Talk:Function (mathematics)/Archive 7

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

Users supporting "correspondence" object to "rule" on the following grounds:
 * A function cannot be said to be a rule because two different rules may define the same function.
 * Describing a function as a rule implies that a function is necessarily given by an algorithm, which would only be true in certain kinds of mathematical constructivism but not in the classical approach.

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

Furthermore, supporters of "rule" note that that the term does not necessarily imply "algorithm". Thus, $$x\mapsto f(x)$$ is a rule whenever f is a function. Describing a function as a rule is not tautological because the intuitive meaning of "rule" is familiar to the general reader. The objection that different rules may describe the same function can be addressed by saying that "a function given by a rule", or "a function is often given by a rule", which is sufficient for a lede. Tkuvho (talk) 07:46, 16 February 2012 (UTC)
 * Add me to the supporters of correspondence and anti-supporters of rule, for basically the reasons you outline. People think they know what it means when you say that it is a rule, and are generally wrong; better to confuse than to mislead. Spivak's "We will not even begin with a proper definition" is fine for a textbook but not very encyclopedic. —David Eppstein (talk) 07:51, 16 February 2012 (UTC)
 * It is better to build upon a student's intuition than "to confuse". I have not seen any serious evidence that calling a function a rule has misled students.  After all, complexity theory is typically taught much letter than pre-calculus.  I agree with you that defining a function as an algorithm may be inappropriate. But thinking that it is better to confuse than to build on the student's intuitions is a Bourbakist error.  Tkuvho (talk) 08:11, 16 February 2012 (UTC)
 * Oppose changing to rule. A function is not a rule. A function is a correspondence. If somebody can thnik of a better phrasing or a different word from correspondence fine. Saying rule is not that better phrasing or word. I might as well say a colour is a type of pigment or an mammal is a type of cow. Dmcq (talk) 08:36, 16 February 2012 (UTC)
 * Allow me to repeat that according to WP:LEADSENTENCE, the first sentence of the lead should give a concise definition: where possible, one that puts the article in context for the nonspecialist. In the first sentence, relation or correspondence is appropriate. The interpretation as a rule fits more naturally in the third paragraph, which starts out with "There are many ways to describe or represent a function." If needed, this paragraph can be moved closer to the top, before the definitions of argument, value, domain and range. Isheden (talk) 08:45, 16 February 2012 (UTC)


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


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


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


 * You see this difficulty in Frege's description of a function as something "unsaturated" (incomplete -- it needs a value assigned to the empty place in order to complete it). This was taken up by Russell in his 1903 where he considered f. Halmos observes the same issue:
 * "The connotations of activity suggested by the synonyms [map, mapping, transformation, correspondence, operator; i.e. " f sends or maps or tansforms x onto y" (Halmos 1970:30)] listed above make some scholars dissatisfied with the definition according to which a function does not do anything but merely is. This dissatisfaction is reflected in a different use of the vocabulary: function is reserved for the undefined object that is somehow active, and the set of ordered pairs that we have called the function is then called the graph of the function." (Halmos 1970:30).
 * So what exactly IS a function? My take is that most mathematicians consider the nomothetic "function" to be a passive [object? entity? structure?]: just a pairing + ordering relation presented in an effective format suitable for a target agent, and it need not be symbolic (discrete): e.g. an analog device like a slide rule or an X-Y plotting device (e.g. two pins in a gridded board with a string around them + a pencil, to draw ellipses.) Bill Wvbailey (talk) 17:16, 16 February 2012 (UTC)


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


 * I agree with you just enough to keep trying to clarify what I see is the issue (formation rule F versus assignment rule R). I don't have an answer yet. Since I'd prefer the word "assigns" (action) or "assignment" (outcome) that's what I'll use: Our choice here is whether the symbol string " function " indicates an active process, a passive (static) object, both in various contexts, or neither. For example: "a function assigns exactly one output to every input" is much different than: "a function is the assignment [correspondence, association] of exactly one output to every input". In the first case the "function" is an agent always following a generalized formation rule F that is usually, but not always, under guidance of a specific assignment rule R. In the second case the function is the outcome of the process, a static "object" (table, algorithm, formula, Halmos' telephone book). In this latter case we need to the "function's" (aka graph's) formation rule F so as agents we can use the "object". So just what is a "function"? An agent or an outcome? This is almost identical to the debate around Gurevich's stance that an algorithm is the programmed Turing machine as opposed to a symbol-string to be instantiated in a Turing machine. No wonder students get confused. Bill Wvbailey (talk) 20:01, 16 February 2012 (UTC)
 * The "function is a rule" language works better in the classroom, especially when it comes to talk of "applying" a function. -- 202.124.74.119 (talk) 05:07, 17 February 2012 (UTC)


