice every morning in deciding between tea and coffee. Tkuvho (talk) 15:32, 21 February 2012 (UTC)

I think we are off track again, my only objection (for now) is to the use of the word "correspondence" (particularly in input/output context) in the lead and I had thought the idea was to round up some modern references, cross reference them for the article and then summarize for the lead. Is that not the case?Selfstudier (talk) 15:29, 21 February 2012 (UTC)

Diversion Definitions(Gowers); also I still approve of Gowers characterization of functions in the Princeton Companion (he is a Fields medallist after all)Selfstudier (talk) 15:58, 21 February 2012 (UTC)

Tkuvho: you didn't answer my question. Selfstudier and I disagree, but I am sure he understands just what it is we disagree about. Rick Norwood (talk) 16:02, 21 February 2012 (UTC)

It's worth mentioning, I think, that "constructivism" is not taught as such, so constructivists are in fact classically trained mathematicians who have switched sides, let's say, and that their number continues to rise in such circumstances ought to give pause. It is very difficult to grow the number rapidly when not taught and then later, there is all the usual pressure to conform, publish and whatnot.If one considers that the original conception was for a rock upon which to found mathematics, then the project has to be deemed a failure...Selfstudier (talk) 16:17, 21 February 2012 (UTC)

All very true, but the pages of Wikipedia are not the platform from which to launch a revival of constructionism. Rick Norwood (talk) 17:04, 21 February 2012 (UTC)

Agree Wikipedia is not a WP:CRYSTAL ball and I see no indication that constructivists will ever be anything but a small minority. Most people regard all real numbers as 'existing' whether or not there is a rule to construct them. We don't consider the real line to have holes in for the numbers we have no rule for generating the digits of, Dmcq (talk) 17:27, 21 February 2012 (UTC)

An actual example of a definable function that has no rule would be one that returns the digits of Chaitin's constant. We simply don't know what the sequence is and will never have an actual rule. I'm sure constructivists will be happy to say the constant doesn't even exist! Dmcq (talk) 17:37, 21 February 2012 (UTC)

I just realized I don't even know what was there before "correspondence", was it "rule" or something else? Is there a way to go back and look at all previous versions?Selfstudier (talk) 17:58, 21 February 2012 (UTC)

Just click the historyy tab and go to some date. Clicking on the date will show that version. You can also step one change at a time when viewing a particular version, or if you click on cur or prev in the history then go forward or back through the diffs one at a time. A binary chop in the history is best first though and can sometimes be quicker than trying to spot what you want in the comments. Dmcq (talk) 18:18, 21 February 2012 (UTC)

Thanks, I will try it...Selfstudier (talk) 18:40, 21 February 2012 (UTC)

Does anyone object to naming the "modern" definition as "the Dirichlet-Bourbaki definition"? Selfstudier (talk) 18:40, 21 February 2012 (UTC)

Hmmm, silence....references are here Dirichlet BourbakiSelfstudier (talk) 19:31, 21 February 2012 (UTC)

Or hereSelfstudier (talk) 19:44, 21 February 2012 (UTC)

In all my researches I've never encountered the "Dirichlet-Bourbaki definition". Bourbaki maybe (one of the references referred to the Bourbaki definition of 1939, never mind the well-known definitions predate this by at least 20 years), but I've never seen Dirichlet's name used in any way shape or form relative to a generalized definition of the notion of "function". My search: http://www.google.com/search?q=Dirichlet-Bourbaki+definition+of+function&btnG=Search+Books&tbm=bks&tbo=1 turned up the same 2 or 3 sources yours did. My "take" from the itsy-bitsy google precis is that this may be an unusual, narrow usage specific to the "education researcher", in particular to the needs and concerns of the hypothetical secondary educator "Edith". Because of this, and the fact that I need to pay $$$ to access the journals or I have to walk up the hill to get to the local U-library, I'm reluctant to support a "Dirichlet-Bourbaki" moniker (at this time, but am always open to further evidence). Bill Wvbailey (talk) 01:39, 22 February 2012 (UTC)

It might be that Dirichlet is included there because he was the the first to introduce the notion of arbitrary correspondence and I see that in the Wiki article for him it says in the lead that "he is credited with being one of the first mathematicians to give the modern formal definition of a function". Bourbaki extended the arbitrariness to sets and apart from the quoted definition (which implies a rule) also gave the ordered pair definition (which doesn't).

More to the point there does seem to be a kind of gap in our article history between "Von Neumann's set theory 1925" and "Since 1950" (under the 1950 heading it speaks for one sentence about Zermelo's set theory (as modified in 1922) being the source of "a" modern definition, which seems to relate back to a previous section "Zermelo's set theory (1908) modified by Skolem (1922)" and which does not seem to me to be the modern definition and then it refers back once again to "Since 1950" and speaks about a terminology change. Our good friend Bourbaki slots into this sequence rather neatly and we ought to recall that there were one or two wars at the relevant times). What do you think?Selfstudier (talk) 10:34, 22 February 2012 (UTC)

One other thing, not sure if we are looking at the same set of sources, I count quite a few over quite some time and not just confined to secondary educators but even if that were the case (everybody having followed someone's lead in the past) they all believe that the textbook modern definition (quoted correctly by many of them) is that which is contained in our sources and they all refer to it as the Dirichlet Bourbaki definition......Selfstudier (talk) 11:11, 22 February 2012 (UTC)

I finally found an excellent reference History in Mathematics Education The ICMI Study (456 pages, sorry) which traces what happened in the educational arena from Dirichlet Bourbaki (they call it Cauchy Dirichlet Bourbaki)through the New Math mess and the full circle back to functional dependancy. Of course, this has only indirect relevance for what goes on in advanced math beyond the university (which I think is partly a function (sic) of when you happened to learn these concepts)Selfstudier (talk) 14:18, 22 February 2012 (UTC)

An editor continues to insert unsourced claims based on his misunderstanding of Morash. Tkuvho (talk) 13:56, 22 February 2012 (UTC)

I really don't think we need mention choice functions in an introduction. There are enough citations about functions not being rules and it's pretty obvious anyway otherwise we'd have real problems when talking about function spaces. Dmcq (talk) 14:24, 22 February 2012 (UTC)

You may be right about choice functions in the introduction. As far as rules go, your problem arises because you interpret "rule" as "algorithm". Otherwise there is no problem with function spaces. Each member f of such a space is determined by the rule $x\mapsto f(x)$. Tkuvho (talk) 14:30, 22 February 2012 (UTC)

You either accept an axiom or not, no rule required; where did "rule as algorithm" originate?Selfstudier (talk) 14:36, 22 February 2012 (UTC)

The foundational framework is not really the issue here. An algorithm is an explicit procedure that some hold could be programmed on a computer; clearly a non-measurable function cannot be programmed on a computer. But a rule does not have to be an algorithm. That's why Spivak and Stewart can define a function as given by a rule. Tkuvho (talk) 14:46, 22 February 2012 (UTC)

