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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 [1] 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)

Just acknowledge the correspondence-rule controversy?

How about a sentence in the lead—but not starting the lead—of the form, "Functions are often thought of as rules associating elements of their domains to elements of their codomains, but many mathematicians reject this intuition on technical grounds"? Heck, we could even make "technical grounds" hot, linking to a discussion below.—PaulTanenbaum (talk) 18:45, 7 March 2012 (UTC)

Domain and co-demain are not introduced till very near the end of the lead, in the third paragraph. Formal definitions of functions give the set of inputs (the domain), the set of paired inputs and outputs, and another set known as the codomain in which the outputs are constrained to fall. NewbyG ( talk) 19:16, 7 March 2012 (UTC)

OK, OK, I proposed something "of the form." So maybe like, "Functions are often thought of as rules associating elements of one set with elements of another (or possibly the same) set..."—PaulTanenbaum (talk) 22:08, 7 March 2012 (UTC)

Rule is mentioned in a correct form in the first paragraph a couple of times and there's a bit saying what you say in the Introduction section. Do we really need to labour the wrongness point? Dmcq (talk) 22:24, 7 March 2012 (UTC)

Heavens no! I wasn't suggesting a further discussion of the whole "rule" question than what's in the lead. Instead, a candidate replacement that might satisfy the "rule" proponents without overly incensing the "correspondence" crowd.—PaulTanenbaum (talk) 22:29, 7 March 2012 (UTC)

...and the "bit in the introduction" that Dmcq refers to is exactly the "discussion below" to which I envision "technical grounds" linking to.—PaulTanenbaum (talk) 22:32, 7 March 2012 (UTC)

I just don't see why we should start saying things like 'are often thought of' in the lead if we don't have to. Rule is used in ways that gives the right idea and it avoids saying anything like rule is wrong in the lead. It is better to be positive rather than wishy washy I think. 23:09, 7 March 2012 (UTC)

There is not actually a controversy, apart from on this talk page. Mathematicians are perfectly fine with what a function is, and they don't spend time debating it. The only difficulty is that when they do say "rule" they don't mean it in the same sense that a naive reader would read it. The reader would think of the "rule" as something like x² while a mathematician would think of the "rule" as an arbitrary correspondence, in other words just a synonym for function.

Separately, if we did put "are often thought of" in the article, I am afraid the "weasel word" police may show up and complain about it. But the first paragraph does include the word "rule" already, it is not as if this word has been expunged from the article. — Carl (CBM · talk) 01:02, 8 March 2012 (UTC)

I agree with Carl, it is not a controversy. The problem with defining a function as a rule is simply that the meaning of "rule" is not clearly defined, so this definition, although intuitively appealing, is not logically precise. I intend to revise the introduction to address this issue. After that, we can work on the lead so that it reflects the main ideas. Isheden (talk) 08:28, 8 March 2012 (UTC)

To elaborate on Paul's idea, we could We don't ave to include the whole article in the lead. Dmcq (talk) 23:12, 8 March 2012 (UTC)mention in the introduction that functions have been alternatively defined as rules, relations, or correspendences. All three are richly documented in the literature. Surely we are not in a position to take it upon ourselves to rule (excuse the pun) on the correctness of either one of the terms commonly found in the literature. Tkuvho (talk) 20:18, 8 March 2012 (UTC)

And then do we say that other people say that is wrong or do we just leave it as okay? I think it is better the way it is. Dmcq (talk) 23:12, 8 March 2012 (UTC)

My point is precisely that "rule" is not incorrect. In fact, it is more correct than "correspondence", which is problematic on at least two counts: (1) it implies a 1-1 relation; (2) it obfuscates the crucial issue of the asymmetry between domain and range. Tkuvho (talk) 21:20, 10 March 2012 (UTC)

A correspondence does not imply one to one. Why do you think the one to one is always put in when one to one is meant? I think the 'input' and 'output' distinguishes the values well enough. Dmcq (talk) 23:25, 10 March 2012 (UTC)

No matter how a function is defined, it comprises three things: 1) a set of inputs, 2) a set of outputs, and 3) a way of associating each input with a unique output. Intuitively, the third component is a rule that assigns a unique output to each input. Mathematicians actually think of this triple as a (left-total and right-unique) correspondence, but this is by no means intuitive to anyone else. I think there is no way around mentioning sets in the lead paragraph. Therefore, I'd like to propose the following lead paragraph:

In mathematics, a function is a correspondence from a set of inputs to a set of outputs that associates each input with exactly one output. Intuitively, a function from a set X to a set Y is a rule that assigns to each element x in X a unique element y in Y. The output of a function f for an input x is denoted by f(x) (read "f of x"). For example, a function f from the reals to the reals may be given by the rule f(x) = 2x that associates any real number with the number twice as large; if x = 5 then f(x) = 10. Two different rules define the same function from X to Y if they make the same associations, for example f(x) = 3x−x defines the same function as f(x) = 2x. Isheden (talk) 22:37, 16 March 2012 (UTC)

With all the other mentions of rule it sounds like it really is a rule except that two different rules can define the same function. The sets sounds okay except I think people could easily get the impression that every item in the output has a corresponding input. In fact perhaps a better example could be used to show that's not true, I think well chosen examples can avoid a lot of unnecessarily careful wording in the lead. Dmcq (talk) 23:08, 16 March 2012 (UTC)

To avoid that impression, one could take as example instead $f(x)=x^{2}$ (for which the output is nonnegative) or $f(x)=\sin x$ (for which the output is between -1 and 1). To avoid excessive use of the word "rule", the last sentence could be changed to "Functions with the same sets of inputs and outputs and that make the same associations are considered equal. Thus, the two functions f(x) = 3x−x and f(x) = 2x, both from the reals to the reals, are equal." Possibly it is better to discuss this after domain and codomain has been introduced, though. Isheden (talk) 08:15, 17 March 2012 (UTC)

I think that last sentence could be removed okay, it is covered properly in the simple intro or whatever it is called now. And using $x^{2}$ would cure the problem about the one to one. If we remove the other mention of rule as well and just leave the intuitive sentence you get:

In mathematics, a function is a correspondence from a set of inputs to a set of outputs that associates each input with exactly one output. Intuitively, a function from a set X to a set Y is a rule that assigns to each element x in X a unique element y in Y. The output of a function f for an input x is denoted by f(x) (read "f of x"). For example, the function f from the reals to the reals given by f(x) = x² associates any real number with its square; if x = 5 then f(x) = 25.

I think the statement about the intuitive meaning is a bit long and clunky though. Dmcq (talk) 13:10, 17 March 2012 (UTC)

How about this:

In mathematics, a function is a correspondence from a set of inputs to a set of outputs that associates each input with exactly one output. Intuitively, a function from a set X of inputs to a set Y of outputs is a rule that assigns to each input x a unique output y. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). For example, the function f from the reals to the reals given by f(x) = x² associates any real number with its square; if x = 5 then f(x) = 25. In this case, the output is always nonnegative. Isheden (talk) 13:32, 17 March 2012 (UTC)

Sounds fine to me. Dmcq (talk) 13:39, 17 March 2012 (UTC)

Since no one else commented on this, I assumed there were no objections, but I was wrong. Comments from other editors would be appreciated. Isheden (talk) 15:19, 18 March 2012 (UTC)

New introduction

I have rewritten the introduction in an attempt to clarify some of the issues that have been discussed here for quite a while. I've cut some sentences that for various don't really fit in the introduction, although they might be relevant somewhere else in the article:

• A function can also be called a map or a mapping. Some authors, however, use the terms "function" and "map" to refer to different types of functions. Other specific types of functions include functionals and operators.
• Sometimes, especially in computer science, the term "range" refers to the codomain rather than the image, so care needs to be taken when using the word.
• indeed, in some cases it may be impossible to define such an algorithm. For example, certain choice functions postulated through the Axiom of Choice have no explicit rule.
• A function of two or more variables is considered in formal mathematics as having a domain consisting of ordered pairs or tuples of the argument values. For example Sum(x,y) = x+y operating on integers is the function Sum with a domain consisting of pairs of integers. Sum then has a domain consisting of elements like (3,4), a codomain of integers, and an association between the two that can be described by a set of ordered pairs like ((3,4), 7). Evaluating Sum(3,4) then gives the value 7 associated with the pair (3,4).
• A family of objects indexed by a set is equivalent to a function. For example, the sequence 1, 1/2, 1/3, ..., 1/n, ... can be written as the ordered sequence <1/n> where n is a natural number, or as a function f(n) = 1/n from the set of natural numbers into the set of rational numbers.

