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 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 . 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 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 . 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 [http://www.maa.org/pubs/Calc_articles/ma001.pdf Evolution), I think the reference is either in here or on the main page, I know I got it here to start withSelfstudier (talk) 23:48, 22 February 2012 (UTC)
- Rick Norwood I think you are making a completely false argument here; in the first place most students are not as far as I know using Wikipedia as primary or sole form of tuition (at least I would hope not). From what you say, you appear to have done some type of study to investigate some number of students background education including as to what type of function definition they have been given (presumably by qualified educators such as yourself). Then further you presumably have identified a subset of students who have been given a different education more in keeping with your conception thereof and found that they have no (less?) trouble in making the transition to abstract mathematics. I would certainly like to hear more about this study which appears to come to a conclusion precisely the opposite of many others. I also note that Wikipedia refers to the well ordering principle as being about positive integers whereas the well ordering theorem is given as referring to every set (by fiat, since it relies on AC)Selfstudier (talk) 23:37, 22 February 2012 (UTC)
- I have in front of me now a university (year 2) text from the UK (not a text book, a text prepared by the university said to be first published in 1997 and reprinted various times through 2005 and it says (in a big box):
- A function f is defined by specifying:
- (a) a set X, called the domain of f;
- (b) a set Y, called the codomain of f;
- (c) a rule, or process, that associates with each x in X a unique y in Y; we write y = f(x) and call f(x) the image of x under f.
- followed by all the usual yada yada
- Now it could well be that the system in the UK is slightly different from that in the US but it ought not to be that much different.The above is copyright so I can't go into more detail, I will however try to get some data from other universities as nowadays they are putting a lot of their material online publicly.
Selfstudier (talk) 00:43, 23 February 2012 (UTC)
- Thanks, that's very useful. There are two issues here that should not be confused: (1) whether or not the lede here should use "rule" in the first sentence; and (2) whether the there is a legitimate (majority or minority) view that it is correct to say that the function is defined by a "rule". Now issue (1) has been endlessly debated in this page without yielding an agreement. As far as issue (2) is concerned, however, there can be no doubt that there is such a legitimate view, as represented by Spivak, Stewart, the UK curriculum cited above, as well as half a dozen users in this page. For this reason, claims to the contrary in the article should be deleted, regardless of what the first sentence says. Tkuvho (talk) 08:56, 23 February 2012 (UTC)
- I have the impression that the other parts of the page besides the lead are being amended somewhat to reflect all these discussions in here and I had thought that the idea was to complete that with suitable references and sources and then to amend the lead so that what follows is merely an expansion of it. Whatever is put there at the end perhaps ought to include the words "is defined by" so that we get away from the "is a" argument once and for all....Selfstudier (talk) 10:07, 23 February 2012 (UTC)
- That's a good idea. At any rate, the reductionist intepretation of "rule" should be resisted as not representing a consensus and contrary to the interest of readability. Tkuvho (talk) 10:14, 23 February 2012 (UTC)
- Tkuvho: If you agree that this is a good idea, why don't you contribute constructively instead of wp:edit warring? What's the difference between "merely implicit" and "not explicit"? I think you are the one that should provide a reference that supports the claim "are therefore given by a rule that is merely implicit". Isheden (talk) 11:39, 23 February 2012 (UTC)
- We are bound by WP:RS and WP:ORIGINAL. So I have stuck in something conforming more closely to what the reliable source talking about rules said. Also see what 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:
- 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 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)