Yes, I agree with you, As I was explaining above, I suspect that what has happened is that for education purposes even up to university level,it has been decided by the (majority of) education gods that ordered pairs, triples etc will only confuse things and had best be left for later (which for most people translates as "never").Selfstudier (talk) 14:53, 22 February 2012 (UTC)

The ordered pairs seems to be a separate issue. What makes you think they don't appear? When you have a simple-minded function like y=x^2, its graph will be defined by ordered pairs. This surely appears in most calculus courses. Tkuvho (talk) 14:56, 22 February 2012 (UTC)

It's not the idea of an ordered pair per se (although there are some problems with that as well) it is whether or not you present "function" as a set theoretic construct (this is what I meant by ordered pair) or instead, more intuitively, via the rule or process idea. Personally, I have no difficulty with the latter conception and I think very little is lost by it for most purposes (as Gowers says, you can always check if you aren't sure)Selfstudier (talk) 15:02, 22 February 2012 (UTC)

With hindsight (wonderful thing, hindsight)you can see why we have the section "Von Neumann's set theory 1925"....Selfstudier (talk) 15:29, 22 February 2012 (UTC)

Just to emphasize the main point that Eppstein ignored: An algorithm is an explicit procedure that some hold could be programmed on a computer; clearly a non-measurable function cannot be programmed on a computer. But a rule does not have to be an algorithm. That's why Spivak and Stewart can define a function as given by a rule. Tkuvho (talk) 16:32, 22 February 2012 (UTC)

RE "no known rule": Observe the very peculiar quote from S.F. Lacroix [1797-1800] a couple sections down: I don't know what to make of it. It's in commentary so it's not O.R. but I'm not sure I agree with Domingus's commentary. I'd have to see the original. BillWvbailey (talk) 16:53, 22 February 2012 (UTC)

It sounds from this that rule is being used in a vacuous or circular way and doesn't actually mean anything. Saying a function is defined by the rule $x\mapsto f(x)$ does not mean anything if rule is not separately defined, here it seems rule is being used as a synonym for function so neither is defined. At least saying an algorithm or relation actually means something. Dmcq (talk) 17:08, 22 February 2012 (UTC)

On the contrary, the word "rule" has intuitive meaning for the reader that helps him understand the concept, which is precisely why Spivak and Stewart, some of the best math writers we have, use it. Correspondence, meanwhile, has a very different meaning for a general reader, which implies both 1-1 and also symmetry between domain and range, so certainly much more confusing than "rule" which clearly implies a direction and therefore introduces the asymetry needed here. Learning is never a formal system, and a reader can only build upon his intuitions. The word "rule" is sufficiently versatile to encompass the intuitive meaning already familiar to the general reader, and a more technical meaning Spivak and Stewart rely upon. At any rate the claim about functions not given by any rule is unsourced. It may not be given by an explicit rule/algorithm, but can certainly be described as being given by a non-explicit rule $x \mapsto f(x)$. Now certainly Bourbaki will not agree to any of this. Tkuvho (talk) 17:31, 22 February 2012 (UTC)

Well I'd prefer relation in the lead okay but as to rule I think this piece [4] in the paragraph about 'One often hears that such a function is or is given by a rule...' and ending with '..that the word "rule" is not to be stripped of the last vestige of its customary meaning' about sums up what I think of the use of rule above. Dmcq (talk) 17:47, 22 February 2012 (UTC)

I'd prefer relation to correspondence (historically this was "rule of correspondence") even though it is IMHO a quite unnecessary generalization, simply explain function and leave relation (which is an increase in abstraction and requires qualification)to later. Actually, I could suffer almost anything except correspondence...:-)Selfstudier (talk) 18:10, 22 February 2012 (UTC)

Tkuvho, I think it's not quite right to say that Bourbaki will not agree; I am guessing a bit here but since in fact Bourbaki gives a version with an implied rule AND the version without a rule, I suspect a similar debate to this one was going on then and the set theorists "won" the argument and not content with that set about trying to get the entirety of mathematics for education reduced to set theoretic principles ultimately leading to the New Math debacleSelfstudier (talk) 18:21, 22 February 2012 (UTC)

Tkuvho: Please explain what you mean by "non-explicit rule". It is true that some mathematicians call a set of ordered pairs a "rule", but that is not the understanding someone reading this article will take away if we say every function follows a rule. Rather, they will take away the false impression that if it doesn't have a rule, it isn't a function. That idea was part of mathematics two hundred years ago, and was still controversial one hundred years ago, but today the word function is used in a more general sense. For example, according to the well-ordering principle, a well-ordering of the real numbers exists, even though none has been given by any "rule". It is easy to prove that, if by a rule you mean an instruction of some kind, for how to find the output for a given input, functions exist for which no set of instructions exist. When I teach a first course in Math Reasoning, this is a point that always comes up, and students who have been taught that a function is a "rule" have trouble making the conceptual jump to abstract mathematics. Rick Norwood (talk) 22:19, 22 February 2012 (UTC)

RE the notion of "function" as it relates to Bourbaki and category theory, etc: I've ordered a book coming in a few days that may help. The snippet-view looked worth the money, plus the relevant paper had a huge bibliography. (No, it's not the $265 book.) It's Marlow Anderson, Victor Katz and Robin Wilson, editors, 2009 Who Gave You the Epsilon? & Other Tales of Mathematical History, Mathematical Association of America, ISBN 978-0-88385-569-0. In particular on page 14ff there's an essay by Israel Kleiner "Evolution of the Function Concept: A Brief Survey"; on page 25-26 the snippet view presents the Bourbaki definition, followed by an thing about the notion of "function" in category theory. Bill Wvbailey (talk) 23:06, 22 February 2012 (UTC~)

That Kleiner review is available here [http://www.maa.org/pubs/Calc_articles/ma001.pdf Evolution), I think the reference is either in here or on the main page, I know I got it here to start withSelfstudier (talk) 23:48, 22 February 2012 (UTC)

Rick Norwood I think you are making a completely false argument here; in the first place most students are not as far as I know using Wikipedia as primary or sole form of tuition (at least I would hope not). From what you say, you appear to have done some type of study to investigate some number of students background education including as to what type of function definition they have been given (presumably by qualified educators such as yourself). Then further you presumably have identified a subset of students who have been given a different education more in keeping with your conception thereof and found that they have no (less?) trouble in making the transition to abstract mathematics. I would certainly like to hear more about this study which appears to come to a conclusion precisely the opposite of many others. I also note that Wikipedia refers to the well ordering principle as being about positive integers whereas the well ordering theorem is given as referring to every set (by fiat, since it relies on AC)Selfstudier (talk) 23:37, 22 February 2012 (UTC)

I have in front of me now a university (year 2) text from the UK (not a text book, a text prepared by the university said to be first published in 1997 and reprinted various times through 2005 and it says (in a big box):

A function f is defined by specifying:

(a) a set X, called the domain of f;

(b) a set Y, called the codomain of f;

(c) a rule, or process, that associates with each x in X a unique y in Y; we write y = f(x) and call f(x) the image of x under f.

followed by all the usual yada yada

Now it could well be that the system in the UK is slightly different from that in the US but it ought not to be that much different.The above is copyright so I can't go into more detail, I will however try to get some data from other universities as nowadays they are putting a lot of their material online publicly.