Also, perhaps dependent variable and independent variable should be mentioned in the definition. Feel free to continue editing the introduction. Isheden (talk) 23:30, 8 March 2012 (UTC)

References

Can we name this section something else, like "intuition" or something? "Introduction" is a bad name for a Wikipedia article section, because the unnamed first section is supposed to be the introduction (see e.g. Wikipedia:Manual of Style/Lead section). —David Eppstein (talk) 00:11, 9 March 2012 (UTC)

The section has been renamed, although I would have suggested "Intuitive notions" instead of "The language of functions", since terminology is described in more detail in the section Definition. Isheden (talk) 14:14, 9 March 2012 (UTC)

---

RE dependent and independent variable: Above, Dmcq and I agreed on the following wording, but I haven't pushed it because of what I later found in Tarski 1946:

"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".

Tarski 1946 (Dover edition 1995) comes down harshly on the use of these words:

"The functional relation is there ["in many elementary textbooks of algebra"] characterized as a relation between two "variable" quantitites or numbers: the "independent variable" and the "dependent variable", which depend upon each other in so far as a change of the first effects a change of the second. Definitions of this kind should no longer be employed today . . . they are remains of a period in which one tried to distinguish between "constant" and "variable" quantities . . .. He who desires to comply with the requirements of contemporary science and yet does not wish to break away completely from tradition, may, however, retain the old terminology and use, beside the terms "argument value" and "function value", the expressions "value of the independent variable" and "value of the dependent variable" ". (p. 99-100)

I haven't been able to make good sense of this (it's a translation from Polish), but it has stopped me from pushing the proposed wording. (I do like the first sentence, though). Maybe someone can shed some light on what Tarksi's point might be or shed other light on this issue. BillWvbailey (talk) 16:24, 10 March 2012 (UTC)

This is discussed also in Bloch 2011 on p. 133: "It is often mistakenly thought that "f(x)" is the name of the function because x is a "variable," rather than a specific element of the domain. In reality, however, there is no such thing as a variable in a function. (...) Historically, following Descartes, mathematicians have often used letters such as x, y and z to denote "variables," and letters such as a, b and c to denote "constants," but from a rigorous standpoint there is no such distinction." Based on that, it would not make sense to mention dependent and independent variables in the definition. Isheden (talk) 20:56, 10 March 2012 (UTC)

Looking at how 'variable' is use it seems to occur mainly in 'functions of a real variable' or 'of a complex variable' or 'of more than one variable'. Those are very common usages in books. It doesn't seem to appear in actual definitions. I suppose one could put in something about what Bloch says but I can't see actually removing the usages like those as accomplishing anything very useful. Dmcq (talk) 22:43, 10 March 2012 (UTC)

I agree that these usages do not need to be removed, but the question is whether dependent and independent variables should be introduced in the article. It seems to me it is better to stick to argument and value. Isheden (talk) 22:57, 10 March 2012 (UTC)

Fine by me, I'd have tried to do something about that before now if I'd a strong feeling about it! Dmcq (talk) 23:15, 10 March 2012 (UTC)

As a grad student I thought that mentioning Descartes' convention about constants and variables, and making the distinction between dependent and independent variables, was trivial. But since I've been a teacher I've changed my mind. There are countless times where, without it ever being stated explicitly in the book, the student needs to understand that y' means the derivative of the dependent variable with respect to the independent variable, and the independent variable is sometimes x and sometimes t and sometimes something else entirely. They need to understand that if y = x^a then y' = ax^(a-1) but if y = a^x then y' = ln(a) a^x. In differential equations, the theorem about uniqueness of solutions to y' = g(x,y,t) is best understood by remembering that both g and partial derivatives with respect to the dependent variable(s) must be continuous. This really is valuable information. What I've discovered is that mathematics is not just axiom, definition, theorem, and proof, like I thought in grad school. There are countless conventions that are part of the culture of mathematics, without which mathematics becomes incomprehensible. Rick Norwood (talk) 13:03, 11 March 2012 (UTC)

I agree with User:Rick Norwood. One further point: Ethan Bloch's comment quoted above is seriously misguided. The distinction between constants and variables is a fundamental one. Some of the most important foundational frameworks cannot even be formulated without such a distinction. I looked through Bloch's publication record and it confirmed my expectation that he does not seem to have much (if any) background in logic. His comment can be safely ignored. Tkuvho (talk) 17:08, 11 March 2012 (UTC)

But what about Tarski's take on it (re the quote just above)? Actually the "constant" versus "variable" issue may be a red herring. IMHO the concern is more the usage of "independent variable" and "dependent variable". BillWvbailey (talk) 19:03, 11 March 2012 (UTC)

The distinction of dependent and independent variable is obviously a very useful one in explaining functions. We don't have to be afraid of Tarski even if he meant what you think he meant, of which I am not completely certain. Tkuvho (talk) 19:48, 11 March 2012 (UTC)

All recent edits have been well-intentioned, but the results are hard for the layperson to read, so I'm going to take another shot at it. Rick Norwood (talk) 13:00, 18 March 2012 (UTC) The main change I've made is to move all the set theory to the second paragraph. I'm done for now. Thanks, Dmcq, for your help with the formating. Rick Norwood (talk) 13:23, 18 March 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)

Just for clarity, when the educators refer to the Dirichlet-Bourbaki definition what they mean is the combination of the set theoretic Bourbaki (1939) with the Dirichlet "rule of correspondence" (ie the Spivak etc).Selfstudier (talk) 13:58, 20 March 2012 (UTC)

I haven't typed in the Bourbaki definition of "correspondence" yet. It's a ordered triple defined in terms of a "graph" and called gamma [Γ]. [put in wrong place, moved it to below under Bourbaki 1970]

Right, that's the one the educators don't concern themselves with (or maybe don't even know about), the "higher" math version...Selfstudier (talk) 10:26, 21 March 2012 (UTC)

My take on it is that here is where the "modern" definition of function comes from (i.e. an ordered triple). But I don't know the historical thread -- when/where did this ordered-triple business first appear?

I think maybe what we have is 3 cases, the educators version of a modern definition ie the school/uni version (the one that refers to a rule and subject of all the angst on here) and what they call the Dirichlet-Bourbaki definition, a sort of poor man's or watered down triple and I think this became common (in education) by some sort of process following Bourbaki 1939 so baking in the usual glacial progress in education, maybe it's common around early 1950's onwards, say (a guess on my part).

Then we have at least 2 other cases, the "proper" triple (post 1970) and the ordered pair (longer history) versions which are both valid definitions but tend to be used in different areas and are not usually taught at lower levels.Selfstudier (talk) 18:07, 21 March 2012 (UTC)

Ok, here we go, the School Mathematics Study Group in 1960 (same time frame as new math so in fact it took about 20 years from Bourbaki 1939)defined function as follows:-

"Let A and B be sets and let there be a given rule which assigns exactly one member to B to each member of A. The rule, together with the set A, is said to be a function and the set A is said to be its domain. The set of all members of B actually assigned to members of A by the rule is said to be the range of the function."