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


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


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


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


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


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


 * If I type function into Wikipedia search I end up on a disambiguation page that says "Function (mathematics), an abstract entity that associates an input to a corresponding output according to some rule".Selfstudier (talk) 12:13, 17 February 2012 (UTC)
 * That's slightly better than saying it is a rule. I think it would be even better if it just said an abstract entity that associates an output with each input. Dmcq (talk) 12:31, 17 February 2012 (UTC)


 * OK I think I see where you are coming from now, you appear to want to cater for cases where there is no "rule" other than association eg 2 lists (array). As you can see from my use of the word array, I tend not to regard such a thing as a function; since there is nothing to compute with, why would you need to call this a function?Selfstudier (talk) 13:26, 17 February 2012 (UTC)
 * See Gandalf61's comment above for a few other examples of mathematical functions that cannot be described by any simple rule. Isheden (talk) 13:47, 17 February 2012 (UTC)


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


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


 * See my comment above, you also have the idea that the key or only part is merely the association between input and output which is fine if everyone would standardize on that, however I can see why this gives rise to cognitive dissonance on the part of many (myself included)Selfstudier (talk) 13:40, 17 February 2012 (UTC)
 * Computation is irrelevant to functions, they just are. If you produce a sorted list using binary sort or quicksort you still end up with a sorted list. The function is the same. In computing one would have different procedures binary_sort(list) or quicksort(list) but as far as functions go all that happens is for each list you have a sorted list. Dmcq (talk) 13:45, 17 February 2012 (UTC)
 * In fact function and procedure epitomise the difference. A person's function may be to ensure the doorknowbs are shiny. The procedure might be for him to get metal polich and cloth and shine them. As far as the functionary is concerned the output s shiny doorknobs. Dmcq (talk) 13:54, 17 February 2012 (UTC)


 * I am talking about the difference between lists with a fixed number of entries and lists where new entries are capable of being added (computed with a rule). As I said, you are defining functions in a particular kind of way that is not universally accepted. In practice, one will always specify (well define) a function and it will be perfectly obvious what the intention is. I would cover the general situation that you appear to be describing with some other word, relation probably.Selfstudier (talk) 14:03, 17 February 2012 (UTC)
 * Could you point to somewhere that defines them differently in the way you're thinking of thanks. Dmcq (talk) 14:18, 17 February 2012 (UTC)


 * This is as good an approach to the question as any...do you have a favourite?Selfstudier (talk) 14:26, 17 February 2012 (UTC)
 * I don't see how any of that is particularly relevant to this discussion. That's an introductory text where they spend all day and still don't end up with a proper definition. This is an encyclopaedia. Dmcq (talk) 15:57, 17 February 2012 (UTC)


 * I thought that it answered the question of why we are having this discussion rather well, actually...Selfstudier (talk) 16:03, 17 February 2012 (UTC)
 * I saw a sort of working definition where rule, relation or correspondence could be used according to the author. I did not see a definition. The text went on and on and is far too long as a basis of a summary for a lead sentence. It did not talk about computation being needed or that a function only applied when large of infinite numbers of inputs were involved. Dmcq (talk) 16:12, 17 February 2012 (UTC)


 * I must have misunderstood you, is it that you simply want a source using "rule" rather than "relation" or "correspondence", that is quite straightforward since the vast majority of educational establishments use such a definition. The other part I am not too clear about is the level, the link I gave was at Calc 1, are you looking for something more technical?(I had thought the lead in Wiki articles was supposed to be aimed low, so to speak).Selfstudier (talk) 16:24, 17 February 2012 (UTC)
 * Look at pp. 153-155 for an alternate basic-level explanation. It defines function in terms of a relation as well. Isheden (talk) 16:36, 17 February 2012 (UTC)