Selfstudier (talk) 00:43, 23 February 2012 (UTC)

Thanks, that's very useful. There are two issues here that should not be confused: (1) whether or not the lede here should use "rule" in the first sentence; and (2) whether the there is a legitimate (majority or minority) view that it is correct to say that the function is defined by a "rule". Now issue (1) has been endlessly debated in this page without yielding an agreement. As far as issue (2) is concerned, however, there can be no doubt that there is such a legitimate view, as represented by Spivak, Stewart, the UK curriculum cited above, as well as half a dozen users in this page. For this reason, claims to the contrary in the article should be deleted, regardless of what the first sentence says. Tkuvho (talk) 08:56, 23 February 2012 (UTC)

I have the impression that the other parts of the page besides the lead are being amended somewhat to reflect all these discussions in here and I had thought that the idea was to complete that with suitable references and sources and then to amend the lead so that what follows is merely an expansion of it. Whatever is put there at the end perhaps ought to include the words "is defined by" so that we get away from the "is a" argument once and for all....Selfstudier (talk) 10:07, 23 February 2012 (UTC)

That's a good idea. At any rate, the reductionist intepretation of "rule" should be resisted as not representing a consensus and contrary to the interest of readability. Tkuvho (talk) 10:14, 23 February 2012 (UTC)

Tkuvho: If you agree that this is a good idea, why don't you contribute constructively instead of wp:edit warring? What's the difference between "merely implicit" and "not explicit"? I think you are the one that should provide a reference that supports the claim "are therefore given by a rule that is merely implicit". Isheden (talk) 11:39, 23 February 2012 (UTC)

We are bound by WP:RS and WP:ORIGINAL. So I have stuck in something conforming more closely to what the reliable source talking about rules said. Also see what [5] about making rule mean whatever they wanted it to mean. Dmcq (talk) 11:42, 23 February 2012 (UTC)

Do people agree with the sentence in the first paragraph 'Two different rules define the same function if they make the same associations, for example f(x) = 3x−x defines the same function as f(x) = 2x.' If so how can we go on to say a function is a rule? Dmcq (talk) 11:54, 23 February 2012 (UTC)

That sentence was one of the reasons why I wanted initially just to delete the whole first para; although it is true, I don't think it is a particularly crucial aspect of function, merely a side note somewhere further down the pageSelfstudier (talk) 12:08, 23 February 2012 (UTC)

Also per above comments, I don't think "is a" is the right way to go although many would say it as a convenient usage. As per Gowers, "is a" straightaway implies "a thing" (noun) and fails to capture the "process" (verb) aspect. Although I appreciate that in some ways this is what a lot of the discussion has all been about, maybe we can avoid the argument with something like "is defined by" or "is given by" (Also Gowers is not the only one to assert that no one definition can capture the entirety of the function concept)Selfstudier (talk) 12:23, 23 February 2012 (UTC)

@Dmcq: we've been through this numerous times already. A function is not a rule; a function is given by a rule. Tkuvho (talk) 12:24, 23 February 2012 (UTC)

Here is a rule:

$f(x) = 0 \text { if } f(x) \ne 0\, ; f(x) = 1 \text { if } f(x) = 0$

Does that define a function ? I don't think it does because there is no association that satisfies that rule. But if you don't have a definition of what a function is, independent of the concept of a rule, then how do you decide which rules do or do not define a valid function ? Gandalf61 (talk) 12:36, 23 February 2012 (UTC)

The above university definition evades (avoids) this problem, it is the educators preferred procedure these days so as to avoid the situation that occurred previously (ie virtual zero retention of the definition, cognitive dissonance with prior conceptions, so called "met-befores" and the failure of a what is definition as a predictor of ability to work constructively with function concepts)Selfstudier (talk) 12:46, 23 February 2012 (UTC)

If you could somehow identify "pure mathematician" at age 5, segregate him(her) and thereafter indoctrinate him in set theory and abstract mathematics etc etc you might be able to get away with that, problem is that there are many people studying mathematics not in that category ie applied mathematicians, physicists, computer science people, even graphics artists, are we to subject these poor souls to the drama?Selfstudier (talk) 12:55, 23 February 2012 (UTC)

I just tried google with 'function is given by a rule' and I got surprisingly few hits, 7 for web and 6 for books, none of which really seemed to be defining what a function was except 'A constructive function is given by a rule for computing its values'. Dmcq (talk) 13:00, 23 February 2012 (UTC)

Since we are googling again try "the function f defined by" which is more a reflection of what is actually going on...Selfstudier (talk) 13:13, 23 February 2012 (UTC)

What Google says is beside the point. This is an old, old discussion, and to bring it up now without awareness of the history behind it accomplishes nothing. Nor does it help, as above, to cite Freshman and Sophemore textbooks. I've already provided a citation for the fact that this is a point calculus textbooks often get wrong or sweap under the rug. Math majors usually do not encounter the idea of a function which exists but cannot be specified by a rule until their third year in college, and they usually encounter it in a course called "Mathematical Reasoning" or something to that effect. But it is settled mathematics, except for a few finitary logicians, who reject the Axiom of Choice, and who may deserve a mention further down in the article. But the vast majority of mathematicians today accept Zermelo/Fraenkle with choice, and if you accept the Axiom of Choice, that Axiom states that there exist functions (called choice functions) without any rule. Some choice functions can be given by a rule, most cannot, even in theory, be given by a rule, unless yhou call "x maps to f(x)" a rule (some books do but it is not what the word "rule" usually means). This article is not the place to argue that mathematicians should reject the Axiom of Choice. Rick Norwood (talk) 13:14, 23 February 2012 (UTC)

I don't understand, what has AC to do with what we are talking about? Are you just referring to your single example of "choice functions" not having a "rule"?Selfstudier (talk) 13:25, 23 February 2012 (UTC)

In any case we are making a mountain out of a molehill, the article itself is going to cover all pov's supported by refs and sources which will only leave the problem of the lead as some kind of summary of the article.Selfstudier (talk) 13:44, 23 February 2012 (UTC)