Selfstudier (talk) 22:48, 21 March 2012 (UTC)

By and large then, the "school/uni" definition hasn't changed that much since then as the above isn't that much different to a couple of current examples given in hereSelfstudier (talk) 22:59, 21 March 2012 (UTC)

Don't know why I haven't looked at it before now, the Wikipedia page for Codomain explicitly refers in its second para of the lead to the triple as the "modern definition" (and the talk pages have an amusing (prior)rehash of our discussions but excluding any discussion of the rule version)Selfstudier (talk) 11:59, 24 March 2012 (UTC)

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)

In order to be able to write up something on the (Dirichlet)Bourbaki, I need to resolve the business about Dirichlet first. He is often mentioned in relation to his 1829 when he produces the Dirichlet function being the one that then led to many other such functions being produced but the function definition is 1937 and I found another version via Towards a Definition of a Function which gives this in French with a translation to English by the authors and credits a French translation of Youshkevitch's 1976 "The concept of function up to the middle of the 19th century" Arch. Hist. Ex. Sci. 16 (1976) 37–85 as follows:

"Let a and b be two fixed numbers and let x be a variable quantity that takes successively all the values between a and b.If for each given x, a unique finite y corresponds to it in the way that, when x moves continuously along the interval between a and b, y=(x) varies progressively too, then y is said to be the continuous function of x over this interval. For this, it is not obligatory at all either that y, on all over the interval, would depend on x according to the same unique law, or that it would be represented by a relation expressed with the help of some mathematical operations"

which appears to be a somewhat lengthier version of the Kleiner/Luzin version.

Lakatos gives Hankel as the source of the "Dirichlet definition" although Steiner notes (p 12):

"The malaise in the understanding and use of the function concept around this time can be gathered from the following account by Hankel (in 1870) concerning the "function concept as it appears in the 'better textbooks of analysis' (Hankel’s phrase): One [text] defines function in the Eulerian manner; the other that y should change with x according to a rule,without explaining this mysterious concept; the third defines them as Dirichlet; the fourth does not define them at all; but everyone draws from them conclusions that are not contained therein""

For better for worse, true or not and whether or not disputed by Lakatos, nearly everybody else seems to accept the above as the first "modern"(ish) definition of function and Dirichlet somehow gets the credit (in mathematics misattribution is more the norm than the exception as far as I can see :-)Selfstudier (talk) 11:14, 28 March 2012 (UTC)

Dirichlet is a translation of the German pagesSelfstudier (talk) 12:12, 28 March 2012 (UTC)

That account by Hankel is really very funny, thanks very much :-) No wonder we still have problems with it. Dmcq (talk) 12:38, 28 March 2012 (UTC)

No wonder indeed:-) Apart from the restriction to an interval, the key new thought in Dirichlet's version was that of an arbitrary correspondence (and the cause of much argument later)Selfstudier (talk) 13:28, 28 March 2012 (UTC)

Bourbaki's 1970 definition

The following is from N. Bourbaki, 1970, Elements of Mathematics, Theory of Sets, soft-cover edition of 1st printing 1968, Springer Verlag, 2004, Heidelberg Germany, ISBN 3-540-22525-0.

"3. CORRESPONDENCES §4. Functions" [§3.4, p. 81]:

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

"If f is a function, F its graph, and x an element of the domain of f, the relation y = f(x) is then equivalent to (x, y)

Also on page 81 appear the definitions of "mapping of A into B" (written agreeably A →^f B , with the correct typography putting the f over the center of the arrow) and "transformation". Here is "transformation:"

"A function f defined on A is said to transform x into f(x) for all (x ∈ A); f(x) is called the transform of x by f or (by abuse of language) the image of x under f. "(p. 81)

and "f is also said to be defined on A and to take its values in B ". (page 81).

Typography is important; before doing anything with this we should ensure it's as the "authors" meant it. Bill Wvbailey (talk) 21:58, 5 March 2012 (UTC)

"3. Correspondences

"GRAPHS AND CORRESPONDENCES

"DEFINITION 1: G is said to be a graph if every element of G is an ordered pair . . ." (p. 75)

"DEFINITION 2: A correspondence between a set A and a set B is a triple

Γ = (G, A, B)

"where G is a graph such that pr₁G ⊂ A and pr₂G ⊂ B. G is said to be a graph of Γ, A is the source, and B is the target of Γ.

"If (x,y)∈G we say that "y corresponds to x in the correspondence Γ". For each x ∈ pr₁G the correspondence Γ is said to be defined at x, and pr₁G is called the domain of Γ; each y ∈ pr₂G is said to be a "value taken by" Γ, and pr₂G is called the "range of" Γ." (page 75, I think; typography should be checked).

I'm using the limited google-view and not the text itself. So I don't know exactly what pr₁G and pr₂G indicate; the usage indicates it's the first and second element of the "graph"'s ordered pair. BillWvbailey (talk) 23:28, 20 March 2012 (UTC)

Essentially, that's right, the "precise" explanation of the pr1/2 is a few pages back but can be safely ignored for our purposes, I have the French version if there is anything you want to checkSelfstudier (talk) 10:14, 23 March 2012 (UTC)

So, by now we have enough on the Bourbaki 1939/1970 structuralist type set theory approach to function ie the triple. Yes? If so, we should include something in the History because as I said before there is a Bourbaki gap in the whole thing. I can work something up if you like....Selfstudier (talk) 12:44, 27 March 2012 (UTC)

By "Bourbaki structuralist type set theory" I mean doing mathematics set theoretically as opposed to reducing mathematics to set theory if you see what I mean...Selfstudier (talk) 12:54, 27 March 2012 (UTC)

Yes, go ahead and add something into the history, esp since you pointed out this gap in the first place and you have some references and knowledge that I'm lacking. I know little about contemporary set theory; my Suppes and Halmos that go back to the 1960s-1970s contain nothing about triples. Exactly where did the "triple" notion came from, i.e. did it first appear in someone's paper, say in the 1930's -- or was it cut out of whole cloth by Bourbaki for pedantic reasons, say "structural purity", in the 1950's? What we'd like is a "thread". There's also the minor element of confusion because Bourbaki lists the Γ or the F first, then the pairs, and the reader needs to be alerted to this anomaly. [This reminds me of the pedantry that surrounds the "formal (symbolic, syntactic) definition" of a Turing machine. It adds nothing to the theory and just baffles students.]

Frankly, I'm confused about Bourbaki's influence. Neither my Grattan-Guinness 2000 The Search for Mathematical Roots 1870-1940, nor my Paolo Mancosu 1998 From Brouwer to Hilbert: the Debate on the foundations of Mathematics in the 1920's mention Bourbaki, and only Weyl of the names associated with Bourbaki is mentioned. The article about Bourbaki portrays it as a pedantic Paris backwater that fizzled out after a while, but whether or not this is an accurate portrayal, Bourbaki apparently left some lasting contributions. But how did those contributions make their entry -- for example, what author(s) first injected them into their "set theory"? BillWvbailey (talk) 15:57, 27 March 2012 (UTC)

I certainly think he was influential in making maths more formal. I can't find a thing about functions in my copy of his 'Elements of the History of Mathematics', which kind of makes me think that perhaps one of those contributors did think of it! Dmcq (talk) 16:23, 27 March 2012 (UTC)

Yes, this was one of the areas that had a long term impact, I did put something about the "thread" below in the Lede section; anyway, I will try to work something up and may be put it in here first for a discussion...Selfstudier (talk) 16:43, 27 March 2012 (UTC)

Bill, "But how did those contributions make their entry -- for example, what author(s) first injected them into their "set theory"?" - this is what I meant about doing mathematics set theoretically, it is not that Bourbaki had any new set theory to offer, it is instead the idea of using set theory in mathematics as a matter of course (rather than reducing mathematics to set theory), a more formal approach to doing mathematics; all Bourbaki notions can be reduced to existing set theorySelfstudier (talk) 14:37, 28 March 2012 (UTC)