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


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

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

Selfstudier (talk) 18:10, 17 February 2012 (UTC)
 * I think the 'special way of matching' in the one Isheden got is more straightforward. I could live with that if people really are dead set against correspondence. The Princeton one just doesn't work except as hand waving. One doesn't turn or transform objects, the original value stays around and is unchanged. Dmcq (talk) 18:20, 17 February 2012 (UTC)


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


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

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

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

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

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

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


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

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


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


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


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


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


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


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


 * I think I am becoming confused as to whether,in the lead, we are seeking to define function or whether we are seeking to explain function. We already have a (long) section titled Definition so I am thinking we must be trying to explain and the fact is there are probably as many ways to do this as texts or courses that you wish to examine....Selfstudier (talk) 16:11, 19 February 2012 (UTC)]
 * WP:LEAD explains what the lead is for. Its main functions are to introduce the topic and summarize the contents. So in fact most of the lead should appear later in the article in expanded form. Dmcq (talk) 16:37, 19 February 2012 (UTC)

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


 * My main objection to correspondence is its static nature, it seems at variance with input and output which imply some sort of process (rule)....Selfstudier (talk) 16:59, 19 February 2012 (UTC)
 * I think what you're talking about is the application of a function. A function is pretty much static,, a graph of a sine function just is, it doesn't do anything. A car doesn't go anywhere unless the engine is switched on and iit is driven (well not normally anyway!) Dmcq (talk) 17:44, 19 February 2012 (UTC)


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

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


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


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


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


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

This whole debate seems to result from a confusion of "rule" with "algorithm." But a rule need not be given by an algorithm -- consider the characteristic function of the rationals, where f(x) = 1 if x is rational, and 0 otherwise. That's obviously a rule, but there is no algorithm to do that. Since the vast majority of books say "a function is a rule" and "rule" in that sense is considered broadly, Tkuvho's suggestion is a very reasonable one. I really dislike "a function is a correspondence" because:
 * 1) "correspondence" is even less clear to the reader than "rule"
 * 2) it's easy to misread "correspondence" as implying all functions are injective
 * 3) if you formalise "correspondence," you wind up defining a function to be its graph (i.e. {(x, f(x))|x in D}), which makes the phrase "graph of a function" rather strange. -- 202.124.75.33 (talk) 23:52, 19 February 2012 (UTC)


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


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


 * a function is a special type of relation where
 * A function is a "well-behaved" relation
 * A function is a relation
 * function is a useful tool
 * function is a relationship
 * A function is a preset formula
 * a function is a rule
 * A function is a group of statements that is execute
 * function is a generator;
 * A function is a rule
 * a function is a type of procedure or routine
 * A function is a subroutine
 * function is any algorithm or subroutine
 * A FUNCTION IS A PROGRAMMING UNIT
 * A function is a block of organized, reusable code
 * A function is a mathematical process that uniquely relates
 * a function is a mapping of some domain onto some range
 * a function is a method
 * A function is a specific relation
 * If you're going to go on google it looks like relation is the one that is most definitive and it is used in the top entries. I could try this on your 'vast majority' of books too but I'm pretty certain rule is not the winner. I think your counting of people is also a bit awry. Dmcq (talk) 00:31, 20 February 2012 (UTC)


 * Google searches (in quotes) 1)"A function is a rule" 19.4 million 2)"A function is a relation" 2.5 million and 3) "A function is a correspondence" 1.4 million Seems about right to me, sure you can argue that it isn't correct but that's not a decision for us to makeSelfstudier (talk) 00:51, 20 February 2012 (UTC)
 * I emphatically disagree with your position that a majority of internet hits should overrule mathematical correctness. —David Eppstein (talk) 01:09, 20 February 2012 (UTC)
 * My position? Don't blame me, I didn't start it....:-) Anyway,we disagree about what is "correct" regardless of Google, right?Selfstudier (talk) 01:14, 20 February 2012 (UTC)