I'm not sure what you meant by try "the function f defined by". Of course many functions are defined by a rule. We have that in the lead paragraph as a perfectly reasonable thing to do. Actually what worries me is the number of school textbooks that seem to be saying a function is a set of ordered pairs, that is only done at a high level in some books about set theory and most definitions require a domain and codomain. Dmcq (talk) 13:56, 23 February 2012 (UTC)

Selfstudier: I only mentioned the Axiom of Choice because some editors on this page seemed to need an example where a function exists but a rule does not, and the Axiom of Choice provides a simple example. There are, of course, countless other examples. In any case, let me again express my hope that everything that needs to be said has been said. Rick Norwood (talk) 14:02, 23 February 2012 (UTC)

Dmcq: Rather than worrying exactly about the definition I was trying to provide a sense of what is actually going on, in other words show the definition (whatever it might be) actually being used (you could also look at arxiv papers for a higher level view). Domain and codomain is pretty standard nowadays, I would have thought (that is the way you get the arbitrary sets into the picture without causing too much confusion).I read that some (many?) education authorities (boards? not sure what you call them) have in their guidelines the "old" (new math) type requirements and that there are still even books from that era reprinted today and of course there is no accounting for teachers who don't teach from the text.Selfstudier (talk) 14:20, 23 February 2012 (UTC)

To reiterate that point that seems to be lost on some editors: reliable sources refer to functions (that's all functions, including choice ones) as "rule". Namely, the rule $x\mapsto f(x)$ In their own private platonist world there may be no room for such things, but in the literature there is, as well as for half a dozen editors who have expressed themselves on this page. Tkuvho (talk) 14:36, 23 February 2012 (UTC)

And that's why there is the statement saying it isn't completely correct and showing why complete with citations. There is no citations showing the definition in the article is wrong. Personally I think there is quite enough about the problem in the article, we don't need yet more bits saying lots of people saying a function is a rule and other people saying that is wrong. Dmcq (talk) 14:46, 23 February 2012 (UTC)

Tkuvho I suppose you could couch the argument as platonist/formalist versus the education authorities but I am not sure that it helps any; what ought to be clear by now is that there is something about a function definition that seems to require a large amount of subsequent explanation for effect and that regardless, the subject, as in the past, continues to cause confusion in the minds of students. (You can find a similar and somewhat related debate in respect of the epsilon delta definition (limit of a function))Selfstudier (talk) 14:47, 23 February 2012 (UTC)

I agree with that. I think the issue is precisely whether the writing of the article should be influenced by what educators say about the effectiveness of this or that approach, rather than derivations "from first principles" (by wiki editors) of what should be the most accomplished platonic or formal definition. Tkuvho (talk) 14:52, 23 February 2012 (UTC)

Well, in an encyclopedia, provided we abandon the notion that an entry has to be short, I think we ought to be able to have our cake and eat it; I don't know who it is that actually visits this page or even which countries they might be from, probably there are a lot of students but that's just a guess on my part. I would be surprised if upper level undergraduates and post graduates were coming here (for the definition) but who knows? And I guess educators of one sort or another might well want to make use of some of the material hereSelfstudier (talk) 15:04, 23 February 2012 (UTC)

Certainly the page should not be geared toward educators, but rather toward those being educated. wiki guidelines indicate that a page should be accessible to as wide an audience as possible. In the case of a math page this needs to be suitably interpreted but certainly our guide should be comprehensibility rather than comprehensiveness, as well as building upon existing intuitions rather than ignoring them for the sake of a platonic ideal. Tkuvho (talk) 15:22, 23 February 2012 (UTC)

This one is amusing, having given the high level definition, the lecturer (in public lecture notes) goes on to say:

"It is probably safe to say that most people do not think of functions as a type of relation which is a subset of the Cartesian product of two sets. A function is like a machine which takes inputs, x and makes them into a unique output, f (x). Of course, that is what the above definition says with more precision. An ordered pair,(x, y) which is an element of the function or mapping has an input, x and a unique output, y,denoted as f (x) while the name of the function is f. “mapping” is often a noun meaning function. However, it also is a verb as in “f is mapping A to B”. That which a function is thought of as doing is also referred to using the word “maps” as in: f maps X to Y . However, a set of functions may be called a set of maps so this word might also be used as the plural of a noun. There is no help for it. You just have to suffer with this nonsense." Selfstudier (talk) 15:45, 23 February 2012 (UTC)

That's funny. Who is this? Now I am not exactly sure which nonsense he is referring to. Is it perhaps the idea that a function is an (immutable, platonist) relation or better correspondence, and don't you dare think that is it is given by a black box that's actually doing something or, perish the thought, moving somewhere? Tkuvho (talk) 15:52, 23 February 2012 (UTC)

I'll just give you the link to the notes and let you sort it out.

I quite like the black box/function machine idea, at any rate it ought to be helpful if you are familiar with some basic computing/programmingSelfstudier (talk) 16:00, 23 February 2012 (UTC)

The phrase "Of course, that is what the above definition says with more precision" or some variation of it, seems to pop up quite frequently and you have to wonder why, if it is so precise, it needs to be explained that it is and why the definition is often followed by a sometimes lengthy commentary...Selfstudier (talk) 16:12, 23 February 2012 (UTC)

Von Neumann

There is something wrong in the sub-section on Von Neumann's set theory, and I'm not sure how to fix it. "I-objects and I-objects" ??? Rick Norwood (talk) 17:16, 21 February 2012 (UTC)

Fixed with this edit it seems. NewbyG ( talk) 21:30, 21 February 2012 (UTC)

Yes, you are correct. The direct quote, italics in the original, is as follows: "We are concerned with I-objects, II-objects, the two distinct objects A and B, and the two operations [x, y] and (x, y). The following axioms hold." [etc, etc, cf van Heijenoort page 398]. Bill Wvbailey (talk) 00:30, 22 February 2012 (UTC)

Thanks. Rick Norwood (talk) 13:39, 22 February 2012 (UTC)

Dedekind's notion of "function"

(Notice the * footnote in the following, and the world "law"). Dedekind has become a favorite of mine, in particular his 1887 Nature and Meaning of Numbers, Dover Publications, Inc, NY, ISBN 0-486-21010-3, also this is in the public domain at http://books.google.com/books?id=PywPAAAAIAAJ&printsec=frontcover&dq=Dedekind+The+Nature+and+Meaning+of+Numbers&hl=en&sa=X&ei=owdFT_HjHaTx0gG-0Nn2Aw&ved=0CDYQ6AEwAA#v=onepage&q=Dedekind%20The%20Nature%20and%20Meaning%20of%20Numbers&f=false .(I mark up my books so I paid $8.95 for the facsimile). Here is his definition in II. TRANSFORMATION OF A SYSTEM:

"21. Definition.* [*See Dirichlet's Vorlesungen uber Zahlentheorie, 3rd edition, 1879, § 163.] By a transformation [Abbildung] φ of a system S we understand a law according to which to every determinate element s of S there belongs a determinate thing which is called the transform of s and denoted by φ(s); we say also that φ(s) corresponds to the element s; that φ(s) results or is produced from s by the transformation φ, that s is transformed into φ(s) by the transformation φ" [etc. What follows is a difficult further elucidation that relies upon his notions developed earlier about T being a part of S, etc] (Page 50 in the Dover edition facsimile of the Open Court Publishing Company's edition of 1901)

This warrants more research. Bill Wvbailey (talk) 15:40, 22 February 2012 (UTC)

Interesting. With Dirichlet Vorlesungen_über_ZahlentheorieSelfstudier (talk) 16:25, 22 February 2012 (UTC)

RE Dirichlet and Dedekind -- There's a translation into English, but damned if it doesn't end at §144; the section referenced by Dedekind is from the 3rd edition at §163! Translator John Stillwell must have used the 1st or 2nd edition (ouch! cf the wiki article Vorlesungen_über_Zahlentheorie there is an issue here). Bill Wvbailey (talk) 23:06, 22 February 2012 (UTC)

---

For only $256 (marked down from $320) this book can be yours -- it contains a wealth of historical analysis by various authors:

I. Grattan-Guiness editor, 2005, Landmark Writings in Western Mathematics, Esevier B.V., Amsterdam, The Netherlands, ISBN: 0-444-50871-6.

I bumped into this in Chapter 20 re S. F. Lacroix [1797-1800]:

"Any quantity the value of which depends on one or more quantities is said to be a function of these latter, whether or not it is known which operations are necessary to go from them to the former (p.1). But the example given for a function in which the necessary operations are not known is the root of a fifth degree equation. In fact this definition means that a function had to be defined explicitly or implicitly by using the usual mathematical operations. As in Euler ("§ 14.2) it is assumed that any function has a power-series expansion" [commentatary by Joao Caramalho Domingus, p. 281]

Unfortunately most of what I can access with the limited views available to me is the word "function" in the context of analysis. Which does add a dimension to this discussion -- that the notion from analysis may be somewhat different than that from abstract mathematics. Bill Wvbailey (talk) 16:33, 22 February 2012 (UTC)

Yes, I think that's right, analytically (as in calculus) mostly function means (or at least meant) formula, rate of change, a dynamic computational pov whereas algebra has more or less been a process of ever increasing abstraction and generalization (just look at geometry from Klein onwards).Selfstudier (talk) 17:58, 22 February 2012 (UTC)

Fourier thought that every function could be given by a Fourier series. Kroniker thought that the idea that some sets were uncountable was nonsense. But for the last hundred years, barring a few people who reject the Axiom of Choice, the issue is pretty much settled. If someone wants to add the views of finatary logicians to the article, I have no objections.Rick Norwood (talk) 22:10, 22 February 2012 (UTC)

Not to worry. This has to do more with the history:

RE the odd definition of Lacroix: When I read it I thought of the choice-function issue: we don't know how to do it, but we're certain there's a "choice" that satisfies the equation/formula/specification. E.g. random trial and error produces the assignment. It's fascinating that this problem appeared even then. BillWvbailey (talk) 23:06, 22 February 2012 (UTC)

Yup, mathematicians were happily using choice (mostly without realising it) well before it dawned on Zermelo and co that that was what they were doing and therefore requiring drastic action to avoid more embarrassments and what better way than just to stick on another axiom...:-)Selfstudier (talk) 23:43, 22 February 2012 (UTC)

@User:Wvbailey: Thanks for all the historical quotes. This is fascinating material. To be sure, when Lacroix referred to "implicit rule" he wasn't referring to the same "randomness" as that associated with the axiom of choice. After all, the roots of a polynomial are determined perfectly well in ZF without C. However, it is not likely to get very far with Lacroix's view because of a phenomenon Bloor referred to as "the knife-edge insistence that the only legitimate mathematics are the mathematics we are currently doing". Tkuvho (talk) 10:12, 23 February 2012 (UTC)

I hadn't heard that from Bloor, spot on I think; mathematics is the only subject that strives for this impression; after all, if you have rigour and everything is proved, then everything must be OK, no? Unfortunately it's not that simple...Selfstudier (talk) 10:31, 23 February 2012 (UTC)

That's David Bloor I believe, though I don't vouch the the exact wording. Tkuvho (talk) 11:59, 23 February 2012 (UTC)

David Bloor, Knowledge and social imagery, page 129: "This is the knife-edge insistence that a style of thinking only deserves to be called mathematics in so far as it approximates to our own." Tkuvho (talk) 12:22, 23 February 2012 (UTC)

There is no randomness in the Axiom of Choice. And while there are plenty of cynical quotes, some by major mathematicians, this subject is no longer being debated in the mathematical community. To say that they just tossed in the Axiom of Choice because it avoided the issue is like saying physics just tossed in the Hiesenberg uncertainty principle because it avoided the issue. Some people do say that, but a Wikipedia article on elementary physics is not the venue to argue the case. Rick Norwood (talk) 13:20, 23 February 2012 (UTC)

I am not sure what you are referring to. Who is tossing out the axiom of choice? I think what Bloor is referring to is a certain Platonist attitude that makes mathematicians think that, for example, there is a notion of a function independent of our intuitions and the knife-edge insistence that this is what we are going to teach to students no matter what. Tkuvho (talk) 13:52, 23 February 2012 (UTC)

You repeatedly suggest that I'm making this all up, or insisting on an unreasonably abstract notion of function, when in fact everyone who teaches upper division math courses knows and understands everything discussed on this page. There are a few mathematicians (there were more in the 1920s) who reject the Axiom of Choice. But all mathematicians, whether they accept the Axiom of Choice or not, know that one implication of that axiom is the existance of functions that cannot be defined by a rule. There is nothing new to be said on the subject, at least not on the pages of Wikipedia. If someone had something new to say on the subject it would rate publication in a journal article. Rick Norwood (talk) 16:04, 23 February 2012 (UTC)

Absolutely; and when they get around to adding further axioms for large cardinals and who knows what else (do you follow FOM?)the only thing that's sure is that I will not be alone in rejecting them...Selfstudier (talk) 16:33, 23 February 2012 (UTC)

Dirichlet's notion of a function.

I hope this subject is now settled, but just in case, here is a quote from the article "Functions" in The Encyclopedic Dictionary of Mathematics. I don't cite it in the article because Wikipedia preferences secondary sources, which we already have, to tertiary sources.

"Dirichlet considered a function of x in [a,b] in his paper (1837)concerning representations of 'completely arbitrary functions' and stated that there was no need for the relation beween y and x to be given by the same law throughout an interval, nor was it necessary that the function be given by mathematical formulas. A function was simply a correspondence in which the values of one variable determined values of another."

"Today, the word 'function' is used generally in mathematics in the same sense as a mapping or, which is the same thing, a univalent correspondence."