What they were teaching during all this

So I'm thinking that if we expand Dirichlet a bit, insert Bourbaki 1939 and Bourbaki 1970 in the History then have a section called Education (or something similar) that explains what the educators response to all these developments was because eg one response to Bourbaki 1939(which included the ordered pair definition as well as a "relation" version) was apparently first to produce a version that seems to be a kind of blend of both of the Bourbaki definitions and the correspondence aspect(for which they credit Dirichlet) and culminating in a "rule" version around the time of NewMath (ModernMath in UK): "Let A and B be sets and let there be a given rule which assigns exactly one member to B to each member of A. The rule, together with the set A, is said to be a function and the set A is said to be its domain. The set of all members of B actually assigned to members of A by the rule is said to be the range of the function." Selfstudier (talk) 13:54, 28 March 2012 (UTC)

The ordered pair of course still had it's own independent existence and some students were instead being taught thatSelfstudier (talk) 14:42, 28 March 2012 (UTC)

I will just make some indicative notes here and expand/tidy them as I go:-

The confusion surrounding function at the time of Halmos was reflected as pressure in the math community (Klein, 1908, "not simply a mathematical method, but the heart and soul of mathematical thinking") and then in education:

US, 1916 National Committee on Mathematical Requirements(NCMR), 1923 NCMR report, Chapter 7 "The Function Concept in Secondary School Mathematics"

30's, Progressive Education Association (PEA) Committee on the Functionality of Mathematics in General Education (function as one of nine "life" topics) and Joint Commission of the MAA and NCTM "the student should acquire understandings of the concept of variables, dependency, and the generality and power of the function concept"

Meanwhile, in mathematics (set theory/new math areas) and in college, definition becoming more abstract (then ordered pair/Bourbaki) so secondary education lagging behind.

Selfstudier (talk) 15:08, 28 March 2012 (UTC)

Early 50's Committee on School Mathematics (the UICSM), "to improve the teaching of mathematics to pre-college students, for the benefit of universities, so as to help overcome the gap between school mathematics and that at the university"

Beberman 1958 (UICSM prject head),"The semantics notion that a noun ought to have a referent has led us to give precise descriptions of relations and functions. The customary vagueness that surrounds the word 'function' in conventional courses vanishes when a student realizes that a function is an entity, a set of ordered pairs in which no two elements have the same first component."

1959 May &Van Engen "We have presented several different points of view from which functions (and relations) may be considered. We may describe them as sets of pairs, sets of points, tables, correspondences, or as mappings. We may emphasize the rule or we may concentrate attention on the set...The modern point of view (of function) is not contradictory to any of them, but unifies them all."

Selfstudier (talk) 15:16, 28 March 2012 (UTC)

1960 SMSG (newmath publisher) "Let A and B be sets and let there be a given rule which assigns exactly one member to B to each member of A. The rule, together with the set A, is said to be a function and the set A is said to be its domain. The set of all members of B actually assigned to members of A by the rule is said to be the range of the function."

I don't know what people are making of this but it seems to me that the educators, far from having solved the problem,were themselves confused!Selfstudier (talk) 15:22, 28 March 2012 (UTC)

1970 Bourbaki mark II, what I need to try find out is how/where this made it's appearance at the upper level, it's pretty clear that the rule idea is a precursor (with reference to range as image rather than codomain)Selfstudier (talk) 17:09, 28 March 2012 (UTC)

1980 Malik "the necessity of teaching the modern definition of function at the school level is not at all obvious and most of the instructors feel that pedagogical considerations were ignored while designing the course content and the mode of presentation." and "We note that the definition of function as an expression or formula representing a relation between variables is for calculus or pre-calculus; is a rule of correspondence between reals for analysis, and a set theoretic definition with domain and range is required to study topology."

1986 Markovits "...the situation is today that the set definition has been taught in schools for about 25 years, and no one disputes the central importance of the concept, whatever the arguments about its definition. The natural question, therefore, arises: do the students 'understand' the new definition?"

Selfstudier (talk) 15:31, 28 March 2012 (UTC)

---Edit conflict . . .

Yes go ahead and approach it the way you see fit. Perhaps "Education since 1960", i.e. the evolution how functions are taught in the secondary schools especially since 1960, maybe in the calculus (e.g. I have my Thomas 1960 3rd edition that I used in college). The introduction of set theory (see Thomas, below) seems to be a historical "fork in the road" like those taken by the analysists vs logicists (I'm thinking of Dedekind, Frege and Russell here).

The following is from George B. Thomas Jr 1960 Calculus and Analytic Geometry 3rd Edition, Addison-Wesley Publishing Company, Inc, Reading MA, LCCCN: 60-5015. Previous editions: 1950, 1953, pages 16-22:

"DEFINITION. A function is a set of ordered pairs of numbers (x, y) such that to each value of the first variable (x) there corresponds a unique value of the second variable (y).

"EXAMPLE 1. Let the domain of x be the set {0, 1, 2, 3, 4}, and to each value of x let us assign the number y = x². The function so defined is the set of pairs

" { (0, 0), (1,1), (2, 4), (3, 9), (4, 16)}

"One memember of this set is the pair (2, 4). We also say that (2, 4) "belongs to" the function."

Thomas goes on to define the function over an interval, then defines graph [traditional plotting]:

"We must resort to this so-called "set-builder" notation in this instance, since the number of elements belonging to the set is infinite and we canno possibly enumerate all the elements explicitly as we did in [the previous] example."

Thomas goes on to state "A function may be denoted by a single letter, say f" (p. 18). Then in the same paragraph he defines "independent" and "dependent" variables, "domain", and "range", and "Similarly, the range of a function is the image of its domain. We also say that the function maps its domain onto its range." (p. 19)

The "rule" notion rears its ugly head, with a novel twist: It now gets fascinating. Thomas baldly states, without exception (see his following remark), that "A function is determined by the domain and by any rule that tells what image in the range is to be associated with each element of the domain. For, once the domain and the rule are given, the set of all ordered pairs (x, y) can, at least in theory be computed by a machine . . ."(p. 21). He has a nice drawing of a "function machine" to compute f(x) = x² + 5. In the "remark" under this paragraph he adds:

"Remark. Another way that a machine can be used to give the value of a function at a particular x is to store in the machine's "memory" a complete table of the pairs (x, y) that constitute the function. . . Or, instead of a complete table, we may store in the machine a partial set of function-values,m and compute others from these by interpolation. In practice, the memory of a computer has limited capacity . . .. and when the values are given by a table, we may still say that there is a "rule" for determining the value of the function at a given x -- the rule being "look it up in the table." This is just what we do in evaluating logarithms, for example, though we shall later learn how logarithms are computed from series . . . (p. 22).