The more relevant thing is not Internet hits (which nobody actually suggested), but Google Books hits, since a book is more likely to be a WP:RS. In fact we cannot apply our own definition of "mathematical correctness" as some would suggest, but we need to follow the books. In general, the books say: So "relation" in fact the most popular, and "correspondence" the least. If we restrict attention to books published by Springer, Addison-Wesley, OUP, CUP, Wiley, and McGraw-Hill (which are even more likely to be a WP:RS), we get "rule" on top by a very slim margin: Basically, the books support "rule" or "relation" but not correspondence. -- 202.124.75.245 (talk) 07:50, 20 February 2012 (UTC)
 * "A function is a relation" 36,000
 * "A function is a rule" 28,100
 * "A function is a mapping" 16,400
 * "A function is a correspondence" 8,900
 * "A function is a rule" 266
 * "A function is a relation" 252
 * "A function is a mapping" 87
 * "A function is a correspondence" 54
 * User:Selfstudier wrote above: I will join the "rule" supporters who by my count now have the edge. I similarly support the "rule" wording which is the most helpful to novices and used by Spivak, therefore arguably mathematically correct.  However, the assertion that "rule" supporters "now have the edge" is somewhat optimistic.  In the current discussion, only Selfsdudier, myself, and an IP (again, I suggest registering with a more stable user name) support "rule".  Unless there is further input by interested editors, pursuing this further would not be consistent with wiki guidelines for resolving content disputes.  Tkuvho (talk) 07:55, 20 February 2012 (UTC)
 * Since Wikipedia is not a democracy (WP:DEM), voting is pointless. And instead of endless debates about the article lead, I think the way forward is to rework the Definition section in the article. The lead should merely reflect what is being expanded later in the article. Isheden (talk) 08:41, 20 February 2012 (UTC)
 * I'm more happy with the content than with the lead at present, I agree with you provided that you don't simply intend a transfer of the whole of this argument to a different place in the article...Selfstudier (talk) 10:41, 20 February 2012 (UTC)
 * A relation and a correspondence are practically the same thing, it is like the difference between 'a function from X to Y' and 'a function'. Correspondence is slightly more accurate but I'm happy with relation. As to google you really need to check the actual references, their counts can be quite widely out and as shown by the ones I listed they can refer to lots of things besides mathematics. Also Wikipedia says we should use the most reliable sources in WP:RS not all sources. Also note that 'function is a relation' even if it worked would not get three of the six relation entries in my list ' a function is a special type of relation ', 'A function is a "well-behaved" relation', and 'A function is a specific relation'. As to counts on checking the references I have found google counts can very easily be out by a factor of ten even when there are only a few thousand responses, and the figures can go down if you say OR or up if you say AND. This is why I checked the references the way I did.
 * What you say here is correct,(binary)relation will always need to be qualified in order to arrive at function since it is a generalization and for purposes of this article and the lead in particular, an unnecessary one as reference to the existence of a generalized version can be made at the foot of the article for those interested in pursuing the matter further.Selfstudier (talk) 10:35, 20 February 2012 (UTC)
 * If you want to check the books properly do something like 'function is a' and actually check the first few pages of returns. Chuck away any that are irrelevant and then actually look at what is left and the standard of the books they were in. Dmcq (talk) 08:51, 20 February 2012 (UTC)


 * The last Google Books search above was restricted to a particular set of reliable publishers: Springer, Addison-Wesley, Oxford University Press, Cambridge University Press, Wiley, and McGraw-Hill. The phrase "a function is a specific ..." occurs only 9 times with those publishers (and only 2 of those occurrences relate to mathematics), so it isn't really a contender.
 * The advantage of "relation" over "correspondence" is that "relation" has no hint of suggesting injectivity. However, my vote is still for "rule." -- 202.124.73.189 (talk) 10:52, 20 February 2012 (UTC)

When all is said and done, we have one group seeking to define function as ordered pair and another as rule. In some sense both are correct and this is reflected in sources although I question whether novices would have the mathematical maturity to cope with the higher abstraction of ordered pair. Also many sources simply fudge the issue with something along the lines of "a rule.....assigns......set A to setB blah". To me this suggests that we ought to go with something simple in the lead for the casual reader and cover the actual situation as reflected in sources for the main article.Selfstudier (talk) 10:59, 20 February 2012 (UTC)
 * I think it is prudent to reflect, in the article section Definition, how function is defined in various sources before attempting to formulate a concise definition for the first sentence of the lead. Isheden (talk) 11:14, 20 February 2012 (UTC)