Rick Norwood (talk) 13:39, 23 February 2012 (UTC)

Have you not been following the discussion? this has already been dealt with above...Selfstudier (talk) 13:48, 23 February 2012 (UTC)

The point is that this standard reference work cites the Dirichlet definition, and twice uses the word "correspondence". Rick Norwood (talk) 16:06, 23 February 2012 (UTC)

Correspondence

Right, if we follow the link we arrive at a pageful of differing usages, the first being "In general mathematics, correspondence is an alternative term for a relation between 2 sets" and if you then follow relation we get "a relation as defined by the triple (X,Y,G) is sometimes referred to as a correspondence instead".

One can only assume that the purpose of having the word correspondence (used only by set theorists in the context we are talking about) and the link chain is so that the casual reader will go these pages and having read them will fall asleep and forget why he came to the function page in the first place.

If we mean relation,why don't we just say so instead of defaulting to correspondence? Selfstudier (talk) 23:53, 23 February 2012 (UTC)

Because it's important to be precise here. A function may be defined by a triple (X,Y,G), where G is a subset of X × Y. But a relation is usually defined simply as a subset of X × Y, without the extra specification of what X and Y are. So if we say "a function is a relation" rather than "a function is a correspondence" then we lose track of what Y is, and we lose the ability to distinguish surjective functions from other functions. —David Eppstein (talk) 23:57, 23 February 2012 (UTC)

Sorry, don't follow, are you saying the linked Wikipedia pages are wrong?~~ — Preceding unsigned comment added by Selfstudier (talk • contribs) 00:11, 24 February 2012 (UTC)

If we wish to use relation in the lead, what do you say that we need to put in order to do that? Putting correspondence is just a link chasing exercise of no value. I have already said that I would prefer neither in the lead but if I must suffer one of them, then I would prefer relation which is the word customarily used.Selfstudier (talk) 00:24, 24 February 2012 (UTC)

Why not put relation and link it to the Definition section? — Preceding unsigned comment added by Selfstudier (talk • contribs) 00:54, 24 February 2012 (UTC)

The disambiguation page correspondence (mathematics) reads: "In general mathematics, correspondence is an alternative term for a relation between two sets. Hence a correspondence of sets X and Y is any subset of the Cartesian product X×Y of the sets." This matches the definition of a relation simply as a set of ordered pairs. Shouldn't it rather read something like: "In general mathematics, a correspondence from X to Y is an ordered triple (X,Y,R), where R is a relation from X to Y, i.e. any subset of the Cartesian product X×Y." Isheden (talk) 09:43, 24 February 2012 (UTC)

"If we mean relation, why don't we just say so instead of defaulting to correspondence?" Because "relation" has little meaning to a naive reader, and "correspondence" has pretty much the right meaning - an association between the inputs and outputs. There's no need to chase links. (There is a separate issue about the domain and codomain, which David Eppstein was bringing up.) If we were not trying to make the lede accessible, we would indeed just say "relation", because that's what a function is, modulo info about the domain and codomain. — Carl (CBM · talk) 00:59, 24 February 2012 (UTC)

Relations have a similar problem to functions in their definitions. Some people think a relation is just a set of ordered pairs. If you look at the section binary relation#Is a relation more than its graph? you'll see a bit more about this. This is one of the problems with Wikipedia being an encyclopaedia, we've got to represent all the major points of view and can't have an integrated viewpoint between articles. If you use correspondence for the triple and only refer to a relation as in a relation from X to Y then there's no ambiguity. Dmcq (talk) 01:15, 24 February 2012 (UTC)

I don't see how one can say that correspondence gives the right meaning to a naive reader, once you follow the link and assuming you are naive, I think you just give up....Waffling on about how it gives the right impression is just wishful thinking, if anything it gives the wrong impression altogether and on looking back in here I see others have said something similar.Selfstudier (talk) 09:59, 24 February 2012 (UTC)

Dmcq Relation from X to Y is absolutely fine with me (and most mathematicians), the other language is used in a narrow context which is explained (at length) further down the page (as well as on the other page by "A special case of this difference in points of view applies to the notion of function....")..duh!Selfstudier (talk) 09:59, 24 February 2012 (UTC)

This pov is (completely) supported by the Bloch reference (avoiding the cumbersome triple etc), do we need more citations..? Selfstudier (talk) 10:25, 24 February 2012 (UTC)

The text follows Bloch fairly closely in distinguishing between "a function" and "a function from A to B". The book considered it important to make the distinction and then talks about the use of function on its own as meaning "a function from A to B" where the domain and codomain are understood from the context. If we were to put that into the lead we'd probably have to mention the domain and codomain much earlier, perhaps we could just say from one set to another and later give names to these sets. Note Bloch says it will not suffice to write only "let f be a function" without specifying the domain and codomain, unless the domain and codomain are known from the context. We've done practically exactly what he has said not to do in he interest of trying to simplify the lead and have made it all right by using the word correspondence. Dmcq (talk) 11:33, 24 February 2012 (UTC)

The lead does not need to be simplified it requires to be a summary of the article content as you have previously stated and I don't agree that you have "made it all right" by introducing a set theoretic term with a complex technical meaning in relation to functions.Selfstudier (talk) 11:48, 24 February 2012 (UTC)

"Correspondence" is not a set theoretic term to most people; the natural language meaning is sufficiently close to the mathematical meaning that a naive reader doesn't need to look it up to read the first paragraph. That is the main benefit of "correspondence" over "relation". — Carl (CBM · talk) 12:15, 24 February 2012 (UTC)

This is like saying that rule and relation are not mathematical terms; apart from that, the purpose of links is so that you follow them (this is not the same as "looking it up" and if you follow them the result should be clarification not confusion). I'm beginning to have the thought we should have a page function (set theory) as well as the current page function (mathematics) so that we may clearly identify points of difference. Defining mathematical notions via set theory is not adding anything mathematical to the process merely demonstrating that you don't theoretically need anything else beyond sets for a mathematical description.Selfstudier (talk) 12:27, 24 February 2012 (UTC)

Of course "relation" is a mathematical term. And the set theoretic definition is the mathematical one, there is not a difference apart from the way that the domain and the codomain are specified. Mathematical functions, like all other objects in mainstream mathematics in the last 50 years , are defined using the basic terms of set theory (set, class, pair, member, etc.). In particular, the technical definition of a function in mathematics is simply that it is a set of ordered pairs so that if (x,y) and (x,z) are in the set then y=z, along with information about the domain and the codomain according to the taste of the author. I'm sure everyone in the discussion already realizes this. The question is the best way to express this in the lede; the question is not what the definition actually is. — Carl (CBM · talk) 12:42, 24 February 2012 (UTC)