I don't mean to ignite the "rule" debate again, but these quotes are historical facts. Whether or not the earlier editions of Thomas are available, I dunno. This also points to the other possibility: a sub-article called "Function characterizations" that could contain the "Education" section. Bill Wvbailey (talk) 15:38, 28 March 2012 (UTC)

I think it's good to keep on airing it just so we get to the bottom of it and this seems quite close ie a rule is not the ordered pair, it is just the ordering (as in a lookup table, an array)Selfstudier (talk) 17:02, 28 March 2012 (UTC)

I just realized something important. The Thomas "remark" goes on and I'm going to add this to the above: note the word "computer". What I'm seeing in Thomas is what was happening in the world from 1950 to 1960: the notion of a "program", Turing machine TABLE + short-term memory, aka "finite state machine". Actually I have not found any earlier definitive notion of a finite state machine; it is as if Turing hatched it out of whole cloth. But research in this area exploded in the late 1940's through the 1950's (I see it in my engineering texts and the papers I've collected e.g. Karnaugh's mapping, the Quine-McCluskey technique, etc.). I need to ponder on this a bit more. BillWvbailey (talk) 15:51, 28 March 2012 (UTC)

Good point and also reflected nowadays too, the idea that computers/computing can help/assist to teach the function concept in a certain kind of waySelfstudier (talk) 17:05, 28 March 2012 (UTC)

Bad idea, because clearly the finite nature of a computer precludes it from calculating almost all functions. −Woodstone (talk) 17:15, 28 March 2012 (UTC)

Lol, very mathematical (almost all reals are in (0,1),duh) but computably enumerable does get us all the way to countable infinity (I want to say who cares about the rest but I will get done for O.R:-)Selfstudier (talk) 17:27, 28 March 2012 (UTC)

Anyway, lest we take off at a tangent, the current crop of suggestions are only to make a better use of the technology eg graphing capabilities, as far as I know there is no intention to rewrite mathematical foundations!Selfstudier (talk) 17:50, 28 March 2012 (UTC)

Post 1960 is where we start to see textbooks presenting functions as a special type of relation (and at the same time blurring (confusing) the issue as to ordered pair or triple. I think it is pretty clear by 80's that the prevailing opinion in education is that the ordered pair is not cutting it (disconnect with prior experience, too abstract) whereas the rule or relation version is also not cutting it but by reason of the number of additional concepts introduced therebySelfstudier (talk) 18:30, 28 March 2012 (UTC)

Now I will have to firm it up but as far as I can see from the materials I have gathered, it seems that as of end 20th century the position has not really altered in the sense that once again there are many calls for the function concept, there has been another NewMath episode (called NewNewMath or fuzzymath)that has had to be resolved somehow and the students seem just as confused as they ever were judging by the studies being done...Selfstudier (talk) 23:15, 28 March 2012 (UTC)

Lead paragraph

For reasons discussed above (section "Just acknowledge the correspondence-rule controversy?") I'd like to propose the following lead paragraph:

In mathematics, a function is a correspondence from a set of inputs to a set of outputs that associates each input with exactly one output. Intuitively, a function from a set X of inputs to a set Y of outputs is a rule that assigns to each input x a unique output y. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). For example, the function f from the reals to the reals given by f(x) = x² associates any real number with its square; if x = 5 then f(x) = 25. In this case, the output is always nonnegative.

Comments appreciated. Isheden (talk) 16:00, 18 March 2012 (UTC)

http://mathoverflow.net/questions/30381/definition-of-function — Preceding unsigned comment added by 88.82.206.110 (talk) 18:10, 18 March 2012 (UTC)

Have a look at the definition section here, it covers all that and more. That sort of stuff is just unsuitable for the lead. Dmcq (talk) 23:12, 18 March 2012 (UTC)

Isheden: I'm sorry you chose to revert my work, though I like your lead paragraph well enough. My main goal was to leave the set theory, the real numbers, and anything else that is apt to confuse the lay reader out of the first sentence. But I am baffled by your comment that my version was "not accurate at all". Please explain. Rick Norwood (talk) 00:18, 19 March 2012 (UTC)

Sorry for that. I just thought it was better to take a step back and discuss things before going further with the lead. Avoiding sets in the lead paragraph may or may not have pedagogical value to the reader, but in my view it is not possible to give an accurate description (even informal) of a function without mentioning sets, since a function consists of three things: a set of inputs, a set of outputs, and a "rule of correspondence" or set of ordered pairs. Restricting a function to only the last component is not accurate, and if we choose not to be accurate we could say that a function is a machine that takes an input and returns a corresponding output. Although a mathematician might think of a correspondence as a triple containing all three components, the lay person will not have this intuition and will just focus on the relation between inputs and outputs. The question is what level we should assume that the reader has. At what level are people familiar with the set of the real numbers? Isheden (talk) 06:58, 19 March 2012 (UTC)

Johnny von Neumann famously said something to the effect that calling a bunch of numbers a set does not improve it. And, in my experience, people only understand the real numbers when they take real and complex analysis. Most people will understand "real numbers" to mean numbers that are really, truely numbers, and wonder why we need to emphasize that numbers are "real". In short, I don't think it is wrong to mention the correspondence in the first sentence, the domain in the second sentence, and the codomain in the second paragraph. But, as I said, your version of the intro is fine. Rick Norwood (talk) 12:06, 19 March 2012 (UTC)

A substantive improvement over the existing;I would rather "informally" than 'intuitively' but no matter.Selfstudier (talk) 12:57, 19 March 2012 (UTC)

For comparison, this was how the lead paragraph evolved from my suggestion above:

In mathematics, a function is a correspondence between inputs and outputs, with the property that for each input there is exactly one output. The input is often (but not always) called x, the output often called y, and the function is often written y = f(x) (read "y equals f of x"). One example is the function f(x) = x² which inputs any real number and outputs its square; if x = 5 then f(x) = 25.

My comments are as follows:

• Here, correspondence is used as a word in ordinary language. It could be exchanged for "machine" or "black box" and does not reflect the meaning of correspondence in mathematics, which is a triple on the form (X, Y, F).
• "Correspondence between inputs and outputs" could be interpreted as a one-to-one relation.
• Specifying "for each" input has no clear purpose if there is no set of inputs.
• The fact that a function from X to Y is loosely described in terms of a "rule of assignment" in many sources is not mentioned. If we have to avoid the word "set" in the lead paragraph, we cannot possibly give an accurate definition of a function (unless the reader interprets "correspondence" in the mathematical sense). There is no word in ordinary English that captures the full mathematical meaning of function, but a "rule" that "assigns" an output to every input is an attempt that is often used in lower-level texts.
• Saying that something is often but not always called something indicates that there is a debate about what to call it.
• Dmcq suggested using the example f(x) = x² to avoid the impression that every output has a corresponding input. This point is lost here.

Perhaps it is a good idea to avoid the term "real numbers" in the first paragraph. How about an example with more intuitive sets of inputs and outputs? I saw the example of a function that assigns to each person his/her biological mother in a book on abstract mathematics. Isheden (talk) 17:08, 19 March 2012 (UTC)

I don't think there is a problem with real numbers. The problem is the 'real', if it was chopped out and just 'number' there it might make it more accessible to a younger audience. Dmcq (talk) 17:16, 19 March 2012 (UTC)

Or, how about integer inputs and outputs? The output would still always be nonnegative. Isheden (talk) 18:03, 19 March 2012 (UTC)

Integers are unnecessarily specific and restrictive. We don't want to leave readers with the false impression that there's something special about integers that is required to make functions work. I think it's ok in this context just to say "number" without specifying what kind of number is intended (as they all work the same way here anyway). —David Eppstein (talk) 18:12, 19 March 2012 (UTC)

I changed it back to "integer" before reading this thread. My concern is that we do not want to encourage reader to think that a rule on its own defines a function; we do need to be correct enough to specify a domain. But the bigger problem is that "number" is not really well-defined on its own. It could mean an integer, a real, an ordinal, an element of a finite field, an octonian, ... So I find it very unsatisfying to try to say we have a function whose domain consists of "numbers". — Carl (CBM · talk) 18:18, 22 March 2012 (UTC)

While the discussion may or may not be ongoing as to precise details, I don't think anyone is going to object overmuch if we include right away the proposal as it standsSelfstudier (talk) 17:58, 19 March 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)

In the first definition of a function in the definitions section the given citation is for function defined via a relation (ie XxY then a subset R -> function as graph)Selfstudier (talk) 10:23, 29 March 2012 (UTC)

That book uses the definition of a relation as a subset of a given Cartesian product rather than just any old set of ordered pairs so I think it is okay, but I think you are right, a clearer example would be better. In effect his relation is always a relation from X to Y as it says at the start of the definitions section. Dmcq (talk) 11:45, 29 March 2012 (UTC)