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


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


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


 * There seemed to be a few duplicate books there, for instance James Stewart books with a rule occur a couple of times but we'll ignore that. As far as I can make out that gives 7 for a rule and 14 for a relation. Dmcq (talk) 13:17, 20 February 2012 (UTC)
 * Personally I prefer correspondence because it doesn't give people the wrong idea from a familiar word like rule, they know they have to read on a bit. Also it is right mathematically. It is terribly hard to get rid of a wrong idea, unlearning is much harder than learning something new, that's why one should try to be reasonably accurate to start with. Dmcq (talk) 13:29, 20 February 2012 (UTC)

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


 * I have just carefully reread the lead as it currently stands, and it seems pretty good to me. Selfstudier's view of set theory ("who cares") is original research.  The google search is meaningless.  Mathematics articles should be based on standard textbooks. Rick Norwood (talk) 13:38, 20 February 2012 (UTC)
 * I agree about textbooks. The more difficult issue here is that "function" is used at all levels of math, from elementary school to postgraduate. If we look at sufficiently low-level books, we will find all sorts of strange things - the same would be true in chemistry, or quantum physics, or any other technical area, where low-level books routinely stretch the truth, sacrificing accuracy for the appearance of accessibility. If we look at books at the upper undergraduate level and above, there is great uniformity in the definition of a function: it's a relation R such that if xRy and xRz then y=z (there is another minor issue whether a relation includes info on its domain and codomain). So in the main body of this article we also use this definition. But in the very first sentence, it may be too much to say that a function is a relation, so we try to make it a little more friendly. In doing so, we should not give up on being accurate, or give a description that a naive reader would be likely to misunderstand.  &mdash; Carl (CBM · talk) 13:46, 20 February 2012 (UTC)


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

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

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

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


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

Create subsection with formal symbolism?
[[cc'd down from above to start a new thread]

Carl, I looked for your definition in the article: "a relation R such that if xRy and xRz then y=z" but I'm not seeing it. If anyone's interested here's a source. From Suppes 1972:57:
 * "DEFINITION 1. A is a relation ↔ (∀x)(x ∈ A → (∃y)(∃y)(x = )).
 * The  is the standard Kuratowski definition of the ordered pair cf page 32. From Suppes 1972:86-87:
 * "DEFINITION 39. f is a function ↔ f is a relation & (∀x)(∀y)(∀z) (x f y & x f z → y = z).
 * "DEFINITION 40. f(x) = y ↔ [(E!z) (x f z) & x f y] V [-(E!z) (x f z) & y = 0]
 * (The symbol 0 indicates the empty set cf p. 14). The formal definition of domain D is on p. 59 and for range ℛ on page 60:
 * "DEFINITION 3. DA = {x: (∃y)(x A y)}
 * "DEFINITION 4. ℛA = {y: (∃x)(x A y)}

I find it useful to have all of these in one place, and to have them as simple and as formal as possible, so there's no questions about their interpretation. BillWvbailey (talk) 16:10, 20 February 2012 (UTC)
 * The article says it in English: "a necessary condition is that every element in the domain is the first element in exactly one ordered pair." In particular, if we have both (x,y) and (x,z) then y=z. &mdash; Carl (CBM · talk) 16:42, 20 February 2012 (UTC)


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

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


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

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

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

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


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


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

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


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


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


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

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

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

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


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

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

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


 * In view of the intense discussion, I believe there is a need for various levels of abstraction in this article. I have attempted to write a more concise definition without introducing to much terminology based on two other sources. Please let me know if you have further concerns. Unfortunately I cannot access the book page you mentioned. Isheden (talk) 14:13, 21 February 2012 (UTC)
 * A definition should not start of by saying what something isn't. That should be left till later.


 * The paragraph


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


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


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


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

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


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


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

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

Cannot be given by any rule?
An editor claims to have found a function not given by any rule. The claim is unsourced. Tkuvho (talk) 13:41, 21 February 2012 (UTC)
 * Furthermore the claim is contradicted by Spivak and Stewart (leading calculus textbook today), who define a function as a rule. Tkuvho (talk) 13:42, 21 February 2012 (UTC)
 * I hope we will be able to have a clearer view of things when we have more refs (DMCQ has added a couple already and I have some others somewhere)Selfstudier (talk) 13:52, 21 February 2012 (UTC)