Here you go doing it again, when you want to emphasize the technical side, you do that but then when we talk about the lead you start saying that it isn't technical and that it doesn't matter that there IS a difference. I give up.Selfstudier (talk) 12:59, 24 February 2012 (UTC)

For the lede, I think it is better to avoid the word "relation" and I think that "correspondence" is better, because the naive meaning matches the mathematical one. But the definition that we allude to in the first paragraph should be exactly the definition that I gave in my previous comment. The current first sentence, "In mathematics, a function is a correspondence that associates each input with exactly one output.", seems to do that in my opinion. — Carl (CBM · talk) 13:02, 24 February 2012 (UTC)

I think we might be better off taking off the link from correspondence in the lead. The English meaning is close enough. If it was just linked in the definition section that would be plenty good enough. People can get confused by following unnecessary links about precise meaning when something straightforward is okay, is there some article about people doing this and ending up somewhere completely different after two hours having forgotten about and not getting past the first line of what they were looking up? Dmcq(talk) 13:25, 24 February 2012 (UTC)

I think that's ok, but the problem Selfstudier pointed out in the beginning of the discussion is that the information given in the linked page is confusing. Let me raise again the proposal that the definition on correspondence (mathematics) should be changed to something like "In general mathematics, a correspondence from X to Y is an ordered triple (X,Y,R), where R is a relation from X to Y, i.e. any subset of the Cartesian product X×Y." Isheden (talk) 13:49, 24 February 2012 (UTC)

Removing the link is fine with me; the reason for using the word in the first place is that most people can just use the normal meaning for it and come away with the right idea. — Carl (CBM · talk) 14:15, 24 February 2012 (UTC)

Okay done and I've updated correspondence (mathematics) as well. Dmcq (talk) 14:22, 24 February 2012 (UTC)

Thanks! One more thing: The present "precise" definition actually defines only a general correspondence in the first sentence, deferring the property many-to-one to the sentence after. Since I don't have access to Bloch, I don't know how this issue is dealt with there, but I think this property should be stated in the first sentence. Isheden (talk) 14:54, 24 February 2012 (UTC)

Lead as Summary of Article

OK, so now we have 3 possibilities:

1)Rule etc (supported by cites but accepted as being at lower level (up to mid-level university, let's say)

2)Relation (high level, supported by cites, standard mathematical usage, requires some qualification eg "special type of")

3)Correspondence (high level, cites?, set theoretic usage, requires convoluted explanations)

Things like association, relationship?, appear to have no support I assume because they are not considered "accurate" enough.

It seems to me that if we can't agree on one word, then we have to put something that includes all three of the above possibilities? — Preceding unsigned comment added by Selfstudier (talk • contribs) 11:40, 24 February 2012 (UTC)

What exactly is the concern about the current language of the first paragraph, which does include both "correspondence" and "rule"? — Carl (CBM · talk) 12:13, 24 February 2012 (UTC)

I have explained that already....Selfstudier (talk) 12:18, 24 February 2012 (UTC)

The long, long sections above are somewhat hard to follow. A one or two sentence explanation of exactly what your concerns are might help refocus the discussion, and it is necessary before I can try to respond to them. My reading of the sections above is that a clear opinion against "a function is a rule" developed by Feb 19, and then the discussion fragmented and began to include things unrelated to the lede sentence. At this point it is in TLDR territory. But simultaneously we arrived at some compromise language that does use the word "rule" to show how a rule can define a function. — Carl (CBM · talk) 12:33, 24 February 2012 (UTC)

Leave the first paragraph as it is now. It is fine. Gandalf61 (talk) 12:51, 24 February 2012 (UTC)

I don't know if time is ripe to begin this discussion again. In my view the introduction should be reworked first, starting out with an intuitive view of what a function is and then describing why the intuitive view of a machine or rule is not quite satisfactory to a mathematician, if possible without discussing any pathological examples. However, if we are to decide on the lead, I think the first four paragraphs of this old revision [6] could serve as a good example. A separate question in my view is whether there should be a first sentence with a concise definition of what a function is or not. Regarding this question I would say we have discussed enough and it's time to suggest a complete first sentence. Isheden (talk) 13:40, 24 February 2012 (UTC)

In case anyone wants to change the first sentence, that is... Isheden (talk) 16:13, 24 February 2012 (UTC)

leave the first paragraph as it is now. It is fine. Rick Norwood (talk) 16:00, 24 February 2012 (UTC)

Blunt, honest opinion: It's weasily. Come on: a static object with input and output? Nah. But not bad enough to shoot, meaning the first two sentences are okay. If it were left up to me I'd get the rule/example out of there (move it down to next sentence). In its place I'd add something to the lead about the old-fashioned notion of functional dependance as in "Thus, the output f(x) is said to be a function of (dependent upon) the input x. Often the output f(x) is assigned a variable-symbol such as y, and x is called the "independent variable"; [right up front I want the notion of "implication" in there i.e. x --> y [cf the Suppes formal definition, but in newbie words]]. Then, in the lead or shortly thereafter I'd also be totally in-yer-face up front about the historicity (I looked that up, it's a real word) of functions being specified by "rules" and "laws", but now in replaced in with the abstract modern usage of an object formed by association [etc]. Bill Wvbailey (talk) 17:14, 24 February 2012 (UTC)

How about after the first sentence a little explanation as in "The output of a function is completely dependent on its input, each input determines a corresponding output'. Dmcq (talk) 18:14, 24 February 2012 (UTC)

Yes, I like it. Below I added the notion of the input value being independent of the output value. Certainly in table lookups this is true, but maybe there are subtle issues around this. I'd also add the Tarski, notice the reversal of the words "input" and "output" in the first sentence to parallel Dmcq's sentence. The many-one correspondence could be removed and the original (now struck) reintroduced:

In mathematics, a function[1] also known as a functional relation[2] or many-one correspondence[3] associates exactly one output to each input. While the output of a function is completely dependent on its input, each input is independent of and determines its corresponding output. ¶

In mathematics, a function[1] also known as a functional relation[2] is a correspondence that associates exactly one output to each input. While the output of a function is completely dependent on its input, each input is independent of and determines its corresponding output. ¶

The output of a function f with input x is denoted f(x) (read "f of x"). [etc]

[2] is the reference from Tarski 1946:32 [Dover 9th printing 1995]. [3]Observes (i) that the correspondence is dyadic [two-placed] and (ii) that in old texts functions are sometimes called "one-many" and reverse the order of the relation [Tarski claims that this is true in relations theory, but that was in 1946].