Lede

I reworded some changes to the lede. Using the word "rule" is good, but we shouldn't include the "intuitively" sentence. That sentence is both too pedagogical and too weasel-worded. But most importantly it gives a naive reader the wrong idea. A function in general cannot be thought of as a "rule" in the ordinary sense. Instead mathematicians have to think of a "rule" as a function, which is the opposite of what the sentence I removed suggested. The lede should not encorage the common mistake of thinking that a function is a rule in the usual sense; adding "intuitively" doesn't make that better IMHO. — Carl (CBM · talk) 01:38, 21 March 2012 (UTC)

Now we're back to the original situation that the word "rule" is being mentioned in the lead without any introduction. In my view, some kind of link between the first sentence and the example with a "rule" {{{1}}} is needed. My bet is that the reader thinks of this as an equation, not a rule, and wonders why a function is given by a rule after the first sentence introduced a function as a kind of correspondence. The main difficulty with the wording in the lead is that we want to use plain language to explain the concept, whereas there is no word in ordinary English that fully captures the concept of a function. The words correspondence and relation cannot be used in the intuitive sense either, as earlier discussions have demonstrated. Isheden (talk) 08:15, 21 March 2012 (UTC)

Since we are all having a go I have reverted to Isheden version and amended it a bit. To be honest, I think we are getting a bit confused about our audience here; we say it will confuse a "naive" reader but then say "mathematicians have to think...",which is it? if we define "naive" as high school/university, the likelihood is that they have all been given a definition that includes "rule" or "process". We should just accept that we are never going to have it perfect (whatever that is) for all audiences. — Preceding unsigned comment added by Selfstudier (talk • contribs) 11:01, 21 March 2012 (UTC)

I checked a few words in the Oxford Advanced American Dictionary and it seems to me that a "connection", "relation", or "link" between A and B are ordinary words that capture part of the meaning of a function quite well:

• connection [countable]: something that connects two facts, ideas, etc., "Scientists have established a connection between cholesterol levels and heart disease."
• relation [countable]: the way in which two or more things are connected, "the relation between rainfall and crop yields"
• link: a connection between two or more people or things, "evidence for a strong causal link between exposure to sun and skin cancer"

However, "correspondence", "rule", or "association" might give slightly different associations:

• correspondence [countable, uncountable]: a connection between two things; the fact of two things being similar, "There is a close correspondence between the two extracts."
• rule [countable]: a statement of what is possible according to a particular system, for example the grammar of a language, "the rules of grammar"
• association [countable, uncountable]: a connection or relationship between people or organizations, "his alleged association with terrorist groups"

Interesting. Isheden (talk) 11:03, 21 March 2012 (UTC)

Maybe this sort of analysis is how "rule of correspondence" came to be?Selfstudier (talk) 11:22, 21 March 2012 (UTC)

Gowers again (second answer)Selfstudier (talk) 11:18, 21 March 2012 (UTC)

A similar check for various suggested verbs gives the following:

• assign: to say that something has a particular value or function, or happens at a particular time or place, "Assign a different color to each different type of information."
• correspond: to be similar to or the same as something else, "The British job of Lecturer corresponds roughly to the U.S. job of Associate Professor."
• associate: to make a connection between people or things in your mind, "I always associate the smell of baking with my childhood." Isheden (talk) 11:31, 21 March 2012 (UTC)
• transform: to change the form of something, "The photochemical reactions transform the light into electrical impulses."
• map: to link a group of qualities, items, etc. with their source, cause, position on a scale, etc., "Grammar information enables students to map the structure of a foreign language onto their own." Isheden (talk) 11:57, 21 March 2012 (UTC)
• relate: show or make a connection between two or more things, "In the future, pay increases will be related to productivity."
• link: if something links two things, facts, or situations, or they are linked, they are connected in some way, "Exposure to ultraviolet light is closely linked to skin cancer." Isheden (talk) 12:02, 21 March 2012 (UTC)

There are a number of meanings for words not just the first one. Correspondence is used in the sense of for instance 'Every right carries a corresponding responsibility' so is pretty close to an actual meaning for a maths term. If you think you can phrase it easier for say a 12 year old though then please have a go. Dmcq (talk) 12:00, 21 March 2012 (UTC)

Of course there are various meanings and which one to pick is subjective. I took the ones that in my view provided the closest meaning in each case. Isheden (talk) 12:04, 21 March 2012 (UTC)

Actually, your example contains the adjective "corresponding": matching or connected with something that you have just mentioned, "A change in the money supply brings a corresponding change in expenditure." Isheden (talk) 13:50, 21 March 2012 (UTC)

Didn't take long for a reversion once it actually got into the article:-) Notice that the present version does NOT say that a function IS A rule....Selfstudier (talk) 12:18, 21 March 2012 (UTC)

Your version is fine. Tkuvho (talk) 12:21, 21 March 2012 (UTC)

Now we have edits being made without any comment or explanation at all....:-(Selfstudier (talk) 12:30, 21 March 2012 (UTC)

The present version effectively now includes (informally) the ordered triple, as well as the ordered pair (also informally), which is as it should be...Selfstudier (talk) 12:40, 21 March 2012 (UTC)

As stated above several times, in modern mathematics it is clear that not every function can be described by a rule. Therefore we should not use the word rule in the definition. Not even informally, to avoid any misunderstanding. The word "assigment" does not have this drawback. −Woodstone (talk) 13:56, 21 March 2012 (UTC)

As stated above many times, including by key WPM editors such as Slawomir Bialy, there is no problem with describing a function as a rule. Namely, a function f is given by the rule $x\mapsto f(x)$. Given that over a third of the monographs on the subject do describe a function as a "rule" rather than either a correspondence or a relation, it would be presumptuous to assume that there is actually something wrong with "rule". Tkuvho (talk) 14:03, 21 March 2012 (UTC)

The difficulty here is the distinction between thinking of a function as a rule (which IMHO is unlikely to actually get you into trouble in practice) and formalizing the definition. I think the present version goes out of its way to avoid the "IS A rule" problem although I agree that it does not constitute a formal definition (which we have later on). The other problem is the number of authoritative sources using the word rule in their definitions and we are not able to just simply ignore that. Tkuvho, we are NOT describing a function as a rule....:-)Selfstudier (talk) 14:12, 21 March 2012 (UTC)

Multivalued function Muddy the waters a bit more..:-)Selfstudier (talk) 14:36, 21 March 2012 (UTC)

That concept is mentioned in the section "Generalizations". Isheden (talk) 14:38, 21 March 2012 (UTC)

I have one source claiming that this, when called "multifunction" , has superseded ~Bourbaki's correspondence...:-)Selfstudier (talk) 14:43, 21 March 2012 (UTC)

I see three different approaches to introducing the concept of a function in the lead:

1) Description based on the intuitive meaning in everyday language:

• A function is a relation between a set of inputs and a set of outputs with the property that each input is related to exactly one output
• A function is a connection between a set of inputs and a set of outputs that links each input to exactly one output
• Informally we can think of a function as a machine that for each object that is put in spits a corresponding object out

2) Definitions that avoid introducing more general mathematical concepts:

• A function from X to Y is a "rule" (actually, a set of ordered pairs (x, y)) that assigns a unique output to each input
• A function is an ordered triple of sets (X, Y, F), where F is a set of ordered pairs (x, y), and a necessary condition is that every element in X is the first element in exactly one ordered pair

3) Mathematical definition as special case of more general concepts:

• A function from X to Y is a left-total, right-unique/single-valued relation from X to Y
• A function is a correspondence (X, Y, F) where F is a left-total, right-unique/single-valued relation from X to Y

The question is basically what level of sophistication we should assume where in the article. Isheden (talk) 15:29, 21 March 2012 (UTC)

I can't help thinking that the trouble all starts with "In mathematics, a function is...."

1/ For all real x, let f(x) = x squared

2/ Let f: R to R be defined by f(x) = x squared

3/ Assume f:R to R :x to x squared

4/ f = (X,Y,G) with X,Y = R, G = {(x,x squared), x in R}

U see the problem, they all the same but more or less off-putting to different audiences.