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

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


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

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


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


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


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


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


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


 * Let me ask you a meta-question, Tkuvho: Do you understand the history behind this discussion, and never-the-less place yourself in the camp of those who reject the Axiom of Choice? Or is this all new to you? Rick Norwood (talk) 15:25, 21 February 2012 (UTC)
 * Why, I use the axiom of choice every morning in deciding between tea and coffee. Tkuvho (talk) 15:32, 21 February 2012 (UTC)
 * I think we are off track again, my only objection (for now) is to the use of the word "correspondence" (particularly in input/output context) in the lead and I had thought the idea was to round up some modern references, cross reference them for the article and then summarize for the lead. Is that not the case?Selfstudier (talk) 15:29, 21 February 2012 (UTC)

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

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


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

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


 * Agree Wikipedia is not a WP:CRYSTAL ball and I see no indication that constructivists will ever be anything but a small minority. Most people regard all real numbers as 'existing' whether or not there is a rule to construct them. We don't consider the real line to have holes in for the numbers we have no rule for generating the digits of, Dmcq (talk) 17:27, 21 February 2012 (UTC)
 * An actual example of a definable function that has no rule would be one that returns the digits of Chaitin's constant. We simply don't know what the sequence is and will never have an actual rule. I'm sure constructivists will be happy to say the constant doesn't even exist! Dmcq (talk) 17:37, 21 February 2012 (UTC)


 * I just realized I don't even know what was there before "correspondence", was it "rule" or something else? Is there a way to go back and look at all previous versions?Selfstudier (talk) 17:58, 21 February 2012 (UTC)
 * Just click the historyy tab and go to some date. Clicking on the date will show that version. You can also step one change at a time when viewing a particular version, or if you click on cur or prev in the history then go forward or back through the diffs one at a time. A binary chop in the history is best first though and can sometimes be quicker than trying to spot what you want in the comments. Dmcq (talk) 18:18, 21 February 2012 (UTC)
 * Thanks, I will try it...Selfstudier (talk) 18:40, 21 February 2012 (UTC)

Does anyone object to naming the "modern" definition as "the Dirichlet-Bourbaki definition"? Selfstudier (talk) 18:40, 21 February 2012 (UTC)
 * Hmmm, silence....references are here Dirichlet BourbakiSelfstudier (talk) 19:31, 21 February 2012 (UTC)
 * Or hereSelfstudier (talk) 19:44, 21 February 2012 (UTC)


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


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

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


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


 * I finally found an excellent reference History in Mathematics Education The ICMI Study (456 pages, sorry) which traces what happened in the educational arena from Dirichlet Bourbaki (they call it Cauchy Dirichlet Bourbaki)through the New Math mess and the full circle back to functional dependancy. Of course, this has only indirect relevance for what goes on in advanced math beyond the university (which I think is partly a function (sic) of when you happened to learn these concepts)Selfstudier (talk) 14:18, 22 February 2012 (UTC)
 * An editor continues to insert unsourced claims based on his misunderstanding of Morash. Tkuvho (talk) 13:56, 22 February 2012 (UTC)
 * I really don't think we need mention choice functions in an introduction. There are enough citations about functions not being rules and it's pretty obvious anyway otherwise we'd have real problems when talking about function spaces. Dmcq (talk) 14:24, 22 February 2012 (UTC)
 * You may be right about choice functions in the introduction. As far as rules go, your problem arises because you interpret "rule" as "algorithm".  Otherwise there is no problem with function spaces.  Each member f of such a space is determined by the rule $$x\mapsto f(x)$$.   Tkuvho (talk) 14:30, 22 February 2012 (UTC)
 * You either accept an axiom or not, no rule required; where did "rule as algorithm" originate?Selfstudier (talk) 14:36, 22 February 2012 (UTC)
 * The foundational framework is not really the issue here. An algorithm is an explicit procedure that some hold could be programmed on a computer; clearly a non-measurable function cannot be programmed on a computer.  But a rule does not have to be an algorithm.  That's why Spivak and Stewart can define a function as given by a rule.  Tkuvho (talk) 14:46, 22 February 2012 (UTC)