Tarski writes the lead sentence twice with slightly different wording (except I've reversed the x* and y* to confrom to contemporary usage re note [3] directly above) (all from page 99):

"The function f assigns (or correlates) the value y* to the argument value x*

"or

"y* is that value of the function f which corresponds to (or is correlated with) the argument value x*"

So if we were to use the first of these together with the notions of input and output, we'd have an active-voice definition:

In mathematics, a function[1] also known as a functional relation[2] or many-one correspondence[3] assigns (associates) exactly one output to each input. While the output of a function is completely dependent on its input, each input is independent of and determines its corresponding output. ¶

Passive-voice version:

In mathematics, a function[1] also known as a functional relation[2] or many-one correspondence[3] is an assignment (association) of exactly one output to each input. While the output of a function is completely dependent on its input, each input is independent of and determines its corresponding output. ¶

BillWvbailey (talk) 21:04, 24 February 2012 (UTC)

I think I like the version "In mathematics, a function[1] also known as a functional relation[2] is a correspondence that associates exactly one output to each input. While the output of a function is completely dependent on its input, each input is independent of and determines its corresponding output" best and it leaves out the one-many/many-one business. Dmcq (talk) 21:15, 24 February 2012 (UTC)

Agreed. Yeah, the one-many/many-one business is a can of worms. It's interesting, isn't it that besides "corresponds" Tarski also used the word "correlates" and "assigns". Let's see if anyone else weighs in, e.g. should anything more be added to the lead sentence or is this enough to get the ball rolling? BillWvbailey (talk) 21:36, 24 February 2012 (UTC)

Two definitions mixed up

In the first definition of a function in the definitions section the first definition is fairly close to the triple definition in the second paragraph and the second one is the same as the definition in the third paragraph. So it has mixed up citations. I'll take out 'in set theory especially' in that third one as it is obvious there are a number of people elsewhere who do it.

I think the first and third paragraphs should be merged and the first citation moved to the other definition. Possibly the expression 'a function from X to Y' should be separated out and then the first definition can be put with that. Should 'a function from X to Y' go first or after the two 'function' definitions? Dmcq (talk) 01:46, 25 February 2012 (UTC)

I have read Bloch's function definition now and I think it's good. A proper definition of a function includes the domain and codomain (unless known from the context) together with the set of ordered pairs, so all definitions are basically equivalent. Regarding the order, I think Bloch's exposition can serve as an example. It might be better to mention the words relation and correspondence only after the function definitions, since they are more general. I don't know exactly how you'd like to restructure, but please go ahead with it. Isheden (talk) 10:37, 25 February 2012 (UTC)

Okay done so. I better go off and do something in the real world now ;-) Dmcq (talk) 14:40, 25 February 2012 (UTC)

[Edit conflict:] As Isheden notes please feel free to do what you think is best. My very first reaction to both the first (Intro) and second sections (Defintion) is that they have a lot of nice information in them, but they're intimidating. They're long (too long IMHO) and/or they lack visual structure. My personal preference is bulleting with bold-face summary such as ● Contemporary terminology, ● Function specified by a rule, ● Set-theoretic definition, ● Multiple processes may create the same function (or whatever) etc etc or the equivalent that indicates "now we're going to talk about xxx", even if xxx is only a sentence or two. If I were going to tackle this (big job) I'd sandbox it to find the sections' structures first and proceed from there. BillWvbailey (talk) 14:57, 25 February 2012 (UTC)

History stuff

Bourbaki definition

This is derived from Andrew, Katz, Wilson 2009 Who Gave you the Epsilon p. 25 written by Israel Kleiner, which in turn is derived from [3] U. Bottazzini, 1986, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass, Springer-Verlag,:

"In 1939, Bourbaki gave the following definition of a function ([3], p. 7):

"Let E and F be two sets, which may or may not be distinct. A relation between a variable element x of E and a variable element y of F is called a functional relation in y if, for all x ∈ E, there exists a unique y ∈ F which is in the given relation with x.

"We give the name of function to the operation which in this way associates with every element x ∈ E the element y ∈ F which is in the given relation with x, and the function is said to be determined by the given functional relation. Two equivalent functional relations determine the same function." (p. 25)

"Bourbaki then also gave the definition of a function as a certain subset of the cartesian product E x F. This is, of course, the definition of functions as a set of ordered pairs. ¶ All of these 'modern' general definitions of function were given in terms of sets, and hence their logic must receive the same scrutiny as that of set theory" (p. 25)

translations of Dedekind definition

In the Dover edition facsimile of Beman's translation, as published in the Open Court Publishing Company's edition of 1901, the "Authorized Translation". Here is his definition in II. TRANSFORMATION OF A SYSTEM:

"21. Definition.* [*See Dirichlet's Vorlesungen uber Zahlentheorie, 3rd edition, 1879, § 163.] By a transformation [Abbildung] φ of a system S we understand a law according to which to every determinate element s of S there belongs a determinate thing which is called the transform of s and denoted by φ(s); we say also that φ(s) corresponds to the element s; that φ(s) results or is produced from s by the transformation φ, that s is transformed into φ(s) by the transformation φ" [etc. What follows is a difficult further elucidation that relies upon his notions developed earlier about T being a part of S, etc] (page 50)

Here is the translation on page 18 of Andrew, Katz, Wilson 2009 Who Gave you the Epsilon p. 25 written by Israel Kleiner:

"As early as 1887, Dedekind gave a fairly “modern” definition of the term “mapping” [23, p. 75]:

"By a mapping of a system S a law is understood, in accordance with which to each determinate element s of S there is associated a determinate object, which is called the image of s and is denoted by φ(s); we say too, that φ(s) corresponds to the element s, that φ(s) is caused or generated by the mapping φ out of s, that s is transformed by the mapping φ into φ(s)." (derived from ref. 23, p. 75: D. Rüthing. "Some Definitions of the Concept of Function from Joh. Bernoulli to N. Bourbaki," Math. Intelligencer 6:4 (1984) 72-77.)

The word "Abbildung" means, generally, a figure as in an illustration, image, chart, graph, picture in a text, also a 2-dimensional representation e.g. map, as in a road map. As to what it meant to a German mathematician in the late 1800's? My supposition is it would be "transformation" as Beman wrote, rather than "mapping" as used in the contemporary set-theoretic sense, but that doesn't mean that e.g. Zermelo et. al. didn't adopt the word "mapping". As I'd have to pay $34 for a pay-to-view of this, or march up to the library on the hill to discover where Rüthing derived his translation, this will have to wait.

Dirichlet's definition

This translation is suspect. No date as to when published. It comes from German to Russian to English. Also derived from Andrew, Katz, Wilson 2009 Who Gave you the Epsilon p. 20 written by Israel Kleiner

y is a function of a variable x, defined on the interval a < x < b, if to every value of the variable ‘’x’’ in this interval there corresponds a definite value of the variable y’’. Also, it is irrelevant in whatway this correspondence is established [19]” (Klein 1989:10, derived from 19. N. Luzin, “Function” (in Russian), The Great Soviet Encyclopedia, v. 59 (ca. 1940), pp. 314-334 ).

Bill Wvbailey (talk) 14:57, 25 February 2012 (UTC)