It seems to me that in an encyclopedia we ought to avoid any pedantry or heavy duty/loaded math notation other than in a section carrying a health warning...Selfstudier (talk) 16:26, 21 March 2012 (UTC)

My general rule about the level for the introduction to a topic is to aim it at people who could reasonably be expected to be dealing with the concept within the next year. So I think aiming as far as possible at a 12 year old whilst staying reasonably encyclopaedic by staying more truthful than a noddy introduction is about right. Dmcq (talk) 21:52, 21 March 2012 (UTC)

Lol, seems reasonable but is it doable? How about set, we could call a set a collection, I suppose (given that we haven't defined set)..hmmmm Selfstudier (talk) 23:07, 21 March 2012 (UTC)

It seems utterly doable with the first proposed phrasing above:

" A function is a relation between a set of inputs and a set of outputs with the property that each input is related to exactly one output".

This is both simple and accurate. −Woodstone (talk) 01:59, 22 March 2012 (UTC)

Well, that is what it says now, except it says correspondence instead of relation, the earlier discussion was about whether one is considering relation/correspondence as taking their normal English meanings or their strict mathematical meaningSelfstudier (talk) 12:14, 22 March 2012 (UTC)

Relation would be preferred over correspondence if the everyday meaning is taken into account. Is anyone keen on sticking to the words "correspondence" and "associates" in the lead sentence? Otherwise I intend to changed to the proposal supported by Woodstone above. Isheden (talk) 20:12, 27 March 2012 (UTC)

That might stir things up again to no useful purpose; if we are just to have simple English, then perhaps "relationship" might do?Selfstudier (talk) 17:21, 28 March 2012 (UTC)

"Relation" and "relationship" are synonyms, but I would say relationship is more typically between persons rather than between objects. Connection or link would also do, however relation is in addition a mathematical term used in this context. Isheden (talk) 20:08, 28 March 2012 (UTC)

Relates? "..a function relates a set of inputs..." (I am trying to avoid both correspondence and relation because of the implied mathematical meanings, you already know that I am not enamoured of correspondence but I will suffer it for a quiet life:-)Selfstudier (talk) 22:29, 28 March 2012 (UTC)

One thing I am finding in trawling through the history/education stuff is that the words correspondence and relation have both been associated with both the ordered pair and rule/set type definitions and I cannot tell whether they were being used in the ordinary sense of the words or not (I think so but I am not certain). As you can imagine this appears to have contributed to the general confusion...Selfstudier (talk) 12:23, 29 March 2012 (UTC)

There seem to be two viewpoints: 1) that the lead should be simple, leaving sets, ordered pairs, the set of real numbers, and so on for the formal definition. 2) That the lead should have all of these concepts introduced in the first few sentences. The second version is up as I write, but that may change in the next few minutes. Oddly (or so it seems to me) the set of those who want to introduce the word "set" early on to be mathematically accurate seems to have a non-empty intersection with the set of those who want to introduce the word "rule" early on, even though they acknowledge that "rule" is mathemaically inaccurate (unless you introduce contortions under which the word "rule" doesn't mean what people will think it means).

I spent some time working on a rewrite that I thought was mathematically accurate but easily accessable, but it was reverted almost as soon as it was posted, so I'm not going to do that again. I'm perfectly content with the lead as it is now. On the 20th, things seemed to settle down, but yeterday and today things have been very busy.. It does seem to me that a bunch of mathematicians ought to be able to find some process that will produce a stable lead.

Rick Norwood (talk) 12:38, 22 March 2012 (UTC)

For myself, I am not unhappy with the way it is now although it could possibly be "touched up" a little. As I understood it, the key point about the lead was it being a reasonable summary of the main article as well as being reasonably accessible. So maybe aiming at a 12 year old for the lead of an article of this sort might be a bit ambitious, I think we need to assume some level of familiarity with math while not assuming mathematical maturity.

On that basis, I would have thought set was OK but set theory not, ordered pair yes, ordered triple maybe not, I'm not too sure about "real numbers" (in fact, I am not that keen on specific examples so early on).Selfstudier (talk) 13:01, 22 March 2012 (UTC)

However, I don't dislike the "example" represented pictorially at the right.....Selfstudier (talk) 13:24, 22 March 2012 (UTC)

What do you guys think about introducing a function as a set of ordered pairs in the second sentence, together with the picture that is presently in the Definition section, and only after this mentioning that the set of ordered pairs can be thought of as a rule? Isheden (talk) 20:18, 27 March 2012 (UTC)

I had thought that the ordered pairs was intended to be conveyed by the first sentence,the formal triple is being left out on accessibility grounds and instead a reference to the watered down triple is made. I don't myself have any objection to the picture but you should note that it has a specified codomain....Selfstudier (talk) 22:53, 27 March 2012 (UTC)

Most people learn by examples. The skill of learning from an abstract definition is a pretty high-level skill. If the definition is just being tweeked, and a stable definition will shortly result, all is well. My concern is that these "tweeks" will continue ad infinitum. Rick Norwood (talk) 13:26, 22 March 2012 (UTC)

Well the lead at this instant of time gets a definite thumbs up from me. And I think that first paragraph is very good at its job. Dmcq (talk) 16:59, 22 March 2012 (UTC)

No rest for the wicked...after some to and fro the example has been edited out altogether; I was wondering whether we might not adapt the picture at right so as to effectively show the same thing as was the intention of the excised example.Selfstudier (talk) 18:56, 22 March 2012 (UTC)

Oh I like examples, some people learn that way. Besides which some of our readers are blind so a picture isn't much help as a total substitute for text. Dmcq (talk) 19:12, 22 March 2012 (UTC)

Less I am misunderstood, I also like examples and sometimes I even like them at the beginning, depending....I'm not sure what you mean here, are you saying that removing the example will affect a blind person more than someone who is sighted? ie that you would like the example back? (I take it you are not saying that we should not improve the picture?)Selfstudier (talk) 19:20, 22 March 2012 (UTC)

I think it's important to have an example in the lead. Can we come up with one that can be defined in a way that satisfies the most pedantic editors and also doesn't involve complicated notation, concepts of real analysis or foundations of mathematics unfamiliar to most 12-year-olds? I would have thought the number-squaring example was good enough for that but apparently not. —David Eppstein (talk) 19:36, 22 March 2012 (UTC)

There is an example in the lead, in the third paragraph; I guess you mean you want one in the first paragraph..Selfstudier (talk) 19:43, 22 March 2012 (UTC)

I think the key here is an example in which it is clear to everone what the set of inputs and the set of outputs are. How about moving the illustration from the section Definition up to the lead and explain it there in detail? Perhaps also mention the ordered pairs definition before saying that the set of ordered pairs can be thought of as a "rule of assignment"? Isheden (talk) 21:21, 22 March 2012 (UTC)

There is not room in the lede to explain anything in detail. — Carl (CBM · talk) 21:32, 22 March 2012 (UTC)

OK, that is of course correct. How about just referring to said figure? At least the sets are very clear in that figure even for people who might not be confortable with the set of the real numbers. Isheden (talk) 21:43, 22 March 2012 (UTC)

How about we have a new para 2, beginning "Some examples:" and then a few, 3 or 4 examples including one which would be in a picture as well?Selfstudier (talk) 19:52, 22 March 2012 (UTC)

In what section? One very brief example could be OK for the lede, but not a list of several or even one long example. The lede is meant to be a short summary of the article, not including detailed explanation on its own. — Carl (CBM · talk) 21:32, 22 March 2012 (UTC)

Well, there were 3, if you include the picture and the one you took out so I am only suggesting a rearrangement of what was there anywaySelfstudier (talk) 22:39, 22 March 2012 (UTC)