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


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


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

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


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


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


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


 * On the contrary, the word "rule" has intuitive meaning for the reader that helps him understand the concept, which is precisely why Spivak and Stewart, some of the best math writers we have, use it. Correspondence, meanwhile, has a very different meaning for a general reader, which implies both 1-1 and also symmetry between domain and range, so certainly much more confusing than "rule" which clearly implies a direction and therefore introduces the asymetry needed here.  Learning is never a formal system, and a reader can only build upon his intuitions.  The word "rule" is sufficiently versatile to encompass the intuitive meaning already familiar to the general reader, and a more technical meaning Spivak and Stewart rely upon.  At any rate the claim about functions not given by any rule is unsourced.  It may not be given by an explicit rule/algorithm, but can certainly be described as being given by a non-explicit rule $$ x \mapsto f(x)$$.  Now certainly Bourbaki will not agree to any of this.  Tkuvho (talk) 17:31, 22 February 2012 (UTC)
 * Well I'd prefer relation in the lead okay but as to rule I think this piece  in the paragraph about 'One often hears that such a function is or is given by a rule...' and ending with '..that the word "rule" is not to be stripped of the last vestige of its customary meaning' about sums up what I think of the use of rule above. Dmcq (talk) 17:47, 22 February 2012 (UTC)
 * I'd prefer relation to correspondence (historically this was "rule of correspondence") even though it is IMHO a quite unnecessary generalization, simply explain function and leave relation (which is an increase in abstraction and requires qualification)to later. Actually, I could suffer almost anything except correspondence...:-)Selfstudier (talk) 18:10, 22 February 2012 (UTC)
 * Tkuvho, I think it's not quite right to say that Bourbaki will not agree; I am guessing a bit here but since in fact Bourbaki gives a version with an implied rule AND the version without a rule, I suspect a similar debate to this one was going on then and the set theorists "won" the argument and not content with that set about trying to get the entirety of mathematics for education reduced to set theoretic principles ultimately leading to the New Math debacleSelfstudier (talk) 18:21, 22 February 2012 (UTC)

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

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


 * That Kleiner review is available here Evolution), I think the reference is either in here or on the main page, I know I got it here to start with[[User:Selfstudier|Selfstudier] (talk) 23:48, 22 February 2012 (UTC)


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


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


 * A function f is defined by specifying:
 * (a) a set X, called the domain of f;
 * (b) a set Y, called the codomain of f;
 * (c) a rule, or process, that associates with each x in X a unique y in Y; we write y = f(x) and call f(x) the image of x under f.


 * followed by all the usual yada yada


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

Selfstudier (talk) 00:43, 23 February 2012 (UTC)
 * Thanks, that's very useful. There are two issues here that should not be confused: (1) whether or not the lede here should use "rule" in the first sentence; and (2) whether the there is a legitimate (majority or minority) view that it is correct to say that the function is defined by a "rule".  Now issue (1) has been endlessly debated in this page without yielding an agreement.  As far as issue (2) is concerned, however, there can be no doubt that there is such a legitimate view, as represented by Spivak, Stewart, the UK curriculum cited above, as well as half a dozen users in this page.  For this reason, claims to the contrary in the article should be deleted, regardless of what the first sentence says.  Tkuvho (talk) 08:56, 23 February 2012 (UTC)


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


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


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


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


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


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


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


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


 * Here is a rule:
 * $$f(x) = 0 \text { if } f(x) \ne 0\, ; f(x) = 1 \text { if } f(x) = 0 $$
 * Does that define a function ? I don't think it does because there is no association that satisfies that rule. But if you don't have a definition of what a function is, independent of the concept of a rule, then how do you decide which rules do or do not define a valid function ? Gandalf61 (talk) 12:36, 23 February 2012 (UTC)


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


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


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


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

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


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


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


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

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

Dmcq: Rather than worrying exactly about the definition I was trying to provide a sense of what is actually going on, in other words show the definition (whatever it might be) actually being used (you could also look at arxiv papers for a higher level view). Domain and codomain is pretty standard nowadays, I would have thought (that is the way you get the arbitrary sets into the picture without causing too much confusion).I read that some (many?) education authorities (boards? not sure what you call them) have in their guidelines the "old" (new math) type requirements and that there are still even books from that era reprinted today and of course there is no accounting for teachers who don't teach from the text.Selfstudier (talk) 14:20, 23 February 2012 (UTC)
 * To reiterate that point that seems to be lost on some editors: reliable sources refer to functions (that's all functions, including choice ones) as "rule". Namely, the rule $$x\mapsto f(x)$$  In their own private platonist world there may be no room for such things, but in the literature there is, as well as for half a dozen editors who have expressed themselves on this page.   Tkuvho (talk) 14:36, 23 February 2012 (UTC)
 * And that's why there is the statement saying it isn't completely correct and showing why complete with citations. There is no citations showing the definition in the article is wrong. Personally I think there is quite enough about the problem in the article, we don't need yet more bits saying lots of people saying a function is a rule and other people saying that is wrong. Dmcq (talk) 14:46, 23 February 2012 (UTC)