Anyway, consider me neutral on this issue, I can live with an early example...or not; as I indicated before,left to my own devices, for this subject matter, I would not have gone for an early specific example but I can live with that if that's what everyone wantsSelfstudier (talk) 23:19, 22 March 2012 (UTC)

Another Diversion Gowers Latest Views on Function — Preceding unsigned comment added by Selfstudier (talk • contribs) 20:12, 22 March 2012 (UTC)

Surely there must be some acceptable way we can refer to a real number that is accessible to a 12 year old? Children use numbers like 1.2 all the time but I don't believe they are often told they are real numbers. If we have to explain real number before using something like 1.2 we're in trouble with making the start of the lead of maths articles readable by children being introduced to things like functions or trigonometry. Dmcq (talk) 23:56, 22 March 2012 (UTC)

It's a long time since I was in school, what exactly does a 12 year old know, mathematically speaking? I'm trying to envision some 12 year old saying "Mom, I just have to go look up function on the internet for school tomorrow..." and here we are trying to explain definitions that we don't care to introduce to university students for fear that it will all be too much for them,lolSelfstudier (talk) 00:29, 23 March 2012 (UTC)

http://en.wikiversity.org/wiki/Function  :-) Selfstudier (talk) 00:38, 23 March 2012 (UTC)

Well we document the controversy so we have a definition of a function from X to Y plus two definitions of a function!

Dr Math: All About Functions talks about functions and only says number rather than real number. Their What is e? says 'real number' right at the start but they don't seem to have anything really talking about real numbers and I think e comes after functions. I had a look at a look at a couple of curricula but wasn't able to work out exactly in what order things were done normally.

I think if we had a project page, say 'assumed knowledge'?, showing how much we could assume at various stages it would help avoid vicious loops of reference chasing and awful dead ends within Wikipedia. Perhaps then we could then assume that if we said 'real number' in the first paragraph here then when they clicked on real number they would get something at the appropriate level in its first paragraph and then come back here. Dmcq (talk) 11:00, 23 March 2012 (UTC)

Regarding the example on Dr. Math, it is clear that we could let go of all technicalities related to sets and describe a function as a machine that takes an input and spits out an output, or as a set of rules for turning inputs to outputs. But then of course the description is quite far from how a mathematician would describe a function. Another way of looking at it would be to ask what bare minimum of basic mathematical concepts you need to give a mathematically satisfactory description of a function. I would say you need to understand (or look up using a corresponding wikilink) what sets and ordered pairs are. On the other hand, you don't need to know the mathematical definitions of correspondence and relation. If these terms are used without introduction, knowing their meaning in everyday language must be sufficient. Isheden (talk) 12:02, 23 March 2012 (UTC)

I've been thinking about this a little bit; just because the ordered pair definition requires less concepts doesn't mean that you should then favour that one, particularly if you are talking about 12 year olds. There are less concepts because things are not included in it that are included in the other so while the mathematically mature can fill in the gaps on auto the same doesn't apply elsewhere.Selfstudier (talk) 15:59, 23 March 2012 (UTC)

Like most of the people above, I favor an example early on, and y = x^2 is the example most people are familiar with. As for the question of what students are taught in school, my students are taught something like this: a function is an equation whose graph that obeys the vertical line rule. That's not too bad, though apt to rankle those who learned it a different way, and of course it implies that "function" means "real valued functions of a real variable". We proabably need to mention the "vertical line rule", since most students today will wonder where it is if we don't, but further down in the article, please.

As for "rule", the people who like it seem to like it a lot. What good it does to call a set of ordered pairs a "rule" I don't know. The only reason for introducing "rule" at all is to make things easy at the expense of accuracy. Calling a set of ordered pairs a "rule" (which some books do) makes things harder, because that is not what people think of when they read the word "rule". Rick Norwood (talk) 12:30, 23 March 2012 (UTC)

If you look up at the history section above, this becomes (a bit)clearer; if you are accustomed to the ordered pair version then the rule version will appear peculiar (as well as bringing in the codomain on top of that).

It would appear that since 1960, students have been taught either the ordered pair or the version with a rule (which is just a watered down triple that will actually take you quite far without causing difficulties).

More recently, it would appear that the rule version (watered down triple) and then later on the "proper" triple are being taught more often than before although for logic and set theory you will need to use the ordered pair.

So most educators (certainly this is the case in the UK, the US seems to have more variation, state by state?) appear to have a preference for following Bourbaki (both 1939 and 1970) rather than the longer established ordered pair.

Then what we have is basically at least 2 and a half definitions for function (and that's leaving out some category stuff that tends to express what amounts to the same thing a bit differently)Selfstudier (talk) 13:03, 23 March 2012 (UTC)

You may find a discussion of this "problem" in Rotman's Advanced Modern Algebra (p26, Ch1)(2003) and he says "the informal calculus definition of rule remains, but we will have avoided the problem of saying what a rule is" and then goes on to give the function as graph definition. Of course, while calculus continues to dominate the undergraduate syllabus, this "problem" is likely to persist, although as I have said before you can go a long way before the issue causes difficulties Selfstudier (talk) 10:06, 25 March 2012 (UTC)

Also, one should note the ease of transition from rule version (watered down triple) to formal triple to category theoretic version (essentially the triple as well). This transition is pedagogically appealingSelfstudier (talk) 11:10, 25 March 2012 (UTC)

I also like an example at the top, apart from the claim that a function could have as its domain simply "numbers", since "numbers" doesn't actually describe any particular set. I think that you dislike both "real numbers" and "integers", although I may misunderstand your argument. I think that both are fine, but I removed the example until we can work something out. — Carl (CBM · talk) 12:35, 23 March 2012 (UTC) transi

While I think for most people, numbers means what mathematicians mean by real numbers, I have no strong objection to "real numbers". Rick Norwood (talk) 13:03, 23 March 2012 (UTC)

I still think you can get out of this problem to some extent with the graph, after all you have a continuous curve so it is implicit that there is a value everywhere (Admittedly, this is not much help to a blind person). Do you want to say "real numbers" only or "the set of real numbers"?Selfstudier (talk) 13:18, 23 March 2012 (UTC)

My inclination is to go with "real numbers". Rick Norwood (talk) 14:56, 23 March 2012 (UTC)

Ok, here is some actual data. Or at least, an anecdote. I asked my 13 year old son about this — he is in seventh grade (US), taking Algebra I (which they more commonly take in 8th grade), at a very good public school. He clearly understood what a real number was, even though they hadn't yet done complex numbers, but he said he had only started learning the concept of real numbers this year. They are also learning about functions this year, and he was still hazy on what exactly that meant. So: if we are aiming for 12 year olds, "real number" is too advanced. If we have a more realistic goal of "one year before they get to functions", "real numbers" is still too advanced, but not by so much. —David Eppstein (talk) 03:15, 24 March 2012 (UTC)

Well I had been thinking one year before they get to functions, so that makes me a lot happier about saying real number. I'm not altogether happy with what they get if they click on real number, I can see far too many examples saying things like algebraic and transcendental in that first paragraph and what do they understand by continuum anyway? I'd stick the example with real number back here and think I'll have a good look at the lead of that instead. Dmcq (talk) 10:00, 24 March 2012 (UTC)

You have paragraph 3 to worry about as well, since we have real number and an example there as well; if you are going to have just one example then it should be a good one and I think it would also be good if the picture reflected (x^2?) rather than some other curveSelfstudier (talk) 10:23, 24 March 2012 (UTC)

function spaces: when I think of a function space, I want the domain to equal the codomain, to allow composition of functions. That has been changed to allow the domain to be different from the codomain in a function space. Is this use common? Rick Norwood (talk) 11:30, 24 March 2012 (UTC)

I haven't edited that part, but I don't have the same association of function spaces with composition that you do. I would probably say that any space whose points are functions is a function space. — Carl (CBM · talk) 11:50, 24 March 2012 (UTC)