 * Tkuvho I suppose you could couch the argument as platonist/formalist versus the education authorities but I am not sure that it helps any; what ought to be clear by now is that there is something about a function definition that seems to require a large amount of subsequent explanation for effect and that regardless, the subject, as in the past, continues to cause confusion in the minds of students. (You can find a similar and somewhat related debate in respect of the epsilon delta definition (limit of a function))Selfstudier (talk) 14:47, 23 February 2012 (UTC)
 * I agree with that. I think the issue is precisely whether the writing of the article should be influenced by what educators say about the effectiveness of this or that approach, rather than derivations "from first principles" (by wiki editors) of what should be the most accomplished platonic or formal definition.  Tkuvho (talk) 14:52, 23 February 2012 (UTC)


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


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

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

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


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


 * I'll just give you the link to the notes and let you sort it out.
 * I quite like the black box/function machine idea, at any rate it ought to be helpful if you are familiar with some basic computing/programmingSelfstudier (talk) 16:00, 23 February 2012 (UTC)


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

Dedekind's notion of "function"
(Notice the * footnote in the following, and the world "law"). Dedekind has become a favorite of mine, in particular his 1887 Nature and Meaning of Numbers, Dover Publications, Inc, NY, ISBN 0-486-21010-3, also this is in the public domain at http://books.google.com/books?id=PywPAAAAIAAJ&printsec=frontcover&dq=Dedekind+The+Nature+and+Meaning+of+Numbers&hl=en&sa=X&ei=owdFT_HjHaTx0gG-0Nn2Aw&ved=0CDYQ6AEwAA#v=onepage&q=Dedekind%20The%20Nature%20and%20Meaning%20of%20Numbers&f=false .(I mark up my books so I paid $8.95 for the facsimile). Here is his definition in II. TRANSFORMATION OF A SYSTEM:
 * "21. Definition.* [*See Dirichlet's Vorlesungen uber Zahlentheorie, 3rd edition, 1879, § 163.] By a transformation [Abbildung] φ of a system S we understand a law according to which to every determinate element s of S there belongs a determinate thing which is called the transform of s and denoted by φ(s); we say also that φ(s) corresponds to the element s; that φ(s) results or is produced from s by the transformation φ, that s is transformed into φ(s) by the transformation φ" [etc. What follows is a difficult further elucidation that relies upon his notions developed earlier about T being a part of S, etc] (Page 50 in the Dover edition facsimile of the Open Court Publishing Company's edition of 1901)

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


 * Interesting. With Dirichlet Vorlesungen_%C3%BCber_ZahlentheorieSelfstudier (talk) 16:25, 22 February 2012 (UTC)


 * RE Dirichlet and Dedekind -- There's a translation into English, but damned if it doesn't end at §144; the section referenced by Dedekind is from the 3rd edition at §163! Translator John Stillwell must have used the 1st or 2nd edition (ouch! cf the wiki article Vorlesungen_%C3%BCber_Zahlentheorie there is an issue here). Bill Wvbailey (talk) 23:06, 22 February 2012 (UTC)

---

For only $256 (marked down from $320) this book can be yours -- it contains a wealth of historical analysis by various authors:
 * I. Grattan-Guiness editor, 2005, Landmark Writings in Western Mathematics, Esevier B.V., Amsterdam, The Netherlands, ISBN: 0-444-50871-6.

I bumped into this in Chapter 20 re S. F. Lacroix [1797-1800]:
 * "Any quantity the value of which depends on one or more quantities is said to be a function of these latter, whether or not it is known which operations are necessary to go from them to the former (p.1). But the example given for a function in which the necessary operations are not known is the root of a fifth degree equation. In fact this definition means that a function had to be defined explicitly or implicitly by using the usual mathematical operations. As in Euler ("§ 14.2) it is assumed that any function has a power-series expansion" [commentatary by Joao Caramalho Domingus, p. 281]

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


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

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


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


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


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


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


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


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

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


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

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


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